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Cumulant-Based Stochastic Nonlinear Programming for Variance Constrained Voltage Stability Analysis of Power Systems Antony Schellenberg, Student Member, IEEE, William Rosehart, Member, IEEE, and José A. Aguado, Member, IEEE
Abstract—This paper proposes a Cumulant Method-based solution to solve a maximum loading problem incorporating a constraint on the maximum variance of the loading parameter. The proposed method takes advantage of some properties regarding saddle node bifurcations to create a linear mapping relationship between random bus loading variables and all other system variables. The proposed methodology is tested using a sample system based on the IEEE 30-bus system using random active and reactive bus loading. Monte Carlo simulations consisting of 10 000 samples are used as a reference solution for evaluation of the accuracy of the proposed method. Index Terms—Cumulants, numerical optimization, nonlinear programming, probabilistic methods.
I. INTRODUCTION
P
ROBABILISTIC and stochastic programming introduces random variables and uncertainty into conventional linear and nonlinear programs [1]. The randomness and uncertainty introduced is generally represented with a probability density function (PDF) [2]. Many papers have been published regarding the introduction of random quantities into traditional power system problems, including probabilistic power flow (P-PF) [3] and probabilistic optimal power flow (P-OPF) [4]–[7]. Probabilistic and stochastic OPF problems allow random quantities, such as random bus loading and generation, to be incorporated as part of the problem. Stochastic nonlinear programming (S-NLP) problems use random information directly as part of the optimization process [8]. A static stability problem and solution based on P-PF is presented in [9], including an analysis of constraint violations. The material presented in [10] covers the formulation of P-PF using the Cumulant Method and discusses distribution reconstruction using the Gram–Charlier A series. In [11], a cumulant-based method is presented in which random variables are convolved together, and Laguerre polynomials, rather than
Manuscript received May 16, 2005; revised September 19, 2005. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Paper no. TPWRS-00305-2005. A. Schellenberg and W. Rosehart are with the University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail:
[email protected];
[email protected]) J. A. Aguado is with the University of Malaga, Malaga 29071, Spain (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPWRS.2006.873103
the Gram-Charlier A series, are used for distribution reconstruction. The first-order second-moment method (FOSMM) for solving P-OPF with random loads modeled by Gaussian distributions is presented in [4]. The FOSMM is restricted to the use of Gaussian distributions but can account for covariance between loading variables. A Cumulant Method for P-OPF is presented in [7] and is compared with the FOSMM method from [4]. The Cumulant Method for P-OPF is applied to Gaussian and gamma-distributed bus loading variables in [6]. An S-NLP problem is proposed in [12] to minimize the variance of active power generation at the slack bus when Gaussian random bus loading is considered. This paper proposes a maximum loading problem with independent random bus loading creating an S-NLP type problem. The proposed technique creates an optimization problem that incorporates the variance of the loading parameter directly. The problem finds the settings of the control variables, for example, generator voltage and active power settings, such that the loading level is maximized while ensuring that the variance of the loading factor is less than a pre-specified maximum given uncertainty in the bus loading and the loading direction. Results are computed using Gaussian distributions for random bus loading, and, in all cases, Monte Carlo simulations are used to provide a reference. The proposed Cumulant Method for S-NLP uses properties of cumulants and a linear approximation of the system to map statistical information from known random variables into state variables. The mapping matrix is associated with Jacobian and Hessian matrices in the problem and contains the coefficients for a linear mapping between known and unknown random variables. Once cumulants are computed for the state variables in the problem, Gram–Charlier/Edgeworth expansion theory is used to create the PDF representations based on the cumulants. In a practical sense, information regarding bus loading distributions is calculated based on measured historical data. The mean and variance, as well as higher order statistical data, can be computed based on sample data [2]. The mean describes the expected value that the load will take over a long period of time, while the variance describes how the loading values spread away from the mean. This paper is focused on the impact of uncertainty in bus loading due to imperfections in forecasting mechanisms. In general, a historical record of forecast and actual loading levels is available; this information is used to compute a statistical distribution for the bus loading based on a forecast value.
