Proceedings of the 32nd Hawaii International Conference on System Sciences  1999 Proceedings of the 32nd Hawaii International Conference on System Sciences  1999
Market Monitoring and Leader Follower Incentive Control of Ancillary Services Ali Keyhani The Ohio State University Columbus, Ohio
[email protected]
Jose Cruz, Jr. The Ohio State University Columbus, Ohio
[email protected]
Marwan A. Simaan University of Pittsburgh Pittsburgh, Pennsylvania
[email protected]
decentralized control of future energy systems will be impacted by the following technological development:
Abstract In this paper the problems of market monitoring and control of ancillary services of a future energy systems are presented. We envision that future system operation of electric power systems will evolve into completely unbundled ancillary service markets that are governed by spot price signals. The grid operators need to acquire ancillary services through competitive markets for control of the system operation. With the above vision of the future, this paper presents the control of ancillary services based on a frequency regulation/load following (LFC) market, a load regulating market (RL), and a base load (BL) market. In the LFC market, the units dispatched for generation must have specific response characteristics as determined by the nature of system loads. These units will be controlled by the grid operator. It is envisioned that LFC will be multitime scale and decentralized. In the regulating market (RL), the units dispatched for generation will not participate in LFC. These units are dispatched for specific period of time for regulating system load. The base load (BL) units are dispatched daily to satisfy the base load of the day on a weekly or monthly basis. The locational market power associated with generators participating in LFC and in RL is even more critical than that for generators participating in satisfying the base load, since these units are required for frequency regulation, voltage support and relief of overload conditions. To create an efficient market for these services, we will propose and formulate the use of incentive leaderfollower strategies for the BL,RL and the LFC markets.
1. Introduction The future energy systems will be controlled by multilateral markets and ancillary services markets. The multilateral markets will provide fixed energy over specified period of time and these energy sources can not be controlled for matching the load demand to generation[16]. The primary function of the ancillary services is to provide generation control and the capability to respond to dynamic system conditions. The market monitoring and
1.1. Impact of generation technology In the twenty first century, deregulation of the power industry will become a reality and a competitive market will determine the efficient production of electric energy. In the deregulated market, there will be many new power producers and market makers. In addition, if fuel cell technology is established as a viable technology in power generation, then every consumer has the potential to become a power producer. Although, these changes may take many years to become reality for small users, one would expect that the large industrial and commercial users would move to establish their own generation using the new established micro turbine and combinedcycles gas technologies. Many complex problems need to be studied. With a highly distributed generation system, how should the base load, regulating load and ancillary services be provided? How should the market be monitored for anti competitive behaviors? 1.2. Impact of communication and computer technology Greatly expanded computer instrumentation, control, sensing, and communications capabilities will be utilized in all levels of power system network from generating stations, transmission system substations, distribution systems and customer sites. The distributed network of computer systems are required for interactive, realtime control of generation and bus voltages , over the highly interconnected, geographically dispersed generation and load. To effectively utilize the power of distributed computer instrumentation, the computational algorithms based on distributed models will be necessary to achieve decentralized computer control of the network operations.
1.3. Impact of environmental issues The impact of pollution on mankind and the planet will continue to be a social issue for the next century. To reduce
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the smog and pollution in the cities, the use of electric vehicles will become a reality with large impact on power usage. As the demand for electric energy grows, more power producers and power marketers will enter the energy market. The need for efficient market monitoring to deal with anticompetitive behavior should be addressed. The solution of this problem will assure the efficient production of electric energy and stable operation of the power network.
1.4. Impact of the Spot Pricing Market With implementation of emerging metering technology, the spot price of energy can be send to the customer[2022]. The metering system can control various loads at the customer sites. The customer can be offered a number of variable price schedules based on the time of usage. With the customer in the loop of energy usage and reacting to the spot price market, control of the ancillary services will need to be investigated.
1.5. Impact of the Control Technology. The rapid expansion of computer, sensing, and communication systems has revolutionized the development of multiagent controllers using neural network/fuzzy logic, and rule based systems for control of largescale, uncertain, nonlinear, and time dependent systems. The vertically integrated power companies have relied on central control of system frequency and power flows on the transmission tie lines to neighboring systems for stable operation. The vulnerabilities of centralized control systems have been demonstrated by blackouts of 1967,1977,1978, and 1996. Is it feasible to decompose the power system and identify a number of control areas based on online measurement? What is needed is decentralized online modeling. For decentralized control of generation, online modeling techniques in presence of noise and measurement errors are needed. Algorithms for realtime processing large data set, pattern extractions, correlating information from separate data sets and knowledge acquisition for the development of online adaptive models are essential. The online models will facilitate the development of decentralized control agents and decision support with partial input and output observations. Therefore, what is needed is the development of decentralized multitime scale, multi agent generation control systems for stable operation of the future restructured power system with dispersed generation sources. This is an essential technology for dealing with emergency conditions and orderly brake up of the systems with a prioi defined control areas that are still viable for stable operation.
