Monte Carlo simulation for evaluating retail wheeling effects

Electric Power Systems Research 60 (2002) 137 144 www.elsevier.com/locate/epsr Monte Carlo simulation for evaluating retail wheeling effects A.G. Bakirtzis a, *, Yong-Ha Kim b,1, A.P. Sakis Meliopoulos c,2 a Department of Electrical and Computer Engineering, Aristotle Uni ersity of Thesssaloniki, 540 06 Thessaloniki, Greece b Department of Electrical Engineering, Uni ersity of Inchon, Inchon, South Korea c School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250, USA Received 20 February 2001; received in revised form 26 June 2001; accepted 26 June 2001 Abstract A Monte Carlo simulation method for evaluating retail-wheeling effects on power systems is formulated and solution methods are presented. The effects of wheeling on operating cost, transmission losses, and system security are considered. For a specific operating condition, the effects are quantified by the sensitivity of specific quantities of interest with respect to power wheeling level. Quantities of interest are total operating cost, transmission losses and security, which is quantified with several indices. This model is utilized within a Monte Carlo simulation to calculate probability distribution functions of the incremental effects of wheeling on operating cost, transmission losses, and system security. The model and solution methods are applied on an example power system and the results are presented. 2002 Elsevier Science B.V. All rights reserved. Keywords: Monte Carlo simulation; Wheeling effects 1. Introduction * Corresponding author. Tel.: +30-31-996-383; fax: +30-31-996-302. E-mail addresses: bakiana@eng.auth.gr (A.G. Bakirtzis), yhkim@inchon.ec.kr (Y.-H. Kim), sakis.meliopoulos@ece.gatech.edu (A.P. Sakis Meliopoulos). 1 Tel.: +820-32-760-8434; fax: +820-32-761-5865. 2 Tel.: +1-404-894-2926; fax: +1-404-894-4641. Recent trends in the electric power industry such as deregulation and open access impose new challenges to the operation of power systems. Power wheeling for third parties has become a very important issue. Power wheeling has been defined as power transmitted from power generators to users by using the transmission or distribution facilities of other systems [1]. Power wheeling can be wholesale (bulk) or retail. Bulk wheeling is confined within the transmission system only. Retail wheeling refers to the notion of direct access [2,3], which means that customers are permitted to choose their own electricity suppliers. Here customers include all users, even individual residential consumers. So retail wheeling can engage both the transmission and distribution systems. Distribution systems are characterized by the fact that they are not symmetric threephase systems and they do not operate as balanced three-phase systems. Specifically, they may include three-phase, two-phase and single-phase circuits, threephase and single-phase loads as well as secondary two line circuits and loads. Therefore the single-phase network representation used for evaluation of bulk wheeling is no longer adequate for the study of retail wheeling. Multiphase network analysis and optimization tools are required for the evaluation of retail wheeling. A major concern is power wheeling pricing because the cost of wheeling is complex and its evaluation faces many difficult conceptual problems. Shirmohammadi et al. [4] identify four components in the cost of a transmission transaction, namely, operating, opportunity, reinforcement and existing (embedded) cost, and present methodologies for evaluating them. The first three components are directly caused by a transaction, while the last component represents existing transmission capital and O&M costs which must be allocated to different transactions. Happ [5] describes four embedded cost methodologies, which allocate the existing transmission cost to the different transactions, and two long-run incremental cost methods for the determination of the combined reinforcement and operating cost 0378-7796/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0378-7796(01)00188-2

138 A.G. Bakirtzis et al. / Electric Power Systems Research 60 (2002) 137 144 of a transaction. One of the embedded cost methods, for example, known as the contract path method assumes that the wheeling power flow is confined along a specified path. Since this method does not require power flow analysis, it is computationally fast but may result in errors because of the unrealistic assumption of a contract path. Another embedded cost method known as the boundary flow method computes changes of power flow due to wheeling with a power flow model. A description of the basic technical concepts about cost based transmission pricing is given in Shirmohammadi et al. [6]. Since, due to the existing economies of scale in power transmission, the embedded cost component is the dominant part of the transmission cost, many regulatory and research efforts have been devoted to its assignment to the involved parties [7 9]. The theory underlying wheeling rates, using short-run marginal cost pricing, has been described [10 13]. Short-run marginal cost pricing is based on the effects of wheeling on power system operating cost. On the other hand, power wheeling affects system security. It is conceptually difficult to quantify the effect of wheeling on security. This paper proposes a comprehensive mathematical model for assessing the effects of retail wheeling to a set of power system performance metrics. The model is based on multiphase power flow and optimal power flow analysis tools and an efficient sensitivity computation method for unbalanced three-phase systems. System performance metrics relative to: (a) operating cost, (b) transmission losses, and (c) system security are considered. While the cost and losses are well defined, quantification of system security presents conceptual problems. In this paper we selected to quantify system security with a set of indices. 