T.J. Hammons∗


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  9. [9] A. Majeed, H.A. Karim, N.H. Al Maskati, & S. Sud, Statusof Gulf Cooperation Council (GCC) electricity grid systeminterconnection, Panel presentation at IEEE PES 2004 GM,Denver, CO, USA, 2004, available at, 15–23.AppendixFormulation of Simultaneous Linear Equations forthe Trading PlatformLet us assume that P MW of power is on offer and thatthere are altogether n offers and m bids. Let us alsoassume that a given bidder j receives power from a givenseller i and that this power is Pij as illustrated in Table 7.For n offers, it can be shown that:P11 + P12 + P13 + P14 + · · · + P1m =mj=1P1jP21 + P22 + P23 + P24 + · · · + P2m =mj=1P2j...Pn1 + Pn2 + Pn3 + Pn4 + · · · + Pnm =mj=1Pnj⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(1.0)381Table 7Power Allocation Table1 2 3 4 TOTALBIDSNAM SEB LEC ZESA1 SNEL P11 P12 P13 P1441 P1j2 ZESCO P21 P22 P23 P2441 P2jOFFERS3 EDM P31 P32 P33 P3441 P3j4 ESKOM P41 P42 P43 P4441 P4jTOTAL41 Pi141 Pi241 Pi341 Pi441 PijSimilarly for m bids we get:P11 + P21 + P31 + P41 + · · · + Pn1 =ni=1Pi1P12 + P22 + P32 + P42 + · · · + Pn2 =ni=1Pi2...P1m + P2m + P3m + P4m + · · · + Pnm =ni=1Pim⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(2.0)If OFFER_POWER [i] is the maximum power of-fered by seller i and BID_POWER [j] is the maximumpower requested by bidder j , then the selling and buyingconditions must satisfy the following equations (3.0) and(4.0) respectively, for offers and bids.mi=1 P1i ≤ OFFER_POWER [1]mi=1 P2i ≤ OFFER_POWER [2]...mi=1 Pni ≤ OFFER_POWER [n]⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭(3.0)nj=1 Pj1 ≤ BID_POWER [1]nj=1 Pj2 ≤ BID_POWER [2]...nj=1 Pjm ≤ BID_POWER [m]⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭(4.0)The value of the power Pij at other positions can easilybe evaluated from the qualification criterion. If the offerprice from the seller I is more than the bidder price bythe bidder j, then if follows that Pij = 0.It was initially stated that the cheaper power has tobe shared first equally amongst all the qualified bidders.Therefore, it follows that the cost function (i.e., cost perhour) with respect to the bidder must be minimized toget the optimal solution for all the offers and bids. Thecost function in this case is the product of the price of theoffer power by the power allocated to the buyer from theparticular seller. The unit of the cost function is thereforecents per KWh.The optimization problem for any given buyer at posi-tion j and for n sellers can then be stated as follows:Minimize:Costf = OFFER_PRICE [1] ∗ P1j+ OFFER_PRICE [2] ∗ P2j+ · · · + OFFER_PRICE [n] ∗ PnjSubject to:1. Pij = 0 when equation (1.0) is not satisfied.2. Transmission system constraints and other Pjk ≤ OFFER_POWER [j] Pij ≤ BID_POWER [j]There are two ways of solving the optimization problemof this nature: mathematically using the well-known lin-ear optimization techniques, or by using the market rulesthat have been formulated. The method that was finallyadopted combined the two approaches. The market rulessimplified the task of incorporating system transmissionand wheeling constraints.The proposed algorithm first allocates the powerequally to all the successful bidders. If the power allocatedis more than what the bidder requested, adjustmentsare made to the affected bidder. Then the transmissionconstraints are applied to check for any system violations.If no violation is detected, the solution is accepted. Ifviolation is encountered, the allocated power is adjusteduntil no violation is observed.Using the proposed algorithm, a computer programcalled STEM Program Manager was written to auto-mate and computerize the allocation of offers to successfulbidders.382

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