1.6. Impact of the Software Technology As the cost of memory and storage systems has rapidly decreased, there has been explosion in the development of new programming languages. These languages are based on object oriented programming techniques using many platforms. The C++ language allows for the design and implementation of the test bed system using a true object oriented approach. The objectoriented approach allows for better system partitioning and visualization in solving complex system. Moreover C++ code is relatively easy to maintain, reuse, modify, and allow for a group of programmers to work on separate parts of the code, without the errors multiplying in proportion to the length of the code. Programs written in C++ can be maintained and extended more easily and addition of functionality to the code is relatively straightforward with fewer risks of introducing errors. By using object oriented features, such as inheritance, reusing already written code is made more practical and codes comparable to Fortran in efficiency can in fact be written in C++, along with the added benefits of saving valuable implementation time due to ease of code reusability and maintenance. Nowadays, visual programming packages for C++ are widely available that can save considerable amount of time in designing and implementing graphical user interface for the test bed. The Javabased object oriented programming is a natural way of designing multiagent simulation systems. The objectoriented programming is also a convenient technology for building libraries of reusable software that will facilitate the exchange of agent (component)” The power system consists of thousands of buses that whose power consumption needs to be modeled online for use in security analysis. The object oriented programming technology is a natural approach for development visualization models. Furthermore, This technology will be used to develop a virtual Java based simulation testbed for study of market monitoring of ancillary services. The main objective of the testbed is to develop the conceptual framework for using the grid operator as a leader in providing incentives to the multiple providers in an ancillary services market. We assume that the grid operator buys energy for frequency regulation and load following from one or more energy providers in the ancillary services market. The grid operator provides transmission capacity rights to the energy providers. To allow the grid operator to influence the price, we assume that the transmission rights can be made functions of the generation of the energy providers. The functions will be chosen in such a way that the resulting price for the ancillary services will be close to the full competition price. The methodology will be applied to the ancillary services market wherein the grid operator purchases energy for the purpose of regulating frequency and load following.
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Proceedings of the 32nd Hawaii International Conference on System Sciences  1999 Proceedings of the 32nd Hawaii International Conference on System Sciences  1999
1.7. A New Philosophy of Future Automatic Generation Control We envision that future system operation of electric power systems will evolve into completely unbundled ancillary service markets that are governed by spot price signals. The grid operators need to acquire ancillary services through competitive markets for control of the system operation. The unbundled ancillary services will include frequency regulation/load following and operating reserves. The power associated with ancillary services is critical, since the grid operators must be able to control specific resources for secure and stable operation. For example, when a grid operator loses a 1300 MW unit, the control system needs to react immediately to control the system for stable operation and then the operator needs to adjust the generator set points according to spot price signals. The reaction of energy users to spot price signals will have great impact on ancillary service markets. With the above vision of the future, we propose a frequency regulation/load following (LFC) market, a load regulating market (RL), and a base load (BL) market. In the LFC market, the units dispatched for generation must have specific response characteristics as determined by the nature of system loads. These components of load data sampled on oneminute interval and fiveminute interval are shown in Figure 1. These units will be controlled by the grid operator. It is envisioned that LFC will be multitime scale and decentralized. In the regulating market (RL), the units dispatched for generation will not participate in LFC. These units are dispatched for specific period of time for regulating system load. The base load (BL) units are dispatched daily to satisfy the base load of the day on a weekly or monthly basis. The locational market power associated with generators participating in LFC and in RL is even more critical than that for generators participating in satisfying the base load, since these units are required for frequency regulation, voltage support and relief of overload conditions. To create an efficient market for these services, the use of incentive leaderfollower strategies for the RL and the LFC markets is proposed that will be discussed later. Figure 2 presents the proposed decentralized generation control. The sub grid operators will function in the same manner for the subsystems as the grid operator does for the entire system. References[18] present the modeling and control problems related to these concepts.
Figure 1. Public Service Indiana, Load Data on oneminute interval (top figure), fiveminute interval (bottom) figure.