2. Problem formulation The overall approach is shown in Fig. 1. It is based on a conceptually straightforward Monte Carlo method. A number, N max, of trials are performed. At each trial, a specific state of the system is randomly selected. Subsequently, the operating condition of the system is determined with a full-featured three-phase optimal power flow. At the operating point, the sensitivities of a set of system performance metrics or quantities of interest with respect to power wheeling are computed and stored. The results from the N max trials are utilized to determine the probability distribution function of the incremental effects of wheeling on the selected quantities of interest. The probability distribution functions provide a realistic picture of the effects of wheeling over a wide range of operating conditions. The selected system performance metrics are: (a) cost, (b) losses and (c) several system security indices. The analytical expressions of the selected system performance indices are: 1. Total system operating cost n F 1 = (a i +b i P gi +c i P 2 gi ) (1) i=1 where P gi is the unit i MW output; a i, b i, c i are the quadratic cost coefficients of unit i. 2. Total system transmission losses 2 F 2 = R l I lrms for all circuits l (2) l where R l is the resistance of circuit l; I lrms is the current of circuit l (rms value). 3. Voltage security index F 3 = i W i V i V i,av V i,st 2n (3) Fig. 1. Proposed Monte Carlo simulation for evaluating power wheeling effects. where V i is the voltage magnitude at bus i; V i,av = max min max min 0.5(V i +V i ); V i,st =0.5(V i V i ); W i is a weighting factor for bus i; n is a selected integer. 4. Transmission loading security index

A.G. Bakirtzis et al. / Electric Power Systems Research 60 (2002) 137 144 139 F 4 = W l l S l max 2n S l where S l is the MVA power flow of circuit l; S l max is the MVA rating of circuit l. The effect of power wheeling on system performance is measured by the incremental changes of the above system performance indices due to wheeling (sensitivities) at the operating point of the system. It is tacitly assumed that power system operation is continually optimized. For this reason, the computational procedure consists of an optimal power flow and sensitivity analysis at the optimized operating point. In subsequent paragraphs, the constituent parts of the approach are briefly described. 2.1. Multiphase optimal power flow model The multiphase OPF is formulated as a nonlinear optimization problem of the form: Min f(x, u) (5) S.t. g(x, u)=0 (6) (4) h(x, u) 0 (7) where f(x, u) is the optimization objective, in this case the total system operating cost expressed by Eq. (1), equality constraints (Eq. (6)) are the multiphase power flow equations [14] and inequality constraints (Eq. (7)) are system operating constraints, such as branch flow limits, bus voltage limits, generator reactive limits, etc. Note that the multiphase optimal power flow formulation appears similar to the conventional optimal power flow. The differences are: 1. The state vector x consists of sub-vectors, one for each bus of the system. As an example, a threephase bus contributes a 6 1 sub-vector consisting of three voltage magnitudes and three voltage phase angles, a single phase bus contributes a 2 1 subvector consisting of one voltage magnitude and one voltage phase angle, etc. In addition, for one bus, the slack bus, one of the voltage phase angles (phase A) is assumed known and equal to zero. 2. The power flow equations, g(x, u)=0, are derived from the multiphase network equations: diag(v ) k A k g k ((A k ) T V ) n* =diag(v ) I inj * =S inj where: V is the vector of all system node voltages (phasors); I inj is the vector of all nodal complex current injections; S inj is the vector of all nodal complex power injections; g k is a vector of functions, which describes the I V relationship of component k: I k =g k (V k ), where I k and V k are vectors of component k terminal currents and voltages; A k is the kth component terminal to node incidence matrix with: A k ij = 1, if the jth terminal of component k is connected to node i 0, otherwise 3. The generated power of a unit, P gk is related to the state variables via the equation: P gk =Re{[(A k ) T V ] T [g k ((A k ) T V )]*} 4. The operating constraints, h(x, u) 0, are appropriate expressions. Details for the network equations for the above model can be found in Ref. [14]. Organizationally, we use the composite node concept [14], which enables the efficient application of sparsity techniques, optimal ordering, LU factorization and forward-back substitutions. The defined problem is solved by the successive linear programming method. A brief description of the method follows. Assume that the current operating state of the system is x 0 and the control variable settings are u 0. By linearizing the OPF objective function and operating constraints around the current operating point (x 0, u 0 ), using the method described in the next section, the initial nonlinear problem is converted to a linear problem: Min c T u (8) S.t. A e u=b e (9) A i u b i (10) u min u 0 u u max u 0 (11) where u=u u 0, is the control variable adjustment. The solution of the linear programming problem defined by Eqs. (8) (11) is obtained by two alternative methods: (a) a sparsity coded simplex method with upper bounds and (b) a barrier method based on the primal dual interior point method. The reason for using both of these methods is that the problem under consideration may be of substantially different size depending on the power system size and operating condition. It has been found [15 17] that for relatively small LP problems, the first method outperforms the second while the roles reverse for larger problems. For efficiency, one of these methods is selected after the LP problem size has been determined. At this point it must be clarified that at the current, optimized operating state, the system operating constraints (Eq. (7)) are satisfied and thus the numerical values of the system voltage and transmission loading security indices are low. Superimposing power wheeling on the current operating state will affect the operating cost, the losses and the security indices. The incremental effects of power wheeling on all performance metrics are calculated using sensitivity analysis, described next.

