DYNAMIC ANALYSIS OF NATURAL GAS ROLE ON CAPACITY EXPANSION IN POWER MARKET

Ali Movahednasab, Masoud Rashidinejad, and Amir Abdollahi

References

  1. [1], [4]; HRj denotes required thermal energy for power generation by a technology in Btu/KWh and decreases every year [4]: MCj = (Fj·HRj + O&Mj + CO2j)(1 + rr)y . (3) Investment cost is settled through the construction period and must be recovered during the operation. Valid reports have expressed the investment cost in $/KW [1], [4], which is convertible into $/KWh using life time of each technology in (4). The offers are adjusted using the marginal cost and the investment cost must be recovered 23 under the influence of market condition during the operation period: ICj($/KWh) = ICj($/KW) LTj × 8760 (1 + rr)y . (4) 3.2 Electric Demand Demand of electric energy is modelled by load duration curve (LDC) for base, middle and peak sections, which are supplied by coal-fired, CCGT and GT, respectively; it grows in each section with a constant growth rate every year and the average amount by (5) is offered to the market as market demand. Market price affects the demand via demand elasticity: D(t) = k i=1 δi·Li·eg·y − λ·Δρ. (5) 3.3 Power Market This paper considers an energy-only market with pay-asbid structure, in which the lowest offers are dispatched and receive their offers from the market and market price is equal to average offer. The superiority of this structure in eliminating price spikes persuaded some markets to prefer it over the uniform price. The firms should adjust their offers properly, above the MC and below their prediction of price via the TREND function; they should make a balance between their profit and the chance to win in the market [3], [45]. 3.4 Profitability Clearing the market specifies the generation of each firm, which is applicable in computing their costs and profits. Total generation cost, in (6), is the sum of firm’s expenses for generating electric energy until studied time t: Φj = t 0 Gj·MCjdt (6) By subtracting the generation and investment costs from the income, the total profit of the firms is given by Πj = t 0 Gj·χj − Gj·MCj − CPj·ICjdt (7) Profitability index is defined in (8), as the ratio of profit to generation cost, for normalizing the profits to a same quantity [46]. This parameter is helpful in investing in a technology rather than its profit: PIj = Πj Φj (8) 3.5 Stable State A market can become stable by recovering its generation and investment costs of the firms [45]. This condition is equivalent to PIj = 0, as both costs are considered in PIj. The firms can reach the stable state by offering a price equal to MC plus a multiple of forecasted price [3], named as stable price. 3.6 Capacity Expansion The PIs of firms are converted into investment rate via S-shaped curves in (9), which limit the rate of variations and final values in each firm [46]. The coefficients mj max, αj and βj differ in each technology, but mj is equal to 1 for PIj = 1 in the whole, as indicated in Fig. 6. The coefficient mj is influenced by reliability policy and profitability for providing enough capacity: mj = mj max 1 + e−(αj P Ij −βj ) (9) Figure 6. The coefficient m for different technologies vs. PI. Equation (10) gives investment rate in each technology as a function of demand growth rate and retirement rate of the firms weighted by the coefficient mj: IRj = mj·( ˙Li + ˙REj) (10) Reliability policy in (11) forms an internal loop in launching process, named as launch scale [9] that changes rate of investment in each technology for holding reserve ratio at a proposed level: Res.Rat = TCP − D(t) D(t) (11) The investment rate is converted into capacity after a construction delay. Equation (12) indicates under construction capacity, which is the difference between investment rate and construction rate in each technology. Exploited capacity in (13) is the difference between constructed capacity and retired amount after a life time. A part of GT capacity is converted into CCGT by a change 24 ratio and change delay that influences the pattern of expansion. Operational capacity is declared to the market and creates the main feedback loop in this process; besides, it is used for providing reliability as an internal loop: UCj = t 0 IRj − IRj(t − CTj)dt (12) CPj = t 0 CNj − CNj(t − LTj)dt (13) 3.7 Wind Technology The wind technology competes with other participants in the market, while previous studies subtracted its capacity from the demand [14], [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6], [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors
  2. [3]. The study is followed via three scenarios, namely stable state, natural gas price variation and access of demand to natural gas and electricity. The first scenario enlarges market stability, which is known by recovering generation and investment costs. At the second scenario the effect of natural gas charge variation is analysed in low, medium and high charges and the third scenario analyses the demand access to both natural gas and electricity via seasonal factors. Unlike other researches, the wind participates at the energy market beside the thermal firms, instead of decreasing its capacity from the demand. The applied data are from published reports by EIA about generation and investment costs of different technologies, natural gas hub price, natural gas city gate price and etc., summarized in Appendix A. The rest of paper is organized as follows; Section 2 describes the concept of system dynamics briefly by introducing employed and important tools in this paper. Section 3 explains general model and its different parts. The results of model simulation in defined scenarios are represented in Section 4 and Sections 5 and 6 discuss about the results and pluralize them, respectively. Appendix A summarizes the applied data in this study. 2. Concept of System Dynamics System dynamics was approached by Sterman for analysing complex systems and system thinking in a practical method. Growing the dynamic complexity in business, industrial and social systems increases the role of modelling, predicting and analysing their complex behaviour for understanding its reasons. System dynamics is a method for 21 understanding and analysing the complex behaviours by a set of conceptual tools and modelling methods, which are helpful in simulating the long-run behaviour of a system in different policies and making better decision. Feedback control theories and nonlinear dynamics found the base of system dynamics. For long-run analysis of a system, it is necessary to understand different effective factors and their causal relation. Moreover, identifying feedbacks, delays and other linearity which leads the system to instability and modelling them by stocks and flows is the main art in analysing a system. Simulation is the only reliable way for testing the validity of the models because of complexity of relations among different nonlinear parameters, which makes understanding the behaviour of the model in a long time period impossible. Without simulation techniques, the system hard behaviour can be improved using feedbacks through the real world which is very slow and inefficient due to delays, nonliterary and costs of testing the ideas [46]. 2.1 Causal Diagram For simulating a dynamic system, different tools are needed. Causal loops are important tools for showing the structure of the feedbacks in the system and their effects. A causal diagram, in Fig. 2, consists of arrows which conFigure 2. The causal representation of a variable. Figure 3. The stock and flow variable. Figure 4. Casual diagram of the TREND function. nects related variables together and shows the influences among them. The positive sign on the arrow shows increasing Y by increment of X and negative sign indicates decreasing of Y . 2.2 Stocks and Flows One of the most limitations of casual loops is their inability in capturing the stocks and flows structure of the system. Stock structures are other tools in studying the system dynamics, which accumulate difference between inflow and outflow of a variable as shown in Fig. 3. Equation (1) expresses the relation of stocks, which create inertia in the system and provide memory for it; they are helpful for creating delays in a system by accumulating the difference between the inflow and outflow of a parameter in a process: Y (t) = t 0 X1(τ) − X2(τ)dτ + Y (t0) (1) 2.3 Forecasting Bounded rationality hypothesis (BRH) is a forecasting algorithm formed by adaptive expectation, in which current expectations are related to the current and past values as in (2). Expectations on the value of variables for time T are revised with adjustment rate κ, if forecasted value in previous periods is different from the actual amount [49]: ξe (t, T) = ξe (t, T − 1) + κ[ξ(T − 1) − ξe (t, T − 1)] (2) Sterman has proposed an expectational model based on the system dynamics, called TREND function; he has used needed times for measuring, collecting and analysing data, historic time horizon and required time for perceiving and reacting to variable changes. Figure 4 represents the structure of TREND function, which is usable for estimating fractional growth rate in input variable [46]. 22 Figure 5. Process of capacity expansion in a power market. 3. Model Description Figure 5 represents an overview of developed model. The firms adjust their offers considering their marginal cost and forecasted market price. Offers, existent capacity and average of demand are submitted to the power market for clearing market price and generation amount by each firm. Clearing the market facilitates calculation of profits and generation costs, considering the investment costs. The profits are normalized and converted into investment through some multipliers, which create under construction and generation capacities after some delays. The existent capacities return to the market via offer, which forms main feedback loop in this process. Reserve ratio makes an internal loop by changing the launch scale for providing the proposed reliability level. Hub price of natural gas acts on fuel cost of natural gasbased technologies and affects city gate price via seasonal factors. Details of different parts of the model are as follows. 3.1 Costs Marginal and investment costs are two expenses for generating electricity. The firms settle the marginal cost for generating each MWh of electric energy including fuel, CO2 and O&M costs, which grows with constant rate of return every year as indicated in (3) [1],
  3. [4]; HRj denotes required thermal energy for power generation by a technology in Btu/KWh and decreases every year [4]: MCj = (Fj·HRj + O&Mj + CO2j)(1 + rr)y . (3) Investment cost is settled through the construction period and must be recovered during the operation. Valid reports have expressed the investment cost in $/KW [1], [4], which is convertible into $/KWh using life time of each technology in (4). The offers are adjusted using the marginal cost and the investment cost must be recovered 23 under the influence of market condition during the operation period: ICj($/KWh) = ICj($/KW) LTj × 8760 (1 + rr)y . (4) 3.2 Electric Demand Demand of electric energy is modelled by load duration curve (LDC) for base, middle and peak sections, which are supplied by coal-fired, CCGT and GT, respectively; it grows in each section with a constant growth rate every year and the average amount by (5) is offered to the market as market demand. Market price affects the demand via demand elasticity: D(t) = k i=1 δi·Li·eg·y − λ·Δρ. (5) 3.3 Power Market This paper considers an energy-only market with pay-asbid structure, in which the lowest offers are dispatched and receive their offers from the market and market price is equal to average offer. The superiority of this structure in eliminating price spikes persuaded some markets to prefer it over the uniform price. The firms should adjust their offers properly, above the MC and below their prediction of price via the TREND function; they should make a balance between their profit and the chance to win in the market [3], [45]. 3.4 Profitability Clearing the market specifies the generation of each firm, which is applicable in computing their costs and profits. Total generation cost, in (6), is the sum of firm’s expenses for generating electric energy until studied time t: Φj = t 0 Gj·MCjdt (6) By subtracting the generation and investment costs from the income, the total profit of the firms is given by Πj = t 0 Gj·χj − Gj·MCj − CPj·ICjdt (7) Profitability index is defined in (8), as the ratio of profit to generation cost, for normalizing the profits to a same quantity [46]. This parameter is helpful in investing in a technology rather than its profit: PIj = Πj Φj (8) 3.5 Stable State A market can become stable by recovering its generation and investment costs of the firms [45]. This condition is equivalent to PIj = 0, as both costs are considered in PIj. The firms can reach the stable state by offering a price equal to MC plus a multiple of forecasted price [3], named as stable price. 3.6 Capacity Expansion The PIs of firms are converted into investment rate via S-shaped curves in (9), which limit the rate of variations and final values in each firm [46]. The coefficients mj max, αj and βj differ in each technology, but mj is equal to 1 for PIj = 1 in the whole, as indicated in Fig. 6. The coefficient mj is influenced by reliability policy and profitability for providing enough capacity: mj = mj max 1 + e−(αj P Ij −βj ) (9) Figure 6. The coefficient m for different technologies vs. PI. Equation (10) gives investment rate in each technology as a function of demand growth rate and retirement rate of the firms weighted by the coefficient mj: IRj = mj·( ˙Li + ˙REj) (10) Reliability policy in (11) forms an internal loop in launching process, named as launch scale [9] that changes rate of investment in each technology for holding reserve ratio at a proposed level: Res.Rat = TCP − D(t) D(t) (11) The investment rate is converted into capacity after a construction delay. Equation (12) indicates under construction capacity, which is the difference between investment rate and construction rate in each technology. Exploited capacity in (13) is the difference between constructed capacity and retired amount after a life time. A part of GT capacity is converted into CCGT by a change 24 ratio and change delay that influences the pattern of expansion. Operational capacity is declared to the market and creates the main feedback loop in this process; besides, it is used for providing reliability as an internal loop: UCj = t 0 IRj − IRj(t − CTj)dt (12) CPj = t 0 CNj − CNj(t − LTj)dt (13) 3.7 Wind Technology The wind technology competes with other participants in the market, while previous studies subtracted its capacity from the demand [14], [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology
  4. [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios
  5. [6], [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors [2]. Analysing published data by EIA about city gate price
  6. [7] via X-12-ARIMA time series