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This paper is based on analysis of maximum loading problems at saddle node bifurcation (SNB) points. Therefore, Section II covers background material related to the properties and equations regarding SNBs as well as general maximum loading problems. The proposed method is based on the Cumulant Method, and therefore, Section III outlines the fundamental behavior of this method, and Section IV applies the fundamentals of the Cumulant Method to OPF problems. The details of the proposed problem are presented in Section V. Finally, Sections VI and VII cover numerical results using a 30-bus system and conclusions, respectively. Background information in probability and statistics is outside the scope of this paper. However, [6] contains most of the background material in these areas that is associated with this paper. Additionally, [2] contains further information regarding general probability and statistics. II. STATIC STABILITY ANALYSIS Static stability analysis investigates the behavior of the system as load is slowly increased. The maximum load the system can support is limited by a number of different factors. Generally, SNBs are expected to occur in all PV curves because of the nature of power systems [13]. The work presented in this paper is focused specifically on saddle node-type bifurcations. In the following sections, SNBs and associated optimization-based maximum loading programs are reviewed. Since a saddle node is associated with loss of equilibrium, power flow equations are used to model the system in this paper.
The maximum loading problem can be formulated as
subject to (2) where models the system loading level, and contains all system variables, including power generation (active and reactive), bus voltages, and bus angles. The equality constraints in , are the ac power flow the maximum loading problem, equations, including a loading factor, and the inequality constraints contain any variable limitations as well as functional limitations of the system, such as line loading limits. The above formulation will find a maximum corresponding to an SNB, if certain system conditions are met, specifically, incorporating PQ load models and not enforcing certain constraints, such as reactive power and voltage magnitude limits [16], [17]. III. CUMULANT METHOD
SNBs are generally characterized by a loss of equilibrium. Mathematically, they are associated with a single zero eigenvalue of the Jacobian matrix. An SNB point must satisfy the following three conditions:
Fundamentally, the Cumulant Method maps statistical information from predefined, or known, distributions into other new, or unknown, random variables. Information is mapped through a statistical measure known as cumulants, and, as a result, this method is referred to as the Cumulant Method. The map itself is an approximation based on analysis of the system as linear combination of random variables, and therefore, this section highlights the behavior of random variables and their cumulants when combined in a linear fashion. A new random variable can be created by linearly com. bining other independent random variables, The formation of the new variable from the linear combination of its component variables is mathematically stated in the following manner:
(1a)
(3)
A. Properties of SNBs
(1b) (1c) where is the power flow Jacobian related to PQ buses in the system, and is the critical eigenvector associated with the zero eigenvalue. The three conditions in (1) ensure that has a zero eigenvalue, the associated eigenvector is not trivial, and the critical eigenvalue is purely real, respectively. Often, these equations are solved in conjunction with the standard power flow problem equations. This combined problem is referred to as the direct method [13]. B. Maximum Loading Problems Maximum loading problems have been investigated within the power system analysis community for many years and are well established. These problems are based around finding the maximum amount of load that a given system can support in a specific loading direction [13]–[15]. The loading direction is a vector for controlling how additional load is distributed amongst the system buses.
where is the th coefficient in the linear combination. By asare indesuming that the component variables pendent, simplifications can be made, and it can be shown that the th-order cumulant for can be computed according to the following relationship [6]: (4) is the th-order cumulant for the new random where the is the th-order cumulant of the th comvariable , and ponent random variable. Using the result in (4), cumulants can be determined for a new random variable based on cumulants from the component distributions, provided that the coefficients in the linear combination, i.e., , are known. Once the cumulants for a new random variable are known, the PDF corresponding to the cumulants can be constructed using the Gram–Charlier A series [10], [18]. The Gram–Charlier A series uses a weighted sum of a normal distribution and its derivatives to represent a general PDF, where the weights for the terms are computed using the moments or cumulants for a distribution.
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IV. STOCHASTIC OPF This section outlines the adaption of the Cumulant Method discussed in Section III to stability-based S-NLP problems. The derivations in this section are formulated based on the need to include random bus loading into optimization problems.
