roceedngs of the 0th WSEAS Internatonal Confenrence on ALIED MATHEMATICS, Dallas, Texas, USA, November -3, 2006 365 Impact of Statc Load Modelng on Industral Load Nodal rces G. REZA YOUSEFI M. MOHSEN EDRAM AZAD Unversty, Saveh Branch Tarbat Moalem Unversty Saveh, IRAN Tehran, IRAN Abstract: Ths paper s concerned wth the nodal prces n a compettve electrcty market based on the pool paradng when the ndustral loads are statstcally modeled. Specfcally t addresses the mpact of statc load modelng on the actve and reactve margnal costs, whle loads are varyng. Based on nodal prcng theory of margnal costs, an approach s proposed n ths paper, n whch the mpacts of statc load modelng on margnal costs of both actve and reactve power of ndustral loads are calculated. The proposed algorthm s evaluated on IEEE 4-bus test system. Key-Words: Optmzaton, Statc Load Modelng, Actve and Reactve ower rcng, Margnal Costs. Introducton Recent power ndustry wth dfferent types of power markets have generated the requrement of better ways to deal wth power prcng and all ssues n ths regards. We are focusng on a pool based market n whch the Independent System Operator (ISO has a vast mount of authorty and decson makng power []. Many documents could be found n lteratures about nodal prcng [2-5]. In [6], based on nodal prcng theory, authors addressed transton costs of both actve and reactve powers. Reactve power prcng s also studed n several papers, such as [7, 8]. Whle nearly all mentoned references employ somehow a margned cost based Optmal ower Flow (OF procedure, they gnore the effect of consumer load modelng on the results. Meanwhle, there are so many actvtes and studes on load modelng such as: [9-3]. In ths paper, a new approach s proposed and OF formulaton s extended to consder ndustral load modelng, and varous operatonal constrants are observed. Margnal costs are used for both actve and reactve power, as nodal prces for ndustral customers. Numercal results are shown on IEEE 4-bus test system. 2 roblem Formulaton 2. Statc Load Modelng As motoned before, statc load modelng topc has been worked wdely. For load modelng ssue, ths model s used for ndustral consumers n our paper [0, ]:. Q = Q o. V (2 n whch: = V np ( o nq 0 (Q0: Actve (reactve ower at nomnal voltage. V: Actual per unt voltage magntude. np (nq: senstvty of actve (reactve power to voltage magntude. 2.2 Nodal rcng Nodal rcng theory [2, 3] s employed to calculate short-term prcng of actve and reactve power whle statc load modelng (equ. and equ. 2 s added to the formulaton. In a pool based electrcty market, ISO tres to mnmze the total power generaton costs whle varous operatons should also be observed and n our paper each ndustral load must be modeled as a statc load, represented by equ. and equ. 2. The problem may be formulated as: mn NG θ, V, g, Qg = C ( g (3 s.t. ( θ, V g + d = 0 (4 Q ( θ, V Q + Q = 0 (5 S V g f (, V S mn mn g mn g d θ (6 V V (7 (8 g g g Q Qg Q (9 d = f ( V (0 Q = g V ( d ( If N and NG, represent total number of buses and number of generatng unts, respectvely, and equ. and equ. 2 use for statc load modelng of ndustral
roceedngs of the 0th WSEAS Internatonal Confenrence on ALIED MATHEMATICS, Dallas, Texas, USA, November -3, 2006 366 loads, the optmzaton could be formulated as follows: mn NG θ, V, g, Qg = C ( g s.t. N ( V V Y Cos( θ + δ δ + g d = 0 {,2,..., = N ( V V Y Sn( θ + δ δ + Qg Qd = 0 {,2,..., = mn mn mn g mn g V Q Q d d (2 (3 (4, {,2,..., (5 g Qg np d nq d V V {,2,..., (6 g {,2,..., NG} (7 Qg {,2,..., NG} (8 = d 0.( V {,2,..., NI} (9 = Q 0.( V {,2,..., NI} (20 d n whch: C : Generaton costs of unt. g ( Qg : Actve (reactve power generaton of unt. V : Voltage magntude of bus. Y ( θ : The magntude (angle of element of the system admttance matrx. δ : Voltage phase angle of bus. d ( Qd : Actve (reactve power consumpton of bus. NI : The number of ndustral consumers. mn mn g ( Q g : Mnmum actve (reactve power generaton of unt. g ( Q g : Maxmum actve (reactve power generaton of unt. To demonstrate each generaton cost, a second order polynomal s used: 2 C ( g = α + β. g + γ. g (2 where α, β and γ are constant. The power flow through each branch, n equ. 5, s formulated as: 2 = V V Y Cos θ + δ δ V Y Cosθ (22 ( The Lagrange method s used to solve the optmzaton problem formulated by equ.2 to equ. 20 Based on margnal cost concept [2, 3], the margnal cost (MC at each bus, s calculated as the rato of total producton cost change to the change of ether actve or reactve power: L MC = $/MW (23 L MCQ = $/MVAr (24 Q In ths paper we have focused on ndustral loads and n the numercal results wll show how statc load modelng effects on margnal cost of such loads. 