Area n Spot Price P
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H: Increase generation L: Decrease generation
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Figure 2. Future Decentralized Generation Control
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Proceedings of the 32nd Hawaii International Conference on System Sciences  1999 Proceedings of the 32nd Hawaii International Conference on System Sciences  1999
3. Detailed Technical Rationale 3.1 Diffusing Horizontal Market Power. Recent Federal law mandates the separation of the functions of generation, transmission, and distribution where there is an interstate power flow. There is supposed to be competition in generation but transmission and distribution remain regulated. Federal law mandates equal access to the transmission system for all generation providers. The open access in itself does not insure full competition. Hogan[910] demonstrates through several examples that when there are transmission constraints, it is possible to have an incremental generation of one MW at one bus to block generation of more than one MW at another bus. Thus an increase in use of the network in one portion of the network can cause a reduction of capacity in another portion of the network. In this situation there is a possibility of exploiting network interactions to restrict competition and manipulate prices. Futhermore, Younes and Ilic [1112] through several examples show that a line created a localized sub market of three buses in a 24bus network, although the submarket was still connected to the rest of the network by four unconstrained lines. They also showed how loop flows aggravate the problem. One approach to moving to a greater participation by small entrants to the market is to allow the grid operator, such as the Independent System Operator, to exercise greater authority and provide incentives to the various energy providers connected to the the power systems. This can be cast in the framework of leaderfollower games where the grid operator is the leader[1318]. For example, an incentive model of duopoly with government coordination is presented in references[1618]. It is shown that the government, acting as leader, can induce two companies to achieve perfect competition price while behaving as duopolists in a Cournot fashion. This concept can be adapted for dynamic systems.
3.2 A Principle of Incentives in LeaderFollower Strategies[1319] Suppose that the Leader has a scalar decision variable x and the Follower has a scalar decision variables y. The Leader wishes to minimize an objective function that is influenced not only by its decision variable but also by the decision variable of the follower. Denote this objective function by G(x,y). Similarly, the Follower has an objective function denoted by F(x,y). For simplicity these functions are assumed to be convex. Finally, suppose that the choices of x and y are constrained by the relationship C(x,y) = 0, also assumed to be convex. In general it is not possible to simultaneously minimize G(x,y) and F(x,y) with respect to both variables. The combination of x and y
that minimizes G(x,y) may not be the same combination that minimizes F(x,y). Furthermore the Leader can only choose x and the Follower can only choose y. In spite of the limitations stated above, the Leader wonders what would happen if it had complete control over the choices of both variables x and y, and minimizes G(x,y) with respect to x and y, subject only to the constraint C(x,y) = 0. Suppose that the unique answer to this minimization is given by x = X, y = Y, and p1 = P, where p1 is the Lagrange multiplier in the Lagrangian function L1(x,y,p1) = G(x,y) + p1C(x,y) that the Leader has to minimize. With this result, the Leader wonders how it could induce the follower to choose y =Y. Suppose that the Leader decides to implement a sophisticated strategy whereby its decision variable x is allowed to be a function of y, i.e., x = f(y) where the function f is to be determined by the Leader. A simple example of such a function is x = A(y – Y) + X where A is a constant yet to be determined, and Y is the Follower’s control, desired by the Leader, obtained as explained earlier. We will demonstrate that under some reasonable conditions this strategy will induce the Follower to choose y = Y. We proceed to examine the Follower’s optimization problem. First we form the Lagrangian function L2(x,y,p2,p3) = F(x,y) + p2C(x,y) + p3{x  A(y Y)  X} where p2 and p3 are Lagrange multipliers for appending the equality constraint C(x,y) = 0 and the incentive control constraint x  A(y Y) – X = 0 to F(x,y). Assuming that F and C are differentiable, the differential of L2(x,y,p2,p3) is dL2(x,y,p2,p3) = (M1+ p3)dx + (M2  Ap3)dy + Cdp2 + {x – A(y  Y)  X}dp3
where M1 is the partial derivative of F+p2C with respect to x, and M2 is the partial derivative of F+p2C with respect to y. Now, since C(x,y) = 0, and x  A(y  Y)  X = 0, we have dL2(x,y,z,p2,p3) = (M1+ p3)dx + (M2  Ap3)dy The Leader calculates the values of M1 and M2 at x = X, y = Y, and p2 = P1. If M1 is not equal to zero, the Leader chooses