  7. [9] that changes rate of investment in each technology for holding reserve ratio at a proposed level: Res.Rat = TCP − D(t) D(t) (11) The investment rate is converted into capacity after a construction delay. Equation (12) indicates under construction capacity, which is the difference between investment rate and construction rate in each technology. Exploited capacity in (13) is the difference between constructed capacity and retired amount after a life time. A part of GT capacity is converted into CCGT by a change 24 ratio and change delay that influences the pattern of expansion. Operational capacity is declared to the market and creates the main feedback loop in this process; besides, it is used for providing reliability as an internal loop: UCj = t 0 IRj − IRj(t − CTj)dt (12) CPj = t 0 CNj − CNj(t − LTj)dt (13) 3.7 Wind Technology The wind technology competes with other participants in the market, while previous studies subtracted its capacity from the demand [14], [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6], [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors [2]. Analysing published data by EIA about city gate price [7] via X-12-ARIMA time series [8] gives the seasonal factors as shown in Fig. 7. These factors adapt to additive model that makes the real variable by adding a seasonal factor. Figure 7. The seasonal factors of the city gate natural gas price. 4. Simulation and Results This section analyses the results of simulating different states of natural gas consumption in the power system in three scenarios including firms stability, changes in the natural gas charge and access of demand to the natural gas and electricity. Some results such as the profitability index, reserve ratio, and capacity expansion are represented and compared with the base state in each scenario. The applied parameters in the simulation are from the published data by the EIA on the generation costs by different technologies, natural gas city gate prices and natural gas hub prices. The results represent the statue of the parameters in 1,200 months for indicating the stability of the developed model. 4.1 Base Scenario In this scenario, the firms stay on the stable state by adjusting the offers for recovering the generation and investment costs [45] that is achievable by PIj = 0 in (8), due to its association to these costs. They adjust their offers by adding multiples of the forecasted price to their marginal cost, which are 0.13, 0.083, 0.058 and 0.45 for the coalfired, CCGT, GT and wind, respectively. The technologies with higher investment cost need a greater coefficient for getting to the stability. Figure 8 represents the profitability index of the firms in this scenario, which tends to zero during the studied horizon. The variation of the PI around zero is a motivation for expanding the capacity by different technologies beside the growing demand. Table 1 summarizes the total present profit of the firms in this scenario. The coal-fired earns the most profit by supplying the base load and the CCGT and GT are at the next places. The wind earns the least profit in the stable state. Figure 9 represents the reserve ratio of the power system calculated by (11) for providing the reliability of the power system, which swings around 0.2 with a limited variation between 0.18 and 0.22. The reliability level is achieved by the pattern of generation capacity, shown in Fig. 10. The capacity of the coalfired, CCGT and GT gets to 8.8 × 104 MW, 4.8 × 104 MW and 3.3 × 104 MW for supplying the base, middle and peak loads, respectively and the wind technology expands its capacity to 1,400 MW in the stable state. 25 Figure 8. The profitability index of the firms in the base scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 1 Total Present Profit of the Firms at the Base Scenario Technology Profit ($) Coal-fired 3 × 106 CCGT 0.94 × 106 GT 0.33 × 106 Wind 0.085 × 106 Figure 9. The reserve ratio of the power system in the base scenario. 4.2 Natural Gas Price Variation The second scenario investigates the influence of the variations in the natural gas price on the long-run investment in the capacity expansion. The variations are modelled as low-price, medium-price and high-price outlines, summarized in Table A.1 [6], [48], [38]. The results of the high and Figure 10. The capacity of different technologies in the base scenario. Figure 11. The price of electric energy for different outlines of natural gas price. low prices of natural gas are compared with the medium price as base scenario. Figure 11 represents the price of electric energy for different outlines of natural gas price. The electric price 26 Figure 12. The profitability index of the firms at the high price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 2 Total Present Profit of the Firms at the High Price of Natural Gas Technology Profit ($) Coal-fired 24 × 106 CCGT 4.3 × 106 GT −1.3 × 106 Wind 3.4 × 106 increases by growing the natural gas price and decreases by its decline, compared with the medium price. Figure 12 indicates the profitability index of the firms at the high price of natural gas. The PI increases to 0.04, 0.01 and 0.33 for the coal-fired, CCGT and wind, respectively, but decreases to −0.0002 for the GT, due to the loss of opportunity for generation by this technology. Reducing the heat rate of the GT, increases its PI to zero on month 900 by raising its generation chance and decreases the PI of the coal-fired in Figs. 12(a) and (c). The present profit of the firms changes by their deviation from the stable state, summarized in Table 2. The profit of the coal-fired, CCGT and wind increases , compared with Table 1, but the high price is disadvantageous for the GT and causes its negative profit in Table 2. The average of the reserve ratio does not change significantly at the high price and grows to the average of 0.21 with a pattern same as Fig. 9. Compared with Fig. 10, the Figure 13. The capacity of different technologies at the high price of natural gas. capacity of different firms changes as indicated in Fig. 13; the capacity of the coal-fired and the wind increases to 9 × 104 MW and 7,300 MW, respectively, but it decreases to 4.4 × 104 MW and 3 × 104 MW for the CCGT and GT, which shows the reduced share of the natural gas consumers in the market and the great ratio of capacity expansion by the wind at high prices. Decreasing the charge of the natural gas, drops the electric price in the market, which leads to the loss of the firms at the stable offer, due to irretrievable investment costs. The profitability index of the firms drops to −0.0073, −0.007, −0.0035 and −0.49 for the coal-fired, CCGT, GT and wind in Fig. 14. Table 3 summarizes the present profit of the firms at the low charge of natural gas, which is negative for the whole and results in their loss by offering the stable price. 27 Figure 14. The profitability index of the firms at the low price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 3 Total Present Profit of the Firms at the Low Price of Natural Gas Technology Profit ($) Coal-fired −2.8 × 106 CCGT −1.3 × 106 GT −0.58 × 106 Wind −0.15 × 106 Figure 15. The capacity of different technologies at the low price of natural gas. Low-price natural gas does not influence the reliability of the power system remarkably and decreases the average of the reserve ratio to the amount 0.19 with the behaviour same as Fig. 9. The pattern of generation capacity changes at low charge of natural gas as shown in Fig. 15. The capacity of the CCGT and the GT increases to 4.9 × 104 MW and 3.4 × 104 MW, while the amount of the coal-fired and the wind decreases to 8.7 × 104 MW and 540 MW, respectively. This variation increases the share of technologies in the market that consume the natural gas. 4.3 Natural Gas Consumption by the Demand This section analyses the access to the electricity and natural gas as two separate energy resources by 10% of the demand. The demand switches between these energy resources by comparing the electric market price and natural gas city gate price and choosing the cheapest one. Figure 16 represents the PI of the firms, when 10% of the demand selects between two energy resources. The PI of the coal-fired decreases to −0.033 in Fig. 16a, but it does not change for the other technologies significantly. Table 4 summarizes the present profit of the firms at the selection of the natural gas by the demand. The loss of the profit by the coal-fired is severe, due to the variations in the base demand, which enforces it to increase its coefficient in the offer. The present profit of the other technologies does not change a lot in this scenario, compared with Table 1. The resource selection by the demand enforces the firms to expand the capacity for a discontinuous demand, which is detectable in the capacity of different technologies in Fig. 17. The capacity of the coal-fired, CCGT and GT increases with swings to 10 × 104 MW, 5.8 × 104 MW and 3.7 × 104 MW, respectively. The wind capacity decreases to 1,047 MW in this scenario. 28 Figure 16. The profitability index of the firms at the natural gas consumption by the demand scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 4 Total Present Profit of the Firms at the Natural Gas Consumption by the Demand Scenario Technology Profit ($) Coal-fired −23.1 × 106 CCGT 1 × 106 GT 0.31 × 106 Wind 0.072 × 106 Figure 17. The capacity of different technologies at the natural gas consumption by the demand scenario. Figure 18. The reserve ratio of the power system at the natural gas consumption by the demand scenario. This pattern of capacity has a negative influence on the reserve ratio of the power system for providing the reliability as shown in Fig. 18. The average of the reserve ratio swings around the average amount of 0.35 and varies between 0.15 and 0.6 in this figure. Increasing the access of the demand to both energy resources from 10% creates undesirable effects on the profitability and reserve ratio. The efficiency of the natural gas consumption by the demand is an effective factor for keeping the stability of the electric market. Growing the efficiency of the natural gas consumption to above 50%, restores the stable state of the firms in the market by keeping the PI of the firms at the zero and the reserve ratio on 0.2 with a little swing same as Figs. 8 and 9. High efficient natural gas demands 29 are dismissed form the electric market, which decreases the installed capacity at the stable state and its swings. 5. Discussion The natural gas affects the power market in generation and consumption levels. Based on the hub price of natural gas, three scenarios can be defined including, low, medium and high charges. The access of demand to the natural gas affects the market via the seasonal factors. The stable state (PI = 0) which is considered as the base scenario compensates the generation and investment costs of the firms and keeps the reserve ratio on about 0.2 for providing the reliability. This state is achieved by adding a multiple of the forecasted price to the MC by the firms, which is 0.13, 0.083, 0.058 and 0.45 for the coal-fired, CCGT, GT and wind, respectively; the technologies with higher investment cost adjust their offer with a greater coefficient. The Medium price of natural gas is considered in the base scenario. High-charge natural gas increases the market price and the firms offers, which increases the profitability index of the coal-fired, CCGT and wind from zero to 0.04, 0.01 and 0.33, respectively; however, this charge decreases the generation opportunity for GT and decreases its PI to −0.0002. So, the GT should decrease its coefficient during high price of natural gas for creating generation opportunity. By increasing the natural gas price reaches the capacity of GT and CCGT increases from 4.8 × 104 MW and 3.3 × 104 MW to 4.4 × 104 MW and 3 × 104 MW and the share of natural gas-based technologies in the market decreases. However, increasing the profit creates the opportunity for the coalfired and wind to expand their capacity and increase their share from 8.8 × 104 MW and 1,400 MW at the stable state to 9 × 104 MW and 7,300 MW. The average of reserve ratio grows from 0.2 to 0.21 in this situation. Low-charge natural gas decreases the average price in the market and reduces the PI of coal-fired, CCGT, GT and wind to −0.0073, −0.007, −0.0035 and −0.49, respectively, which results in the loss of the firms. The share of CCGT and GT increases to 4.9 × 104 MW and 3.4 × 104 MW in the market, while the capacity of the coal-fired and wind decreases to 8.7 × 104 MW and 540 MW. The capacity expansion is due to the effects of demand growth rate and retirement rate on the investment rate for providing the proposed reliability level of about 0.19. The charge of natural gas at the consumption level is influenced by seasonal factors, which has additive pattern and its amount is greater in colder months. Natural gas consumption by demand decreases the profitability index of the coal-fired to −0.033 as base supplier. This condition enforces the firms to expand their capacity, while their generation is not consumed by the demand continuously, which hardens recovering the investment. Choosing the cheapest energy resource by the demand keeps the PI of the CCGT, GT and wind at zero and increasing the percentage of demand access decreases the PI of CCGT and GT. The capacity of coal-fired, CCGT, GT and wind reaches to 10 × 104 MW, 5.8 × 104 MW, 3.7 × 104 MW and 1,047 MW. Switching between the energy resources by the demand causes the swing of the reserve ratio, which is resolved by increasing the efficiency of the natural gas consumers. 6. Conclusion This paper analyses the effect of the natural gas on the capacity expansion by the firms in a pay-as-bid energyonly market using the system dynamics. Natural gas-based technologies and demand selection between the electricity and natural gas are two considered tie points between these resources. This subject is studied via three scenarios, namely, (1) the firms stability, (2) changes in the natural gas charge and (3) access of demand to natural gas and electricity. Four generation technologies including coalfired, CCGT, GT and wind participate in an energy-only market using the published data in the reports of EIA. At the first scenario, the firms adjust their offers above the marginal cost for recovering the generation and investment costs as stable state, which is known by PIj = 0. The firms adjust their offers by adding a multiple of the forecasted price to the marginal cost, which is greater for the technologies with higher investment cost. The stable state can recover the costs of the firms and provide the targeted reliability level of the power system. The natural gas price as assumed to fluctuate between low, medium and high prices, where the medium price is applied at the base scenario. High-price natural gas increases the offers of the firms and the market price, which increases the profit of the coal-fired, CCGT and wind, but causes the loss of the profit by the GT, as it loses the opportunity for generation. More profit increment by coalfired and loss of GT reduces the share of natural gas-based technologies in the market at high charges of natural gas. Low-price natural gas decreases the offers and market price to an amount, which cannot recover the investment costs and causes the loss of the whole. The capacity is expanded for supplying the demand and compensating the retirement with a less ramp. The loss of coal-fired and wind in low charge is more severe due to their higher investment costs, which increases the share of natural gas-based technologies in the market. The reserve ratio of the power system does not change remarkably by changing the natural gas price. Selecting the natural gas as resource of energy by a portion of demand causes the loss of profit by the coal-fired and does not influence on the revenue of the rest. This behaviour of demand expands the generation capacity and increases the average of reserve ratio, but creates swing in it. Increasing the per cent of demand access to the natural gas intensifies the unstable condition in the market. The growth of efficiency in the natural gas demand dismisses it from the power market and restores it to the stability. Appendix A Table A.1 summarizes the parameters of the simulation, which are from the published report by EIA [4]. 30 Table A.1 The Parameters of the Simulation Technology Coal-Fired CCGT GT Wind Parameter Fuel Cost 65 ($/ton) Low Price = 3.8 $/MMBtu Thermal Value Medium Price = 4.6 $/MMBtu = 6080 kcal/kg High price = 6 $/MMBtu Heat rate (Btu/KWh) 9,200 6,752 9,289 Heat rate changes (Btu/KWh year) −30 −28 −50 O&M costs ($/MWh) 7.7 3.3 4.3 3.4 CO2 costs ($/MWh) 24 10.5 16 Investment costs ($/MWh) 3.7 3.3 2.3 10.97 Construction time (months) 48 36 24 6 Life time (months) 720 360 360 240 Rate of return (%/year) 5% Peak demand (MW) 1,200 Peak duration 0.2 Middle demand (MW) 1,000 Middle duration 0.6 Base demand (MW) 700 Base duration 0.2 Demand growth rate (%/year) 5% Rated wind speed (m/s) 7 Product wind speed (m/s) 4 Cut out wind speed (m/s) 13 References [1] “Projected Costs of Generating Electricity , International Energy Agency, 2010 Edition. [2] S. Macmillan, A. Antonyuk, and H. Schwind, Gas to Coal Competition in the U.S. Power Sector, International Energy Agency, May, 2013. [3] S.F. Tierney, T. Schatzki, and R. Mukerji, Uniform-Pricing versus Pay-as-Bid in Wholesale Electricity Markets: Does it Make a Difference?, Analysis Group & New York ISO, March 2008. [4] R. Tidball, J. Bluestein, N. Rodriguez, and S. Knoke, Cost and Performance Assumptions for Modeling Electricity Generation Technologies, ICF International Fairfax, Virginia, November 2010, 96–102. [5] Life Data Analysis Reference, Worldwide Headquarters, AZ, USA, May 22, 2015. [6] A. Sieminski, Annual Energy Outlook 2015, U.S. Energy Information Administration, May, 2015. [7] Indepndent Statics and Data Analysis, US Natural Gas City gate Price, U.S. Energy Information Administration, May, 2016. [8] Guide to Seasonal Adjustment with X-12-ARIMA, TSAB, March, 2007. [9] A.S. Cui, M. Zhao, and T. Ravichandran, Market uncertainty and dynamic new product launch strategies: A system dynamics model, IEEE Transactions on Engineering Management, 58(3), 2011, 530–550.
  8. [10] A. Ford, System dynamics and the electric power industry, System Dynamics Review, 13(1), 1997, 57–85.
  9. [11] F. Olsina, F. Garce, and H.J. Haubrich, Modeling long-term dynamics of electricity markets, Energy Policy, 34(12), 2006, 1411–1433.
  10. [12] A. Movahednasab, M. Rashidinejad, and A. Abdollahi, A system dynamics analysis of the long run investment in marketbased electric generation expansion with renewable resources, International Transactions on Electrical Energy Systems, 2017. DOI: 10.1002/etep.2338.
