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This issue includes the following Power Engineering Letters: ○ Optimal Power Flow Using Tabu Search, by T. ...... For more information, contact William J.
M.E. El-Hawary, Editor

Power Engineering Letters T

his section of the magazine offers a vehicle that speeds publication of new results, discoveries, and developments. The section affords authors the opportunity to publish contributions within a few months of submission to ensure rapid dissemination of ideas and timely archiving of developments in our rapidly changing field. Original and significant contributions in applications, case studies, and research in all fields of power engineering are invited. Of specific interest are contributions defining emerging problems and special needs in specific areas. Submit contributions, criticism, and queries to the Power Engineering Letters editor: M.E. El-Hawary, DalTech, Dalhousie University, P.O. Box 1000, Halifax, NS B3J 2X4 Canada, +1 902 494 6198 or +1 902 494 6199, fax +1 902 429 3011, e-mail [email protected].

Editorial Board The following Power Engineering Letters editorial board is responsible for the peer review of all Letters appearing in this section: ● M.E. El-Hawary (editor), Dalhousie University, Canada ● A. Bose, Washington State University, USA ● M.T. Correia de Barros, Universidade Tecnica de Lisboa, Portugal ● A.M. DiCaprio, PJM Interconnection, USA ● A.R. El-Kieb, University of Alabama, USA ● G.N. Ericsson, Svenska Kraftnät, Sweden ● R.K. Green, Jr., Alstom, USA ● T.J. Hammons, University of Glasgow, UK ● S.I. Iwamoto, Waseda University, Japan ● J.H. Jones, Southern Companies Services, USA ● P. Kundur, Power Tech Labs, Canada ● F.N. Lee, University of Oklahoma, USA ● J. Momoh, Howard University, USA ● M.M. Morcos, Kansas State University, USA ● A. Papalexopoulos, ECO International ● M. Poloujadoff, Universite Pierre et Marie Curie, France ● N. Rau, ISO New England Inc., USA ● A. Schwab, Universitat Karlsruhe, Germany ● M. Shahidehpour, Illinois Institute of Technology, USA ● M. Shwehdi, King Fahd University of Petroleum and Minerals, Saudi Arabia ● W.L. Snyder, Jr., Siemens Energy & Automation, Inc., USA ● J.S. Thorp, Cornell University, USA ● S.S. Venkata, Iowa State University, USA ● B.F. Wollenberg, University of Minnesota, USA.

In This Issue This issue includes the following Power Engineering Letters: ● Optimal Power Flow Using Tabu Search, by T. Kulworawanichpong and S. Sujitjorn ● Fault Current Limiter with Fast-Closing Switch, by Jiyan Zou, Jinxiang Chen and Enyuan Dong ● Utilization of Generator Reactive Capability: A Transmission Viewpoint, by Shih-Min Hsu and Henry J. Holley ● Distribution Matching Power Flow: A New Technique for Distribution System State Estimation, by H.B. Sun and B.M. Zhang ● Block Pricing for Electric Power Markets, by Xifan Wang, Xiaohong Guan, and Xiuli Wang ● Handling High R/X Ratios in Meshed Systems with Moderate Heterogeneity, by O.R. Saavedra and A.B. Rodrigues ● Application of a Novel Fuzzy Control Strategy for HVdc Link To Improve Interarea Stability, by T.S. Chung and Da-zhong Fang. IEEE Power Engineering Review, June 2002

Optimal Power Flow Using Tabu Search T. Kulworawanichpong, S. Sujitjorn Author Affiliation: School of Electrical Engineering, Suranaree University of Technology, Nakhon Ratchasima, Thailand. Abstract: Optimal power flow (OPF) is one of the main functions of power system operation and control. Solving such problems requires efficient optimization algorithms. This letter describes the development of an efficient tabu search (TS) algorithm for solving the optimal power flow problem. The developed algorithm was tested against a modified IEEE-RTS-24-bus test system. The results were compared with ones obtained from sequential quadratic programming (SQP) and evolutionary programming (EP) techniques. All approaches give more or less the same optimal solutions. Tabu search is the most efficient technique among these, since it consumes minimum computing time and provides accurate solutions. Keywords: Optimal power flow, tabu search, evolutionary programming. Nomenclature: Total generation cost (R/h) FT Fuel cost of generator I f ( PGi ) Number of generators NG Scheduled active power at bus k Psch , k Scheduled reactive power at bus k Qsch , k Admittance magnitude between buses i and k Yik Admittance angle between buses i and k θik Voltage Magnitude at bus I Vi Voltage angle at bus I δi Number of buses NB PGimin , PGimax Limits on the active power output of generator I QGimin , QGimax Limits on the reactive power output of generator I Vjmin , Vjmax Limits of voltage magnitude at bus j Tkmin , Tkmax Limits of transformer tap position of transformer k Introduction: An optimal power flow problem is concerned with finding the minimum generation cost while balancing the entire power flow [1]. The problem is generally nonconvex and nonlinear. It may have many local minima. Conventional techniques such as linear programming and interior point method [2] have been applied to this problem. These techniques need simplification of the problem to make it linear or convex. The minimum thus found cannot be guaranteed as the global minimum to the entire space of the OPF problem. To be able to incorporate realistic constraints in the OPF problem, some stochastic optimization and artificial intelligence methods such as simulated annealing (SA), artificial neural network (ANN), and evolutionary programming (EP) [3]-[7] have been applied. Searching for an optimal solution is then more efficient and realistic. However, these algorithms normally take extensive computing time compared with the conventional optimization methods. Recently, the TS technique [8-12] has been applied to power system and control problems. Even though the TS algorithm was originally developed as a stochastic optimization technique, it can search for an optimal solution within a short computing time [12]. The work reported herein describes an efficient TS algorithm for solving the OPF problem under combined real and reactive powers. The power components in the OPF problem are commonly decomposed to avoid complexity. This would lead to an unsatisfactory so-

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37

lution, because real and reactive powers interact in the real situations. The more realistic approach is to combine both power components while attempting to optimize the power flow. Optimal solutions can be obtained with fast and efficient TS algorithms, as reported in this letter. Problem Formulation: The OPF problem is in general nonlinear optimization problem in the following form. Minimize f ( x ) Subjected to gi ( x ) = 0

hj ( x ) ≤ 0 l≤x ≤u where f ( x ) is the objective function describing the mathematical formula of the aim. In the OPF problem, f ( x ) is the total generation cost. The equations gi ( x ) = 0 are equality constraints and hj ( x ) ≤ 0 are inequality constraints. The objective function, the equality and the inequality constraints, are described as follows. Objective Function: Total generation cost [2] NG

FT = ∑ f ( PGi ) i=1

(1)

Equality Constraints: Power mismatch equations [2] NB

Psch , k − ∑ | YikVV i k | cos( θik + δk −δi ) = 0 i=1

(2)

and NB

Qsch , k − ∑ | YikVV i k | sin( θik + δk −δi ) = 0. i=1

(3)

Inequality Constraints: Limits of control or state variables [2] PGimin ≤ PGi ≤ PGimax QGimin ≤ QGi ≤ QGimax Vjmin ≤ Vj ≤ Vjmax Tkmin ≤ Tk ≤ Tkmax Vrefmin ≤ Vref ≤ Vrefmax

Figure 1. Modified IEEE-RTS-24-bus test system

(4)

Optimization Methods: To find optimal solutions for the power flow problem, an appropriate optimization method has to be chosen to handle its nonlinear and nonconvex nature. In fact, there is no restriction for making selection. Searching speed and accuracy are mainly the matters of concern. This letter attempts to demonstrate effectivenesses of three different optimization methods, namely sequential quadratic programming (SQP), evolutionary programming (EP), and tabu search (TS) techniques. Only EP and TS methods are reviewed herein, since the SQP method is conventional and commonly known. Evolutionary Programming: This method involves a random search technique. Some researchers [3]-[7] have applied the EP technique of various forms to solve the OPF problem. The solution obtained depends on the suitability measure of the members of selected population. Hence, the effectiveness of the method depends on the computation of this suitability measure. Members of the population are divided into two groups, namely feasible sets ( ψ ) and infeasible sets ( ϑ ). The suitability measure is defined by ; pi ∈ ψ FT ( pi ) WP =  FT ( pi ) + γ ; pi ∈ ϑ 

(5)

where γ represents penalty when the member considered does not have feasible properties. This positive value is added up to increase the suitability measure index. It is likely that an increased index will be discarded along the selection process. The value of γ satisfies: γ ≥ Max FT ( pi ). pi

Figure 2. Fuel-cost vs power of a generator 38

(6)

The EP technique consists of the following steps. 1) An initial sampling of the population pi ∈ ψ is made on the basis of uniform random selection with Np greater than 10 to ensure a normal distribution. IEEE Power Engineering Review, June 2002

Table 1. Generator data and its fuel cost coefficients PGimin (MW)

PGimax (MW)

0.0850

60.0

280.0

0.0945

50.0

250.0

557

0.0575

40.0

340.0

0.00625

720

0.0625

80.0

425.0

0.00765

1315

0.0315

60.0

625.0

Ai

Bi

Ci

Ei

Fi

20

200.0

10.333

0.00889

340

21

240.0

10.833

0.00741

325

3

22

213.1

10.699

0.00533

4

23

250.0

10.500

5

24

196.0

10.243

i

Bus

1 2

Table 2. Optimal total generation costs Minimum Cost Function (ℜ / h)

Average Cost Function (ℜ / h)

Maximum Cost Function (ℜ / h)

Standard Deviation

SQP

9845.1472

10123.3158

10476.3113

169.5489

EP

9788.1987

10012.8832

10274.6049

172.6414

Tabu Search

9950.2741

9998.4981

10146.2382

77.2721

Table 3. Computing time consumed Minimum Computing Time (s) SQP EP Tabu Search

Average Computing Time (s)

Maximum Computing Time (s)

Standard Deviation

709.09

1177.82

1549.12

318.03

1200.01

1656.83

2488.40

435.70

169.50

316.08

462.30

78.54

2) An offspring p ′i is generated through a mutation process in which p ′ij = pij + N ( 0, σ2j ). The term N ( 0, σ2j ) represents a Guassian random variable with zero mean µ = 0, variance of σ2j and mutation scale factor β. 3) Competition and selection of offspring are made. An offspring of minimum suitability measure index is selected and kept. 4) Searching is terminated when the numbers of offspring reach maximum or the minimum value of suitability measure index is attained. Tabu Search: TS is a stochastic optimization method developed especially for solving combinatorial optimization problems with high efficiency [11]. The method can be performed step-by-step as follows. 1) Generate randomly an initial solution ( x 0 ) from a feasible region. Set x 0 is an initial optimal solution x ′′( x ′′= x 0 ). 2) Create randomly a neighborhood N ( x 0 ) around x 0 . Let x ′ is the best solution in the neighborhood set. 3) If f ( x ′ ) > f ( x ′′ ), go to step 4, else set x ′ is the best solution replacing x ′′( x ′′= x ′ ). 4) If x ′ is not in the tabu list, then update the tabu list by inserting x ′ in this list. 5) Check the termination criterion. If the stopping rule is satisfied, then stop and keep the best solution of the last iteration as the optimal solution of the problem, or else go to step 2. Notice that the tabu list and the termination criterion must be appropriately given for each TS problem space. Test System: A modified version of the IEEE-RTS-24-bus standard [13] is used as the test system. Figure 1 illustrates the system that operates at 740 MW, 390 Mvar load level. The installed capacity of the system is 1920 MW. Fuel cost function of each generator connected to the test system is established in the form of valve-point loading [3] and [5] expressed by F ( PGi ) = Ai + Bi PGi + C i PGi2 + | E i sin Fi ( PGimin − PG i )|. IEEE Power Engineering Review, June 2002

