DETERMINATION OF CONGESTION ZONES IN DEREGULATED ELECTRICITY MARKETS USING FUZZY CLUSTERING

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1 DETERMINATION OF CONGESTION ZONES IN DEREGULATED ELECTRICITY MARKETS USING FUZZY CLUSTERING Ying-Yi Hong#, Senior Member, IEEE Chih-Nein Chang- Chien# Kuo-Lung Wu+ and Miin-Shen Yang+ #Department of Electrical Engineering, +Department of Mathematics Chung Yuan Christian University, Chung Li 30, Taiwan tel: ext 4800; fax: Abstract Congestion management is an important task for dealing with congested transmission line flow in deregulated electricity markets. This paper presents a new method for determining congestion zones. The method uses the fuzzy-c-means (FCM) algorithm and a partition separation index (PSI). The nodal prices or the locational marginal prices (LMP) serve as inputs to the FCM for a given cluster (zone) number while the PSI is used to determine an appropriate zone number. The IEEE 30- and 118-bus systems are employed to serve as test systems for showing the applicability of the proposed method. Keywords: Congestion Zone, Deregulation, Fuzzy Clustering 1 INTRODUCTION Deregulation has a great impact on the power industry restructure nowadays. For a deregulated environment, transaction of the electric energy is achieved either in a competitive bidding market or by bilateral contracts. The transmission congestion is most likely to occur when many transactions or scheduled/forced outages exist in the power system. The congestion may be mitigated by proper congestion management [1,,3]. The pricing system (e.g., nodal price or locational marginal price (LMP)) plays an important role in the congestion management [1,4,5]. The congestion management in the California market is achieved by Zonal-Forward Market in which the congestion zone was defined as follows: a congestion zone is a portion of the ISO transmission grid within which transmission congestion is expected to be small and infrequent. Interfaces between zones or control area boundaries consist of paths that are expected to have relatively high congestion management cost [1]. The congestion cost of the interfaces (boundaries) among zones over a certain time period exceeds a specified threshold while the congestion cost of a zone over a certain time period does not exceed a specified threshold. According to this definition, it is difficult to identify the boundaries for constructing several zones in a power system. The reasons are as follows: (1) assigning the thresholds are heuristic and () sometimes the identified boundaries cannot construct loops surrounding the zones. There are many existing papers dealing with congestion management [4, 6-10]. Alomoush proposed fixed transmission right model for the congestion management [4]. Singh presented approaches for dealing with the congestion cost: the pool model and the bilateral model in [6]. An operation decision support software system covering the functionality of transmission dispatch and congestion management system (TDCMS) is proposed in [7] by Shirmohammadi, etc. In [8], both real and reactive power controllers were considered to mitigate the transmission congestion using optimization; three possible curtailment strategies were also discussed. Baran addressed an equitable allocation of congestion relief cost to the bilateral transactions [9]; pricing signals for relieving congestion on lines were presented in [9]. Fu in [10] identified the reactive support and real power loss services in case of congestion. To the present, there is no paper involving determination of the congestion zones using systematic approaches. In this paper, a new approach based on the Fuzzyc-Means (FCM) algorithm [11-13] is presented to identify the congestion zones in a deregulated market. Because the nodal prices or LMPs reflect the transmission operational conditions [14], the historic nodal prices or LMPs at each node serve as inputs for FCM. According to the membership function values, buses with similar nodal prices or LMPs can be easily identified and are clustered as a zone in a system. The magnitude of the center vector of each cluster (zone) can be defined as the Zonal Price [1]. On the other hand, a partition separation index (PSI) is employed to determine an optimal zone number [15]. This optimal zone number also ensures that the bus clustering pattern is the most suitable for the congestion management. The boundaries of congestion zones including frequent large line flows bound to their limits can be easily identified through the FCM associated with the partition separation index. The IEEE 30- and 118-bus systems are used to serve as test systems for showing the applicability of the proposed method. The simulation results show that the proposed method is efficient for determination of the congestion zones in a deregulated electricity market.