  11. [14], [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6], [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors [2]. Analysing published data by EIA about city gate price [7] via X-12-ARIMA time series [8] gives the seasonal factors as shown in Fig. 7. These factors adapt to additive model that makes the real variable by adding a seasonal factor. Figure 7. The seasonal factors of the city gate natural gas price. 4. Simulation and Results This section analyses the results of simulating different states of natural gas consumption in the power system in three scenarios including firms stability, changes in the natural gas charge and access of demand to the natural gas and electricity. Some results such as the profitability index, reserve ratio, and capacity expansion are represented and compared with the base state in each scenario. The applied parameters in the simulation are from the published data by the EIA on the generation costs by different technologies, natural gas city gate prices and natural gas hub prices. The results represent the statue of the parameters in 1,200 months for indicating the stability of the developed model. 4.1 Base Scenario In this scenario, the firms stay on the stable state by adjusting the offers for recovering the generation and investment costs [45] that is achievable by PIj = 0 in (8), due to its association to these costs. They adjust their offers by adding multiples of the forecasted price to their marginal cost, which are 0.13, 0.083, 0.058 and 0.45 for the coalfired, CCGT, GT and wind, respectively. The technologies with higher investment cost need a greater coefficient for getting to the stability. Figure 8 represents the profitability index of the firms in this scenario, which tends to zero during the studied horizon. The variation of the PI around zero is a motivation for expanding the capacity by different technologies beside the growing demand. Table 1 summarizes the total present profit of the firms in this scenario. The coal-fired earns the most profit by supplying the base load and the CCGT and GT are at the next places. The wind earns the least profit in the stable state. Figure 9 represents the reserve ratio of the power system calculated by (11) for providing the reliability of the power system, which swings around 0.2 with a limited variation between 0.18 and 0.22. The reliability level is achieved by the pattern of generation capacity, shown in Fig. 10. The capacity of the coalfired, CCGT and GT gets to 8.8 × 104 MW, 4.8 × 104 MW and 3.3 × 104 MW for supplying the base, middle and peak loads, respectively and the wind technology expands its capacity to 1,400 MW in the stable state. 25 Figure 8. The profitability index of the firms in the base scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 1 Total Present Profit of the Firms at the Base Scenario Technology Profit ($) Coal-fired 3 × 106 CCGT 0.94 × 106 GT 0.33 × 106 Wind 0.085 × 106 Figure 9. The reserve ratio of the power system in the base scenario. 4.2 Natural Gas Price Variation The second scenario investigates the influence of the variations in the natural gas price on the long-run investment in the capacity expansion. The variations are modelled as low-price, medium-price and high-price outlines, summarized in Table A.1 [6], [48], [38]. The results of the high and Figure 10. The capacity of different technologies in the base scenario. Figure 11. The price of electric energy for different outlines of natural gas price. low prices of natural gas are compared with the medium price as base scenario. Figure 11 represents the price of electric energy for different outlines of natural gas price. The electric price 26 Figure 12. The profitability index of the firms at the high price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 2 Total Present Profit of the Firms at the High Price of Natural Gas Technology Profit ($) Coal-fired 24 × 106 CCGT 4.3 × 106 GT −1.3 × 106 Wind 3.4 × 106 increases by growing the natural gas price and decreases by its decline, compared with the medium price. Figure 12 indicates the profitability index of the firms at the high price of natural gas. The PI increases to 0.04, 0.01 and 0.33 for the coal-fired, CCGT and wind, respectively, but decreases to −0.0002 for the GT, due to the loss of opportunity for generation by this technology. Reducing the heat rate of the GT, increases its PI to zero on month 900 by raising its generation chance and decreases the PI of the coal-fired in Figs. 12(a) and (c). The present profit of the firms changes by their deviation from the stable state, summarized in Table 2. The profit of the coal-fired, CCGT and wind increases , compared with Table 1, but the high price is disadvantageous for the GT and causes its negative profit in Table 2. The average of the reserve ratio does not change significantly at the high price and grows to the average of 0.21 with a pattern same as Fig. 9. Compared with Fig. 10, the Figure 13. The capacity of different technologies at the high price of natural gas. capacity of different firms changes as indicated in Fig. 13; the capacity of the coal-fired and the wind increases to 9 × 104 MW and 7,300 MW, respectively, but it decreases to 4.4 × 104 MW and 3 × 104 MW for the CCGT and GT, which shows the reduced share of the natural gas consumers in the market and the great ratio of capacity expansion by the wind at high prices. Decreasing the charge of the natural gas, drops the electric price in the market, which leads to the loss of the firms at the stable offer, due to irretrievable investment costs. The profitability index of the firms drops to −0.0073, −0.007, −0.0035 and −0.49 for the coal-fired, CCGT, GT and wind in Fig. 14. Table 3 summarizes the present profit of the firms at the low charge of natural gas, which is negative for the whole and results in their loss by offering the stable price. 27 Figure 14. The profitability index of the firms at the low price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 3 Total Present Profit of the Firms at the Low Price of Natural Gas Technology Profit ($) Coal-fired −2.8 × 106 CCGT −1.3 × 106 GT −0.58 × 106 Wind −0.15 × 106 Figure 15. The capacity of different technologies at the low price of natural gas. Low-price natural gas does not influence the reliability of the power system remarkably and decreases the average of the reserve ratio to the amount 0.19 with the behaviour same as Fig. 9. The pattern of generation capacity changes at low charge of natural gas as shown in Fig. 15. The capacity of the CCGT and the GT increases to 4.9 × 104 MW and 3.4 × 104 MW, while the amount of the coal-fired and the wind decreases to 8.7 × 104 MW and 540 MW, respectively. This variation increases the share of technologies in the market that consume the natural gas. 4.3 Natural Gas Consumption by the Demand This section analyses the access to the electricity and natural gas as two separate energy resources by 10% of the demand. The demand switches between these energy resources by comparing the electric market price and natural gas city gate price and choosing the cheapest one. Figure 16 represents the PI of the firms, when 10% of the demand selects between two energy resources. The PI of the coal-fired decreases to −0.033 in Fig. 16a, but it does not change for the other technologies significantly. Table 4 summarizes the present profit of the firms at the selection of the natural gas by the demand. The loss of the profit by the coal-fired is severe, due to the variations in the base demand, which enforces it to increase its coefficient in the offer. The present profit of the other technologies does not change a lot in this scenario, compared with Table 1. The resource selection by the demand enforces the firms to expand the capacity for a discontinuous demand, which is detectable in the capacity of different technologies in Fig. 17. The capacity of the coal-fired, CCGT and GT increases with swings to 10 × 104 MW, 5.8 × 104 MW and 3.7 × 104 MW, respectively. The wind capacity decreases to 1,047 MW in this scenario. 28 Figure 16. The profitability index of the firms at the natural gas consumption by the demand scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 4 Total Present Profit of the Firms at the Natural Gas Consumption by the Demand Scenario Technology Profit ($) Coal-fired −23.1 × 106 CCGT 1 × 106 GT 0.31 × 106 Wind 0.072 × 106 Figure 17. The capacity of different technologies at the natural gas consumption by the demand scenario. Figure 18. The reserve ratio of the power system at the natural gas consumption by the demand scenario. This pattern of capacity has a negative influence on the reserve ratio of the power system for providing the reliability as shown in Fig. 18. The average of the reserve ratio swings around the average amount of 0.35 and varies between 0.15 and 0.6 in this figure. Increasing the access of the demand to both energy resources from 10% creates undesirable effects on the profitability and reserve ratio. The efficiency of the natural gas consumption by the demand is an effective factor for keeping the stability of the electric market. Growing the efficiency of the natural gas consumption to above 50%, restores the stable state of the firms in the market by keeping the PI of the firms at the zero and the reserve ratio on 0.2 with a little swing same as Figs. 8 and 9. High efficient natural gas demands 29 are dismissed form the electric market, which decreases the installed capacity at the stable state and its swings. 5. Discussion The natural gas affects the power market in generation and consumption levels. Based on the hub price of natural gas, three scenarios can be defined including, low, medium and high charges. The access of demand to the natural gas affects the market via the seasonal factors. The stable state (PI = 0) which is considered as the base scenario compensates the generation and investment costs of the firms and keeps the reserve ratio on about 0.2 for providing the reliability. This state is achieved by adding a multiple of the forecasted price to the MC by the firms, which is 0.13, 0.083, 0.058 and 0.45 for the coal-fired, CCGT, GT and wind, respectively; the technologies with higher investment cost adjust their offer with a greater coefficient. The Medium price of natural gas is considered in the base scenario. High-charge natural gas increases the market price and the firms offers, which increases the profitability index of the coal-fired, CCGT and wind from zero to 0.04, 0.01 and 0.33, respectively; however, this charge decreases the generation opportunity for GT and decreases its PI to −0.0002. So, the GT should decrease its coefficient during high price of natural gas for creating generation opportunity. By increasing the natural gas price reaches the capacity of GT and CCGT increases from 4.8 × 104 MW and 3.3 × 104 MW to 4.4 × 104 MW and 3 × 104 MW and the share of natural gas-based technologies in the market decreases. However, increasing the profit creates the opportunity for the coalfired and wind to expand their capacity and increase their share from 8.8 × 104 MW and 1,400 MW at the stable state to 9 × 104 MW and 7,300 MW. The average of reserve ratio grows from 0.2 to 0.21 in this situation. Low-charge natural gas decreases the average price in the market and reduces the PI of coal-fired, CCGT, GT and wind to −0.0073, −0.007, −0.0035 and −0.49, respectively, which results in the loss of the firms. The share of CCGT and GT increases to 4.9 × 104 MW and 3.4 × 104 MW in the market, while the capacity of the coal-fired and wind decreases to 8.7 × 104 MW and 540 MW. The capacity expansion is due to the effects of demand growth rate and retirement rate on the investment rate for providing the proposed reliability level of about 0.19. The charge of natural gas at the consumption level is influenced by seasonal factors, which has additive pattern and its amount is greater in colder months. Natural gas consumption by demand decreases the profitability index of the coal-fired to −0.033 as base supplier. This condition enforces the firms to expand their capacity, while their generation is not consumed by the demand continuously, which hardens recovering the investment. Choosing the cheapest energy resource by the demand keeps the PI of the CCGT, GT and wind at zero and increasing the percentage of demand access decreases the PI of CCGT and GT. The capacity of coal-fired, CCGT, GT and wind reaches to 10 × 104 MW, 5.8 × 104 MW, 3.7 × 104 MW and 1,047 MW. Switching between the energy resources by the demand causes the swing of the reserve ratio, which is resolved by increasing the efficiency of the natural gas consumers. 6. Conclusion This paper analyses the effect of the natural gas on the capacity expansion by the firms in a pay-as-bid energyonly market using the system dynamics. Natural gas-based technologies and demand selection between the electricity and natural gas are two considered tie points between these resources. This subject is studied via three scenarios, namely, (1) the firms stability, (2) changes in the natural gas charge and (3) access of demand to natural gas and electricity. Four generation technologies including coalfired, CCGT, GT and wind participate in an energy-only market using the published data in the reports of EIA. At the first scenario, the firms adjust their offers above the marginal cost for recovering the generation and investment costs as stable state, which is known by PIj = 0. The firms adjust their offers by adding a multiple of the forecasted price to the marginal cost, which is greater for the technologies with higher investment cost. The stable state can recover the costs of the firms and provide the targeted reliability level of the power system. The natural gas price as assumed to fluctuate between low, medium and high prices, where the medium price is applied at the base scenario. High-price natural gas increases the offers of the firms and the market price, which increases the profit of the coal-fired, CCGT and wind, but causes the loss of the profit by the GT, as it loses the opportunity for generation. More profit increment by coalfired and loss of GT reduces the share of natural gas-based technologies in the market at high charges of natural gas. Low-price natural gas decreases the offers and market price to an amount, which cannot recover the investment costs and causes the loss of the whole. The capacity is expanded for supplying the demand and compensating the retirement with a less ramp. The loss of coal-fired and wind in low charge is more severe due to their higher investment costs, which increases the share of natural gas-based technologies in the market. The reserve ratio of the power system does not change remarkably by changing the natural gas price. Selecting the natural gas as resource of energy by a portion of demand causes the loss of profit by the coal-fired and does not influence on the revenue of the rest. This behaviour of demand expands the generation capacity and increases the average of reserve ratio, but creates swing in it. Increasing the per cent of demand access to the natural gas intensifies the unstable condition in the market. The growth of efficiency in the natural gas demand dismisses it from the power market and restores it to the stability. Appendix A Table A.1 summarizes the parameters of the simulation, which are from the published report by EIA [4]. 30 Table A.1 The Parameters of the Simulation Technology Coal-Fired CCGT GT Wind Parameter Fuel Cost 65 ($/ton) Low Price = 3.8 $/MMBtu Thermal Value Medium Price = 4.6 $/MMBtu = 6080 kcal/kg High price = 6 $/MMBtu Heat rate (Btu/KWh) 9,200 6,752 9,289 Heat rate changes (Btu/KWh year) −30 −28 −50 O&M costs ($/MWh) 7.7 3.3 4.3 3.4 CO2 costs ($/MWh) 24 10.5 16 Investment costs ($/MWh) 3.7 3.3 2.3 10.97 Construction time (months) 48 36 24 6 Life time (months) 720 360 360 240 Rate of return (%/year) 5% Peak demand (MW) 1,200 Peak duration 0.2 Middle demand (MW) 1,000 Middle duration 0.6 Base demand (MW) 700 Base duration 0.2 Demand growth rate (%/year) 5% Rated wind speed (m/s) 7 Product wind speed (m/s) 4 Cut out wind speed (m/s) 13 References [1] “Projected Costs of Generating Electricity , International Energy Agency, 2010 Edition. [2] S. Macmillan, A. Antonyuk, and H. Schwind, Gas to Coal Competition in the U.S. Power Sector, International Energy Agency, May, 2013. [3] S.F. Tierney, T. Schatzki, and R. Mukerji, Uniform-Pricing versus Pay-as-Bid in Wholesale Electricity Markets: Does it Make a Difference?, Analysis Group & New York ISO, March 2008. [4] R. Tidball, J. Bluestein, N. Rodriguez, and S. Knoke, Cost and Performance Assumptions for Modeling Electricity Generation Technologies, ICF International Fairfax, Virginia, November 2010, 96–102. [5] Life Data Analysis Reference, Worldwide Headquarters, AZ, USA, May 22, 2015. [6] A. Sieminski, Annual Energy Outlook 2015, U.S. Energy Information Administration, May, 2015. [7] Indepndent Statics and Data Analysis, US Natural Gas City gate Price, U.S. Energy Information Administration, May, 2016. [8] Guide to Seasonal Adjustment with X-12-ARIMA, TSAB, March, 2007. [9] A.S. Cui, M. Zhao, and T. Ravichandran, Market uncertainty and dynamic new product launch strategies: A system dynamics model, IEEE Transactions on Engineering Management, 58(3), 2011, 530–550. [10] A. Ford, System dynamics and the electric power industry, System Dynamics Review, 13(1), 1997, 57–85. [11] F. Olsina, F. Garce, and H.J. Haubrich, Modeling long-term dynamics of electricity markets, Energy Policy, 34(12), 2006, 1411–1433. [12] A. Movahednasab, M. Rashidinejad, and A. Abdollahi, A system dynamics analysis of the long run investment in marketbased electric generation expansion with renewable resources, International Transactions on Electrical Energy Systems, 2017. DOI: 10.1002/etep.2338. [13] D. Eager, B.F. Hobbs, and J.W. Bialek, Dynamic modeling of thermal generation capacity investment: Application to markets with high wind penetration, IEEE Transactions on Power Systems, 27(4), 2002, 2127–2137. [14] M. Hasani-Marzooni and S.H. Hosseini, Short-term market power assessment in a long-term dynamic modeling of capacity investment, IEEE Transactions on Power Systems, 28(2), 2013, 626–638.