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The fuel cost coefficients of each generator are given in Table 1. Figure 2 depicts the curvy characteristic of the fuel cost of one generator as an example. To assess the usefulness and advantages of the TS technique, its simulation results are compared with ones obtained from the SQP and the EP optimization methods. Table 2 shows the search results with more or less the same quality. The TS method gives a little bit better standard deviation. The computing time consumed by the TS method is very short, as confirmed by Table 3. Conclusion: To summarize, this letter presents the application of the TS algorithm to the OPF problem. The problem space utilizes combined real and reactive power. This leads to more realistic optimal solutions. The simulation results confirm the effectiveness of the TS method in that it: ● Can escape a deadlock from local minima that other search algorithms cannot or spend too much time to do so ● Uses a very short computing time. The results obtained from this work reveal that the TS algorithm is suitable for nonlinear combinatorial optimization problems such as OPF. Some refinements are still needed in order to obtain more accurate solutions. These issues in terms of power systems are: ● Load forecast error ● Parameter uncertainty. The parameters of concern are those such as system impedances, fuel cost functional coefficients, etc. In order to improve deterministically the solution accuracy, these errors must be rectified through actual measurement and more accurate estimation. An alternative is to apply probabilistic methods in the simulation process, such as using the Metropolis algorithm. Nonetheless, some errors in the present solutions still exist due to randomly generated initial solution, no matter whether the TS or EP method is used. There is a need to concentrate on a better way to generate random numbers and map them from a probabilistic domain to real domain. References [1] H.W. Dommel and W.F. Tinney, “Optimal power flow solutions,” IEEE Trans. Power App. Syst., vol. PAS-87, 1968. [2] A.J. Wood and B.F. Wollenburg, Power Generation Operation and Control. New York: Wiley, 1996. [3] H. Yang, P. Yang, and C. Huang, “Evolutionary programming based economic dispatch for units with nonsmooth fuel cost functions,” IEEE Trans. Power Syst., vol. 11, Feb. 1996. [4] K.P. Wong, “Computational intelligence applications in unit commitment, economic dispatch, and load flow,” in Proc. 4th Int. Conf. Advances in Power Syst. Control, Operation, and Manage., APSCOM-97, Hong Kong, 1997. [5] D.C. Walters and G.B. Sheble, “Genetic algorithm solution of economic dispatch with valve point loading,” IEEE Trans. Power Syst., vol. 8, pp. 1325-1332, Aug. 1993. [6] K. Doan and K.P. Wong, “Artificial intelligence-based machine-learning system for thermal generator scheduling,” IEE Proc. Generation, Transmission, and Distribution, vol. 142, 1995. [7] J.T. Ma and L.L. Lai, “Evolutionary programming approach to reactive power planning,” IEE Proc. Generation, Transmission, and Distribution, vol. 143, no. 4, 1996. [8] H. Mori and Y. Ogita, “Capacitor placement using parallel tabu search in distribution systems, in Proc. 1999 IEEE Int. Conf. Syst., Man, and Cybernetics, vol. 6, pp 521-525, 1999. [9] H. Mori and T. Hayashi, “New parallel tabu search for voltage and reactive power control in power systems,” in Proc. 1998 IEEE Int. Symp. Circuits and Syst., vol. 3, pp. 431-434, 1998. [10] H. Mori and Y. Sone, “Tabu search based meter placement for topological observability in power system state estimation,” in Proc. 1999 IEEE Transmission and Distribution Conf., vol. 1, pp 172-177, 1999. [11] A.H. Mantawy, Y.L. Abdel-Magid, and S.Z. Selim, “Unit commitment” by Tabu Search, IEE Proc. Generation, Transmission, and Distribution, vol. 145, no. 1, 1998. 39

[12] M. Denna, G. Mauri, and A.M. Zanaboni, “Learning fuzzy rules with tabu search: an application to control,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 295-318, Jun. 1999. [13] M.E. El-Hawary, Electric Power Applications of Fuzzy Systems. Piscataway, NJ: IEEE Press, 1998. Copyright Statement: ISSN 0282-1724/02/$17.00 © 2002 IEEE. Manuscript received 5 March 2001, revised 1 December 2001. This paper is published herein in its entirety.

2002 International Conference on Harmonics and Quality of Power 6-9 October, Rio de Janeiro, Brazil The International Conference on Harmonics and Quality of Power (ICHQP) will be held 6-9 October 2002 in Rio de Janeiro, Brazil. The conference provides a forum for electrical engineers and scientists in universities, utilities, and industry to present their work and share information in the area of harmonics and power quality. It is a well-established IEEE PES-sponsored conference that is held biennially by rotation in and outside the United States. All presentations and conference manuscripts will be in English. The conference, which was formerly the International Conference on Harmonics in Power Systems (ICHPS), covers all aspects of power quality, including the following topics: analysis and modeling; devices; loads; flexible, reliable, and intelligent electrical energy delivery systems; measuring and monitoring techniques; sources of disturbances; power converters; traction systems; harmonics; voltage quality; power conditioning; active and passive filters; var compensation; UPS; surge protection devices; grounding systems; phase balancing; standards and recommended practices; diagnostic and expert systems applications; electromagnetic compatibility (EMC); power quality in distribution systems; quality aspects of industrial, commercial and residential consumers; power quality economics and liability; and power quality in a deregulated electricity market. Invitation to Submit Tutorial, Session, and Panel Programs: Attendees are invited to submit one-page summaries of possible tutorials (no longer than half day), technical sessions, or panels (approximately 2 hours in length, and indicating number of panelists or invited papers), which they desire to organize, directly to the conference chair. Indicate the title of the tutorial, session or panel. The deadline date is 12 April 2002. Exhibits: Space consisting of tables or booths is reserved for exhibits and demonstrations by corporations and businesses specializing in power quality related instrumentation, mitigation equipment and software products. Early reservation for exhibit space is recommended. The deadline is 30 August 2002. Other Important Dates: 31 May 2002 - Notification of Acceptance will be mailed to authors. 28 June 2002 Deadline for final manuscripts. Secretariat: Gilson Paulillo, ICHQP 2002 Executive Secretary, Itajuba Federal School of Engineering – EFEI, Power Quality Study Group – GQEE, P.O. Box #50 37500-903, Itajuba, MG, Brazil, +55 35 3629 1312, fax +55 35 3629 1326, e-mail [email protected], Web http://www. ichqp2002.efei.br

Fault Current Limiter with Fast-Closing Switch Jiyan Zou, Jinxiang Chen, Enyuan Dong Author Affiliation: Department of Electrical and Electronic Engineering, Dalian University of Technology, China; Department of Electrical Engineering, Huazhong University of Science and Technology, China. Abstract: A new type of fault current limiter (FCL) with a fast-closing switch is proposed. It is composed of a capacitor bank and a reactor in series. The main control component is a fast-closing switch connected in parallel with the capacitors, which is driven by the electromagnetic repulsion force. The new FCL is shown to be more effectivee with high reliability and low cost. Keywords: Fault current limiter (FCL), series compensation, fast-closing switch. Introduction: The concept of FCL was first proposed in the 1970s [1]. The GTO-based FCL was developed more recently[2] and [4]. It has the same features as series compensation, but the high power GTO array and its complicated synchronous control system in high-voltage applications make it expensive. This paper presents a new scheme of

Figure 1. Basic configuration of FCL

Figure 2. Mode system for FCL analysis

Figure 3. Simulation results 40

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IEEE Power Engineering Review, June 2002

FCL based fast-closing switch instead of GTOs. Its advantages are simple construction and lower cost. Basic Configuration and Operation of the New FCL: Figure 1 shows the basic configuration of the proposed FCL. It consists mainly of the capacitor bank C1, reactor L1, and the fast-closing switch SW1. The low impedance Z1 limits the inrush current flowing through SW1. The overvoltage protecting device ZnO arrester and the bypass switch BPS, which backs up the switch SW1, are also connected in parallel to the capacitors C1. The low impedance Z2 restrains the inrush current when BPS closes. This system operates as follows. The synthetic impedance of the capacitors C1 and the reactor L1 is capacitive. The FCL behaves as a conventional series compensator in the transmission line in normal conditions. When a fault occurs, the switch SW1 turns from the offstate to the on-state at very high speed to bypass the capacitors C1, and the reactor L1 works as the fault current limiter accordingly. The operation principle is simillar with GTO-based FCL with series compensatation by Sugimoto et al. [4]. Compared with the GTO-based FCL, the difference is the control component SW1, which is a mechanical vacuum switch operated by electromagnetic repulsion force. Its operating time can be less than 1 ms [3], which can meet the demands of FCL; the FCL can operate before the short-circuit current reaches the peak of the first wave. In the scheme of the FCL, the switch SW1 only does close, not interrupt, so the vacuum interrupter is simple. Compared with the GTO based FCL, the most advantage is the cost. Besides the expensive GTO arrays, com-