140 A.G. Bakirtzis et al. / Electric Power Systems Research 60 (2002) 137 144 which is the most involved model. For this load, we first write the current voltage equations in the form: Fig. 2. Illustration for the definition of a wheeling transaction. 2.2. Sensiti ity analysis For the purpose of computing the sensitivity of the system performance metrics F 1, F 2, F 3 and F 4 with respect to power wheeling, a parameter must be introduced which uniquely defines power wheeling. This can be done in a number of ways. In this paper, we define the power wheeling variable u to be the amount of real power transported from the seller to the buyer of the wheeling transaction. As shown in Fig. 2 the seller may be a three-phase generator at the transmission (A) or distribution (B) level. Single phase generators at the distribution level may also be handled with the proposed method but are of no practical significance. The buyer may be a three-phase load at the transmission (C) or distribution (D) level or a single-phase load at the distribution level (E). The following four cases can be identified: (a) Generator 3, Load 3, (b) Generator 3, Load 1, (c) Generator 1, Load 3, (d) Generator 1, Load 1. A general and powerful method for computing the sensitivity of the system performance metrics F 1, F 2, F 3 and F 4 with respect to power wheeling has been developed. This method is an extension of the costate method, initially developed for balanced three-phase networks [18 20]. A brief description of the extension of this method to accommodate power wheeling for any of the four cases identified above is presented. First, a control variable representing power wheeling is introduced as follows: For generality, a power wheeling load is defined as a combination of constant impedance load (linear) and constant power load (nonlinear). Here we present the model equations of the nonlinear constant power load, I k =(x p Y p +jx q Y q ) V k where Y p is the real part of the load nominal admittance matrix, Y q is the imaginary part of the load nominal admittance matrix, x p and x q are state variables. The power equation for this load is: u+j u=(a k V ) T (x p Y p jx q Y q )(A k V *) where is a constant of the load (representing the power factor of the load). The above equation is added to the power flow equations. Note that the addition of the above complex equation (or two real equations) is accompanied by the addition of two state variables x p and x q, resulting in a consistent set of equations, i.e. number of equations and unknowns. The control variable u represents the total wheeling power. Wheeling of u+j u from node i to node j is modeled by the simultaneous increase of node j load by u+j u and increase of node i generation by the same amount. Wheeling can take place from any node to any node. Therefore there are n(n 1) combinations of power wheeling for an n-bus system. The effects of all these combinations can be computed by simply computing the sensitivity, S, of the metric under consideration with respect to a load wheeling at all buses, i.e. S i, i=1, n. Then the sensitivity of this metric with respect to a wheeling transaction from bus i to bus j is computed from S i S j. This is the reason that we only need to compute the sensitivities S i. It is also important to note that the sensitivities S i correspond to power wheeling from the slack bus to bus i. Next, any system performance metric, such as F 1, F 2, F 3 and F 4, can be expressed as a function of the system state x and power wheeling variable u, i.e. F(x, u). On the other hand, the states, x, and the power wheeling, u, must satisfy the power flow equations. g(x,u) =0 (12) where x is the system states; u is the power wheeling control variables. Differentiation of the function F(x, u) and the power flow Eq. (12) yields: df(x,u) = F du u + F x x (13a) u g u + x x g u =0 (13b) Upon elimination of the derivatives x/ u, one obtains: df(x,u) = F du u F g 1 g (14) x x u where ( g/ x) 1 is the inverse of the system Jacobian matrix.