  12. [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6], [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors [2]. Analysing published data by EIA about city gate price [7] via X-12-ARIMA time series [8] gives the seasonal factors as shown in Fig. 7. These factors adapt to additive model that makes the real variable by adding a seasonal factor. Figure 7. The seasonal factors of the city gate natural gas price. 4. Simulation and Results This section analyses the results of simulating different states of natural gas consumption in the power system in three scenarios including firms stability, changes in the natural gas charge and access of demand to the natural gas and electricity. Some results such as the profitability index, reserve ratio, and capacity expansion are represented and compared with the base state in each scenario. The applied parameters in the simulation are from the published data by the EIA on the generation costs by different technologies, natural gas city gate prices and natural gas hub prices. The results represent the statue of the parameters in 1,200 months for indicating the stability of the developed model. 4.1 Base Scenario In this scenario, the firms stay on the stable state by adjusting the offers for recovering the generation and investment costs [45] that is achievable by PIj = 0 in (8), due to its association to these costs. They adjust their offers by adding multiples of the forecasted price to their marginal cost, which are 0.13, 0.083, 0.058 and 0.45 for the coalfired, CCGT, GT and wind, respectively. The technologies with higher investment cost need a greater coefficient for getting to the stability. Figure 8 represents the profitability index of the firms in this scenario, which tends to zero during the studied horizon. The variation of the PI around zero is a motivation for expanding the capacity by different technologies beside the growing demand. Table 1 summarizes the total present profit of the firms in this scenario. The coal-fired earns the most profit by supplying the base load and the CCGT and GT are at the next places. The wind earns the least profit in the stable state. Figure 9 represents the reserve ratio of the power system calculated by (11) for providing the reliability of the power system, which swings around 0.2 with a limited variation between 0.18 and 0.22. The reliability level is achieved by the pattern of generation capacity, shown in Fig. 10. The capacity of the coalfired, CCGT and GT gets to 8.8 × 104 MW, 4.8 × 104 MW and 3.3 × 104 MW for supplying the base, middle and peak loads, respectively and the wind technology expands its capacity to 1,400 MW in the stable state. 25 Figure 8. The profitability index of the firms in the base scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 1 Total Present Profit of the Firms at the Base Scenario Technology Profit ($) Coal-fired 3 × 106 CCGT 0.94 × 106 GT 0.33 × 106 Wind 0.085 × 106 Figure 9. The reserve ratio of the power system in the base scenario. 4.2 Natural Gas Price Variation The second scenario investigates the influence of the variations in the natural gas price on the long-run investment in the capacity expansion. The variations are modelled as low-price, medium-price and high-price outlines, summarized in Table A.1 [6], [48], [38]. The results of the high and Figure 10. The capacity of different technologies in the base scenario. Figure 11. The price of electric energy for different outlines of natural gas price. low prices of natural gas are compared with the medium price as base scenario. Figure 11 represents the price of electric energy for different outlines of natural gas price. The electric price 26 Figure 12. The profitability index of the firms at the high price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 2 Total Present Profit of the Firms at the High Price of Natural Gas Technology Profit ($) Coal-fired 24 × 106 CCGT 4.3 × 106 GT −1.3 × 106 Wind 3.4 × 106 increases by growing the natural gas price and decreases by its decline, compared with the medium price. Figure 12 indicates the profitability index of the firms at the high price of natural gas. The PI increases to 0.04, 0.01 and 0.33 for the coal-fired, CCGT and wind, respectively, but decreases to −0.0002 for the GT, due to the loss of opportunity for generation by this technology. Reducing the heat rate of the GT, increases its PI to zero on month 900 by raising its generation chance and decreases the PI of the coal-fired in Figs. 12(a) and (c). The present profit of the firms changes by their deviation from the stable state, summarized in Table 2. The profit of the coal-fired, CCGT and wind increases , compared with Table 1, but the high price is disadvantageous for the GT and causes its negative profit in Table 2. The average of the reserve ratio does not change significantly at the high price and grows to the average of 0.21 with a pattern same as Fig. 9. Compared with Fig. 10, the Figure 13. The capacity of different technologies at the high price of natural gas. capacity of different firms changes as indicated in Fig. 13; the capacity of the coal-fired and the wind increases to 9 × 104 MW and 7,300 MW, respectively, but it decreases to 4.4 × 104 MW and 3 × 104 MW for the CCGT and GT, which shows the reduced share of the natural gas consumers in the market and the great ratio of capacity expansion by the wind at high prices. Decreasing the charge of the natural gas, drops the electric price in the market, which leads to the loss of the firms at the stable offer, due to irretrievable investment costs. The profitability index of the firms drops to −0.0073, −0.007, −0.0035 and −0.49 for the coal-fired, CCGT, GT and wind in Fig. 14. Table 3 summarizes the present profit of the firms at the low charge of natural gas, which is negative for the whole and results in their loss by offering the stable price. 27 Figure 14. The profitability index of the firms at the low price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 3 Total Present Profit of the Firms at the Low Price of Natural Gas Technology Profit ($) Coal-fired −2.8 × 106 CCGT −1.3 × 106 GT −0.58 × 106 Wind −0.15 × 106 Figure 15. The capacity of different technologies at the low price of natural gas. Low-price natural gas does not influence the reliability of the power system remarkably and decreases the average of the reserve ratio to the amount 0.19 with the behaviour same as Fig. 9. The pattern of generation capacity changes at low charge of natural gas as shown in Fig. 15. The capacity of the CCGT and the GT increases to 4.9 × 104 MW and 3.4 × 104 MW, while the amount of the coal-fired and the wind decreases to 8.7 × 104 MW and 540 MW, respectively. This variation increases the share of technologies in the market that consume the natural gas. 4.3 Natural Gas Consumption by the Demand This section analyses the access to the electricity and natural gas as two separate energy resources by 10% of the demand. The demand switches between these energy resources by comparing the electric market price and natural gas city gate price and choosing the cheapest one. Figure 16 represents the PI of the firms, when 10% of the demand selects between two energy resources. The PI of the coal-fired decreases to −0.033 in Fig. 16a, but it does not change for the other technologies significantly. Table 4 summarizes the present profit of the firms at the selection of the natural gas by the demand. The loss of the profit by the coal-fired is severe, due to the variations in the base demand, which enforces it to increase its coefficient in the offer. The present profit of the other technologies does not change a lot in this scenario, compared with Table 1. The resource selection by the demand enforces the firms to expand the capacity for a discontinuous demand, which is detectable in the capacity of different technologies in Fig. 17. The capacity of the coal-fired, CCGT and GT increases with swings to 10 × 104 MW, 5.8 × 104 MW and 3.7 × 104 MW, respectively. The wind capacity decreases to 1,047 MW in this scenario. 28 Figure 16. The profitability index of the firms at the natural gas consumption by the demand scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 4 Total Present Profit of the Firms at the Natural Gas Consumption by the Demand Scenario Technology Profit ($) Coal-fired −23.1 × 106 CCGT 1 × 106 GT 0.31 × 106 Wind 0.072 × 106 Figure 17. The capacity of different technologies at the natural gas consumption by the demand scenario. Figure 18. The reserve ratio of the power system at the natural gas consumption by the demand scenario. This pattern of capacity has a negative influence on the reserve ratio of the power system for providing the reliability as shown in Fig. 18. The average of the reserve ratio swings around the average amount of 0.35 and varies between 0.15 and 0.6 in this figure. Increasing the access of the demand to both energy resources from 10% creates undesirable effects on the profitability and reserve ratio. The efficiency of the natural gas consumption by the demand is an effective factor for keeping the stability of the electric market. Growing the efficiency of the natural gas consumption to above 50%, restores the stable state of the firms in the market by keeping the PI of the firms at the zero and the reserve ratio on 0.2 with a little swing same as Figs. 8 and 9. High efficient natural gas demands 29 are dismissed form the electric market, which decreases the installed capacity at the stable state and its swings. 5. Discussion The natural gas affects the power market in generation and consumption levels. Based on the hub price of natural gas, three scenarios can be defined including, low, medium and high charges. The access of demand to the natural gas affects the market via the seasonal factors. The stable state (PI = 0) which is considered as the base scenario compensates the generation and investment costs of the firms and keeps the reserve ratio on about 0.2 for providing the reliability. This state is achieved by adding a multiple of the forecasted price to the MC by the firms, which is 0.13, 0.083, 0.058 and 0.45 for the coal-fired, CCGT, GT and wind, respectively; the technologies with higher investment cost adjust their offer with a greater coefficient. The Medium price of natural gas is considered in the base scenario. High-charge natural gas increases the market price and the firms offers, which increases the profitability index of the coal-fired, CCGT and wind from zero to 0.04, 0.01 and 0.33, respectively; however, this charge decreases the generation opportunity for GT and decreases its PI to −0.0002. So, the GT should decrease its coefficient during high price of natural gas for creating generation opportunity. By increasing the natural gas price reaches the capacity of GT and CCGT increases from 4.8 × 104 MW and 3.3 × 104 MW to 4.4 × 104 MW and 3 × 104 MW and the share of natural gas-based technologies in the market decreases. However, increasing the profit creates the opportunity for the coalfired and wind to expand their capacity and increase their share from 8.8 × 104 MW and 1,400 MW at the stable state to 9 × 104 MW and 7,300 MW. The average of reserve ratio grows from 0.2 to 0.21 in this situation. Low-charge natural gas decreases the average price in the market and reduces the PI of coal-fired, CCGT, GT and wind to −0.0073, −0.007, −0.0035 and −0.49, respectively, which results in the loss of the firms. The share of CCGT and GT increases to 4.9 × 104 MW and 3.4 × 104 MW in the market, while the capacity of the coal-fired and wind decreases to 8.7 × 104 MW and 540 MW. The capacity expansion is due to the effects of demand growth rate and retirement rate on the investment rate for providing the proposed reliability level of about 0.19. The charge of natural gas at the consumption level is influenced by seasonal factors, which has additive pattern and its amount is greater in colder months. Natural gas consumption by demand decreases the profitability index of the coal-fired to −0.033 as base supplier. This condition enforces the firms to expand their capacity, while their generation is not consumed by the demand continuously, which hardens recovering the investment. Choosing the cheapest energy resource by the demand keeps the PI of the CCGT, GT and wind at zero and increasing the percentage of demand access decreases the PI of CCGT and GT. The capacity of coal-fired, CCGT, GT and wind reaches to 10 × 104 MW, 5.8 × 104 MW, 3.7 × 104 MW and 1,047 MW. Switching between the energy resources by the demand causes the swing of the reserve ratio, which is resolved by increasing the efficiency of the natural gas consumers. 6. Conclusion This paper analyses the effect of the natural gas on the capacity expansion by the firms in a pay-as-bid energyonly market using the system dynamics. Natural gas-based technologies and demand selection between the electricity and natural gas are two considered tie points between these resources. This subject is studied via three scenarios, namely, (1) the firms stability, (2) changes in the natural gas charge and (3) access of demand to natural gas and electricity. Four generation technologies including coalfired, CCGT, GT and wind participate in an energy-only market using the published data in the reports of EIA. At the first scenario, the firms adjust their offers above the marginal cost for recovering the generation and investment costs as stable state, which is known by PIj = 0. The firms adjust their offers by adding a multiple of the forecasted price to the marginal cost, which is greater for the technologies with higher investment cost. The stable state can recover the costs of the firms and provide the targeted reliability level of the power system. The natural gas price as assumed to fluctuate between low, medium and high prices, where the medium price is applied at the base scenario. High-price natural gas increases the offers of the firms and the market price, which increases the profit of the coal-fired, CCGT and wind, but causes the loss of the profit by the GT, as it loses the opportunity for generation. More profit increment by coalfired and loss of GT reduces the share of natural gas-based technologies in the market at high charges of natural gas. Low-price natural gas decreases the offers and market price to an amount, which cannot recover the investment costs and causes the loss of the whole. The capacity is expanded for supplying the demand and compensating the retirement with a less ramp. The loss of coal-fired and wind in low charge is more severe due to their higher investment costs, which increases the share of natural gas-based technologies in the market. The reserve ratio of the power system does not change remarkably by changing the natural gas price. Selecting the natural gas as resource of energy by a portion of demand causes the loss of profit by the coal-fired and does not influence on the revenue of the rest. This behaviour of demand expands the generation capacity and increases the average of reserve ratio, but creates swing in it. Increasing the per cent of demand access to the natural gas intensifies the unstable condition in the market. The growth of efficiency in the natural gas demand dismisses it from the power market and restores it to the stability. Appendix A Table A.1 summarizes the parameters of the simulation, which are from the published report by EIA [4]. 30 Table A.1 The Parameters of the Simulation Technology Coal-Fired CCGT GT Wind Parameter Fuel Cost 65 ($/ton) Low Price = 3.8 $/MMBtu Thermal Value Medium Price = 4.6 $/MMBtu = 6080 kcal/kg High price = 6 $/MMBtu Heat rate (Btu/KWh) 9,200 6,752 9,289 Heat rate changes (Btu/KWh year) −30 −28 −50 O&M costs ($/MWh) 7.7 3.3 4.3 3.4 CO2 costs ($/MWh) 24 10.5 16 Investment costs ($/MWh) 3.7 3.3 2.3 10.97 Construction time (months) 48 36 24 6 Life time (months) 720 360 360 240 Rate of return (%/year) 5% Peak demand (MW) 1,200 Peak duration 0.2 Middle demand (MW) 1,000 Middle duration 0.6 Base demand (MW) 700 Base duration 0.2 Demand growth rate (%/year) 5% Rated wind speed (m/s) 7 Product wind speed (m/s) 4 Cut out wind speed (m/s) 13 References [1] “Projected Costs of Generating Electricity , International Energy Agency, 2010 Edition. [2] S. Macmillan, A. Antonyuk, and H. 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  30. [35]–[44] studied the role of natural gas in the power system via optimization problems and solved it by different techniques. However, there is a gap in long-run analysis of natural gas effect on the power system for accessing to a real-time view about this subject. In this regard, system dynamics is applied, which is a new idea in this field. The effect is studied in generation and consumption levels. Four generation technologies including coal-fired, CCGT, GT and wind participate in a pay-as-bid energyonly market, which is chosen by some markets due to elimination of price spikes in this structure [3]. The study is followed via three scenarios, namely stable state, natural gas price variation and access of demand to natural gas and electricity. The first scenario enlarges market stability, which is known by recovering generation and investment costs. At the second scenario the effect of natural gas charge variation is analysed in low, medium and high charges and the third scenario analyses the demand access to both natural gas and electricity via seasonal factors. Unlike other researches, the wind participates at the energy market beside the thermal firms, instead of decreasing its capacity from the demand. The applied data are from published reports by EIA about generation and investment costs of different technologies, natural gas hub price, natural gas city gate price and etc., summarized in Appendix A. The rest of paper is organized as follows; Section 2 describes the concept of system dynamics briefly by introducing employed and important tools in this paper. Section 3 explains general model and its different parts. The results of model simulation in defined scenarios are represented in Section 4 and Sections 5 and 6 discuss about the results and pluralize them, respectively. Appendix A summarizes the applied data in this study. 2. Concept of System Dynamics System dynamics was approached by Sterman for analysing complex systems and system thinking in a practical method. Growing the dynamic complexity in business, industrial and social systems increases the role of modelling, predicting and analysing their complex behaviour for understanding its reasons. System dynamics is a method for 21 understanding and analysing the complex behaviours by a set of conceptual tools and modelling methods, which are helpful in simulating the long-run behaviour of a system in different policies and making better decision. Feedback control theories and nonlinear dynamics found the base of system dynamics. For long-run analysis of a system, it is necessary to understand different effective factors and their causal relation. Moreover, identifying feedbacks, delays and other linearity which leads the system to instability and modelling them by stocks and flows is the main art in analysing a system. Simulation is the only reliable way for testing the validity of the models because of complexity of relations among different nonlinear parameters, which makes understanding the behaviour of the model in a long time period impossible. Without simulation techniques, the system hard behaviour can be improved using feedbacks through the real world which is very slow and inefficient due to delays, nonliterary and costs of testing the ideas [46]. 2.1 Causal Diagram For simulating a dynamic system, different tools are needed. Causal loops are important tools for showing the structure of the feedbacks in the system and their effects. A causal diagram, in Fig. 2, consists of arrows which conFigure 2. The causal representation of a variable. Figure 3. The stock and flow variable. Figure 4. Casual diagram of the TREND function. nects related variables together and shows the influences among them. The positive sign on the arrow shows increasing Y by increment of X and negative sign indicates decreasing of Y . 