Figure 4. Current resonance type test circuit

Figure 5. System model

Figure 6. Simulation results IEEE Power Engineering Review, June 2002

plicated control strategy is avoided, and it can also save money and increase the reliability. Simulation and Experiment: Figure 2 shows the model system for the analysis. The fault is assumed to be a three-phase grounding fault. During the simulation by EMTP, we suggest that after 5 ms when the fault occurs the switch SW1 is operated to bypass the capacitors C1. The simulation results are shown in Figure 3. The effectiveness of the FCL is seen to be good by comparing Figures 3a and 3b. Without the FCL, the fault current peak is 10.88 p.u.; but with the FCL, the fault current peak is restricted to 4.98 p.u., and the fault current wave is smooth and without syntony. To verify the feasibility of the FCL-based fast-closing switch by experiment, a test circuit shown in Figure 4 is designed [5]. In the test, we use a triggered vacuum switch (TVS) instead of the fast-closing switch for convenience. The current limiting effect of FCL is confirmed by comparing the transient waveforms of the fault current without and with FCL operation. Transient Stability Analysis: A simulation of the transient stability of a system with the FCL and fast-closing switch included. Figure 5 shows the simple power system model for analysis. A three-phase short circuit fault is assumed to occur in one line at the end close to the generator. When the fault occurs, the switch SW1 bypasses the series capacitors, and, after 0.1 s, the fault line is opened by the CB operation. Simulation results are shown in Figure 6. The first group of figures composed of Figure 6a and 6c are the results without FCL at 250 MW transmission, and the second group of figures composed of Figure 6b and 6d are the results with the FCL having a 30% compensation degree and 20% limiting rate at 350 MW transmission. Comparing Figures 6a and 6b, we see that without the FCL, the system becomes unstable at the 250 MW transmissions; but with the FCL, the system carrying more than 350 MW is still stable. From Figure 6c, we can find that without FCL, the power output drops when the fault occurs. After the fault is cleared, although they are recovered, the generator becomes unstable at 0.76 s. On the other side, with the FCL, the generator can still output power, because the FCL starts to limit the fault current immediately (Figure 6d). After the fault line is opened, the power output recovers, and the system becomes stable because of the series compensation on the sound line. Conclusion: The effectiveness of the proposed FCL with fast-closing switch is verified by digital simulation and experiments. Compared with the GTO-based FCL, the fast-closing switch-based FCL has more advantages. It is more simple and reliable with low cost. More work will be done in the future and includes detecting fault and sending the operation order to switch in an extremely short time, reliability, and development of practical units of FCL with fast-closing switch. References [1] C.A. Falcome, J.E. Beehler, et al., “Current limiting device: a utility’s need,” IEEE Trans. Power App. Syst., vol. 93, no. 6, pp. 1768-1775, 1974. [2] T. Veda, et al. “Solid state current limiter for power distribution system,” IEEE Trans. Power Delivery, vol. 8, no. 10, pp. 1796-1801, 1993. [3] G. Takamu, et al. “400 class fast-closing current limiting circuit switch for electric power system,” IEEE Trans. Power Delivery, vol. 9, no. 3, 1994. [4] S. Sugimoto, et al. “Principle and characteristics of a fault current limiter with series compensation,” IEEE Trans. Power Delivery, vol. 11, no. 2, pp. 842-847, 1996. [5] Shi Jng, Zou Jiyan, and He Junjia, “Triggered vacuum switch based fault current limiter,” IEEE Power Eng. Rev., vol. 20, p. 51, June 2000. Copyright Statement: ISSN 0282-1724/02/$17.00 © 2002 IEEE. Manuscript received 26 March 2001, revised 1 December 2001. This paper is published herein in its entirety. 41

Utilization of Generator Reactive Capability: A Transmission Viewpoint Shih-Min Hsu, Henry J. Holley Author Affiliation: Transmission Planning, Southern Company Services. Abstract: This letter discusses an alternative method to evaluate the net reactive power delivered to a system by a generator. A commonly accepted transformer model with an off-nominal turns ratio is used to demonstrate the method. The main benefit from performing such an evaluation is to avoid the likelihood of a generator not being able to provide its reactive capability to a system. Keywords: Reactive capability, transformer, off-nominal turns ratio, transmission, generation. Introduction: Generators produce active power (MW) in accordance with the requirements of an automatic generation control system. Because of deregulation in the electric utility industry, many independent power producers find it economically attractive to build generating plants. However, in many cases, improvement of the transmission grid is not keeping pace with the addition of generating plants. Studies show that voltage instability often occurs after key generators reach their reactive capability limit [1] and [2]. Proper reactive power support from

Figure 1. Model of the two-bus evaluation system

Figure 2. Typical generator reactive capability curve 42

generators is crucial in maintaining voltage stability of a transmission grid. Reactive power is required to provide voltage support for the grid, meet the reactive component of the loads and losses, and enable active power to be transferred across the transmission grid from the generator to the loads. One of the most important reactive resources is from generators connected to the transmission grid. For a given generator rated power factor, the reactive capability is a built-in function of the generator. North American Electric Reliability Council (NERC) Planning Standards [3] require that “Generation owners and transmission providers shall work jointly to optimize the use of generator reactive power capability.” The choice of a generator step-up (GSU) transformer and the taps associated with it are important in ensuring that a generator can generate its reactive capability when system conditions demand it. Elements Affecting the Amount of Mvars Delivered to the System: The rated power factors of generators are usually 0.85 or 0.90. A lower power factor generator has a higher reactive capability at its rated MW generation. In general, a standard 0.90 power factor generator costs approximately 6% more than a 0.85 power factor generator. This letter discusses the factors affecting the amount of reactive power delivered to a system with a predetermined generator reactive capability. To illustrate these factors, the model of a generator, a generator step-up transformer, and an infinite bus representing the system is used. Such a two-bus system is shown in Figure 1. There are three main factors that affect the amount of reactive power delivered to the system: ● Reactive power taken by station service loads ● Reactive power losses in the GSU ● Generator terminal voltage. The first two factors are evident from Figure 1, while the third factor is not so obvious and may be overlooked. If the problem is not uncovered in the design stage, potentially a part of the reactive capacity of the generator may be trapped in the machine and consequently affect the voltage regulating function of the generator. Trapped reactive power is defined later. This letter presents a procedure to help designers to identify potential problems so that an inadequate GSU impedance selection and tap setting can be avoided. Flowchart and Calculations: A typical generator reactive power capability curve is shown in Figure 2. The curve provides a set of steady-state limits (gross) at every specific generated MW. These limits are the lagging reactive power limit, Qmax , and the leading reactive power limit, Qmin . This letter discusses only the portion of the curve that applies to lagging power factor operation. The analysis for the generator operating at leading power factors is similar to the discussion at lagging power factors and will not be discussed further. Assume that the generator is generating its maximum MW output. The first factor reducing the reactive power delivered to the system is the reactive power taken by the station service (auxiliary) loads. The station service loads will take about 5% of MW generation at 0.8 power factor for coal-fired steam plants and nuclear plants. With single-cycle gas-fired units and combined-cycle units, station service loads take about 2% of MW generation at maximum MW output. Commonly, station service loads are connected to the generator terminals. This loading is represented by PssL + jQssL in Figure 1. In some cases, station service loads are connected to the system terminals. In Figure 1,

Figure 3. Equivalent circuit of the GSU transformer 0272-1724/02/$17.00©2002 IEEE

IEEE Power Engineering Review, June 2002

PssH + jQssH represents this type of station service loads. It is clear that either QssL or QssH needs to be subtracted from the gross reactive power QG produced by the generator. The calculations necessary to evaluate a particular situation are based on the equivalent circuit of the GSU. This equivalent circuit is shown by Figure 3. To simplify the discussion, the off-nominal tap setting, a, of the GSU is initially set to be one (a = 1). Other values of a are considered later. Generators are designed to be operated within a range of voltage limits, usually 5% higher and 5% lower than the rated generator voltage. However, such voltage limits may become more restricted by the voltage levels at the station auxiliary system. If the maximum workable generator voltage is V1 , the scheduled voltage at the high voltage side of the GSU isV2 , and the impedance of the GSU is R + jX , then the current can be calculated as follows: I1 =

V1 ∠δ − V2 , Z∠90 °−φ

(1)

where Z = R 2 + X 2 and φ = tan −1 ( R / X ). The complex power injection at the low voltage side of the GSU is S 1 = V1 I 1* =

V12 VV ∠90 °−φ + 1 2 ∠90 °+ δ − φ. Z Z

(2)

Then, the net active power at the low voltage side of the GSU can be obtained as the real part of S 1 . Thus, P1 =

V12 VV sin φ + 1 2 sin( δ − φ). Z Z

(3)

Since all values except δ are known, δ can be calculated as follows:  Z  V2  δ = sin −1   P1 − 1 sinφ  + φ. Z  V1V2   

(4)

The reactive power at the low-voltage side of the GSU can be obtained as: Q1 =

V12 VV cos φ − 1 2 cos( δ − φ). Z Z

(5)

If the value of QG is greater than the generator capability Qmax , then QG must be set to this limit and V1 and δ must be re-evaluated for this changed circumstance. Since P1 and Q1 are known in this case, the generator terminal voltage V1 can be expressed in terms of its real and imaginary parts as V1 = E 1 + jE 2 ,

Figure 4. Flowchart of the proposed calculations

(6)

where E2 =

P1 X − Q1 R , V2

(7)

and E1 =

(

V2 + V22 − 4 E 22 − P1 R − Q1 X 2

).

(8)

Then V1 = E 12 + E 22 , and IEEE Power Engineering Review, June 2002

(9) Figure 5. Generator terminal voltage versus gross reactive power in p.u. 43

Table 1. Voltage V1 and reactive powers in per-unit with varying tap a

V1

QG

QssL

Qloss

Q2

QTrapped

1.05

1.043

1.83

0.0354

0.5445

1.2501

0

1.025

1.05

1.4909

0.0354

0.4864

0.969

0.3391

1.00

1.05

0.9824

0.0354

0.4297

0.5173

0.8476

0.975

1.05

0.448

0.0354

0.396

0.0166

1.382

0.95

1.05

-0.114

0.0354

0.3891

-0.5389

1.83

Table 2. Voltage V1 and reactive powers in per-unit with varying tap for a GSU of higher impedance QssL

Qloss

Q2

1.7066

0.0354

0.6265

1.0448

0.1234

1.05

1.3046

0.0354

0.5601

0.7092

0.5254

1.05

0.8828

0.0354

0.5099

0.3375

0.9472

0.975

1.05

0.4395

0.0354

0.4787

-0.0747

1.3905

0.95

1.05

-0.0269

0.0354

0.4698

-0.5321

1.83

a

V1

1.05

1.05

1.025 1.00

QG

QTrapped

E  δ = tan −1  2  .  E1 

(10)

Reactive power losses of the GSU transformer can be calculated as

(V1 cos δ − V2) + (V1 sin δ) 2

Qloss = I 12 X =

Z2

2

X.

(11)

After calculating the Qloss, the reactive power delivered to the system can be obtained as follows: Q2 = QG − QssL − Qloss − QssH .