2 BACKGROUND OF ZONAL-FORWARD MARKET.1 Basis of Zonal-Forward Market This section describes the zonal-forward market for congestion management in CA, USA [1]. This will serve as a basis for the proposed method. A fundamental element of the zonal-congestion management model is the use of LMP [1] which will be used for calculating the congestion cost. Transmission congestion is divided into the following two categories. (1) The first category is referred to as inter-zonal congestion and occurs primarily on transmission interfaces between congestion zones. () The second one is referred to as intra-zonal congestion. Therefore, by definition and design, the transmission interfaces between zones experience major congestion, whereas congestion zones are network partitions that experience minor internal congestion [1].. Forward Market Inter-Zonal Congestion Management The inter-zonal congestion management method uses a commercial model, DC Optimal Power Flow (OPF), which has proven to be robust, accurate and efficient. The OPF is used to adjust the preferred or revised schedules, in both day-ahead and hour-ahead scheduling processes, to mitigate congestion: Objective function- to minimize the net cost of redispatch as determined by Schedule Coordinators (SC) incremental and decremental adjustment bids. Controls- all resources with adjustment bids (generators/loads/imports/exports). Constraints- the limits on control variables, the nodal active power flow balance equations, the inter-zonal interface active power flow inequality constraints, and the real power balance equation for each SC..3 Forward Market Intra-Zonal Congestion Management Intra-zonal congestion is managed in real time by the system operator using reliability must-run units and submitted bids [1]. An AC OPF is suggested; however, the ISO does not manage currently intra-zonal congestion in the forward market but only in real-time: Objective function- to minimize the absolute MW of re-dispatch, taking into account the net cost of redispatch, as determined by the submitted incremental and decremental adjustment bids. Controls- all resources that voluntarily submit adjustment bids plus MVAR controls. Constraints- the limits on control variables, the nodal active and reactive power flow balance equations, the inter-zonal interface active power flow inequality constraints, and all intra-zonal constraints (MW/MVA line flow limits and voltage limits, etc). 3 THE PROPOSED METHOD 3.1 General Description This paper addresses a new approach to determine congestion zones, which construct the intra-zone and inter-zone model in Sec.. The proposed method takes advantage of all nodal prices or LMPs at all buses and avoids examining only the congestion cost heuristically. The proposed method is achieved by both FCM and PSI. The FCM algorithm provides a method for partitioning N vectors, X i s, into C clusters. The dimension of X i is determined from the historic data. In the c-th cluster, there is a corresponding center vector V c. One can use the membership function value ci as a weighting factor between V c and X i to determine the clustering pattern [11~13]. The FCM can only be achieved by a fixed/given C. The supplemental index should be used to determine an appropriate C for a problem. The problem for finding an optimal C is usually called cluster validity [15,16]. The PSI uses a normalized partition coefficient and an exponential separation measure to determine the most appropriate C [15]. 3. Determination of Congestion Zones In this paper, the FCM algorithm is used to determine the congestion zones. The symbol X i represents the nodal prices (LMPs) for the i-th bus. The symbol N denotes the total bus number in the power system. The dimension of X i depends on the historic data, e.g., transaction periods and days, etc. The reasons for adopting the nodal prices (LMPs) as the inputs for the FCM are as follows. (a) The LMP is currently considered a very useful signal for pricing in some successful markets, e.g. PJM (Pennsylvania, New Jersey, and Maryland) market in the U.S.A. [5]. (b) The nodal price reflects the system operation conditions [14]: - The source (sink) buses generally have low (high) nodal price values. - Compared with normal conditions, the nodal price values become higher once the contingencies or the line flow congestion occur. - The nodal price value at a generator bus is higher (lower) than its offer price if its dispatched MW generation reaches its upper (low) limit. On the other hand, Bezdek defined a function J(U,V) as an objective in the FCM algorithm [1]: N C J, V m ci X i V c, 1 m (1) i 1c 1 When the value of J(U,V) is minimized, the N data vectors can be partitioned into C clusters. Appendix A provides the algorithmic steps to determine the unknowns: ci and the vector V c.