2.2 Stocks and Flows One of the most limitations of casual loops is their inability in capturing the stocks and flows structure of the system. Stock structures are other tools in studying the system dynamics, which accumulate difference between inflow and outflow of a variable as shown in Fig. 3. Equation (1) expresses the relation of stocks, which create inertia in the system and provide memory for it; they are helpful for creating delays in a system by accumulating the difference between the inflow and outflow of a parameter in a process: Y (t) = t 0 X1(τ) − X2(τ)dτ + Y (t0) (1) 2.3 Forecasting Bounded rationality hypothesis (BRH) is a forecasting algorithm formed by adaptive expectation, in which current expectations are related to the current and past values as in (2). Expectations on the value of variables for time T are revised with adjustment rate κ, if forecasted value in previous periods is different from the actual amount [49]: ξe (t, T) = ξe (t, T − 1) + κ[ξ(T − 1) − ξe (t, T − 1)] (2) Sterman has proposed an expectational model based on the system dynamics, called TREND function; he has used needed times for measuring, collecting and analysing data, historic time horizon and required time for perceiving and reacting to variable changes. Figure 4 represents the structure of TREND function, which is usable for estimating fractional growth rate in input variable [46]. 22 Figure 5. Process of capacity expansion in a power market. 3. Model Description Figure 5 represents an overview of developed model. The firms adjust their offers considering their marginal cost and forecasted market price. Offers, existent capacity and average of demand are submitted to the power market for clearing market price and generation amount by each firm. Clearing the market facilitates calculation of profits and generation costs, considering the investment costs. The profits are normalized and converted into investment through some multipliers, which create under construction and generation capacities after some delays. The existent capacities return to the market via offer, which forms main feedback loop in this process. Reserve ratio makes an internal loop by changing the launch scale for providing the proposed reliability level. Hub price of natural gas acts on fuel cost of natural gasbased technologies and affects city gate price via seasonal factors. Details of different parts of the model are as follows. 3.1 Costs Marginal and investment costs are two expenses for generating electricity. The firms settle the marginal cost for generating each MWh of electric energy including fuel, CO2 and O&M costs, which grows with constant rate of return every year as indicated in (3) [1], [4]; HRj denotes required thermal energy for power generation by a technology in Btu/KWh and decreases every year [4]: MCj = (Fj·HRj + O&Mj + CO2j)(1 + rr)y . (3) Investment cost is settled through the construction period and must be recovered during the operation. Valid reports have expressed the investment cost in $/KW [1], [4], which is convertible into $/KWh using life time of each technology in (4). The offers are adjusted using the marginal cost and the investment cost must be recovered 23 under the influence of market condition during the operation period: ICj($/KWh) = ICj($/KW) LTj × 8760 (1 + rr)y . (4) 3.2 Electric Demand Demand of electric energy is modelled by load duration curve (LDC) for base, middle and peak sections, which are supplied by coal-fired, CCGT and GT, respectively; it grows in each section with a constant growth rate every year and the average amount by (5) is offered to the market as market demand. Market price affects the demand via demand elasticity: D(t) = k i=1 δi·Li·eg·y − λ·Δρ. (5) 3.3 Power Market This paper considers an energy-only market with pay-asbid structure, in which the lowest offers are dispatched and receive their offers from the market and market price is equal to average offer. The superiority of this structure in eliminating price spikes persuaded some markets to prefer it over the uniform price. The firms should adjust their offers properly, above the MC and below their prediction of price via the TREND function; they should make a balance between their profit and the chance to win in the market [3], [45]. 3.4 Profitability Clearing the market specifies the generation of each firm, which is applicable in computing their costs and profits. Total generation cost, in (6), is the sum of firm’s expenses for generating electric energy until studied time t: Φj = t 0 Gj·MCjdt (6) By subtracting the generation and investment costs from the income, the total profit of the firms is given by Πj = t 0 Gj·χj − Gj·MCj − CPj·ICjdt (7) Profitability index is defined in (8), as the ratio of profit to generation cost, for normalizing the profits to a same quantity [46]. This parameter is helpful in investing in a technology rather than its profit: PIj = Πj Φj (8) 3.5 Stable State A market can become stable by recovering its generation and investment costs of the firms [45]. This condition is equivalent to PIj = 0, as both costs are considered in PIj. The firms can reach the stable state by offering a price equal to MC plus a multiple of forecasted price [3], named as stable price. 3.6 Capacity Expansion The PIs of firms are converted into investment rate via S-shaped curves in (9), which limit the rate of variations and final values in each firm [46]. The coefficients mj max, αj and βj differ in each technology, but mj is equal to 1 for PIj = 1 in the whole, as indicated in Fig. 6. The coefficient mj is influenced by reliability policy and profitability for providing enough capacity: mj = mj max 1 + e−(αj P Ij −βj ) (9) Figure 6. The coefficient m for different technologies vs. PI. Equation (10) gives investment rate in each technology as a function of demand growth rate and retirement rate of the firms weighted by the coefficient mj: IRj = mj·( ˙Li + ˙REj) (10) Reliability policy in (11) forms an internal loop in launching process, named as launch scale [9] that changes rate of investment in each technology for holding reserve ratio at a proposed level: Res.Rat = TCP − D(t) D(t) (11) The investment rate is converted into capacity after a construction delay. Equation (12) indicates under construction capacity, which is the difference between investment rate and construction rate in each technology. Exploited capacity in (13) is the difference between constructed capacity and retired amount after a life time. A part of GT capacity is converted into CCGT by a change 24 ratio and change delay that influences the pattern of expansion. Operational capacity is declared to the market and creates the main feedback loop in this process; besides, it is used for providing reliability as an internal loop: UCj = t 0 IRj − IRj(t − CTj)dt (12) CPj = t 0 CNj − CNj(t − LTj)dt (13) 3.7 Wind Technology The wind technology competes with other participants in the market, while previous studies subtracted its capacity from the demand [14], [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6], [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors [2]. Analysing published data by EIA about city gate price [7] via X-12-ARIMA time series [8] gives the seasonal factors as shown in Fig. 7. These factors adapt to additive model that makes the real variable by adding a seasonal factor. Figure 7. The seasonal factors of the city gate natural gas price. 4. Simulation and Results This section analyses the results of simulating different states of natural gas consumption in the power system in three scenarios including firms stability, changes in the natural gas charge and access of demand to the natural gas and electricity. Some results such as the profitability index, reserve ratio, and capacity expansion are represented and compared with the base state in each scenario. The applied parameters in the simulation are from the published data by the EIA on the generation costs by different technologies, natural gas city gate prices and natural gas hub prices. The results represent the statue of the parameters in 1,200 months for indicating the stability of the developed model. 4.1 Base Scenario In this scenario, the firms stay on the stable state by adjusting the offers for recovering the generation and investment costs [45] that is achievable by PIj = 0 in (8), due to its association to these costs. They adjust their offers by adding multiples of the forecasted price to their marginal cost, which are 0.13, 0.083, 0.058 and 0.45 for the coalfired, CCGT, GT and wind, respectively. The technologies with higher investment cost need a greater coefficient for getting to the stability. Figure 8 represents the profitability index of the firms in this scenario, which tends to zero during the studied horizon. The variation of the PI around zero is a motivation for expanding the capacity by different technologies beside the growing demand. Table 1 summarizes the total present profit of the firms in this scenario. The coal-fired earns the most profit by supplying the base load and the CCGT and GT are at the next places. The wind earns the least profit in the stable state. Figure 9 represents the reserve ratio of the power system calculated by (11) for providing the reliability of the power system, which swings around 0.2 with a limited variation between 0.18 and 0.22. The reliability level is achieved by the pattern of generation capacity, shown in Fig. 10. The capacity of the coalfired, CCGT and GT gets to 8.8 × 104 MW, 4.8 × 104 MW and 3.3 × 104 MW for supplying the base, middle and peak loads, respectively and the wind technology expands its capacity to 1,400 MW in the stable state. 25 Figure 8. The profitability index of the firms in the base scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 1 Total Present Profit of the Firms at the Base Scenario Technology Profit ($) Coal-fired 3 × 106 CCGT 0.94 × 106 GT 0.33 × 106 Wind 0.085 × 106 Figure 9. The reserve ratio of the power system in the base scenario. 4.2 Natural Gas Price Variation The second scenario investigates the influence of the variations in the natural gas price on the long-run investment in the capacity expansion. The variations are modelled as low-price, medium-price and high-price outlines, summarized in Table A.1 [6], [48], [38]. The results of the high and Figure 10. The capacity of different technologies in the base scenario. Figure 11. The price of electric energy for different outlines of natural gas price. low prices of natural gas are compared with the medium price as base scenario. Figure 11 represents the price of electric energy for different outlines of natural gas price. The electric price 26 Figure 12. The profitability index of the firms at the high price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 2 Total Present Profit of the Firms at the High Price of Natural Gas Technology Profit ($) Coal-fired 24 × 106 CCGT 4.3 × 106 GT −1.3 × 106 Wind 3.4 × 106 increases by growing the natural gas price and decreases by its decline, compared with the medium price. Figure 12 indicates the profitability index of the firms at the high price of natural gas. The PI increases to 0.04, 0.01 and 0.33 for the coal-fired, CCGT and wind, respectively, but decreases to −0.0002 for the GT, due to the loss of opportunity for generation by this technology. Reducing the heat rate of the GT, increases its PI to zero on month 900 by raising its generation chance and decreases the PI of the coal-fired in Figs. 12(a) and (c). The present profit of the firms changes by their deviation from the stable state, summarized in Table 2. The profit of the coal-fired, CCGT and wind increases , compared with Table 1, but the high price is disadvantageous for the GT and causes its negative profit in Table 2. The average of the reserve ratio does not change significantly at the high price and grows to the average of 0.21 with a pattern same as Fig. 9. Compared with Fig. 10, the Figure 13. The capacity of different technologies at the high price of natural gas. capacity of different firms changes as indicated in Fig. 13; the capacity of the coal-fired and the wind increases to 9 × 104 MW and 7,300 MW, respectively, but it decreases to 4.4 × 104 MW and 3 × 104 MW for the CCGT and GT, which shows the reduced share of the natural gas consumers in the market and the great ratio of capacity expansion by the wind at high prices. Decreasing the charge of the natural gas, drops the electric price in the market, which leads to the loss of the firms at the stable offer, due to irretrievable investment costs. The profitability index of the firms drops to −0.0073, −0.007, −0.0035 and −0.49 for the coal-fired, CCGT, GT and wind in Fig. 14. Table 3 summarizes the present profit of the firms at the low charge of natural gas, which is negative for the whole and results in their loss by offering the stable price. 27 Figure 14. The profitability index of the firms at the low price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 3 Total Present Profit of the Firms at the Low Price of Natural Gas Technology Profit ($) Coal-fired −2.8 × 106 CCGT −1.3 × 106 GT −0.58 × 106 Wind −0.15 × 106 Figure 15. The capacity of different technologies at the low price of natural gas. Low-price natural gas does not influence the reliability of the power system remarkably and decreases the average of the reserve ratio to the amount 0.19 with the behaviour same as Fig. 9. The pattern of generation capacity changes at low charge of natural gas as shown in Fig. 15. The capacity of the CCGT and the GT increases to 4.9 × 104 MW and 3.4 × 104 MW, while the amount of the coal-fired and the wind decreases to 8.7 × 104 MW and 540 MW, respectively. This variation increases the share of technologies in the market that consume the natural gas. 4.3 Natural Gas Consumption by the Demand This section analyses the access to the electricity and natural gas as two separate energy resources by 10% of the demand. The demand switches between these energy resources by comparing the electric market price and natural gas city gate price and choosing the cheapest one. Figure 16 represents the PI of the firms, when 10% of the demand selects between two energy resources. The PI of the coal-fired decreases to −0.033 in Fig. 16a, but it does not change for the other technologies significantly. Table 4 summarizes the present profit of the firms at the selection of the natural gas by the demand. The loss of the profit by the coal-fired is severe, due to the variations in the base demand, which enforces it to increase its coefficient in the offer. The present profit of the other technologies does not change a lot in this scenario, compared with Table 1. The resource selection by the demand enforces the firms to expand the capacity for a discontinuous demand, which is detectable in the capacity of different technologies in Fig. 17. The capacity of the coal-fired, CCGT and GT increases with swings to 10 × 104 MW, 5.8 × 104 MW and 3.7 × 104 MW, respectively. The wind capacity decreases to 1,047 MW in this scenario. 28 Figure 16. The profitability index of the firms at the natural gas consumption by the demand scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 4 Total Present Profit of the Firms at the Natural Gas Consumption by the Demand Scenario Technology Profit ($) Coal-fired −23.1 × 106 CCGT 1 × 106 GT 0.31 × 106 Wind 0.072 × 106 Figure 17. The capacity of different technologies at the natural gas consumption by the demand scenario. Figure 18. The reserve ratio of the power system at the natural gas consumption by the demand scenario. This pattern of capacity has a negative influence on the reserve ratio of the power system for providing the reliability as shown in Fig. 18. The average of the reserve ratio swings around the average amount of 0.35 and varies between 0.15 and 0.6 in this figure. Increasing the access of the demand to both energy resources from 10% creates undesirable effects on the profitability and reserve ratio. The efficiency of the natural gas consumption by the demand is an effective factor for keeping the stability of the electric market. Growing the efficiency of the natural gas consumption to above 50%, restores the stable state of the firms in the market by keeping the PI of the firms at the zero and the reserve ratio on 0.2 with a little swing same as Figs. 8 and 9. High efficient natural gas demands 29 are dismissed form the electric market, which decreases the installed capacity at the stable state and its swings. 5. Discussion The natural gas affects the power market in generation and consumption levels. Based on the hub price of natural gas, three scenarios can be defined including, low, medium and high charges. The access of demand to the natural gas affects the market via the seasonal factors. The stable state (PI = 0) which is considered as the base scenario compensates the generation and investment costs of the firms and keeps the reserve ratio on about 0.2 for providing the reliability. This state is achieved by adding a multiple of the forecasted price to the MC by the firms, which is 0.13, 0.083, 0.058 and 0.45 for the coal-fired, CCGT, GT and wind, respectively; the technologies with higher investment cost adjust their offer with a greater coefficient. The Medium price of natural gas is considered in the base scenario. High-charge natural gas increases the market price and the firms offers, which increases the profitability index of the coal-fired, CCGT and wind from zero to 0.04, 0.01 and 0.33, respectively; however, this charge decreases the generation opportunity for GT and decreases its PI to −0.0002. So, the GT should decrease its coefficient during high price of natural gas for creating generation opportunity. By increasing the natural gas price reaches the capacity of GT and CCGT increases from 4.8 × 104 MW and 3.3 × 104 MW to 4.4 × 104 MW and 3 × 104 MW and the share of natural gas-based technologies in the market decreases. However, increasing the profit creates the opportunity for the coalfired and wind to expand their capacity and increase their share from 8.8 × 104 MW and 1,400 MW at the stable state to 9 × 104 MW and 7,300 MW. The average of reserve ratio grows from 0.2 to 0.21 in this situation. Low-charge natural gas decreases the average price in the market and reduces the PI of coal-fired, CCGT, GT and wind to −0.0073, −0.007, −0.0035 and −0.49, respectively, which results in the loss of the firms. The share of CCGT and GT increases to 4.9 × 104 MW and 3.4 × 104 MW in the market, while the capacity of the coal-fired and wind decreases to 8.7 × 104 MW and 540 MW. The capacity expansion is due to the effects of demand growth rate and retirement rate on the investment rate for providing the proposed reliability level of about 0.19. The charge of natural gas at the consumption level is influenced by seasonal factors, which has additive pattern and its amount is greater in colder months. Natural gas consumption by demand decreases the profitability index of the coal-fired to −0.033 as base supplier. This condition enforces the firms to expand their capacity, while their generation is not consumed by the demand continuously, which hardens recovering the investment. Choosing the cheapest energy resource by the demand keeps the PI of the CCGT, GT and wind at zero and increasing the percentage of demand access decreases the PI of CCGT and GT. The capacity of coal-fired, CCGT, GT and wind reaches to 10 × 104 MW, 5.8 × 104 MW, 3.7 × 104 MW and 1,047 MW. Switching between the energy resources by the demand causes the swing of the reserve ratio, which is resolved by increasing the efficiency of the natural gas consumers. 