(12)

Note that if the station service loads are connected to the low (high) voltage side of GSU, then, QssH = 0(QssL = 0 ). When the off-nominal tap position is not unity, then V2 needs to be replaced by V2 / a in all of the above equations. The flowchart for the proposed procedure is shown in Figure 4. An example is presented to illustrate the simplicity of the proposed procedure. Numerical Illustration: A generator of 313.3 MVA, 0.85 power factor with rated voltage of 18 kV is connected to a 230 kV system through a GSU transformer with the following data: ● OA/FA/FA: 190/253/315 MVA ● R + jX = 0.00129 + j0.05119 on 100 MVA system base ● Rated voltages = 18/230 kV (nominal) ● Available tap positions: 241.5/235.75/230/224.25/218.5 kV (Tap 1 = 1.05; Tap 2 = 1.025; Tap 3 = 1; Tap 4 = 0.975; Tap 5 = 0.95) ● Station service loads: P ssL = 5.9 MW and Q ssL = 3.54 Mvar. Assuming that the MW generation is at its rated MW, 295 MW, then the maximum reactive production limit (gross) is 183 Mvar. The maximum permissible generator terminal voltage is 105% of its rating, and the system voltage is set to be 1.01 p.u. The values for V1 , the reactive power, in per-unit, shown by Figure 1 and the amount of reactive power trapped in the machine are tabulated in Table 1 at the taps specified above. The amount of reactive power trapped in a machine is calculated by subtracting the gross reactive power QG from the value of Qmax at that MW output. If the machine is absorbing reactive power, then the

44

trapped reactive power is taken to be Qmax . Table 1 shows that it is possible to utilize the full reactive capability of the machine only when a tap value of 1.05 is used. Other tap positions will potentially trap an amount from 34 Mvars to its reactive production limit. Table 2 shows the values obtained with the GSU having a higher impedance than was used for Table 1. Let the value of GSU impedance be R + jX = 0.001562 + j0.061941 on 100 MVA system base. With this higher impedance GSU transformer, reactive power is trapped in the generator at every tap position; in other words, the generator terminal high-voltage limit will be reached before its reactive capability is reached. When system conditions change while holding the same scheduled system voltage, the generator terminal voltage and gross reactive power production change as well. Also, it would be helpful to use a plot with the generator terminal voltage versus the gross reactive power production/absorption to demonstrate graphically the numerical example above. As shown in Figure 5, the V-Q characteristics for all five taps for the lower GSU impedance and the 1.05 tap for the higher impedance GSU are included. Also, the values of Qmax , Qmin , Vmax , and Vmin are used as the boundaries for such a plot. It is clear that with lower impedance the V-Q characteristic has a steeper slope than it does with the higher impedance [4]. It implies that a GSU transformer with lower impedance has a potential to better utilize its reactive power capability. Conclusions: The amount of reactive power that a generator can deliver to a system is reduced by the amount of reactive power taken by the station service loads and the reactive power losses of the GSU transformer. There is a tendency to trap vars in a generator as the GSU impedance increases because of the high-voltage limitation of the generator. It is important to perform calculations such as those demonstrated in this letter in order to obtain an adequate coordination. Such calculations should be performed at the design stage. References: [1] P. Kundur, Power System Stability and Control, New YorkMcGraw-Hill, chp. 14, pp. 967. [2] Powertech Labs, “Voltage stability studies for southern company services,” EPRI TR-109490, Final Report, Dec. 1997. [3] NERC Engineering Committee, NERC Planning Standards, I, D, pp. 19-21, Sep. 1997. [4] J.D. Gregory and T.A. Higgins, “Parametters effecting generating unit var capability,” Electrical Equipment System Committee, Edison Electric Institute, San Diego, CA, 14-16 Feb., 1983. Copyright Statement: ISSN 0282-1724/02/$17.00 © 2002 IEEE. Manuscript received 1 May 2001, revised 1 December 2001. This paper is published herein in its entirety.

2003 Industrial and Commercial Power Systems Technical Conference 4-7 May 2003 St. Louis, Missouri, USA The IEEE Industry Applications Society 2002 Industrial and Commercial Power Systems Technical Conference will be held 4-7 May 2003 at the Hyatt Regency Union Station, St. Louis, Missouri, USA. For more information, contact William J. Moylan, Moylan Engineering Assoc., Inc., 39325 Plymouth Road, Suite 103, Livonia, MI 48150 USA, +1 734 591 4000, e-mail w.j.moylan@ieee. org.

IEEE Power Engineering Review, June 2002

Distribution Matching Power Flow: A New Technique for Distribution System State Estimation

matches the predicted load data with reasonable modifications. In this letter, such distribution power flow is called as DMPF. The equations of DMPF are developed as follows. If the predicted loads at the leaf nodes are not accurate enough, we will observe an imbalance of power flow at the root node r, which can be described as:

H.B. Sun, B.M. Zhang Author Affiliation: Department of Electrical Engineering, Tsinghua University, Beijing 100084, China. Abstract: To avoid difficulty in determining weights for real-time measurement and predicted load data in the conventional WLS-based distribution system state estimation, a novel method called distribution matching power flow (DMPF) is proposed in this letter. The method is implemented by two steps: first state estimation for the main network and then DMPF calculation for radial subsystems. In the method, state estimation for radial subsystems can be converted into a DMPF calculation. The equivalent relationship between DMPF calculation and radial distribution state estimation, which lays a theoretical foundation for DMPF, is derived. A practical forward/backward sweep algorithm for DMPF is given. Numerical results show the efficiency of DMPF calculation for distribution state estimation. Keywords: Distribution matching power flow, state estimation, distribution system. Introduction: Because a distribution network is large in scale, it is impossible to telemeter the whole distribution system outright. In general, only power flow in some distribution trunk lines and some loads for important customers can be measured. In order to carry out state estimation, predicted load data is needed to complement the serious lack of real-time measurements. In the traditional WLS-based state estimation method, the predicted load data are treated as pseudo measurements and assigned to low weights. The disadvantages of this approach are that it is difficult to assign proper weights for real-time measurements and pseudo measurements. A large diversity for weights will cause some numerical problems in calculation, while small diversity will result in a low precision of state estimation for the main distribution network. Moreover, convergence and numerical stability of the distribution state estimation algorithm will deteriorate for a large-scale distribution system with low measurement redundancy, except a large error caused by pseudo measurements. Novel Method for Distribution System State Estimation: The method proposed here includes two steps: 1) Perform a state estimation for the main network by using real-time measurements; 2) Calculate the power flow for the remaining radial subsystems by using DMPF technology. As a result, a coincident power flow solution for whole distribution system is obtained. In the method, based on actual measurement, a state estimation for main distribution network can be done using an existing algorithm [1]-[3]. After a state estimation for the main network is done, there are n individual radial subsystems left in the whole distribution system. In n radial subsystems, the voltages (V&1 , V&2 ,..., V&n ) and the power injections ( S& 1 , S& 2 ,..., S& n ) at root nodes (r1 , r2 ,..., rn ) are determined by state estimation for main network. So, the remaining problem is how to solve power flow for each subsystem provided injection power and voltage at root node are known. DMPF Technology: Consider a radial distribution subsystem with N load nodes as shown in Figure 2. Let C be the aggregate of all load nodes and ( PD i , QD i ) be the load power of node i. The voltage V&r and the power injection ( Pr , Qr ) of root node have been calculated by the state estimation for main network. Suppose that there are N M ( < N ) loads having real-time power measurements, and let C M be the aggregate of these load nodes. In this case, the other N-NM loads needed to be predicted. Because no redundant real-time measurement exists and, at the same time, real-time measurement is much more accurate than predicted data, it is reasonable to suppose that the real-time measurement is exactly relative to predicted data. Thus, it is expected that the power flow of the subsystem matches strictly all of those real-time measurements and the result of state estimation done on the main network and IEEE Power Engineering Review, June 2002

 ∆PΣ = Pr − ∑ PD i − PL(V , θ) i ∈C   ∆ QD i − QL(V , θ)  QΣ = Qr − ∑ i ∈C 

(1)

where ( ∆PΣ , ∆QΣ ) is the boundary mismatch and ( PL, QL ) is the power loss of the distribution subsystem. Each load at the leaf nodes will share the mismatch by distribution factors α i and β i for P and Q, respectively, and we have:  ∑ α i = 1  i ∈C   β =1 i ∑  i ∈C

(2)

For the node in set C M , the corresponding factor is zero, i.e., α i = 0 β = 0  i

i ∈C M . (3)

Thus, for the distribution subsystem, the load flow equations become: ( PD i + α i ⋅ ∆PΣ ) + Π i (V , θ) = 0 (Q + β ⋅ ∆Q ) + QI (V , θ) = 0 i = 1,..., N , i Σ i  Di

(4)

where ( Π i , QI i ) is the sum of branch power flows leaving node i. Once the distribution factor vector ( α , β) is specified, a unique DMPF can be determined. Using (4), we can derive the node voltage equations of the subsystem as follows: YV& = I&(V& ).

(5)

Figure 1. Main network and n subsystems

Figure 2. Radial distribution subsystem

0272-1724/02/$17.00©2002 IEEE

45

Figure 3. Simple test subsystem

Figure 4. Numerical result of DMPF calculation

Here, Y is the nodal admittance matrix of subsystem. Thus, it is quite similar to distribution power flow [4] in that the equations (4) can be solved by the forward/backward sweep algorithm, as follows: 1) Initialization:( α , β ) is specified in accordance with (2) and (3), all elements of vector V& are initialized as Vr , ( ∆PΣ , ∆QΣ ) is initialized as (0,0), k = 0. 2) Modify node load by ( PD + α ⋅ ∆PΣ( k ) , QD + β ⋅ ∆QΣ( k ) ). 3) V& ( k ) is known, power flows of branches are calculated by a forward sweep. 4) By using power flow of branches just calculated, voltages V& ( k + 1 ) are calculated by a backward sweep. 5) Use V& ( k + 1 ) to solve ( ∆PΣ( k + 1 ) , ∆QΣ( k + 1 ) ) by (1). 6) Check whether |V& ( k + 1 ) − V& ( k )| is smaller than a given tolerance ε? If yes, stop; otherwise, k = k + 1, return to step 2. Determination of Optimal Distribution Factors for Boundary Mismatch: The solution of (4) largely depends on the distribution factors ( α , β ). In order to get an optimal solution, optimal distribution factors should be determined. If the power injection ( Pr , Qr ) at the root node is taken as real-time measurements and the predicted load data as pseudo-measurements, then there exists one redundant measurement in radial subsystem. Therefore, we can get the optimal state of the subsystem using the conventional WLS-based state estimation. Suppose that residuals for real-time measurements are zero; then only residuals appear in the predicted pseudo measurements. As a result, from the WLS-based state estimation point of view, the modifications of predicted load data ( α ⋅ ∆PΣ , β ⋅ ∆QΣ ) in (4) are the residuals of load pseudo measurements. If neglecting the variation of system loss, WLS-based state estimation with equality constraints (1), (2), (3), and (4) can be solved by the Kuhn-Tucker condition directly, and the optimal distribution factors can be derived as follows:   1   α i =  wPD i ⋅ ∑ w i ∈( C − C M ) PD i      1 β i =  wQ D ⋅ ∑ i   i ∈( C − C M ) w Q D i  

α i = 0  = 0 β i

       

−1

−1

Table 1. Numerical results of DMPF calculation under different boundary mismatch Boundary Mismatch

i ∈C − C M 0%

(6)

i ∈C M , (7)

where ( wPD , wQ D ) is the weight of load pseudo measurement at the i i node i, which should be given before state estimation. 46

Obviously, corresponding to state estimation with specified weights of load pseudo measurements, there exists a unique DMPF solution with an optimized distribution factor ( α , β ) calculated by (6) and (7). Therefore, the optimal distribution factors for DMPF can be determined as follows. 1) Estimate the variance of predicted data errors and take the reciprocal of square of the variance as weight of the pseudo measurement. 2) Determine distribution factors by (6) and (7). With a reasonable distribution power flow estimated, DMPF avoids the difficulty to determine different weights between real-time measurement and pseudo measurement. Test Results: The computer used here is a SUN ULTRA 10 workstation. The convergence tolerance of the forward/backward sweep algorithm proposed above is 0.0001pu. To understand the principle of DMPF, a simple test subsystem is shown in Figure 3. Here the power injection (15.7 MW + j3.12 Mvar) and voltage (23.0 kV) of root node 1 are calculated by state estimation for the main network, one real-time measurement (9.8 MW + j1.68 Mvar) is located at an important load node 5, (1.84 MW + j0.18 Mvar), (0.98 MW + j0.14 Mvar), and (1.79 MW + j0.18 Mvar) are predicted load data of leaf nodes 2, 3, and 4, respectively. Figure 4 shows the numerical result of DMPF calculation for the test subsystem in Figure 3. The power flow result matches strictly the real-time measurements and injection power of root node, only with reasonable modifications to the predicted load data. Table 1 lists the numerical results of DMPF calculation for five test distribution subsystems under a different boundary mismatch. In this table, the percentage of boundary mismatch is relative to original total load of the subsystem. From the table, it is seen that when the boundary mismatch is zero, the convergence of the DMPF calculation is fast. With the increase of boundary mismatch, the convergence speed of DMPF slows down a little. But even with a large boundary mismatch (up to 100%), the calculation of DMPF can also converge reliably. When the boundary mismatch is zero, the results calculated by the DMPF method are entirely the same as calculated by the general distribution power flow method. Such numerical results verify the uniqueness of the solution of DMPF. Conclusions: A novel method for distribution system state estimation is proposed in this letter. This method calculates a state estimate for the main network in the first step, and then a DMPF calculation for the radial subsystems is done in the second step. Using the DMPF technique, the difficulty in determining the different weights between real-time measurement and predicted load data is avoided. The equivalent relationship between DMPF and radial distribution state estimation, which is the theoretical foundation of DMPF, is derived. Numerical results show the efficiency of DMPF calculation for distribution state estimation. The DMPF technique will be implemented in a commercial distribution management system, and applied to several distribution control centers in China. References: [1] C.N. Lu, J.H. Teng, and W.H.E. Liu, “Distribution system state estimation,” IEEE Trans. Power Syst., vol. 10, pp. 229-240, Feb. 1995.