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A = M2/M1 With this choice of A by the Leader, the resulting differential of the Follower’s Lagrangian function evaluated at x = X, y = Y, p2 = P1, and p3 = M1 yields zero! Since F and C are convex, this first order condition is sufficient to guarantee that y = Y is the unique solution for the minimization of F subject to the constraint of C = 0. If M1 = 0, the Leader’s decision variable does not affect the Follower’s Lagrangian function at the Leader’s desired operating point and the incentive strategy can not induce the Follower to choose y = Y. Thus, except for the case where M1 = 0, the incentive strategy of the Leader is effective. The concept of incentive strategy can be extended to situations that involve several leaders and several followers. Static as well as dynamic models can also be considered. In the static case, a multileader multifollower problem would first require that the leaders and followers decide on the nature of the solution to be implemented within each group. This could be cooperative such as the Pareto solution or noncooperative such as the Nash solution. Let us, for illustration purposes, assume that there are N (instead of one) Leaders in the incentive problem discussed earlier and that the constraint C=0 has been removed for simplicity. Let the decision variables of the leaders be denoted by x1, x2,.., xN respectively and assume, as before, that the Follower has a decision variable y. If the Leaders wish to implement a Pareto solution among them, then this would necessitate minimizing an objective function that is a convex combination of the individual objective functions. That is: G(x1,…,xN,y) = a1G1(x1,..,xN,y) + a2G2(x1,..,xN,y) + ……+ aNGN(x1,..,xN,y) where a1,a2,..,aN are nonnegative constants which satisfy a1+a2+…aN=1. Suppose that the unique answer to this minimization is given by x1=X1, x2=X2,….,xN=XN, and y=Y. If, on the other hand, the Leaders wish to implement a Nash solution among them, then as a first step, the noncooperative nature of this solution would require that the Follower’s decision variable be assigned, or allocated, to one or more of the Leaders. If, for instance, it were allocated to leader #1, then the Nash solution would necessitate solving for X1, X2, ….XN, and Y that will satisfy the inequalities: Gk(X1,.,Xk,.,XN,Y) < Gk(x1,.,Xk,.,XN,y)
for k=1
Gk(X1,.,Xk,.,XN,Y) < Gk(X1,.,xk,.,XN,Y) for k=2,….N.
Whichever solution (Pareto or Nash) the Leaders decide to implement among themselves, it can then be achieved by collectively using incentive strategies xI = fI(y), as discussed earlier, to induce the Follower to choose y = Y. In the simple case of affine strategies, these could take the form: xi = Ai(y – Y)+ Xi ,
for i=1,…,N
The constants Ai will be determined in a manner similar to what was described earlier. Using the incentive strategies, the leaders will be able to force the followers’ optimal solution to coincide with their desired solution X1, X2,...,XN,Y. An interesting aspect of the above solutions, in the general case, is that the incentive strategies will depend on how the Followers are assigned, or distributed, among the Leaders. This opens the possibility of forming different coalitions between Leaders and Followers, and that Leaders may have preferences as to which Followers they would like to have in their coalition. Furthermore, in the special case where all Leaders have the same objective functions, the Pareto and Nash solutions will be the same, but the incentive strategies used by the Leaders will depend on how the Followers are assigned to the various Leaders’ coalitions. Dynamic games are usually characterized by a system whose state vector evolves according to a differential equation that is controlled simultaneously by the leaders and followers. In the case of one leader and one follower, this equation typically takes the form:
dx = f ( x, u L , u F , t ) dt where x(t) is the state variable, uL(t) and uF(t) are the controls of the leader and follower respectively and t is time. Assume that the leader and follower wish to minimize integrated objective functions over a finite time horizon in the form: T
∫
J L (u L , u F ) = G L ( x, u L , u F , t )dt 0
and T
∫
J F (u L , u F ) = G F ( x, u L , u F , t )dt . 0
Following a similar analysis as in the static case, the leader first minimizes J L (u L , u F ) as if it has complete control over the choices of u L and u F . Let U L and U F be the unique functions that minimize J L (u L , u F ) . Now the
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leader wants to induce the follower to choose u F = U F . To achieve this, the leader will implement a strategy u L = h(u F ) where the function h is to be determined by the leader such that the minimization of J F (h(u F ), u F ) by the follower will yield u F = U F . As in the static case, a simple example of the function is h u L = A(u F − U F ) + U L where A is an appropriate function to be determined. In the dynamic case, however, openloop and feedback strategies need to be considered and the function A will be different in each of these cases. In the open loop case, the controls uF and uL and A will all be functions of time only, whereby in the feedback case, these will be functions of time and the state x(t).
dominant provider DP which owns a larger share of the generating units of the assumed power market with two providers, and another provider IP representing other power producers. Let us assume the power market model has “n” buses and “m” lines. PDP and PIP are the nvectors of loads at each of the n buses. PGDP and PGIP are the nvectors of generators at each of the n buses. Y denotes the nvector of net injections at each of the n buses. Pij(min) and Pij(max) are the lower and upper bounds on the real power line flows.