6. Conclusion This paper analyses the effect of the natural gas on the capacity expansion by the firms in a pay-as-bid energyonly market using the system dynamics. Natural gas-based technologies and demand selection between the electricity and natural gas are two considered tie points between these resources. This subject is studied via three scenarios, namely, (1) the firms stability, (2) changes in the natural gas charge and (3) access of demand to natural gas and electricity. Four generation technologies including coalfired, CCGT, GT and wind participate in an energy-only market using the published data in the reports of EIA. At the first scenario, the firms adjust their offers above the marginal cost for recovering the generation and investment costs as stable state, which is known by PIj = 0. The firms adjust their offers by adding a multiple of the forecasted price to the marginal cost, which is greater for the technologies with higher investment cost. The stable state can recover the costs of the firms and provide the targeted reliability level of the power system. The natural gas price as assumed to fluctuate between low, medium and high prices, where the medium price is applied at the base scenario. High-price natural gas increases the offers of the firms and the market price, which increases the profit of the coal-fired, CCGT and wind, but causes the loss of the profit by the GT, as it loses the opportunity for generation. More profit increment by coalfired and loss of GT reduces the share of natural gas-based technologies in the market at high charges of natural gas. Low-price natural gas decreases the offers and market price to an amount, which cannot recover the investment costs and causes the loss of the whole. The capacity is expanded for supplying the demand and compensating the retirement with a less ramp. The loss of coal-fired and wind in low charge is more severe due to their higher investment costs, which increases the share of natural gas-based technologies in the market. The reserve ratio of the power system does not change remarkably by changing the natural gas price. Selecting the natural gas as resource of energy by a portion of demand causes the loss of profit by the coal-fired and does not influence on the revenue of the rest. This behaviour of demand expands the generation capacity and increases the average of reserve ratio, but creates swing in it. Increasing the per cent of demand access to the natural gas intensifies the unstable condition in the market. The growth of efficiency in the natural gas demand dismisses it from the power market and restores it to the stability. Appendix A Table A.1 summarizes the parameters of the simulation, which are from the published report by EIA [4]. 30 Table A.1 The Parameters of the Simulation Technology Coal-Fired CCGT GT Wind Parameter Fuel Cost 65 ($/ton) Low Price = 3.8 $/MMBtu Thermal Value Medium Price = 4.6 $/MMBtu = 6080 kcal/kg High price = 6 $/MMBtu Heat rate (Btu/KWh) 9,200 6,752 9,289 Heat rate changes (Btu/KWh year) −30 −28 −50 O&M costs ($/MWh) 7.7 3.3 4.3 3.4 CO2 costs ($/MWh) 24 10.5 16 Investment costs ($/MWh) 3.7 3.3 2.3 10.97 Construction time (months) 48 36 24 6 Life time (months) 720 360 360 240 Rate of return (%/year) 5% Peak demand (MW) 1,200 Peak duration 0.2 Middle demand (MW) 1,000 Middle duration 0.6 Base demand (MW) 700 Base duration 0.2 Demand growth rate (%/year) 5% Rated wind speed (m/s) 7 Product wind speed (m/s) 4 Cut out wind speed (m/s) 13 References [1] “Projected Costs of Generating Electricity , International Energy Agency, 2010 Edition. [2] S. Macmillan, A. Antonyuk, and H. Schwind, Gas to Coal Competition in the U.S. Power Sector, International Energy Agency, May, 2013. [3] S.F. Tierney, T. Schatzki, and R. Mukerji, Uniform-Pricing versus Pay-as-Bid in Wholesale Electricity Markets: Does it Make a Difference?, Analysis Group & New York ISO, March 2008. [4] R. Tidball, J. Bluestein, N. Rodriguez, and S. Knoke, Cost and Performance Assumptions for Modeling Electricity Generation Technologies, ICF International Fairfax, Virginia, November 2010, 96–102. [5] Life Data Analysis Reference, Worldwide Headquarters, AZ, USA, May 22, 2015. [6] A. Sieminski, Annual Energy Outlook 2015, U.S. Energy Information Administration, May, 2015. [7] Indepndent Statics and Data Analysis, US Natural Gas City gate Price, U.S. Energy Information Administration, May, 2016. [8] Guide to Seasonal Adjustment with X-12-ARIMA, TSAB, March, 2007. [9] A.S. Cui, M. Zhao, and T. 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  31. [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6],
  32. [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors [2]. Analysing published data by EIA about city gate price [7] via X-12-ARIMA time series [8] gives the seasonal factors as shown in Fig. 7. These factors adapt to additive model that makes the real variable by adding a seasonal factor. Figure 7. The seasonal factors of the city gate natural gas price. 4. Simulation and Results This section analyses the results of simulating different states of natural gas consumption in the power system in three scenarios including firms stability, changes in the natural gas charge and access of demand to the natural gas and electricity. Some results such as the profitability index, reserve ratio, and capacity expansion are represented and compared with the base state in each scenario. The applied parameters in the simulation are from the published data by the EIA on the generation costs by different technologies, natural gas city gate prices and natural gas hub prices. The results represent the statue of the parameters in 1,200 months for indicating the stability of the developed model. 4.1 Base Scenario In this scenario, the firms stay on the stable state by adjusting the offers for recovering the generation and investment costs [45] that is achievable by PIj = 0 in (8), due to its association to these costs. They adjust their offers by adding multiples of the forecasted price to their marginal cost, which are 0.13, 0.083, 0.058 and 0.45 for the coalfired, CCGT, GT and wind, respectively. The technologies with higher investment cost need a greater coefficient for getting to the stability. Figure 8 represents the profitability index of the firms in this scenario, which tends to zero during the studied horizon. The variation of the PI around zero is a motivation for expanding the capacity by different technologies beside the growing demand. Table 1 summarizes the total present profit of the firms in this scenario. The coal-fired earns the most profit by supplying the base load and the CCGT and GT are at the next places. The wind earns the least profit in the stable state. Figure 9 represents the reserve ratio of the power system calculated by (11) for providing the reliability of the power system, which swings around 0.2 with a limited variation between 0.18 and 0.22. The reliability level is achieved by the pattern of generation capacity, shown in Fig. 10. The capacity of the coalfired, CCGT and GT gets to 8.8 × 104 MW, 4.8 × 104 MW and 3.3 × 104 MW for supplying the base, middle and peak loads, respectively and the wind technology expands its capacity to 1,400 MW in the stable state. 25 Figure 8. The profitability index of the firms in the base scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 1 Total Present Profit of the Firms at the Base Scenario Technology Profit ($) Coal-fired 3 × 106 CCGT 0.94 × 106 GT 0.33 × 106 Wind 0.085 × 106 Figure 9. The reserve ratio of the power system in the base scenario. 4.2 Natural Gas Price Variation The second scenario investigates the influence of the variations in the natural gas price on the long-run investment in the capacity expansion. The variations are modelled as low-price, medium-price and high-price outlines, summarized in Table A.1 [6], [48], [38]. The results of the high and Figure 10. The capacity of different technologies in the base scenario. Figure 11. The price of electric energy for different outlines of natural gas price. low prices of natural gas are compared with the medium price as base scenario. Figure 11 represents the price of electric energy for different outlines of natural gas price. The electric price 26 Figure 12. The profitability index of the firms at the high price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 2 Total Present Profit of the Firms at the High Price of Natural Gas Technology Profit ($) Coal-fired 24 × 106 CCGT 4.3 × 106 GT −1.3 × 106 Wind 3.4 × 106 increases by growing the natural gas price and decreases by its decline, compared with the medium price. Figure 12 indicates the profitability index of the firms at the high price of natural gas. The PI increases to 0.04, 0.01 and 0.33 for the coal-fired, CCGT and wind, respectively, but decreases to −0.0002 for the GT, due to the loss of opportunity for generation by this technology. Reducing the heat rate of the GT, increases its PI to zero on month 900 by raising its generation chance and decreases the PI of the coal-fired in Figs. 12(a) and (c). The present profit of the firms changes by their deviation from the stable state, summarized in Table 2. The profit of the coal-fired, CCGT and wind increases , compared with Table 1, but the high price is disadvantageous for the GT and causes its negative profit in Table 2. The average of the reserve ratio does not change significantly at the high price and grows to the average of 0.21 with a pattern same as Fig. 9. Compared with Fig. 10, the Figure 13. The capacity of different technologies at the high price of natural gas. capacity of different firms changes as indicated in Fig. 13; the capacity of the coal-fired and the wind increases to 9 × 104 MW and 7,300 MW, respectively, but it decreases to 4.4 × 104 MW and 3 × 104 MW for the CCGT and GT, which shows the reduced share of the natural gas consumers in the market and the great ratio of capacity expansion by the wind at high prices. Decreasing the charge of the natural gas, drops the electric price in the market, which leads to the loss of the firms at the stable offer, due to irretrievable investment costs. The profitability index of the firms drops to −0.0073, −0.007, −0.0035 and −0.49 for the coal-fired, CCGT, GT and wind in Fig. 14. Table 3 summarizes the present profit of the firms at the low charge of natural gas, which is negative for the whole and results in their loss by offering the stable price. 27 Figure 14. The profitability index of the firms at the low price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 3 Total Present Profit of the Firms at the Low Price of Natural Gas Technology Profit ($) Coal-fired −2.8 × 106 CCGT −1.3 × 106 GT −0.58 × 106 Wind −0.15 × 106 Figure 15. The capacity of different technologies at the low price of natural gas. Low-price natural gas does not influence the reliability of the power system remarkably and decreases the average of the reserve ratio to the amount 0.19 with the behaviour same as Fig. 9. The pattern of generation capacity changes at low charge of natural gas as shown in Fig. 15. The capacity of the CCGT and the GT increases to 4.9 × 104 MW and 3.4 × 104 MW, while the amount of the coal-fired and the wind decreases to 8.7 × 104 MW and 540 MW, respectively. This variation increases the share of technologies in the market that consume the natural gas. 4.3 Natural Gas Consumption by the Demand This section analyses the access to the electricity and natural gas as two separate energy resources by 10% of the demand. The demand switches between these energy resources by comparing the electric market price and natural gas city gate price and choosing the cheapest one. Figure 16 represents the PI of the firms, when 10% of the demand selects between two energy resources. The PI of the coal-fired decreases to −0.033 in Fig. 16a, but it does not change for the other technologies significantly. Table 4 summarizes the present profit of the firms at the selection of the natural gas by the demand. The loss of the profit by the coal-fired is severe, due to the variations in the base demand, which enforces it to increase its coefficient in the offer. The present profit of the other technologies does not change a lot in this scenario, compared with Table 1. The resource selection by the demand enforces the firms to expand the capacity for a discontinuous demand, which is detectable in the capacity of different technologies in Fig. 17. The capacity of the coal-fired, CCGT and GT increases with swings to 10 × 104 MW, 5.8 × 104 MW and 3.7 × 104 MW, respectively. The wind capacity decreases to 1,047 MW in this scenario. 28 Figure 16. The profitability index of the firms at the natural gas consumption by the demand scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 4 Total Present Profit of the Firms at the Natural Gas Consumption by the Demand Scenario Technology Profit ($) Coal-fired −23.1 × 106 CCGT 1 × 106 GT 0.31 × 106 Wind 0.072 × 106 Figure 17. The capacity of different technologies at the natural gas consumption by the demand scenario. Figure 18. The reserve ratio of the power system at the natural gas consumption by the demand scenario. This pattern of capacity has a negative influence on the reserve ratio of the power system for providing the reliability as shown in Fig. 18. The average of the reserve ratio swings around the average amount of 0.35 and varies between 0.15 and 0.6 in this figure. Increasing the access of the demand to both energy resources from 10% creates undesirable effects on the profitability and reserve ratio. The efficiency of the natural gas consumption by the demand is an effective factor for keeping the stability of the electric market. Growing the efficiency of the natural gas consumption to above 50%, restores the stable state of the firms in the market by keeping the PI of the firms at the zero and the reserve ratio on 0.2 with a little swing same as Figs. 8 and 9. High efficient natural gas demands 29 are dismissed form the electric market, which decreases the installed capacity at the stable state and its swings. 5. Discussion The natural gas affects the power market in generation and consumption levels. Based on the hub price of natural gas, three scenarios can be defined including, low, medium and high charges. The access of demand to the natural gas affects the market via the seasonal factors. The stable state (PI = 0) which is considered as the base scenario compensates the generation and investment costs of the firms and keeps the reserve ratio on about 0.2 for providing the reliability. This state is achieved by adding a multiple of the forecasted price to the MC by the firms, which is 0.13, 0.083, 0.058 and 0.45 for the coal-fired, CCGT, GT and wind, respectively; the technologies with higher investment cost adjust their offer with a greater coefficient. The Medium price of natural gas is considered in the base scenario. High-charge natural gas increases the market price and the firms offers, which increases the profitability index of the coal-fired, CCGT and wind from zero to 0.04, 0.01 and 0.33, respectively; however, this charge decreases the generation opportunity for GT and decreases its PI to −0.0002. So, the GT should decrease its coefficient during high price of natural gas for creating generation opportunity. By increasing the natural gas price reaches the capacity of GT and CCGT increases from 4.8 × 104 MW and 3.3 × 104 MW to 4.4 × 104 MW and 3 × 104 MW and the share of natural gas-based technologies in the market decreases. However, increasing the profit creates the opportunity for the coalfired and wind to expand their capacity and increase their share from 8.8 × 104 MW and 1,400 MW at the stable state to 9 × 104 MW and 7,300 MW. The average of reserve ratio grows from 0.2 to 0.21 in this situation. Low-charge natural gas decreases the average price in the market and reduces the PI of coal-fired, CCGT, GT and wind to −0.0073, −0.007, −0.0035 and −0.49, respectively, which results in the loss of the firms. The share of CCGT and GT increases to 4.9 × 104 MW and 3.4 × 104 MW in the market, while the capacity of the coal-fired and wind decreases to 8.7 × 104 MW and 540 MW. The capacity expansion is due to the effects of demand growth rate and retirement rate on the investment rate for providing the proposed reliability level of about 0.19. The charge of natural gas at the consumption level is influenced by seasonal factors, which has additive pattern and its amount is greater in colder months. Natural gas consumption by demand decreases the profitability index of the coal-fired to −0.033 as base supplier. This condition enforces the firms to expand their capacity, while their generation is not consumed by the demand continuously, which hardens recovering the investment. Choosing the cheapest energy resource by the demand keeps the PI of the CCGT, GT and wind at zero and increasing the percentage of demand access decreases the PI of CCGT and GT. The capacity of coal-fired, CCGT, GT and wind reaches to 10 × 104 MW, 5.8 × 104 MW, 3.7 × 104 MW and 1,047 MW. Switching between the energy resources by the demand causes the swing of the reserve ratio, which is resolved by increasing the efficiency of the natural gas consumers. 6. Conclusion This paper analyses the effect of the natural gas on the capacity expansion by the firms in a pay-as-bid energyonly market using the system dynamics. Natural gas-based technologies and demand selection between the electricity and natural gas are two considered tie points between these resources. This subject is studied via three scenarios, namely, (1) the firms stability, (2) changes in the natural gas charge and (3) access of demand to natural gas and electricity. Four generation technologies including coalfired, CCGT, GT and wind participate in an energy-only market using the published data in the reports of EIA. At the first scenario, the firms adjust their offers above the marginal cost for recovering the generation and investment costs as stable state, which is known by PIj = 0. The firms adjust their offers by adding a multiple of the forecasted price to the marginal cost, which is greater for the technologies with higher investment cost. The stable state can recover the costs of the firms and provide the targeted reliability level of the power system. The natural gas price as assumed to fluctuate between low, medium and high prices, where the medium price is applied at the base scenario. High-price natural gas increases the offers of the firms and the market price, which increases the profit of the coal-fired, CCGT and wind, but causes the loss of the profit by the GT, as it loses the opportunity for generation. More profit increment by coalfired and loss of GT reduces the share of natural gas-based technologies in the market at high charges of natural gas. Low-price natural gas decreases the offers and market price to an amount, which cannot recover the investment costs and causes the loss of the whole. The capacity is expanded for supplying the demand and compensating the retirement with a less ramp. The loss of coal-fired and wind in low charge is more severe due to their higher investment costs, which increases the share of natural gas-based technologies in the market. The reserve ratio of the power system does not change remarkably by changing the natural gas price. Selecting the natural gas as resource of energy by a portion of demand causes the loss of profit by the coal-fired and does not influence on the revenue of the rest. This behaviour of demand expands the generation capacity and increases the average of reserve ratio, but creates swing in it. Increasing the per cent of demand access to the natural gas intensifies the unstable condition in the market. The growth of efficiency in the natural gas demand dismisses it from the power market and restores it to the stability. Appendix A Table A.1 summarizes the parameters of the simulation, which are from the published report by EIA [4]. 