Subsystem

No. of Nodes

50%

100%

No. of CPU Time No. of Iterations (ms) Iterations

No. of Iterations

A

16

3

0.9

4

5

B

33

4

3.3

5

5

C

10

5

1.2

7

9

D

44

3

3.1

5

5

E

70

5

8.5

6

7

IEEE Power Engineering Review, June 2002

[2] M.E. Baran and A.W. Kelley, “A branch-current-based state estimation method for distribution systems,” IEEE Trans. Power Syst., vol.10, pp.483-491, Feb. 1995. [3] H.B. Sun, B.M. Zhang, and N.D. Xiang, “Method of distribution state estimation based on branch power,” Automation of electric power systems, (in Chinese), vol. 22, pp. 12-16, Aug. 1998. [4] D. Shirmohammadi, H.W. Hong, A. Semlyen, and G.X. Luo, “A compensation-based power flow method for weakly meshed distribution and transmission networks,” IEEE Trans. Power Syst., vol. 3, pp. 753-762, May 1988. Copyright Statement: ISSN 0282-1724/02/$17.00 © 2002 IEEE. Manuscript received 16 July 2001, revised 1 December 2001. This paper is published herein in its entirety.

2002 Conference on Power Systems and Communication Systems Infrastructures for the Future 23-27 September, Beijing, China The Conference on Power Systems and Communication Systems Infrastructures for the Future will be held in Beijing, China on 23-27 September 2002. Certain technological infrastructures are critical for the well being of modern societies. During the past several years, electric power networks, communication networks, and computer networks have become so intimately interlinked that it is necessary to consider these infrastructures in an integrated framework. When catastrophic events such as floods, fires, earthquakes, or hurricanes occur, the survivability and integrity of these infrastructures is of paramount importance. The conference brings together researchers, manufacturers of infrastructure hardware and software systems, and institutions involved in emergency management at national and international levels, in order to exchange views on what should be the direction of developments in these critical disciplines. It is hoped that through the synergy among the experts in these fields the directions for the development of integrated infrastructures for the future will become clearer. Manufacturers of power, communication, and control equipment are expected to participate in the conference by showcasing their most modern products. A workshop on wide area measurements in power systems is being planned in conjunction with the conference. The conference Web site, http://www.cris.vt.edu, provides additional details about the conference organization and facilities. If you have an interest in participating in this conference, please contact A.G. Phadke, Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061 USA, +1 540 231 7029, fax +1 540 231 4303, e-mail [email protected].

Block Pricing for Electric Power Markets Xifan Wang, Xiaohong Guan, Xiuli Wang Author Affiliation: Xian Jiaotong University, Xian, China. Abstract: Many problems are associated with the widely accepted current instant bidding rule. The inherent causes of the problems are that some fundamental characteristics of electric power production are not carefully considered in designing the market bidding rules when applying spot price theory. In this letter, the shortcomings of the instant bidding rule are studied. Based on this study, the block bidding rule and the associated market scheme are presented. By comparing the characteristics of the two bidding rules, the letter shows that the market with the block bidding rule is more efficient than those with the instant bidding rule. Furthermore, the proposed market scheme is more suitable for electric power production and more convenient for operating power markets. Keywords: Electric power industry deregulation, market design, electric power market auction, and bidding rule. Introduction: Up to now, almost every electric power market design is based on spot pricing theory, developed by Schweppe, et al. in 1988 [1]. The key idea of the theory is that the spot price in a perfect market should be the market-clearing price equal to the marginal production cost of electric energy and determined by the demand-supply relationship. Since the demand changes with time, so do the spot prices, which would send economic signals to market participants to achieve optimal allocation of resources. There are some important issues that need to be addressed in the practice of spot pricing and auction mechanism for electric power markets. First, should the market be settled by uniform clearing prices or individual bid prices (pay-as-bid)? Based on the principle of fair competition, “the same price for the same quality” should be applied to every product. However what is the quality of electric energy? It is generally accepted that the prices at load peaks should be higher than those in the valleys. Do all generators produce energy with the same quality at the same peak period and is it fair to award a uniform clearing price? Secondly, is it necessary for a Genco to submit 24 or 48 bid curves on an hourly or half-hourly basis for the next day? The operation of thermal generators requires a certain degree of continuity, and there may be complicated physical constraints on unit commitment of thermal units involving a lot of additional resources and costs. Bidding on each time instance may increase the uncertainty on operating thermal units, which is neither good for the Genco nor for entire system economy and security. Finally, the supply bids should be based on cost analysis of the Gencos. Generally, the average cost of a thermal unit decreases below its 80-90% nominal capacity. However in most power markets, the bid price versus power is required to be monotonically increasing (at least non-decreasing). This differentiation between cost and bid price would not send correct price signals for the participating Gencos to improve their efficiency but may encourage them to concentrate on price speculation and opportunist gaming. The inherent causes of the above problems are that some fundamental characteristics of electric power production are not carefully considered in designing the bidding rules when applying the spot price theory. Here the problem does not stem from spot price theory but from how the spot prices are created and how the bidding and settlement rules are designed. Shortcomings of Instant Pricing: Market design problems have surfaced in many electric power markets based on spot pricing theory world-wide. It would be worthwhile to review the spot pricing theory and the conditions for its applications first. From [1, Sec. 6.1.2], the spot energy price is defined as ρ( t ) =

IEEE Power Engineering Review, June 2002

0272-1724/02/$17.00©2002 IEEE

∂GFM [ g( t )] , ∂d( t )

(1) 47

where GFM [ g( t )] is the total fuel and maintaining costs of the entire system at time t in dollars, g( t ) is the total power production at the time t in MWh,and d( t ) is the total system demand at time t in MWh. To maximize the total social welfare, ρ( t ) should satisfy the following condition [1, Sec. 6.1.8 and 6.1.9]: ρ( t ) =

∂GFM [ g( t )] = γ F ( t ) + γ M ( t ). ∂g( t )

(2)

This says that the spot price should equal to the marginal production cost at time t, where γ F ( t ) and γ M ( t ) reflect the fuel and maintenance cost, respectively. These two equations are the basis of the spot pricing theory for the current market design. However, two assumptions are implicitly made here, which would result in major problems encountered in many markets. First, it is assumed in (1) that the production cost is only a function of the time t but not related to the other time instants. This assumption does not conform to the reality of electric power production, since above 70% of the generation resources in the world are thermal, including nuclear units, which have very complicated procedures to commit or decommit possibly with additional startup and shutdown costs. The operating constraints have a significant influence on production costs. Currently, almost all power markets require Gencos to bid at each time instant, and the markets are cleared independently at each time instantaccording to the demand-supply balance without considering the operating constraints and the inherent difficulty to store electric energy. This would in turn create difficulties in bidding for Gencos and randomness of unit commitment and thus cause a negative impact on market operation. Secondly, in (2), all the generating units are assumed as one equivalent unit. However the production cost GFM [ g( t )] in (2) should be replaced by GFM [ g j ( t )], ( j = 1, 2, ⋅⋅⋅ ), reflecting different units. Since the characteristics of the production cost function of different unit j are different, their impacts on the MCP at time t are also different. This is illustrated in Figure 1, where Genco A and B provide the load LP jointly with other Genco’s at time t. According to the current rule, the energy

provided by Genco A and B is settled by the same MCP ρ( t ) at time t. However, these two Gencos are very different in terms of production cost and contribution to the entire system. Genco A provides base loads with low cost but little load-following capability. Theoretically, Genco A should not be awarded at the same price as Genco B. Otherwise, there would actually exist cross-subsidies from Genco A to Genco B and tying [2], which will definitely decrease market efficiency. Therefore, the major shortcomings of the instant bidding rule are: 1) it is difficult for Genco’s to bid and difficult for the system operator to determine unit commitment; 2) principle of fair competition and the same price for the same quality is not implemented, resulting in lower market efficiency. It should be noted that the above assumptions were also mentioned in [1], but no further research was carried out. Block Bidding Rule: Based on the analysis of the last section and the principle of “the same price for the same quality,” the block bidding rule is proposed. The key idea of this bidding rule is to divide demand into blocks with continuous time segments, to auction and balance demand by blocks, and to form the MCP for each block. As shown in Figure 2, the forecasted load curve is divided into l1 , l2 , ⋅⋅⋅ , LP continuous production segments with corresponding time span h1 , h2 , ⋅⋅⋅ , hP . The MCP for each block is not only a function of the loading level of this block but also the time span of this block. The energy produced in different blocks may have different production costs and different utility. Generally, when h1 > h2 > ⋅⋅⋅ > hP , the MCP for each block has the following relationship: ρ b ( h1 ) < ρ b ( h2 ) ρ 2 > ρ 1 . The above discussion reveals that the purchase cost for satisfying the demand in the market with the block bidding rule is lower than that with the instance bidding rule. Therefore, with the block bidding rule, it is possible for the consumers to get cheaper power, and, thus, the power market would become more efficient [2]. It is worthwhile to discuss how blocks should be divided and their relationship to market efficiency. Suppose the load curve is differentiable. Theoretically, the total load can be divided into as many blocks as wished so that each block of power can be provided by only one Genco, and the MCP for each block would be the bid price of a particular Genco. In this case, the settlement for the market with the block-bidding rule is equivalent to “pay-as-bid” for a certain time span. From the previous discussion, it is not difficult to conclude that the finer the blocks are divided into the lower the total purchase cost and higher efficiency of the market. However, the finer division means that the energy of a block will be provided by fewer units. This would be constrained by ramping capability of the units. In fact, ramping across time usually needs the contributions of multiple Gencos, which should be proportional to their energy contributions and paid at the same price. Therefore, the block division should depend on the ramping capability of the system. Conclusions: This letter analyzes the bidding rule of the current power markets and points out the shortcomings of the current instant bidding rule and its inconsistence with the principle of fair competition. The root of these shortcomings lies in the inconsistence between the bidding rule and the continuity features of electric energy production. Based on the analysis, the block-bidding rule is presented with the following features in comparison with the existing instance bidding rule: ● Consistence with the features of electric energy production and consumption, convenience for supply bidding and system operation; ● Competition on the same quality of electric energy and better market fairness; ● Lower purchase cost favorable for consumers to obtained cheaper energy; ● Higher market efficiency and better social welfare, ● Convenience to determine production schedules. Therefore, the block-bidding rule deserves further investigation in order to deal with the market design issues encountered world-wide. IEEE Power Engineering Review, June 2002