3.3 Diffusing Horizontal Market Power In the literature, it has been pointed out that although there is equal access to the transmission network, there remains substantial opportunity to exercise horizontal market power. Individual transmission line constraints can prevent additional generation from some buses and thus prevent free competition to prevail. Under the present rules of the grid operation, generation providers may purchase transmission capacity rights as transmission congestion contracts (TCC). Thus a provider that holds a TCC on a specific transmission line may receive revenues corresponding to the capacity associated with the TCC times the price differential between the buses connected to the transmission line. In the process of optimizing its total profits an energy provider may cause some transmission lines to be congested. Hogan[9,10] has modeled this bulk power market, without leadership of the grid operator, as an oligopoly. The resulting market price is higher than the competitive benchmark price. A goal of the grid operator is to design incentives so that the resulting market prices correspond to perfect competition prices in the ideal case. If the perfect competition prices can not be achieved, the market prices should be as close to the perfect competition prices as possible. The market power model proposed here seeks to define the strategy that a grid operator should implement for achieving a virtually perfect competition market. If each energy provider uses the same optimization procedure while assuming that the other energy providers have fixed strategies, the equilibrium market price will be a Cournot equilibrium. The challenge for the grid operator is to design the incentives so that the Cournot equilibrium prices are equal to the perfect competition prices. A simplified model of two energy providers will be considered. One energy provider will be assumed to be a
B( .) and C( .) are the benefit and cost functions for load and generation. The benefit function B(.) depends on the energy sold and on the price of the energy. It is the area under the demand curves. The cost function C(.) represents the area under the supply curves. TD is a contract on transmission capacity rights that defines a vector of net loads and pays p’TD for DP. TI is similarly defined for IP. The grid operator will be considered as the leader and the two energy providers will be the followers in a leaderfollower formulation. The control variables for the grid operator are TD and TI. The control variables for the energy providers are their generations and loads. A powerful type of control for the grid operator is to allow TD and TI to be incentive functions of the generations and the loads of the energy providers. Affine functions will be considered for simplicity, i.e., TD and TI will be linear combinations of constants and follower control variables with proportionality constants. The optimization problem for DP is to maximize ( BDP(PDP)  CDP(PGDP)) + p’TD with respect to PDP, PGDP . This maximization is subject to the assumption that the strategy of IP is fixed and that the grid operator provides the value of TD or its functional dependence on DP’s control variables. Furthermore, the maximization is subject to power flow problem [512]:
[Y] − [A T ][Pline flow ] = 0
∑ (P
DP
− PG DP ) + ∑ (PIP − PG IP ) = Y
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[3] Keyhani "Onestepahead load forecasting for online application." IEEE Power Apparatus and Systems 75.1 (July/Aug. 1975)
(Net input balance equation) A denotes the (m by n) incidence matrix .. There is a similar optimization formulation for the IP player, under the assumption that the control strategy of DP is fixed, and the grid operator provides TI. The above two problems need to be solved simultaneously for the scheduled transmission systems conditions. The equilibrium solution provides the optimum amount of power to be generated by DP and IP to share in meeting the total load. It also provides the resulting market prices at the n buses. The leader’s challenge is to design TD and TI so that the resulting price vector is as close to the perfect competition price vector. The DP and IP optimizations can be carried out using nonlinear programming. The first step is to form Lagrangian functions, where p is the Lagrange multiplier vector for the net input balance equality constraint. Thus p appears linearly in two places in each of the Lagrangian functions. One is in p’TD (or p’TI) and another is in the appended equality constraint. The KuhnTucker conditions for the simultaneous maximization of the two Lagrangian functions will contain the parameters in the design of TD and TI. The grid operator will choose the parameters in such a way that the KuhnTucker conditions both for DP and for IP are satisfied when the price vector equals the ideal perfect competition price vector. 4. Conclusion This paper presents a proposed framework for market monitoring and leader follower incentive control of ancillary services. The proposed control formulated based on developing a frequency regulation/load following market, a load regulating market and a base load market. The above market will be controlled by independent system operator acting as a leader with a predefined incentive function that all market players must use to compute the cost of their energy to be offered to the respective markets.