30 Table A.1 The Parameters of the Simulation Technology Coal-Fired CCGT GT Wind Parameter Fuel Cost 65 ($/ton) Low Price = 3.8 $/MMBtu Thermal Value Medium Price = 4.6 $/MMBtu = 6080 kcal/kg High price = 6 $/MMBtu Heat rate (Btu/KWh) 9,200 6,752 9,289 Heat rate changes (Btu/KWh year) −30 −28 −50 O&M costs ($/MWh) 7.7 3.3 4.3 3.4 CO2 costs ($/MWh) 24 10.5 16 Investment costs ($/MWh) 3.7 3.3 2.3 10.97 Construction time (months) 48 36 24 6 Life time (months) 720 360 360 240 Rate of return (%/year) 5% Peak demand (MW) 1,200 Peak duration 0.2 Middle demand (MW) 1,000 Middle duration 0.6 Base demand (MW) 700 Base duration 0.2 Demand growth rate (%/year) 5% Rated wind speed (m/s) 7 Product wind speed (m/s) 4 Cut out wind speed (m/s) 13 References [1] “Projected Costs of Generating Electricity , International Energy Agency, 2010 Edition. [2] S. Macmillan, A. Antonyuk, and H. 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  37. [44] studied the role of natural gas in the power system via optimization problems and solved it by different techniques. However, there is a gap in long-run analysis of natural gas effect on the power system for accessing to a real-time view about this subject. In this regard, system dynamics is applied, which is a new idea in this field. The effect is studied in generation and consumption levels. Four generation technologies including coal-fired, CCGT, GT and wind participate in a pay-as-bid energyonly market, which is chosen by some markets due to elimination of price spikes in this structure [3]. The study is followed via three scenarios, namely stable state, natural gas price variation and access of demand to natural gas and electricity. The first scenario enlarges market stability, which is known by recovering generation and investment costs. At the second scenario the effect of natural gas charge variation is analysed in low, medium and high charges and the third scenario analyses the demand access to both natural gas and electricity via seasonal factors. Unlike other researches, the wind participates at the energy market beside the thermal firms, instead of decreasing its capacity from the demand. The applied data are from published reports by EIA about generation and investment costs of different technologies, natural gas hub price, natural gas city gate price and etc., summarized in Appendix A. The rest of paper is organized as follows; Section 2 describes the concept of system dynamics briefly by introducing employed and important tools in this paper. Section 3 explains general model and its different parts. The results of model simulation in defined scenarios are represented in Section 4 and Sections 5 and 6 discuss about the results and pluralize them, respectively. Appendix A summarizes the applied data in this study. 2. Concept of System Dynamics System dynamics was approached by Sterman for analysing complex systems and system thinking in a practical method. Growing the dynamic complexity in business, industrial and social systems increases the role of modelling, predicting and analysing their complex behaviour for understanding its reasons. System dynamics is a method for 21 understanding and analysing the complex behaviours by a set of conceptual tools and modelling methods, which are helpful in simulating the long-run behaviour of a system in different policies and making better decision. Feedback control theories and nonlinear dynamics found the base of system dynamics. For long-run analysis of a system, it is necessary to understand different effective factors and their causal relation. Moreover, identifying feedbacks, delays and other linearity which leads the system to instability and modelling them by stocks and flows is the main art in analysing a system. Simulation is the only reliable way for testing the validity of the models because of complexity of relations among different nonlinear parameters, which makes understanding the behaviour of the model in a long time period impossible. Without simulation techniques, the system hard behaviour can be improved using feedbacks through the real world which is very slow and inefficient due to delays, nonliterary and costs of testing the ideas [46]. 2.1 Causal Diagram For simulating a dynamic system, different tools are needed. Causal loops are important tools for showing the structure of the feedbacks in the system and their effects. A causal diagram, in Fig. 2, consists of arrows which conFigure 2. The causal representation of a variable. Figure 3. The stock and flow variable. Figure 4. Casual diagram of the TREND function. nects related variables together and shows the influences among them. The positive sign on the arrow shows increasing Y by increment of X and negative sign indicates decreasing of Y . 2.2 Stocks and Flows One of the most limitations of casual loops is their inability in capturing the stocks and flows structure of the system. Stock structures are other tools in studying the system dynamics, which accumulate difference between inflow and outflow of a variable as shown in Fig. 3. Equation (1) expresses the relation of stocks, which create inertia in the system and provide memory for it; they are helpful for creating delays in a system by accumulating the difference between the inflow and outflow of a parameter in a process: Y (t) = t 0 X1(τ) − X2(τ)dτ + Y (t0) (1) 2.3 Forecasting Bounded rationality hypothesis (BRH) is a forecasting algorithm formed by adaptive expectation, in which current expectations are related to the current and past values as in (2). Expectations on the value of variables for time T are revised with adjustment rate κ, if forecasted value in previous periods is different from the actual amount [49]: ξe (t, T) = ξe (t, T − 1) + κ[ξ(T − 1) − ξe (t, T − 1)] (2) Sterman has proposed an expectational model based on the system dynamics, called TREND function; he has used needed times for measuring, collecting and analysing data, historic time horizon and required time for perceiving and reacting to variable changes. Figure 4 represents the structure of TREND function, which is usable for estimating fractional growth rate in input variable [46]. 22 Figure 5. Process of capacity expansion in a power market. 3. Model Description Figure 5 represents an overview of developed model. The firms adjust their offers considering their marginal cost and forecasted market price. Offers, existent capacity and average of demand are submitted to the power market for clearing market price and generation amount by each firm. Clearing the market facilitates calculation of profits and generation costs, considering the investment costs. The profits are normalized and converted into investment through some multipliers, which create under construction and generation capacities after some delays. The existent capacities return to the market via offer, which forms main feedback loop in this process. Reserve ratio makes an internal loop by changing the launch scale for providing the proposed reliability level. Hub price of natural gas acts on fuel cost of natural gasbased technologies and affects city gate price via seasonal factors. Details of different parts of the model are as follows. 3.1 Costs Marginal and investment costs are two expenses for generating electricity. The firms settle the marginal cost for generating each MWh of electric energy including fuel, CO2 and O&M costs, which grows with constant rate of return every year as indicated in (3) [1], [4]; HRj denotes required thermal energy for power generation by a technology in Btu/KWh and decreases every year [4]: MCj = (Fj·HRj + O&Mj + CO2j)(1 + rr)y . (3) Investment cost is settled through the construction period and must be recovered during the operation. Valid reports have expressed the investment cost in $/KW [1], [4], which is convertible into $/KWh using life time of each technology in (4). The offers are adjusted using the marginal cost and the investment cost must be recovered 23 under the influence of market condition during the operation period: ICj($/KWh) = ICj($/KW) LTj × 8760 (1 + rr)y . (4) 3.2 Electric Demand Demand of electric energy is modelled by load duration curve (LDC) for base, middle and peak sections, which are supplied by coal-fired, CCGT and GT, respectively; it grows in each section with a constant growth rate every year and the average amount by (5) is offered to the market as market demand. Market price affects the demand via demand elasticity: D(t) = k i=1 δi·Li·eg·y − λ·Δρ. (5) 3.3 Power Market This paper considers an energy-only market with pay-asbid structure, in which the lowest offers are dispatched and receive their offers from the market and market price is equal to average offer. The superiority of this structure in eliminating price spikes persuaded some markets to prefer it over the uniform price. The firms should adjust their offers properly, above the MC and below their prediction of price via the TREND function; they should make a balance between their profit and the chance to win in the market [3],
  38. [46]. 2.1 Causal Diagram For simulating a dynamic system, different tools are needed. Causal loops are important tools for showing the structure of the feedbacks in the system and their effects. A causal diagram, in Fig. 2, consists of arrows which conFigure 2. The causal representation of a variable. Figure 3. The stock and flow variable. Figure 4. Casual diagram of the TREND function. nects related variables together and shows the influences among them. The positive sign on the arrow shows increasing Y by increment of X and negative sign indicates decreasing of Y . 2.2 Stocks and Flows One of the most limitations of casual loops is their inability in capturing the stocks and flows structure of the system. Stock structures are other tools in studying the system dynamics, which accumulate difference between inflow and outflow of a variable as shown in Fig. 3. Equation (1) expresses the relation of stocks, which create inertia in the system and provide memory for it; they are helpful for creating delays in a system by accumulating the difference between the inflow and outflow of a parameter in a process: Y (t) = t 0 X1(τ) − X2(τ)dτ + Y (t0) (1) 2.3 Forecasting Bounded rationality hypothesis (BRH) is a forecasting algorithm formed by adaptive expectation, in which current expectations are related to the current and past values as in (2). Expectations on the value of variables for time T are revised with adjustment rate κ, if forecasted value in previous periods is different from the actual amount [49]: ξe (t, T) = ξe (t, T − 1) + κ[ξ(T − 1) − ξe (t, T − 1)] (2) Sterman has proposed an expectational model based on the system dynamics, called TREND function; he has used needed times for measuring, collecting and analysing data, historic time horizon and required time for perceiving and reacting to variable changes. Figure 4 represents the structure of TREND function, which is usable for estimating fractional growth rate in input variable [46]. 22 Figure 5. Process of capacity expansion in a power market. 3. Model Description Figure 5 represents an overview of developed model. The firms adjust their offers considering their marginal cost and forecasted market price. Offers, existent capacity and average of demand are submitted to the power market for clearing market price and generation amount by each firm. Clearing the market facilitates calculation of profits and generation costs, considering the investment costs. The profits are normalized and converted into investment through some multipliers, which create under construction and generation capacities after some delays. The existent capacities return to the market via offer, which forms main feedback loop in this process. Reserve ratio makes an internal loop by changing the launch scale for providing the proposed reliability level. Hub price of natural gas acts on fuel cost of natural gasbased technologies and affects city gate price via seasonal factors. Details of different parts of the model are as follows. 3.1 Costs Marginal and investment costs are two expenses for generating electricity. The firms settle the marginal cost for generating each MWh of electric energy including fuel, CO2 and O&M costs, which grows with constant rate of return every year as indicated in (3) [1], [4]; HRj denotes required thermal energy for power generation by a technology in Btu/KWh and decreases every year [4]: MCj = (Fj·HRj + O&Mj + CO2j)(1 + rr)y . (3) Investment cost is settled through the construction period and must be recovered during the operation. Valid reports have expressed the investment cost in $/KW [1], [4], which is convertible into $/KWh using life time of each technology in (4). The offers are adjusted using the marginal cost and the investment cost must be recovered 23 under the influence of market condition during the operation period: ICj($/KWh) = ICj($/KW) LTj × 8760 (1 + rr)y . (4) 3.2 Electric Demand Demand of electric energy is modelled by load duration curve (LDC) for base, middle and peak sections, which are supplied by coal-fired, CCGT and GT, respectively; it grows in each section with a constant growth rate every year and the average amount by (5) is offered to the market as market demand. Market price affects the demand via demand elasticity: D(t) = k i=1 δi·Li·eg·y − λ·Δρ. (5) 3.3 Power Market This paper considers an energy-only market with pay-asbid structure, in which the lowest offers are dispatched and receive their offers from the market and market price is equal to average offer. The superiority of this structure in eliminating price spikes persuaded some markets to prefer it over the uniform price. The firms should adjust their offers properly, above the MC and below their prediction of price via the TREND function; they should make a balance between their profit and the chance to win in the market [3], [45]. 3.4 Profitability Clearing the market specifies the generation of each firm, which is applicable in computing their costs and profits. Total generation cost, in (6), is the sum of firm’s expenses for generating electric energy until studied time t: Φj = t 0 Gj·MCjdt (6) By subtracting the generation and investment costs from the income, the total profit of the firms is given by Πj = t 0 Gj·χj − Gj·MCj − CPj·ICjdt (7) Profitability index is defined in (8), as the ratio of profit to generation cost, for normalizing the profits to a same quantity [46]. This parameter is helpful in investing in a technology rather than its profit: PIj = Πj Φj (8) 3.5 Stable State A market can become stable by recovering its generation and investment costs of the firms [45]. This condition is equivalent to PIj = 0, as both costs are considered in PIj. The firms can reach the stable state by offering a price equal to MC plus a multiple of forecasted price [3], named as stable price. 3.6 Capacity Expansion The PIs of firms are converted into investment rate via S-shaped curves in (9), which limit the rate of variations and final values in each firm [46]. The coefficients mj max, αj and βj differ in each technology, but mj is equal to 1 for PIj = 1 in the whole, as indicated in Fig. 6. The coefficient mj is influenced by reliability policy and profitability for providing enough capacity: mj = mj max 1 + e−(αj P Ij −βj ) (9) Figure 6. The coefficient m for different technologies vs. PI. Equation (10) gives investment rate in each technology as a function of demand growth rate and retirement rate of the firms weighted by the coefficient mj: IRj = mj·( ˙Li + ˙REj) (10) Reliability policy in (11) forms an internal loop in launching process, named as launch scale [9] that changes rate of investment in each technology for holding reserve ratio at a proposed level: Res.Rat = TCP − D(t) D(t) (11) The investment rate is converted into capacity after a construction delay. Equation (12) indicates under construction capacity, which is the difference between investment rate and construction rate in each technology. Exploited capacity in (13) is the difference between constructed capacity and retired amount after a life time. A part of GT capacity is converted into CCGT by a change 24 ratio and change delay that influences the pattern of expansion. Operational capacity is declared to the market and creates the main feedback loop in this process; besides, it is used for providing reliability as an internal loop: UCj = t 0 IRj − IRj(t − CTj)dt (12) CPj = t 0 CNj − CNj(t − LTj)dt (13) 3.7 Wind Technology The wind technology competes with other participants in the market, while previous studies subtracted its capacity from the demand [14], [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds
  39. [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6], [38],
  40. [49]: ξe (t, T) = ξe (t, T − 1) + κ[ξ(T − 1) − ξe (t, T − 1)] (2) Sterman has proposed an expectational model based on the system dynamics, called TREND function; he has used needed times for measuring, collecting and analysing data, historic time horizon and required time for perceiving and reacting to variable changes. Figure 4 represents the structure of TREND function, which is usable for estimating fractional growth rate in input variable [46]. 22 Figure 5. Process of capacity expansion in a power market. 3. Model Description Figure 5 represents an overview of developed model. The firms adjust their offers considering their marginal cost and forecasted market price. Offers, existent capacity and average of demand are submitted to the power market for clearing market price and generation amount by each firm. Clearing the market facilitates calculation of profits and generation costs, considering the investment costs. The profits are normalized and converted into investment through some multipliers, which create under construction and generation capacities after some delays. The existent capacities return to the market via offer, which forms main feedback loop in this process. Reserve ratio makes an internal loop by changing the launch scale for providing the proposed reliability level. Hub price of natural gas acts on fuel cost of natural gasbased technologies and affects city gate price via seasonal factors. Details of different parts of the model are as follows. 3.1 Costs Marginal and investment costs are two expenses for generating electricity. The firms settle the marginal cost for generating each MWh of electric energy including fuel, CO2 and O&M costs, which grows with constant rate of return every year as indicated in (3) [1], [4]; HRj denotes required thermal energy for power generation by a technology in Btu/KWh and decreases every year [4]: MCj = (Fj·HRj + O&Mj + CO2j)(1 + rr)y . (3) Investment cost is settled through the construction period and must be recovered during the operation. Valid reports have expressed the investment cost in $/KW [1], [4], which is convertible into $/KWh using life time of each technology in (4). The offers are adjusted using the marginal cost and the investment cost must be recovered 23 under the influence of market condition during the operation period: ICj($/KWh) = ICj($/KW) LTj × 8760 (1 + rr)y . (4) 3.2 Electric Demand Demand of electric energy is modelled by load duration curve (LDC) for base, middle and peak sections, which are supplied by coal-fired, CCGT and GT, respectively; it grows in each section with a constant growth rate every year and the average amount by (5) is offered to the market as market demand. Market price affects the demand via demand elasticity: D(t) = k i=1 δi·Li·eg·y − λ·Δρ. (5) 3.3 Power Market This paper considers an energy-only market with pay-asbid structure, in which the lowest offers are dispatched and receive their offers from the market and market price is equal to average offer. The superiority of this structure in eliminating price spikes persuaded some markets to prefer it over the uniform price. The firms should adjust their offers properly, above the MC and below their prediction of price via the TREND function; they should make a balance between their profit and the chance to win in the market [3], [45]. 