Acknowledgments: The research presented in this letter is supported by a Key Project 59937150 of the National Natural Science Foundation of China. The second author is also supported by the National Outstanding Young Investigator Grant 6970025, the National Natural Science Foundation of China, and EPRI/DoD Complex Interactive Networks Initiative under Contract WO 8333-03. References: [1] F.C. Schweppe, M.C. Caramanis, R.D Tabors, and R.E. Bohn, Spot Pricing of Electricity. Norwell, MA: Kluwer, 1988. [2] R.S. Pindyck and D.L. Rubinfeld, Microeconomics. Englewood Cliffs, NJ: Prentice-Hall, 1995. Copyright Statement: ISSN 0282-1724/02/$17.00 © 2002 IEEE. Manuscript received 21 August 2001, revised 1 December 2001. This paper is published herein in its entirety.

2002 Large Engineering Systems Conference on Power Engineering 26-29 June 2002 Halifax, Nova Scotia, Canada The 2002 Conference on Power Engineering is being held in Halifax, Nova Scotia, Canada, as part of the Large Engineering Systems Conference series in the period 26-29 June 2002. The conference will provide an international forum for the participants to share knowledge, experience, and new ideas and to discuss recent developments and practical applications in power engineering. The venue will be the Sheraton Hotel. Sessions will be held on topics related to new developments in the application, operating experience, field testing, theory, design, control and analysis, in all areas of power systems control, operation and planning, power plant instrumentation and control, and electrical equipment. Emphasis is on utilizing intelligent systems techniques, such as artificial neural networks, fuzzy systems, genetic algorithms, evolutionary techniques, knowledge-based or expert systems, machine learning systems, case-based or model-based reasoning, human machine interface, and other intelligent systems techniques. For more information, contact the LESCOPE ’02 secretariat at Large Engineering Systems, P.O. Box 25041, Halifax, NS B3M 4H4, Canada, +1 902 443 2400, fax +1 902 445 5110, e-mail F.El-hawary @ieee.org, Website http://is.dal.ca/~lescope/.

49

Handling High R/X Ratios in Meshed Systems with Moderate Heterogeneity O.R. Saavedra, A.B. Rodrigues Author Affiliation: Power System Group, Electrical Engineering Department, Federal University of Maranhão, São Luís, MA, Brazil. Abstract: A new property of the fast decoupled load flow (FDLF) for meshed homogeneous r/x ratios and moderate nonuniform r/x ratio networks with the presence of one or some isolated abnormally high r/x ratios is stated. This property shows that a critical branch has a local effect, preserving the rest of the matrix and suggests a modification for the traditional BX fast decoupled load flow. From this modification, the convergence characteristic of the FDLF is preserved for these extreme cases. Validation tests are included using the standard IEEE 118 system and a Brazilian 308-bus low-voltage underground distribution system. Keyword: Newton’s method, decoupling, load flow. Introduction: The fast decoupled load flow was proposed in 1974 by Stott and Alsaç [1] and makes use of the same algorithm as the decoupled Newton method [2]: angle and voltage corrections are calculated in an alternate way, but the matrices H and L are replaced by constant approaches of B′ and B′′, respectively. The hypotheses assumed for the FDLF deduction are valid in very high voltage networks, where the r/x ratios are small, typically less than 20%. Furthermore, Stott and Alsaç notice that neglecting the resistance in B′ results in better performance. The explanation for the mechanism that governs the decoupled method’s behavior remained unknown for several years, until the publication of a theoretical unified explanation [4]. In this work, four properties related to the decoupling were stated. From these properties, we can derive the BX and XB versions of decoupled methods [3]. Decoupling: From [4], the solution for load flow by the Newton method can be written in an equivalent form as follows: ∆P  H   ∆Q MH −1 ∆P  =  0 −   

N   ∆θ  , Leq   ∆V 

(1)

where diag(G 2 / B ) and diag( X −1 ) are diagonal matrices; each (i, i ) element is given by gi2 / bi and 1 / x i , respectively. C is the bus incidence matrix. 2) For most practical purposes (i.e., meshed systems with a moderate non-uniform r/x ratio), Leq can be nicely approximated by (5). Therefore, for both cases 1) and 2), Leq computed at V = 1 p.u., and θ = 0 can be obtained by replacing the susceptances bkm by the inverse of the corresponding reactances 1 x km . Thus, the BX fast decoupled load flow can be derived: ∆θ = [ B ′]−1 ∆P(V , θ) θ ← θ + ∆θ, and (6) ∆V = [ B ′′]−1 ∆Q(V , θ) V ← V + ∆V ,

where matrix B′′ is composed by the inverse of reactances. However, difficulties remain when the assumed hypothesis is not satisfied. For instance, strongly heterogeneous meshed networks cannot be modeled adequately by (5), leading to poor behavior of the fast-decoupled load flow. Local Heterogeneity Property: Let us consider a meshed network with r/x homogeneous ratios equal to η. If we include a critical branch with ratio η′, where η′ >> η, then the new Leq matrix at V = 1 and θ = 0 becomes a matrix with local heterogeneity. Let Lo , M o , N o and , H o be matrices of the original homogeneous system, computed at V = 1 and θ = 0. Then, after the inclusion of a critical branch in nodes k and m, matrix Leq becomes: Leq = ( Lo + ∆L ) − ( M o + ∆M )( Z$ o + ∆Z$ )( N o + ∆N )

t , Leq = Lo + η2 H o + δkm ekm ekm

(2)

Property 1 gives the theoretical basis for using an intermediate update of V and θ. In other words, this property states, for example, that calculating the mismatches ∆Q − MH −1 ∆P at (V , θ) is equivalent to calculating the mismatches ∆Q at (V , θ + H −1 ∆P ). Property 2 shows that (1) can be solved in a decoupled three-step way without ignoring the submatrices N and M. Property 3 allows the definition of the two-step decoupled method, as follows: 1) Angle corrections: ∆θ = H −1 ∆P(V , θ).

(3)

2) Voltage corrections: ∆V = L−eq1 ∆Q(V , θ + H −1 ∆P ).

(4)

Property 4 states that the equivalent matrix Leq is as sparse as the matrix L either for radial or meshed systems with uniform r/x ratios. In other words: 1) For a radial or meshed systems with uniform r/x ratios, Leq computed at V = 1 and θ = 0 is given by: Leq = −C t [ diag( B ) + diag(G 2 / B )]C = C t diag( X −1 )C = B 50

(5)

(8)

$ ∆N are the perturbation matrices where Z$ o = [ H o ]−1 and ∆L, ∆M, ∆Z, originated by the inclusion of a critical branch. After algebraic handling,

where Leq is a matrix defined as: Leq = L = MH −1 N .

(7)

(9)

where δkm = − bkm +

  z$eqo η2 − 2 ηgkm  km + 1 $z $   z

 o z$ o  2  z$eq km + eq km  + gkm z$    2

(10)

In these expressions, bkm and gkm are the susceptance and conductance of the critical branch, respectively; ekm is a vector with 1 and -1 in −1 positions k and m, respectively; z$ = bkm − z$eqo km is a scalar, and z$eqo km is the equivalent impedance seen from nodes k - m (making an analogy with matrix Z). Expression (9) shows that heterogeneity of the critical branch affects only its neighborhood. In other words, (9) shows that the homogeneous matrix is modified only in positions related to the critical branch. The contributions (diagonal and off-diagonal elements) are given by (10). The effect of heterogeneity is propagated locally without affecting the rest of the matrix. If the critical branch has the same r/x ratio of the rest of network, thenrkm = ηx km ; thus, Leq assumes the well-known form given by (5). The property given by (9) naturally suggests a modification of the fast-decoupled load flow in order to deal with abnormal high r/x ratios. In this case, the positions of B′′ related to the critical branch are given by (9), while for the rest of the system, B′′ is given by (5). Tests have been performed, and results are presented next.

0272-1724/02/$17.00©2002 IEEE

IEEE Power Engineering Review, June 2002

Table 1. Test results with varying resistance of max[r/x] branch System

r-scale

r/x-max

BX

BX-M

1.0

0.4

7/6

7/6

3.0

1.4

7/6

7/6

5.0

2.3

7/7

7/6

10.0

4.7

9/9

7/6

20.0

9.4

11/10

7/6

30.0

14.2

11/11

7/6

1.0

1.6

6/5

6/5

3.0

5.0

7/7

7/6

5.0

8.4

12/11

7/6

10.0

16.9

22/21

7/5

20.0

33.8

33/33

6/5

30.0

50.7

39/39

6/5

IEEE 118

308-bus

Table 2. Test results with varying reactance of max[r/x] branch System

IEEE 118

308-bus

x-scale

r/x-max

BX

BX-M

1.0

0.4

7/6

7/6

0.5

0.9

7/6

7/6

0.1

4.7

15/14

7/7

0.05

9.4

27/27

8/8

0.03

15.7

44/43

9/8

0.01

47.3

nc

0.001

473.4

10/9

0.0001

4734.8

10/9

10/9

1.0

1.6

6/5

6/5

0.5

3.3

6/5

6/5

0.1

16.9

13/12

8/7

0.05

33.8

23/22

9/8

0.03

56.4

35/34

10/9

0.01

169.3

nc

10/9

0.001

1693.0

11/10

0.0001

16930.0

11/10

Numerical Results: In order to validate the property discussed earlier, numerical results of tests performed with the modified FDLF (called BX-M) are presented here. The load flow tolerance has been 0.01 MW/Mvar. The shunts have been modeled as suggested in [5]. In Tables 1 and 2, the performance of the BX and of a modified version of FDLF (BX-M), respectively, are illustrated. Tests have been performed with the IEEE 118 systems and a 308-bus low-voltage underground distribution system. In these tests, the higher r/x ratio has been altered by increasing the resistance (Table 1) and by decreasing the reactance (Table 2), respectively. In both cases, the modification was found to be robust. By using the property, the good performance of BX FDLF method can be extended for cases with the presence of some high r/x ratios. Notice that no significant additional effort is required for calculating (9). z$eqo km is computed by using the previously available triangular factor of H o ( B′ ). IEEE Power Engineering Review, June 2002

Conclusions: We presented a new property of the FDLF for homogeneous network (or moderate heterogeneous networks) with one or some branches with abnormally high r/x ratios. From this property, we have suggested a modification of the traditional BX FDLF method. Results obtained from tests validate the property. The modified BX method can be used for solving problems with moderate heterogeneity and to handle isolated high r/x ratios. If no abnormal r/x ratios are presented, the traditional BX method can be invoked in an adaptive way. Acknowledgments: The authors wish to acknowledge the support from Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq and Banco do Nordeste do Brasil, BNB, Brazil. References: [1] B. Stott and O. Alsac, “Fast decoupled load flow,” IEEE Trans. Power App. Syst., vol. 93, pp. 859-869, 1974. [2] B. Stott, “Decoupled newton load flow,” IEEE Trans. Power App. Syst., vol. 91, pp. 1955-1969, 1972. [3] R.A.M. Van Amerongen, “A general-purpose version of the fast decoupled load flow,” IEEE Trans. Power Syst., vol. 4, pp. 760 -770, May 1989. [4] A. Monticelli, A.V. Garcia, and O.R. Saavedra, “Fast decoupled load flow: hypothesis, derivations and testing,” IEEE Trans. Power Syst., vol. 5, pp. 1425-1431, Nov. 1990. [5] O.R. Saavedra, A.V. Garcia, and A. Monticelli, “The representation of shunt elements in fast decoupled load flow,” IEEE Trans. Power Syst., vol. 9, pp. 1434-1440, Aug. 1994. Copyright Statement: ISSN 0282-1724/02/$17.00 © 2002 IEEE. Manuscript received 24 July 2001, revised 1 December 2001. This paper is published herein in its entirety.