3 For example, X 1 =(1,3), X =(1.5,3.), X 3 =(3,1.5) and X 4 =(3,1). If ci converges to the following values with C=: i i i i 4 c 1 c X 1 and X have membership values ( and ) close to one in the first row; therefore, X 1 and X belong to the first cluster. Similarly, X 3 and X 4 have membership values ( and ) close to one in the second row; therefore, X 3 and X 4 belong to the second cluster. In our problem, i=1,,,n (bus number) and c=1,,, C where C is determined by PSI; the dimension of X i is 4 depending on the historic data available. As illustrated above, according to the membership function values, the buses with similar nodal prices or LMPs can be easily identified and these buses are clustered as a zone in a system. A supplemental benefit of using the FCM algorithm is that FCM provides a center vector for each cluster. The meaning of the center vector V c is that all X i s in the c-th cluster are coherent to V c, which acts as a representative for all corresponding X i s. This paper presents that the average value of all elements in V c denotes the zonal price for the c-th zone. 3.3 Determination of Optimal C For a given C, the N sets of X i s can be partitioned into C clusters without difficulty. However, an optimal (appropriate) C* should exist in both the mathematical and physical viewpoints. The problem for finding optimal C* is usually called cluster validity [15,16]. The authors proposed a new validity index for fuzzy clustering called a partition separation index (PSI) which uses two factors of a normalized partition coefficient and an exponential separation measure for each cluster and then aggregate these two factors to sum up all clusters to create the PS validity index [15]. The PSI for cluster (zone) c is defined as follows: N PS(c)= ci / M exp( min Vc V k / T ) () i 1 ck N where k=1,, C, M max ci and 1cCi 1 C Vc V c 1 T. V is the mean of all V c s. Finally, C the PS validity index is defined as follows: C PS(C) = PS( c) (3) c1 The large PS(C) value implies that the corresponding cluster c is compactness inside and separation between other (C-1) clusters. Therefore, the optimal integer C* can be obtained as follows: C* max PS( C) (4) C N 3.4 Algorithmic Steps Seven algorithmic steps for obtaining optimal C* in Eq. (4) and congestion zones are summarized as follows: Step 1: Let C=; in other words, assume that there are congestion zones initially. Step : Solve Eq. (1) to obtain ci and V c (i=1,, N and c=1,, C). Step 3: Identify zones with membership values ci. Step 4: Calculate PS(C) using Eqs. () and (3). Step 5: C=C+1. Step 6: If C N, go to Step ; otherwise, go to Step 7. Step 7: C*=max{ PS(), PS(3), PS(4), }. 4 SIMULATION RESULTS A DC optimal power flow (OPF) approach [1,14] is used for solving the deregulated electricity market in this paper. The nodal prices or LMPs can be obtained from the dual variables of the corresponding DC power flow equations. The IEEE 30- and 118-bus systems serve as examples to illustrate the simulation results. 4.1 IEEE 30-bus System N sets of historic data (different load patterns and bilateral contracts) should be considered for determination of congestion zones. In this paper, 4 transaction periods with different load levels are considered as the historic data. Figure 1 illustrates the 4 load levels. Table 1 shows the generator data. The original IEEE 30-bus system data is at the 4 th transaction period. A bilateral contract between buses 8 and 14 is considered. Congestion occurs frequently in lines between buses 4-1 and After solving 4 DC OPFs, 30 4-dimensional vectors of nodal prices are used as input for FCM. It is found that the number of occurrence of congestions for the line between buses 4-1 is 3 (out of 4) and for both lines between buses 4-1 and 9-10 are 11. There are 10 cases without congestion. Table illustrates the nodal prices for the IEEE 30-bus system in the 16 th transaction period. The possible values of C are, 3, 4, and 5. Figures -5 demonstrate the clustering results for identifying the congestion zones with various C s. As shown in Figure, lines between buses 4-1 and 9-10 locate on the boundaries of identified zones: the first zone covers buses 1~9, 11, 5~30 while the remaining