5. Bibliography [1]Abur, A., A. Keyhani, and H. Bakhtiari. "Autoregressive filters for the identification and replacement of bad data in power system state estimation." IEEE Transactions on Power Systems 2.3 (Aug. 1987): 552560. [2] Keyhani, A "Dynamic system load generation control using variable pressure steam generators." IEEE Transactions on Power Apparatus and Systems 75.2 (Nov./Dec. 1975)
[4] Keyhani and ElAbiad ,A"Realtime modeling techniques for application to automatic generation control." IEEE Transactions on Power Apparatus and Systems 95.1 (Nov./Dec. 1976). [5] Keyhani, A., A. Abur, and S. Hao. "Evaluation of power flow techniques for personal computers." IEEE Transactions on Power Systems 4.2 (Aug. 1989) [6] Keyhani, A., S. Hao, and W.R. Wagner. " A Rule Based Approach for construction A local network model for decentralized voltage control." Electric Power Systems Research Journal 16 (1989) [7] Keyhani, A and Mir, S.M. "Online weathersensitive and industrial group bus load forecasting for microprocessorbased applications." IEEE Transactions on Power Apparatus and Systems 102.12 (Dec. 1983): 38683876. [8] Wagner, W.R., A. Keyhani, S. Hao, and T.C. Wong. “A RuleBased Approach to Decentralized Voltage Control.” IEEE Transactions on Power Systems 5.2 (May 1990) :64351. [9] Hogan, William W. “Contract Networks for Electric Power Transmission.” Journal of Regulatory Economics 4.3 (1992): 21142. [10] Hogan, William W “A Market Power Model with Strategic Interaction in Electricity Networks.” The Energy Journal 18.4 (1997). [11] Ilic, M.D. “PerformanceBased Value of Transmission Services for Competitive Energy Management.” Proc. 26th North American Power Symposium (NAPS), Kansas State Univ., Manhattan, KS, 2627 Sept., 1994. [12] Ilic, M., E.H. Allen, and Z. Younes. “Transmission Scarcity: Who Pays?” The Electricity Journal July 1997: 3849. [13] Basar, Tamer and Jose B. Cruz, Jr. “Concepts and Methods in MultiPerson coordination and Control.” Optimization and Control of Dynamic Operational Research Models. Ed. S.G. [15] Cruz, J.B. Jr. "Stackelberg Strategies for Hierarchical Control of Large Scale Systems," Proc. Second Workshop on HierarchicalControl, Warsaw, Poland, June 1978. 341355 (Invited). [16] Cruz, J.B. Jr. "Survey of LeaderFollower Concepts in Hierarchical DecisionMaking," Proc. Fourth International Conf. on Analysis and Optimization of Systems, Le Chesnay, France, December 1980. 384396 (Invited). [17]Simaan, M. and J.B. Cruz, Jr. "Additional Aspects of the Stackelberg Strategy in NonzeroSum Games," Journal of Optimization Theory and Applications 11.6 (June 1973)
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[18] Simaan, M. and J.B. Cruz, Jr "On the Stackelberg Strategy in NonzeroSum Games." Journal of Optimization Theory and Applications 11.5 (May 1973) [19] Simaan, M. and J.B. Cruz, Jr "A Stackleberg Solution for Games with Many Players." IEEE Trans. on Automatic Control AC18 (June 1973) [20] Schweppe, F.C., M.C. Caramanis, and R.D. Tabors. “Evaluation of Spot Price Based Electricity Rates.” IEEE Transaction of Power Apparatus and Systems PAS104.7 (July 1985). [21] Schweppe, F.C., R. Bohn, R. Tabors, and M. Caramanis. Spot Pricing of Electricity. Boston: Kluwer Academic, 1988. [22] Schweppe, F.C., R.D. Tabors, and J.L. Kirtley. “Homeostatic Control for Electric Power Usage.” IEEE Spectrum July 1982 [23] Kassaei, H.R., A. Keyhani, T. Woung, and M. Rahman. “A Hybrid Fuzzy, Neural Network Bus Load Modeling and Predication.” IEEE Transactions on Power Systems PE 021PWRS006 (1998).(in press) [24] Keyhani, A. "Development of an interactive power system research simulator." IEEE Transactions on Power Apparatus and Systems 103.3 (March/April 1984)
[25] Keyhani, A., and A.B. Proca. “A Virtual Testbed for Instruction and Design of Permanent Magnet Machines.” IEEE Transactions on Power Systems PE273PWRS007 (1998).
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