ISAP 2003 31 August - 3 September Myrina, Greece The twelfth Intelligent Systems Application to Power Systems Conference (ISAP 2003) will be held 31 August – 3 September 2003 at the Luxury Hotel Portomyrina Palace in Myrina, Greece, on the Island of Lemnos (http://www.lemnos-isl.gr) in the Northeastern Aegean Sea. Connections are by ferry and by air, daily from Athens. ISAP 2003 is organized by the IEEE Greece Section and National Technical University of Athens Power Systems Laboratory. IEEE PES is a technical cosponsor. The language of the conference is English. Scheduled events include conference registration and a welcome reception, the official conference dinner, visits to various sites, and a companions’ program. For more information, contact the ISAP 2003 secretariat, Electric Energy Systems Laboratory, National Technical University of Athens, +301 0 7723661 or 7723699, fax +301 0 7723659, e-mail isap03@power. ece.ntua.gr, Web http://isap2003.power.ece. ntua.gr.

51

Application of a Novel Fuzzy Control Strategy for HVdc Link To Improve Interarea Stability T.S. Chung, Da-zhong Fang Author Affiliation: Dept. of Electrical Engineering, The Hong Kong Polytechnic University; Dept. of Electric Power and Automation, Engineering, Tianjin University, China, e-mail [email protected]. Abstract: A novel control strategy is developed for HVdc links to improve the interarea stability of interconnected power systems. A fuzzy rule is adopted to smooth the power transition for reducing power “shock” to the power system. The strategy also incorporates adaptive control techniques to prevent tie-link power “chattering.” Simulation results using the proposed control scheme on two typical power systems are presented. Introduction: Figure 1 shows a typical two-area power system in which ac and dc links are used to transmit power between areas. For such operation, transient stability and oscillatory stability problems may limit the available transfer capacity. This letter describes a novel control approach developed for high voltage dc (HVdc) link for damping postdisturbance oscillations and enhancing transient stability. Conventionally, supplementary control is used in HVdc modulation to damp ac/dc system oscillations [1]. The pole placement technique is employed to design the supplementary controller. The difficulty in analysis of eigenvalues and eigenvectors for a very large linearized power system limits this technique’s application. In this letter, an alternative control method is proposed to modulate power order input to HVdc master control for such ac/dc power system in Figure 1. The methodology is based on the characteristics of the ac tie power versus the line phase angle where an area control principle is presented to explain the physical process of ac tie power oscillations. A bang-bang control strategy is utilized to increase the stability deceleration area for stability improvement. Fuzzy rules are adopted in the strategy to control the HVdc power smooth transition for reducing power shocks caused by bang-bang controls. The fuzzy logic approach has an advantage in that it does not require precise numerical values of control inputs and system parameters [2] and [4]. The proposed controller also incorporates adaptive techniques to avoid ac tie-line power chattering

while the tie oscillation is being damped to small amplitude and to modify the oscillatory center angle identically converging to steady the operation point of a postfault system. Stability Control: Electromechanical Oscillations Between Areas: Consider Figure 1. Assume the ac tie lines have an impedance of 0.0055 + j0.055 (p.u.) and the voltages of buses A1 and B1 maintained at 1.0 p.u.. Thus the characteristic of the power through bus A1 sent to area B is plotted in Figure 2, where line angle δA 1 B 1 denotes δA 1 − δB 1 , difference of voltage angles of bus A1 and B1. Here point O is the steady operation point of the ac line. Assume that there exists electromechanical oscillation between area A and area B after a disturbance. For the ac link A1-B1, the oscillation is exhibited both on the phase angle, δA 1 B 1 , and on the power transferred through the ac link. Relative motions of generator rotors between the two areas [7] usually cause an area mode oscillation. To explain the nature of the area mode oscillation, we consider the motion of the center of inertia (COI) of generators in area A (called generator A for simplicity) and the motion of COI of generators in area B (called generator B for simplicity) [3]. The relative motion between the two COIs dominates the oscillation behavior. Under the assumption that the total generations and consumption of power in both areas maintained constant during the oscillation, the motion equations of the two COIs can be expressed by: & A = PA 0 − PAD − PA 1 , δ& = ω Β − ωR M Aω

(1)

and & B = PB 0 + PBD + PB 1 , δ& B = ωB − ωR , M Bω

(2)

where δA , δB , δA , and δB are the angles and speeds of generators A and B; ωR denotes ac synchronous speed; PA 0 and PB 0 are the mechanical powers of generators A and B, which represent the differences between the total generations and the total consumption of the two areas; PAD , PBD , PA 1 , and PB 1 are the dc and ac link powers at each their terminal; PAD 0 and PBD 0 express the constant of PAD and PBD , if they remain unchanged during oscillation. Parameters M A and M B denote inertia constants of the two COIs [3].

Figure 1. Typical two-area power system with ac and dc tie links

Figur 3. AC link power plot at bus A1 versus difference of bus voltage angles

Figure 2. Swing curves of angles and speeds obtained on the four-generator, two-area ac-dc system by software simulation without damping control 52

Figure 4. Configuration of SFLAC

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IEEE Power Engineering Review, June 2002

During oscillations, the power PA 1 will change in value along the plot shown in Figure 3. To show the physical process, one cycle of oscillation is divided into four phases. As shown in Figure 3, phase 1 is for the swing from point O to B. This phase is characterized by ωAB > 0. Considering (1) and (2), as PA 0 − PAD 0 − PA 1 < 0 and & A < 0 and ω & B > 0, i.e., − PB 0 − PBD 0 − PB 1 < 0, it is easy to show that ω & AB < 0, in this phase. At point B, the speed reaches zero (ωAB = 0). ω Phase 2 is for the swing back from B to O. Similarly, it can be shown & AB < 0. that the motion in this phase is characterized by ωAB < 0 and ω Phases 3 and 4 are for the swings from point O to F and F to O, respectively. Because PA 0 − PAD 0 − PA 1 > 0 and − PB 0 − PBD 0 − PB 1 > 0 in the two & AB > 0. As the oscilstages, the motions for them are characterized by ω lation power must transmit through the ac/dc tie link, the variations of δA 1 B 1 should be in the same mode with that of δAB , i.e., at point B (and F), both angles δA 1 B 1 and δAB reach their maximum (and minimum) value synchronously. The above point has been verified on the four-generator test ac/dc test system [1]. Figure 4 shows the simulation results on the four-generator system (see Figure 5) for the verifications. In Figure 4, δAB and δAB denote the difference of the angle and speed between the generators A and B [3]. δ5 , 9 is for the line angle between tie buses 5 and 9, and δ5 , 9 denotes its derivative to time. Bang-Bang Control Strategy: As δA 1 B 1 and δAB swing in the same mode, the characteristics of PA 1 versus δA 1 B 1 shown in Figure 3 can also be used to represent characteristics of PA 1 versus δAB . Hence, in phase 1, the area of ABO with respect to generator A is backward deceleration energy and, with respect to generator B, is forward deceleration energy. In phase 2, the area of ABO is forward acceleration energy for generator A and backward acceleration energy for generator B. Similarly, in phase 3, the area of EFO is forward deceleration energy for generator A and backward deceleration energy for generator B. In phase 4, the area of EFO is backward acceleration energy for generator A and forward

acceleration energy for the generator B. The tie oscillations will be continuously repeated because of the same acceleration and deceleration areas for the four phases in one cycle. For realistic ac tie-line oscillations, in addition to voltage phase angles, voltage amplitudes are also involved in the oscillations. This will make the characteristic shown in Figure 2 vary from time to time associated with oscillations. Hence, the sum of the deceleration areas may not be just equal to the sum of acceleration areas for a realistic power system. If the deceleration areas are larger than acceleration areas the oscillation will be decreased, otherwise it will be amplified. Based on the basic principle presented above, a strategy of bang-bang control of HVdc power for damping post-fault area-mode oscillation is proposed here. As shown in Figure 3, assume that in phase 1 the HVdc power at the rectifier and the inverter is controlled to maintain PAD 0 + ∆PAD and PBD 0 + ∆PBD , respectively. In phase 3, the HVdc power at the rectifier and the inverter is controlled to maintain PAD 0 − ∆PAD and PBD 0 − ∆PBD , respectively. In phases 2 and 4, the HVdc power remains unchanged. Thus, in one cycle, the area of AOADC + AOEHG in Figure 2 denotes the kinetic energy at point O of generator A decreased (of generator B increased if the ac/dc tie power loss is ignored). Hence, after one oscillation at point O, ωAB is decreased. The HVdc control can damp the ac tie oscillation identically if the conditions of ωAB > 0 and the reduction of ωAB at point O maintain. If ∆ PAD is large, the damping will be effective. However, when ∆PAD is large, the bang-bang control nature may cause a big shock to the generator turbine shafts in areas A and B near the HVdc convertors because of sudden changes of output power to the generators. This can in turn cause a reduction in the shaft life. On the other hand, a fixed ∆PAD may cause chattering of the ac tie-line power while the oscillation is being stabilized to small amplitude. In the next paragraph, some techniques are developed to overcome these problems.