4 buses are included in the second zone. It is noted that the lines between buses 6-10 and 4-5 are not congested actually. However, a boundary between zones 1 and cannot be defined without these lines. Therefore, the proposed method helps identify pseudo congestion lines between buses 6-10 and 4-5. Table 3 shows the membership values at different buses for C=. Lines between buses 6-10, 4-5 and 7-8 on the boundaries of zones are not congested in Figure 3. For the same reason, lines between buses 6-10, 15-18, 15-3, 16-17, 4-5 and 7-8 locating on the boundaries are infrequently congested in Figure 4; moreover, lines between buses 6-10, 15-18, 15-3, 16-17, 4-5, 5-7 and 7-8 locating on the boundaries are not reasonable in Figure 5. From the above discussion, the value of C should be from the viewpoint of the power system line flow pattern. Different C s should be further examined by SPI in the viewpoint of mathematics. As shown in Table 4, the value of PS() is the largest. This implies that clustering congestion zones is the most appropriate. The average zonal prices for zones 1 and are 5.17 and 5.74 $/p.u., respectively. Nodal price Nodal price Nodal price Table : The nodal prices for the 16 th transaction period. load ratios transaction periods Zone Figure : Two congestion zones identified with C= Figure 1: Twenty four load levels for the IEEE 30-bus system PG (MW) Offer PG max (MW) PG min (MW) Wheeling Rate ($/p.u.-km) Table 1: Generator data for the IEEE 30-bus system Zone Zone 3 Figure 3: Three congestion zones Identified with C=3

5 Zone 4 Zone Zone 3 Figure 4: Four congestion zones identified with C=4 es Cluster Cluster es Cluster Cluster es Cluster Cluster es Cluster Cluster es Cluster Cluster Table 3: Membership values at different buses for C= PS() PS(3) PS(4) PS(5) Table 4: Different PS validity indices for IEEE 30-bus system PS() PS(3)* PS(4) PS(5) PS(6) * PS(7) PS(8) PS(9) PS(10) Table 5: Different PS validity indices for IEEE 118-bus System Zone Zone 4 Zone 5 Zone 3 Figure 5: Five congestion zones identified with C=5 4. IEEE 118-bus System In this subsection, the IEEE-118 bus system is used to illustrate the simulation results. Twenty-four system operational conditions are studied by DC OPF to obtain nodal prices at each bus. The maximum possible clustering number is 10 ( 118 ). As shown in Table 5, decomposing the system into 3 congestion zones is the most appropriate. Figure 6 shows the partitioned result with 3 zones. The average zonal prices for zones 1, and 3 are 9.67, 6.96, and $/p.u., respectively. 5 CONCLUSION A new method based on the fuzzy-c-means algorithm and the partition separation index is proposed for determination of the congestion zones in a deregulated electricity market. The system buses are partitioned/clustered into several congestion zones according to the membership function values of nodal prices in the FCM. The boundaries of the congestion zones can therefore be identified without difficulties. The partition separation validity index verifies and identifies the most appropriate congestion zone number. The simulation results obtained from the IEEE 30- and 118-bus systems show that the proposed method is applicable and efficient. 6 ACKNOWLEDGEMENT The authors would like to thank the National Science Council, Taiwan, R.O.C. for the financial support to this

6 research under the grant number NSC E BIOGRAPHIES Ying-Yi Hong received the B.S.E.E and M.S.E.E. degrees from the Chung Yuan Christian University (CYCU) and the National Chen Kung University, Taiwan, in 1984 and 1986, respectively. Sponsored by the Ministry of Education of R.O.C., he conducted research in the Department of E.E. at the University of Washington, Seattle, from August 1989 to August He received the Ph.D. degree from the Institute of E.E. in December 1990 from the National Tsing-Hua University, Taiwan. From February 1991 to July 1995, he served as an associate professor in the Department of E.E. at the CYCU. He was promoted to the rank of full professor in August He is an IEEE senior member. His areas of interest