A.G. Bakirtzis et al. / Electric Power Systems Research 60 (2002) 137 144 141 Fig. 3. A 10-bus test system. Define the costate vector xˆ as follows: xˆ T= x g F 1 x In terms of the costate vector xˆ T, Eq. (14) becomes: df(x,u) = F du u xˆ T g (15) u It is important to note that the widely used short run marginal costs can be directly computed from the sensitivities of the total system operating cost with respect to wheeling. 2.3. Monte Carlo simulation Monte Carlo simulation [21,22] has been extensively utilized in power system probabilistic analysis. The basic idea of Monte Carlo simulation is very simple. It simulates a specified system with a reasonable number of random draws of all possible system states according to their probability distributions. The simulation of a randomly selected system condition is done with the use Table 1 Maximum wheeling sensitivities for 10-bus test system of power system analysis methods and subsequent computations of the system performance metrics. The results of the simulation are reported as probability distribution functions of the system performance metrics. The key issues in Monte Carlo simulation are: (1) the number of trials must be large enough to adequately capture all possibilities relative to the application and (2) the analysis methods must be appropriate with respect to the application, i.e. they should accurately compute the quantities of interest. The proposed method uses an accurate computational method for the quantities of interest and a large number of trials. A future improvement of the method will be the use of variance reduction techniques to reduce the number of trials. In the proposed Monte Carlo simulation, each trial defines a system state which may include a combination of circuit and unit outages. Subsequently, an optimal power flow is solved to determine the actual system operating conditions. At this operating point the effects of power wheeling on the quantities of interest are quantified by computing the sensitivities of F 1, F 2, F 3, and F 4 with respect to wheeling between any two buses in the system and the results are stored. After a sufficient number of trials has been performed, probability distributions of the computed sensitivities are generated. The number of probability distributions that can be generated is large, for example, one for each pair of buses in the system and this can be repeated for each one of the system performance metrics F 1, F 2, F 3, and F 4. 3. Evaluation The proposed method has been applied on two test systems. The first test system, shown in Fig. 3, is a 10-bus system which comprises transmission and distribution lines. The proposed method is applied to this system to compute maximum sensitivities of the system performance metrics F 1, F 2, F 3,andF 4 with respect to Objective Producer bus Consumer bus Sensitivities Operating cost (F 1 ) 0 1 3 1.1871 ($/MWHR) 0.2 1 3 1.5888 ($/MWHR) 0.4 1 3 2.0032 ($/MWHR) Transmiss. losses (F 2 ) 0 9 2 0.0513 0.2 9 2 0.0566 0.4 9 2 0.0620 Voltage security index (F 3 ) 0 9 2 0.021648 (per MW) 0.2 9 2 0.040124 (per MW) 0.4 9 2 0.058600 (per MW) T. Loading security index (F 4 ) 0 10 2 0.048160 (per MW) 0.2 10 2 0.052406 (per MW) 0.4 10 2 0.056653 (per MW)

142 A.G. Bakirtzis et al. / Electric Power Systems Research 60 (2002) 137 144 Fig. 4. IEEE RTS-96 reliability test system. The second test system is the RTS-96, a 73-bus system [23]. The system is illustrated in Fig. 4. It consists of three identical IEEE 24-bus RTS interconnected via five tie lines. The proposed Monte Carlo simulation method has been applied to this system without modification (Case A). Figs. 5 and 6 illustrate the inverted cumulative probability distribution function (PDF) of the sensitivity of the performance metrics F 1 and F 2 (Production Cost and Transmission Losses) with respect to power wheeling from bus 119 (producer) to bus 205 (consumer). The x-axes of Figs. 5 and 6 contain the different possible values, x, of the sensitivity of the corresponding performance metric S F = df du treated as random variable, and the y-axis represents the inverted cumulative PDF of random variable S F, i.e. Prob[S F x]. In order to demonstrate the effects of wheeling to distribution level customers, the RTS-96 system has been modified. A radial distribution network, comprising two distribution level buses, 2051 and 2052, (not shown in Fig. 4) has been added to RTS-96 as follows: A 138 kv/12 kv, 10 MVA step down transformer has been added between buses 205 and 2051 and a 5-mile, 12 kv distribution line has been added between buses 2051 and 2052. The effects of transmission of real power from bus 119 to distribution bus 2052 are studied under Case B. Figs. 7 and 8 illustrate the inverted cumulative probability distribution function of the sensitivity of the performance metrics F 1 and F 2 (Production cost and Transmission Losses) with respect to power wheeling from bus 119 (producer) to bus 2052 (consumer). Comparing the results of Figs. 5 and 6 with the ones of Figs. 7 and 8, it is observed that the sensitivities of both the production cost and the transmission losses are much higher for power wheeling to a distribution level bus. Fig. 5. Inverted cumulative PDF of production cost sensitivity w.r.t. wheeling variable (Case A). wheeling of u+j u between any two buses. Table 1 illustrates some typical simulation results. Only the maximum sensitivities are listed, together with the location of the producer/consumer, which results in maximum sensitivities. Note that the maximum impact (sensitivity) for the four performance metrics F 1, F 2, F 3, and F 4 occurs at different buses. Fig. 6. Inverted cumulative PDF of transmission loss sensitivity w.r.t. wheeling variable (Case A).