Table 1. Fuzzy rules Sector A ∆Pmax ( p. u. )

Sector B ∆Pmin( p. u. )

Figure 7. Angular membership functions Figure 5. Eleven-bus, four-generator, two-area ac/dc system

Figure 6. Phase plane for control strategy IEEE Power Engineering Review, June 2002

Figure 8. Variations of δ 59 , and the angle δ 0 using SFLAC for fault 3 -SC-6-(tcl=0.25s)-(1, 6-9) on the four-generator ac-dc test system 53

Development of HVdc Controller: The controller presented in this section is referred to as the stability fuzzy logic adaptive controller (SFLAC). The configuration of the SFLAC is shown in Figure 4. In Figure 4, the notations of δ, δ0 , PDC 0 , and ∆P are for the ones of δA 1 B 1 , δ0 , ∆PAD 0 , and ∆PAD used before. Here, the signal PDC 0 , the dc power flow arrangement, come from the ac/dc system control center. The phase angle δ is the input signal that can be measured by detecting the crossover points of the respective voltage waveforms with respect to synchronized pulse trains [5]. With the development of new technology, precise time-synchronized phasor measurement units are available that offer an alternate way in the measurement of signal δ. The output of the SFLAC is the power order signal to the HVdc master control, as shown in Figure 3. High pulses in the signal ∆δ or ∆ω may interfere with the proper working of the SFLAC. Hence, two low-pass filters are used to wash out the components of high frequency in the signals of ∆δ or ∆ω, which is usually brought in by measurement errors or sudden change of system conditions, such as switching on or tripping out of a large load (or a line). The signals coming out from the filters are de$ The signal of δ0 initially takes the predisturbance noted by ∆δ$ and ∆ω. $ passes value of δ and is adaptively modified by the value of ∆δ$ once ∆ω its maximum or minimum value. This adaptive modification of δ0 makes the SFLAC independent on the operating point of the postfault system. To apply a fuzzy logic approach in smoothing the transition of $ in the form of Cartesian HVdc link power, the variable pair (∆δ$ and ∆ω) phase are transformed to a pair of polar coordinate variables by $2 γ = ∆δ$ 2 + ∆ω

(3)

and $ / ∆δ$ ). α = tan −1 ( ∆ω

(4)

Thus, an ac tie link oscillation can be studied on the phase plane shown in Figure 7. The phases 1, 2, 3, and 4 of a oscillation mentioned $ passing earlier correspond to the trajectory of state of (∆δ$ and ∆ω) through the quadrants 1, 4, 3, and 2, respectively. For the bang-bang control strategy shown earlier, the HVdc link power is switched between the maximum P0 + ∆P and minimum P0 − ∆P. In terms of fuzzy logic, this means that the output of a fuzzy logic controller (FLC) has two linguistic variables referred to as ∆Pmax and ∆Pmin . The input-output relationship of the fuzzy logic controller is given in terms of a very simple fuzzy rule set in Table 1. For damping an area mode oscillation, as shown in Figure 4, the angle α is fuzzified using the membership function shown in Figure 7. The angle α is to fire the fuzzy rules in Table 1 to result output, ∆Preq , of the fuzzy logic controller. Here the output fuzzy set is denoted by { ∆Preq ,µ p } for {( ∆Pmax , µpmax ), ( ∆Pmin , mupmin )}. The µpmax and µpmin , the membership functions in Figure 7, denote the associated membership grades of the linguistic variables ∆Pmax and ∆Pmin . The membership functions are particularly designed in the overlapping regions of sectors A and B, αT1 and αT2, to make the output of the FLC gradually change between the two sectors. Mathematically, the relationship between ∆Ppre and α is ∆Ppre = µ max ( α )∆Pmax + µ min ( α )∆Pmin .

(5)

For example, if α belongs to sector A with a membership of 0.4, then the output fuzzy set will consist of {{∆Pmax , 0.4} and {∆Pmin , 0.6}}. Then the output of the defuzzifier, ∆Ppre , is calculated by ∆Ppre = ( 0.4 − 0.6 )∆P0 ,

Figure 9. Variations of generator angle of δ G 3 − δ G 1 , and δ 59 for fault 3 -SC-6-(tcl=0.25s)-0 in the four-generator ac-dc test system

Figure 10. Variations of ac line power (P59) and total ac-dc line power ( P59 + PDC ) for fault 3 -SC-6-(tcl=0.25s)-(1, 6-9) on the four-generator ac-dc test system 54

(6)

if ∆Pmax = ∆P0 and ∆Pmin = − ∆P0 . Here, ∆P0 is the expected increment of the HVdc power controlled. The variable γ 2 (Figure 6) gives a measurement of the oscillation magnitude. Hence, in designing SFLAC, γ 2 is used to adjust the increment of the HVDC power order adaptively. Suitable constant parameter k (Figure 6) makes the SFLAC operation satisfactory. For large disturbances, the ∆Ppre is amplified greatly but confined by the limit unit. On the other hand, for small oscillations, the ∆Ppre is reduced to control ∆δ approaches to zero identically. Simulation Tests: To evaluate the effect of SFLAC on stability enhancement, the controller was extensively tested using computer simulations on an 11-bus, 4-generator, two-area ac/dc system in [1], shown in Figure 5. The HVdc internal dynamic models adopted here are similar to those used in the supplementary control of the HVdc link in [1], but the maximum short-time current limit is allowed to reach 1.3 times of its normal full load current for transient stability enhancement [1]. The signal of power order from the SFLAC is transformed to current order signal limited by the maximum current, minimum current, and voltage-dependent current in the master control [1]. The system data of the four-generator ac/dc system is just the same as that used in the example of the HVdc link supplementary control in [1, Ch. 17]. All four generators have self-excited dc exciter. The dc link is represented as a monopolar link with a voltage rating of 56 kV and current rating of 3.6 kA. Under the assumption of short time maximum current 4.68 kA (1.3 times of the rating current), simulation results for a three-phase short circuit at bus 6 cleared in 0.25 s without line tripping (3φ-SC-6-(tcl=0.25s)-0), as shown in Figure 8. Simulation results for a three-phase short circuit at bus 6 cleared in 0.25 s, with one line (6-9) tripping (3φ-SC-6-(tcl=0.25s)-(1, 6-9)), as shown in Figures 9 and 10. It is observed that after about 6 s the postfault interarea mode oscillation on the ac tie are stabilized down for the two fault conditions. The oscillation, after 6 s, exhibited on the plot δG 3 − δG 1 in Figure 8 is induced by the local mode oscillations in each area, and, after 10 s, the local mode oscillations are also stabilized down by the SFLAC. Plot of δ0 in Figure 8 and plot P59 + PDC in Figure 10 show that the methods of IEEE Power Engineering Review, June 2002

Table 2 Comparison of critical fault clearing time with and without SFLAC. fault type 3φ-SC-6-0)

tcl without SFLAC 0.32-0.33(s)

tcl with SFLAC 0.37-0.38(s)

adaptive modification of δ0 and ∆Ppre are effective in stabilizing oscillations down. Comparing with the simulation results of the conventional HVdc link supplementary controller in [1], the following aspects are noted. ● SFLAC has a stronger effect on damping the inter area mode post-fault oscillation. ● SFLAC is an adaptive controller, that means it is effective for inter-area mode oscillation with different frequencies. ● In controller design of SFLAC, eigenvalue and eigenvector analysis are not required which is not an easy work for a large ac/dc power system. Extensive tests were also performed to examine the enhancement of transient stability by SFLAC. As examples, tests are done for faults of three-phase short circuit at bus 6 without line tripping (3φ-SC-6-0)) on the four-generator test system. Test results presented in Table 2 demonstrate that SFLAC can enhance transient stability for severe disturbances near ac/dc tie links. Conclusions: A novel HVdc control strategy for oscillatory and transient stability enhancement was developed. An area analysis approach is proposed to explain the physical process of ac tie oscillations. Using the capability of HVdc in rapid control of transmission power, the control strategy is to increase the deceleration area. Techniques such as fuzzy logic and adaptive control are verified to be effective on controlling the HVdc tie power in smooth transition and on avoiding ac tie line power chattering. Simulation results show that the control strat-

2002 International Conference on Energy Integration in Northeast Asia 9-13 September 2002 Irkutsk, Russia The third International Conference on Energy Integration in Northeast Asia will be held 9-13 September 2002 in Irkutsk, Russia. The Energy Systems Institute and the IEEE PES Russian Chapter are sponsoring the conference with support of many Russian and international organizations. The conference major concerns are: international energy markets and their deregulation; promising energy technologies development and impacts; energy security requirements; environmental and social impacts of energy options; economical risks of implemented energy projects; comprehensive analysis of interstate energy integration; effectiveness of interstate energy infrastructure; etc. For more information, contact Nikolai I. Voropai, director of the Energy Systems Institute and PES Chapter chair, 130 Lermontov Street, Irkutsk 664033, Russia, +7 3952 461700, fax +7 3952 461702, e-mail [email protected].

IEEE Power Engineering Review, June 2002

egy provides significant effect in damping area mode oscillations. The simplicity and robustness of the stability fuzzy logic adaptive controller are the attractive features. References: [1] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1993. [2] M. Jamshidi, N. Vadiee, and T.J. Ross, Fuzzy Logic and Control: Software and Hardware Application. Englewood Cliffs, NJ: Prentice-Hall, 1993. [3] Da-zhong Fang , T.S. Chung, Yao Zhang, and Wennan Song, “Transient stability limit conditions analysis using a corrected transient energy function approach,” in Proc. IEEE PES Summer Meeting, Edmonton, AB, Canada, Jul. 1999. [4] T. Hiyama, K. Miyazaki, and H. Satoh, “A fuzzy logic excitation system for stability enhancement of power system with multimode oscillations,” IEEE Trans. Energy Conversion, vol. 11, pp. 125-131, Jun. 1996. [5] G. Missout, J. Beland, G. Beland, and Y. Lafleur, “Dynamic measurement of the absolute voltage angle on long transmission line,” IEEE Trans. Power App. Syst., vol. PAS-100, pp. 4428-4434, Nov. 1981. [6] S.E. Stanton, C. Slivinsky, K. Martin, and J. Nordstrom, “Application of phasor measurement and partial energy analysis in stabilizing large disturbances,” IEEE Trans. Power Syst., vol. 10, pp. 297-302, Feb. 1995. [7] V. Vittal, N. Bhatia, and A.A. Fouad, “Analysis of the inter-area mode phenomenon in power system following large disturbances,” IEEE Trans. Power Syst., vol. 6, pp. 1515-1521, Nov. 1991. Copyright Statement: ISSN 0282-1724/02/$17.00 © 2002 IEEE. Manuscript received 12 September 2001, revised 1 December 2001. This paper is published herein in its entirety.

2002 North American Power Symposium 14-15 October 2002 Tempe, Arizona, USA The 2002 North American Power Symposium (NAPS) will be held 14-15 October 2002 in Tempe, Arizona, which is part of the Phoenix metropolitan area and the location of Arizona State University (ASU). The symposium is hosted by ASU and sponsored by the IEEE Power Engineering Society (PES). The purpose of NAPS is to stimulate advanced scholarly work and more research activity in the field of electric power engineering. This symposium is a forum where advanced students, their academic advisors, and practicing engineers can present the results of their work, discuss the activities of their colleagues, and publish their technical accomplishments with a minimum time delay. NAPS 2002 will include a day and a half of sessions devoted to contributed papers. Papers on all areas of electric power engineering are sought. For more information, contact Gerald T. Heydt, Arizona State University, +1 480 965 8307, fax +1 480 965 0745, e-mail [email protected], \Web http:// ceaspub.eas.asu.edu/naps2002.

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