3 Numercal Results To study the effect of load modelng on margnal cost of ndustral consumers, IEEE 4-bus test system s selected (Fg. Typcal data for ndustral load modelng s selected as [0]: np = 0.8, nq = 6.00. It s assumed that the loads on buses to 4 are ndustral and the other loads kept constant. Two separate tests have been done: ncreasng loads at buses to 4, whle the loads are modeled as constant and Q. Increasng loads at buses to 4 whle the loads are modeled based on equ. and equ.2. Table to Table 4 show the nputs and results n each case, respectvely. Fg. IEEE 4-bus test system To clarfy the mpact of load modelng on margnal costs, based on the results shown n Table 2 to Table 4, Fg. 2 to Fg. 5 are prepared. As far as, np s selected 0.8, margnal costs for actve power are not too senstve to the varaton of voltage at ther buses. Whle nq s equal to 6.00 for an ndustral load, as could be seen n Fg. 2 to Fg. 5, the margnal costs for reactve power have sgnfcantly changed wth load and voltage varatons. So, t s too mportant to consder load models, specally, n electrcty power markets n whch partcpants should bd ther generaton and ther consumptons. 4 Concluson
roceedngs of the 0th WSEAS Internatonal Confenrence on ALIED MATHEMATICS, Dallas, Texas, USA, November -3, 2006 367 ower system ndustry faces wth varous electrcty markets that actve and reactve power prcng has a key rule n such envronment. Usually, the reactve generaton cost s gnored, meanwhle actve loads are modeled as constant. In ths paper we showed how load modelng could change the margnal costs and the power prces at a gven bus of power system. Table : Load varaton at buses to 4 (loads are and Q constant Bus Case Case 2 Case 3 Case 4 Case 5 Case 6 No Q Q Q Q Q Q 3.5.8 4.2 2.6 4.9 2.52 5.6 2.88 6.3 3.24 7 3.6 2 6..6 7.32.92 8.54 2.24 9.76 2.56 0.98 2.88 2.2 3.2 3 3.5 5.8 6.2 6.96 8.9 8.2 2.6 9.28 24.3 0.44 27.6 4 4.9 5 7.88 6.00 20.86 7.00 23.84 8.00 26.82 9.00 29.8 0 (MW, Q (MVar Table 2: Actve and reactve MC based on the loads n Table Bus Case Case 2 Case 3 Case 4 Case 5 Case 6 No MC MC Q MC MC Q MC MC Q MC MC Q MC MC Q MC MC Q 40.6 0.23 40.32 0.26 40.44 0.29 40.54 0.33 40.64 0.50 40.75 0.60 2 40.38 0.2 40.70 0.26 40.95 0.3 4.6 0.36 4.4 0.60 4.65 0.78 3 40.58 0.35 40.92 0.43 4.20 0.52 4.44 0.60 4.72 0.87 42.02.07 4 4.20 0.57 4.58 0.70 4.94 0.84 42.29 0.98 42.67.26 43.09.49 MC ($/MW, MCQ ($/MVar Table 3: Load varaton at buses to 4 (loads are modeled based on equ. and equ.2 Bu Case Case 2 Case 3 Case 4 Case 5 Case 6 No Q Q Q Q Q Q 3.53 2.36 4.23 2.8 4.94 3.26 5.64 3.69 6.34 4.0 7.04 4.40 2 6.5 2.08 7.37 2.46 8.60 2.82 9.82 3.7.04 3.46 2.25 3.65 3 3.60 7.34 6.30 8.6 9.0 9.8 2.7 0.95 24.40.97 27.06 2.53 4 4.96 5.76 7.94 6.68 20.90 7.52 23.86 8.24 26.8 8.84 29.72 9.9 (MW, Q (MVar Table 4: Actve and reactve MC based on the loads n Table 3 Bus Case Case 2 Case 3 Case 4 Case 5 Case 6 No. MC MC Q MC MC Q MC MC Q MC MC Q MC MC Q MC MC Q 40.6 0.26 40.33 0.30 40.45 0.33 40.55 0.48 40.65 0.57 40.76 0.66 2 40.40 0.27 40.72 0.32 40.97 0.37 4.20 0.58 4.44 0.73 4.67 0.86 3 40.59 0.43 40.93 0.52 4.2 0.60 4.48 0.82 4.76 0.99 42.03.4 4 4.22 0.66 4.60 0.79 4.96 0.92 42.32.4 42.7.3 43.09.46 MC ($/MW, MCQ ($/MVar
roceedngs of the 0th WSEAS Internatonal Confenrence on ALIED MATHEMATICS, Dallas, Texas, USA, November -3, 2006 368 Fg. 2 MCq versus Q at Bus No. Fg. 3 MCq versus Q at Bus No. 2
roceedngs of the 0th WSEAS Internatonal Confenrence on ALIED MATHEMATICS, Dallas, Texas, USA, November -3, 2006 369 Fg. 4 MCq versus Q at Bus No. 3 Fg. 5 MCq versus Q at Bus No. 4
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