A.G. Bakirtzis et al. / Electric Power Systems Research 60 (2002) 137 144 143 Fig. 7. Inverted cumulative PDF of production cost sensitivity w.r.t. wheeling variable (Case B). rates for firm, long-term transmission transactions that wish to buy transmission insurance from the wheeling company, the effect of wheeling on all foreseeable future system operating states, including those resulting from system contingencies, must be taken into account. If, for example, the wheeling company charges a transmission transaction according to average, over the foreseeable future operating states, scaled marginal losses, there is a probability that it will under-recover its cost of losses due to the specific transaction. The proposed methodology allows the computation of this probability. It also allows the computation of the appropriate transaction loss liability, so that the probability that the wheeling company under-recovers loss costs is below a certain percentage. While there are many ways to account for wheeling charges, our methodology should be viewed as the tool by which alternative pricing schemes can be evaluated. Acknowledgements This research has been partially supported by a NATO Collaborative Research Grant CRG.960101. References Fig. 8. Inverted cumulative PDF of transmission loss sensitivity w.r.t. wheeling variable (Case B). 4. Conclusions A Monte Carlo simulation method is proposed for evaluating the effects of power wheeling (both bulk and retail) on the operation of an electric power system. The method is capable of computing the probability distribution of the sensitivity of any system performance metric with respect to wheeling between any two system buses. These data quantify the effects of power wheeling on system operating cost, transmission losses, and system security. Marginal system costs can be directly computed from the sensitivities of the operating cost with respect to wheeling. The probability distributions of the sensitivities provide a complete picture of the effects of wheeling over a wide range of operating conditions. These data can be used in setting wheeling rates and planning the operation of the system in the presence of wheeling. In setting spot wheeling rates, the marginal effect of wheeling on the current system operating state must be evaluated. However, in setting in advance wheeling [1] Electric Power Wheeling and Dealing, Congress of the United States, Office of Technology Assessment, Washington, DC, 1989. [2] G. Zorpette, Power and energy, IEEE Spectrum, January 1993, pp. 56 57. [3] S. Hunt, G. Shuttleworth, Unlocking the grid, IEEE Spectrum, July 1996, pp. 20 31. [4] D. Shirmohammadi, C. Rajagopalan, E.R. Alward, C.L. Thomas, Cost of transmission transactions: an introduction, IEEE Trans. Power Syst. 6 (4) (1990) 1546 1560. [5] H.H. Happ, Cost of wheeling methodologies, IEEE Trans. Power Syst. 9 (1) (1994) 147 156. [6] D. Shirmohammadi, X.V. Filbo, B. Gorenstin, M.P. Pereira, Some fundamental technical concepts about cost based transmission pricing, IEEE Trans. Power Syst. 11 (2) (1996) 1002 1008. [7] Y. Tsukamoto, I. Iyoda, Allocation of fixed transmission cost to wheeling transactions by cooperative game theory, IEEE Trans. Power Syst. 11 (2) (1996) 620 629. [8] J.W. Marangon Lima, Allocation of transmission fixed charges: an overview, IEEE Trans. Power Syst. 11 (3) (1996) 1409 1418. [9] J.W. Marangon Lima, M.V.F. Pereira, J.L.R. Pereira, An integrated framework for cost allocation in a multi-owned transmission system, IEEE Trans. Power Syst. 10 (2) (1995) 971 977. [10] M.C. Caramanis, R.E. Bohn, F.C. Schweppe, The costs of wheeling and optimal wheeling rates, IEEE Trans. Power Syst. 1 (1) (1986) 63 73. [11] M.C. Caramanis, N. Roukos, F.C. Schweppe, Wrates: A tool for evaluating the marginal cost of wheeling, IEEE Trans. Power Syst. 4 (2) (1989) 594 605. [12] H.M. Merrill, B.W. Erickson, Wheeling rates based on marginalcost theory, IEEE Trans. Power Syst. 4 (4) (1989) 1445 1450. [13] Y.Z. Li, A.K. David, Wheeling rates of reactive power flow under marginal cost pricing, IEEE Trans. Power Syst. 9 (3) (1994) 1263 1269.

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