3.4 Profitability Clearing the market specifies the generation of each firm, which is applicable in computing their costs and profits. Total generation cost, in (6), is the sum of firm’s expenses for generating electric energy until studied time t: Φj = t 0 Gj·MCjdt (6) By subtracting the generation and investment costs from the income, the total profit of the firms is given by Πj = t 0 Gj·χj − Gj·MCj − CPj·ICjdt (7) Profitability index is defined in (8), as the ratio of profit to generation cost, for normalizing the profits to a same quantity [46]. This parameter is helpful in investing in a technology rather than its profit: PIj = Πj Φj (8) 3.5 Stable State A market can become stable by recovering its generation and investment costs of the firms [45]. This condition is equivalent to PIj = 0, as both costs are considered in PIj. The firms can reach the stable state by offering a price equal to MC plus a multiple of forecasted price [3], named as stable price. 3.6 Capacity Expansion The PIs of firms are converted into investment rate via S-shaped curves in (9), which limit the rate of variations and final values in each firm [46]. The coefficients mj max, αj and βj differ in each technology, but mj is equal to 1 for PIj = 1 in the whole, as indicated in Fig. 6. The coefficient mj is influenced by reliability policy and profitability for providing enough capacity: mj = mj max 1 + e−(αj P Ij −βj ) (9) Figure 6. The coefficient m for different technologies vs. PI. Equation (10) gives investment rate in each technology as a function of demand growth rate and retirement rate of the firms weighted by the coefficient mj: IRj = mj·( ˙Li + ˙REj) (10) Reliability policy in (11) forms an internal loop in launching process, named as launch scale [9] that changes rate of investment in each technology for holding reserve ratio at a proposed level: Res.Rat = TCP − D(t) D(t) (11) The investment rate is converted into capacity after a construction delay. Equation (12) indicates under construction capacity, which is the difference between investment rate and construction rate in each technology. Exploited capacity in (13) is the difference between constructed capacity and retired amount after a life time. A part of GT capacity is converted into CCGT by a change 24 ratio and change delay that influences the pattern of expansion. Operational capacity is declared to the market and creates the main feedback loop in this process; besides, it is used for providing reliability as an internal loop: UCj = t 0 IRj − IRj(t − CTj)dt (12) CPj = t 0 CNj − CNj(t − LTj)dt (13) 3.7 Wind Technology The wind technology competes with other participants in the market, while previous studies subtracted its capacity from the demand [14], [16]. Different effective factors such as generation and investment costs, construction time and life time are considered for analysing long-run behaviour of wind technology [1], [4]. Wind speed, which is modelled by Weibull probability distribution function in (14), perturbs output power of wind technology [5]: Vw(t) = γ η · t η η−1 ·e−( t η ) γ (14) Wind generation in (15) is affected by wind perturbation and restriction of facilities in low and high wind speeds [47]. The CCGT compensates the lack of wind planed generation, due to its continuous generation and fast response, which increases its income in the market [37]: Gw = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ 0, Vw < Vci Vw Vr 3 CPrw, Vci ≤ Vw < Vr CPrw, Vr ≤ Vw < Vco 0, Vw ≥ Vco. (15) 3.8 Natural Gas Natural gas affects the power system in generation and demand parts. This fuel reaches to the consumer with 90% efficiency, while the efficiency decreases to about 35% through the electric generation. The natural gas-based technologies provide their fuel from hubs with a hub price, which is a source of uncertainty in the power systems. Different references have modelled the uncertainty of natural gas hub price by low, medium and high scenarios [6], [38], [48]. Price of natural gas for demand is known by city gate price, which is affected by hub price and seasonal factors [2]. Analysing published data by EIA about city gate price [7] via X-12-ARIMA time series [8] gives the seasonal factors as shown in Fig. 7. These factors adapt to additive model that makes the real variable by adding a seasonal factor. Figure 7. The seasonal factors of the city gate natural gas price. 4. Simulation and Results This section analyses the results of simulating different states of natural gas consumption in the power system in three scenarios including firms stability, changes in the natural gas charge and access of demand to the natural gas and electricity. Some results such as the profitability index, reserve ratio, and capacity expansion are represented and compared with the base state in each scenario. The applied parameters in the simulation are from the published data by the EIA on the generation costs by different technologies, natural gas city gate prices and natural gas hub prices. The results represent the statue of the parameters in 1,200 months for indicating the stability of the developed model. 4.1 Base Scenario In this scenario, the firms stay on the stable state by adjusting the offers for recovering the generation and investment costs [45] that is achievable by PIj = 0 in (8), due to its association to these costs. They adjust their offers by adding multiples of the forecasted price to their marginal cost, which are 0.13, 0.083, 0.058 and 0.45 for the coalfired, CCGT, GT and wind, respectively. The technologies with higher investment cost need a greater coefficient for getting to the stability. Figure 8 represents the profitability index of the firms in this scenario, which tends to zero during the studied horizon. The variation of the PI around zero is a motivation for expanding the capacity by different technologies beside the growing demand. Table 1 summarizes the total present profit of the firms in this scenario. The coal-fired earns the most profit by supplying the base load and the CCGT and GT are at the next places. The wind earns the least profit in the stable state. Figure 9 represents the reserve ratio of the power system calculated by (11) for providing the reliability of the power system, which swings around 0.2 with a limited variation between 0.18 and 0.22. The reliability level is achieved by the pattern of generation capacity, shown in Fig. 10. The capacity of the coalfired, CCGT and GT gets to 8.8 × 104 MW, 4.8 × 104 MW and 3.3 × 104 MW for supplying the base, middle and peak loads, respectively and the wind technology expands its capacity to 1,400 MW in the stable state. 25 Figure 8. The profitability index of the firms in the base scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 1 Total Present Profit of the Firms at the Base Scenario Technology Profit ($) Coal-fired 3 × 106 CCGT 0.94 × 106 GT 0.33 × 106 Wind 0.085 × 106 Figure 9. The reserve ratio of the power system in the base scenario. 4.2 Natural Gas Price Variation The second scenario investigates the influence of the variations in the natural gas price on the long-run investment in the capacity expansion. The variations are modelled as low-price, medium-price and high-price outlines, summarized in Table A.1 [6], [48], [38]. The results of the high and Figure 10. The capacity of different technologies in the base scenario. Figure 11. The price of electric energy for different outlines of natural gas price. low prices of natural gas are compared with the medium price as base scenario. Figure 11 represents the price of electric energy for different outlines of natural gas price. The electric price 26 Figure 12. The profitability index of the firms at the high price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 2 Total Present Profit of the Firms at the High Price of Natural Gas Technology Profit ($) Coal-fired 24 × 106 CCGT 4.3 × 106 GT −1.3 × 106 Wind 3.4 × 106 increases by growing the natural gas price and decreases by its decline, compared with the medium price. Figure 12 indicates the profitability index of the firms at the high price of natural gas. The PI increases to 0.04, 0.01 and 0.33 for the coal-fired, CCGT and wind, respectively, but decreases to −0.0002 for the GT, due to the loss of opportunity for generation by this technology. Reducing the heat rate of the GT, increases its PI to zero on month 900 by raising its generation chance and decreases the PI of the coal-fired in Figs. 12(a) and (c). The present profit of the firms changes by their deviation from the stable state, summarized in Table 2. The profit of the coal-fired, CCGT and wind increases , compared with Table 1, but the high price is disadvantageous for the GT and causes its negative profit in Table 2. The average of the reserve ratio does not change significantly at the high price and grows to the average of 0.21 with a pattern same as Fig. 9. Compared with Fig. 10, the Figure 13. The capacity of different technologies at the high price of natural gas. capacity of different firms changes as indicated in Fig. 13; the capacity of the coal-fired and the wind increases to 9 × 104 MW and 7,300 MW, respectively, but it decreases to 4.4 × 104 MW and 3 × 104 MW for the CCGT and GT, which shows the reduced share of the natural gas consumers in the market and the great ratio of capacity expansion by the wind at high prices. Decreasing the charge of the natural gas, drops the electric price in the market, which leads to the loss of the firms at the stable offer, due to irretrievable investment costs. The profitability index of the firms drops to −0.0073, −0.007, −0.0035 and −0.49 for the coal-fired, CCGT, GT and wind in Fig. 14. Table 3 summarizes the present profit of the firms at the low charge of natural gas, which is negative for the whole and results in their loss by offering the stable price. 27 Figure 14. The profitability index of the firms at the low price of natural gas: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 3 Total Present Profit of the Firms at the Low Price of Natural Gas Technology Profit ($) Coal-fired −2.8 × 106 CCGT −1.3 × 106 GT −0.58 × 106 Wind −0.15 × 106 Figure 15. The capacity of different technologies at the low price of natural gas. Low-price natural gas does not influence the reliability of the power system remarkably and decreases the average of the reserve ratio to the amount 0.19 with the behaviour same as Fig. 9. The pattern of generation capacity changes at low charge of natural gas as shown in Fig. 15. The capacity of the CCGT and the GT increases to 4.9 × 104 MW and 3.4 × 104 MW, while the amount of the coal-fired and the wind decreases to 8.7 × 104 MW and 540 MW, respectively. This variation increases the share of technologies in the market that consume the natural gas. 4.3 Natural Gas Consumption by the Demand This section analyses the access to the electricity and natural gas as two separate energy resources by 10% of the demand. The demand switches between these energy resources by comparing the electric market price and natural gas city gate price and choosing the cheapest one. Figure 16 represents the PI of the firms, when 10% of the demand selects between two energy resources. The PI of the coal-fired decreases to −0.033 in Fig. 16a, but it does not change for the other technologies significantly. Table 4 summarizes the present profit of the firms at the selection of the natural gas by the demand. The loss of the profit by the coal-fired is severe, due to the variations in the base demand, which enforces it to increase its coefficient in the offer. The present profit of the other technologies does not change a lot in this scenario, compared with Table 1. The resource selection by the demand enforces the firms to expand the capacity for a discontinuous demand, which is detectable in the capacity of different technologies in Fig. 17. The capacity of the coal-fired, CCGT and GT increases with swings to 10 × 104 MW, 5.8 × 104 MW and 3.7 × 104 MW, respectively. The wind capacity decreases to 1,047 MW in this scenario. 28 Figure 16. The profitability index of the firms at the natural gas consumption by the demand scenario: (a) coal-fired; (b) CCGT; (c) GT; and (d) wind. Table 4 Total Present Profit of the Firms at the Natural Gas Consumption by the Demand Scenario Technology Profit ($) Coal-fired −23.1 × 106 CCGT 1 × 106 GT 0.31 × 106 Wind 0.072 × 106 Figure 17. The capacity of different technologies at the natural gas consumption by the demand scenario. Figure 18. The reserve ratio of the power system at the natural gas consumption by the demand scenario. This pattern of capacity has a negative influence on the reserve ratio of the power system for providing the reliability as shown in Fig. 18. The average of the reserve ratio swings around the average amount of 0.35 and varies between 0.15 and 0.6 in this figure. Increasing the access of the demand to both energy resources from 10% creates undesirable effects on the profitability and reserve ratio. The efficiency of the natural gas consumption by the demand is an effective factor for keeping the stability of the electric market. Growing the efficiency of the natural gas consumption to above 50%, restores the stable state of the firms in the market by keeping the PI of the firms at the zero and the reserve ratio on 0.2 with a little swing same as Figs. 8 and 9. High efficient natural gas demands 29 are dismissed form the electric market, which decreases the installed capacity at the stable state and its swings. 5. Discussion The natural gas affects the power market in generation and consumption levels. Based on the hub price of natural gas, three scenarios can be defined including, low, medium and high charges. The access of demand to the natural gas affects the market via the seasonal factors. The stable state (PI = 0) which is considered as the base scenario compensates the generation and investment costs of the firms and keeps the reserve ratio on about 0.2 for providing the reliability. This state is achieved by adding a multiple of the forecasted price to the MC by the firms, which is 0.13, 0.083, 0.058 and 0.45 for the coal-fired, CCGT, GT and wind, respectively; the technologies with higher investment cost adjust their offer with a greater coefficient. The Medium price of natural gas is considered in the base scenario. High-charge natural gas increases the market price and the firms offers, which increases the profitability index of the coal-fired, CCGT and wind from zero to 0.04, 0.01 and 0.33, respectively; however, this charge decreases the generation opportunity for GT and decreases its PI to −0.0002. So, the GT should decrease its coefficient during high price of natural gas for creating generation opportunity. By increasing the natural gas price reaches the capacity of GT and CCGT increases from 4.8 × 104 MW and 3.3 × 104 MW to 4.4 × 104 MW and 3 × 104 MW and the share of natural gas-based technologies in the market decreases. However, increasing the profit creates the opportunity for the coalfired and wind to expand their capacity and increase their share from 8.8 × 104 MW and 1,400 MW at the stable state to 9 × 104 MW and 7,300 MW. The average of reserve ratio grows from 0.2 to 0.21 in this situation. Low-charge natural gas decreases the average price in the market and reduces the PI of coal-fired, CCGT, GT and wind to −0.0073, −0.007, −0.0035 and −0.49, respectively, which results in the loss of the firms. The share of CCGT and GT increases to 4.9 × 104 MW and 3.4 × 104 MW in the market, while the capacity of the coal-fired and wind decreases to 8.7 × 104 MW and 540 MW. The capacity expansion is due to the effects of demand growth rate and retirement rate on the investment rate for providing the proposed reliability level of about 0.19. The charge of natural gas at the consumption level is influenced by seasonal factors, which has additive pattern and its amount is greater in colder months. Natural gas consumption by demand decreases the profitability index of the coal-fired to −0.033 as base supplier. This condition enforces the firms to expand their capacity, while their generation is not consumed by the demand continuously, which hardens recovering the investment. Choosing the cheapest energy resource by the demand keeps the PI of the CCGT, GT and wind at zero and increasing the percentage of demand access decreases the PI of CCGT and GT. The capacity of coal-fired, CCGT, GT and wind reaches to 10 × 104 MW, 5.8 × 104 MW, 3.7 × 104 MW and 1,047 MW. Switching between the energy resources by the demand causes the swing of the reserve ratio, which is resolved by increasing the efficiency of the natural gas consumers. 6. Conclusion This paper analyses the effect of the natural gas on the capacity expansion by the firms in a pay-as-bid energyonly market using the system dynamics. Natural gas-based technologies and demand selection between the electricity and natural gas are two considered tie points between these resources. This subject is studied via three scenarios, namely, (1) the firms stability, (2) changes in the natural gas charge and (3) access of demand to natural gas and electricity. Four generation technologies including coalfired, CCGT, GT and wind participate in an energy-only market using the published data in the reports of EIA. At the first scenario, the firms adjust their offers above the marginal cost for recovering the generation and investment costs as stable state, which is known by PIj = 0. The firms adjust their offers by adding a multiple of the forecasted price to the marginal cost, which is greater for the technologies with higher investment cost. The stable state can recover the costs of the firms and provide the targeted reliability level of the power system. The natural gas price as assumed to fluctuate between low, medium and high prices, where the medium price is applied at the base scenario. High-price natural gas increases the offers of the firms and the market price, which increases the profit of the coal-fired, CCGT and wind, but causes the loss of the profit by the GT, as it loses the opportunity for generation. More profit increment by coalfired and loss of GT reduces the share of natural gas-based technologies in the market at high charges of natural gas. Low-price natural gas decreases the offers and market price to an amount, which cannot recover the investment costs and causes the loss of the whole. The capacity is expanded for supplying the demand and compensating the retirement with a less ramp. The loss of coal-fired and wind in low charge is more severe due to their higher investment costs, which increases the share of natural gas-based technologies in the market. The reserve ratio of the power system does not change remarkably by changing the natural gas price. Selecting the natural gas as resource of energy by a portion of demand causes the loss of profit by the coal-fired and does not influence on the revenue of the rest. This behaviour of demand expands the generation capacity and increases the average of reserve ratio, but creates swing in it. Increasing the per cent of demand access to the natural gas intensifies the unstable condition in the market. The growth of efficiency in the natural gas demand dismisses it from the power market and restores it to the stability. Appendix A Table A.1 summarizes the parameters of the simulation, which are from the published report by EIA [4]. 30 Table A.1 The Parameters of the Simulation Technology Coal-Fired CCGT GT Wind Parameter Fuel Cost 65 ($/ton) Low Price = 3.8 $/MMBtu Thermal Value Medium Price = 4.6 $/MMBtu = 6080 kcal/kg High price = 6 $/MMBtu Heat rate (Btu/KWh) 9,200 6,752 9,289 Heat rate changes (Btu/KWh year) −30 −28 −50 O&M costs ($/MWh) 7.7 3.3 4.3 3.4 CO2 costs ($/MWh) 24 10.5 16 Investment costs ($/MWh) 3.7 3.3 2.3 10.97 Construction time (months) 48 36 24 6 Life time (months) 720 360 360 240 Rate of return (%/year) 5% Peak demand (MW) 1,200 Peak duration 0.2 Middle demand (MW) 1,000 Middle duration 0.6 Base demand (MW) 700 Base duration 0.2 Demand growth rate (%/year) 5% Rated wind speed (m/s) 7 Product wind speed (m/s) 4 Cut out wind speed (m/s) 13 References [1] “Projected Costs of Generating Electricity , International Energy Agency, 2010 Edition. [2] S. Macmillan, A. Antonyuk, and H. 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