are power system analysis, power quality analysis and AI applications. Chih-Nein Chang-Chien received his B.S.E.E. and M.S.E.E. from the Feng-Chia University, Taiwan, and CYCU in 1997 and 001, respectively. He has been with CYCU as an instructor since August 001. His research interests include the applications of artificial intelligence and the deregulation. Kuo-Lung Wu received the B.S. degree in mathematics in 1997 and the M.S. degree in applied mathematics in 000 from the CYCU. He is a Ph.D. student and a part-time instructor, Department of Mathematics, CYCU. His research interests include cluster analysis, pattern recognition and neural networks. Miin-Shen Yang received the B.S. degree in mathematics from the CYCU in 1977, the M.S. degree in applied mathematics from the National Chiao-Tung University, Hsinchu, Taiwan, in 1980, and Ph.D. degree in statistics from the University of South Carolina, Columbia, in He is a Professor, Department of Mathematics, CYCU. His current research interests include applications of statistics, fuzzy cluster analysis, pattern recognition and neural network systems. REFERENCES [1] Comprehensive Congestion Management Reform Zonal-Forward Market White Paper, California ISO, April 000. [] Congestion Management Working Group of the NERC Market Interface Committee, Comparison of System Redispatch Methods for Congestion Management, NERC Report, September [3] Market Redispatch Pilot Project Summer 000 Procedure, NERC Report, March 000. [4] M.J. Alomoush and S.M. Shahidehpour, Fixed Transmission Right for Zonal Congestion Management, IEE Proceedings-Generation. Transmission & Distribution, Vol. 146, No. 5, Sep. 1999, pp [5] PJM LMP Implementation Training Course Interpreting Locational Pricing, PJM Interconnection, L.L.C., DOCS #6310, March 5, [6] H. Singh, S. Hao and A. Papalexopoulos, Transmission Congestion Management in Competitive Electricity Markets, IEEE Trans. on Power Systems, Vol. 13, No., May 1998, pp [7] D. Shirmohammadi, B. Wollenberg, A. vojdani, P. Sandrin, M. Pereira, F. Rahimi, T. Schneider and B. Stott, Transmission Dispatch and Congestion Management in the Emerging Energy Market Structures, IEEE Trans. on Power Systems, Vol. 13, No. 4, Nov. 1998, pp [8] R.S. Fang and A.K. David, Trnasmission Congestion Management in an Electricity Market, IEEE Trans. on Power Systems, Vol. 14, No. 3, Aug. 1999, pp [9] M.E. Baran, V. Banunarayanan, and K.E. Garren, Equitable Allocation of Congestion Relief Cost to Transactions, IEEE Trans. on Power Systems, Vol. 15, No., May 000, pp [10] J. Fu and J.W. Lamont, A Combined Framework for Service Identification and Congestion Management, IEEE Trans. on Power Systems, Vol. 16, No. 1, Feb. 001, pp [11] R.J. Hathaway, J. C. Bezdek and Y. Hu, Generalized Fuzzy-c-Means Clustering Strategies Using Lp Norm Distances, IEEE Trans. on Fuzzy Systems, Vol. 8, No. 5,, Oct. 000, pp [1] Bezdek, J.C., Ehrlich, R., and Full, W.: FCM: the Fuzzy c-means Clustering Algorithm, Computers & Geosciences, Vol. 10, No. -3, 1984, pp [13] J.C. Dunn, A Fuzzy Relative of the ISODATA process and its Use in Detecting Compact Well-

7 separated Cluster, J. Cybernet., Vol. 3, 1974, pp [14] Y.Y. Hong and M.T. Weng, Investigation of Nodal Prices in a Deregulated Competitive Market Case Studies, IEEE PowerTech 99, Budapest, Aug. 9-Sep., [15] M.S. Yang and K.L. Wu, A New Validity Index for Fuzzy Clustering, 10 th IEEE International Conference on Fuzzy Systems, Melbourne, Australia, -5 Dec., 001. [16] J.C. Bezdek, Cluster Validity with Fuzzy Sets, J. Cybernet., Vol. 3, No. 3, 1974, pp APPENDIX A Bezdek developed 4 solution steps to achieve the minimum of Eq. (1) as follows [1]: N U ci i1 V c h N U h i1 Step 3: Update ci m X i m 1 c C 1 i N (6) U ci h for all X i, i=1,...,n. U ( h) 1 ci C X i Vc j 1 X i V j m1 (7) Step 4: If U ( h) U ( h1) ci ci, stop; else go to Step. The symbol is the convergence tolerance. Step 1: Estimate a matrix of membership functions: U ( h R U ( h) ) CN ci (5) where h is the iterative index and h=0 initially. Step : Let h=h+1. Compute the center of the c-th cluster: Figure 6: Three congestion zones in IEEE 118-bus system