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matrix of linear mapping coefficients relating bus loading and correspond to the state variables. Specifically, the rows in terms in (4) and can therefore be used to determine the cumulants for state variables based on the cumulants of the loads. B. Cumulant Method for the Maximum Loading Problem
In general, the standard ac power flow equations can be written as follows:
In maximum loading problems, the ac power flow equations can be modeled with the addition of a loading parameter as follows:
(5)
(11)
Usually, system loading, i.e., and , are included in problems as predefined deterministic parameters. The method proposed in this paper examines the interrelationships between bus loading and other variables within power system problems. Therefore, both active and reactive and , are considered in the same fashion as loads, any other state variable within the problem for the purposes of determining these relationships. By including bus loading as a variable, (5) is written in the following compact form:
Grouping power flow equations together as and taking the full derivative in a similar fashion to (8) yields
A. Cumulant Method for OPF
where as
represents the vector of state variables, and
(12) Rearranging (12) and grouping the sults in
and
terms together re-
(13)
(6)
Each of the gradient terms in (13) can be evaluated and replaced in the following way:
is defined
(14a) (14b)
(7) Taking the total derivative of (6) yields the following [12]: (8) is the standard Jacobian matrix The first gradient in (8), for the power flow equations. The second term, however, is not a part of the standard power flow equations and results from treatment of the load as a variable rather than a constant. Several continuation methods [19] incorporate prediction steps that are based on a similar derivation to (8). in (8) reIf loads are modeled as constant power, duces to the negative identity matrix. Therefore, the second term . is reduced to Rewriting (8) using the above observations gives (9) which can be rearranged to form (10) Based on the result in (10), the inverse Jacobian matrix of the power flow equations relates a change in bus loading to a change in state variables at the current operating point. In a general linear combination, the coefficients relate a change in one varican be viewed as a able to a change in another. Therefore,
(14c) where is the standard Jacobian matrix of the power flow equations. Based on substitutions from (14), (13) can be rewritten in the following form: (15) The matrix on the left-hand side of (15) is no longer square, and consequently noninvertible, due to the addition of the extra variable but no new equations. Therefore, the same derivation that was used to arrive at (10) cannot be used with (15). The matrix can be made square by the addition of another equation, but, unfortunately, there is no convenient single equation that can be added to the problem relating the variables in and . To overcome the difficulties associated with the nonsquare matrix, (1a) and (1b) can be added to the problem by imposing the restriction that the maximum loading point be an SNB point. is In addition, a variable corresponding to the eigenvector added to the problem variables. Therefore, new variables have equations. Consequently, been added to the problem with the matrix is now square, and the problem can be re-evaluated with the new equations in place
(16)
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where the variables in the problem are , , , and . The following equation results from taking the full derivative with respect to each of the variables and grouping the terms: (17) where
parameter is less than a predefined constant. During the optimization process, the variance of the loading parameter is computed using the techniques and procedures presented within this to map the variance of independent loads paper by using into the variance of the maximum loading level. The proposed problem is formulated as follows:
is defined as subject to (18)
where is the Hessian matrix of the power flow equations for the load buses with respect to state variables. The final result is obtained by rearranging (17) (19) Similar arguments regarding the result (10) can be made for (19). Essentially, contains linear coefficient information relating changes in to changes in system variables, i.e., the terms in (4), and can therefore be used to map statistical information from bus loading into system variables , the critical eigenvector , and stability margin . C. Application of the Cumulant Method Based on the derivations previously presented, the inverse Jacobian matrix associated with the power flow and direct equations can be used as a matrix of linear coefficients for the combination of random variables. In order to compute statistical information, (4) is used in conjunction with the coefficients from the mapping matrix. is written in the following form: If
.. .
.. .
.. .
..
.
then the th cumulant for the th variable in computed using the following equation:
.. .
(20)
is (21)
where variable.
is the th cumulant for the th random bus loading
(22) is the power flow equations, are where the inequality constraints, and defines the upper bound on the variance of the maximum loading level, The proposed problem is formulated based on the assumption that the system is limited by an SNB. In general, all PV curves exhibit SNBs due to the nature of the power system [13]. B. Solution Techniques The problem is implemented using Matlab as well as the commercial solver SNOPT [20]. Matlab’s CMEX system is used to interact with the SNOPT subroutines. SNOPT is based on a sparse sequential quadratic programming solver that incorporates a limited memory quasi-Newton Hessian updating scheme. As implemented here, SNOPT requires user-supplied evaluations for both nonlinear function values and associated gradients. A numerical differentiation technique, known as the complex-step method [21], is used to supply gradient information to the solver. A matrix inverse operation is used in the creation of the mapping matrix, and consequently, the mapping matrix is dense. Therefore, the gradient vector corresponding to the variance constraint is also dense. However, since the variance constraint only comprises a single row of the matrix associated with the nonlinear functions in the problem, the gradient matrix is still very sparse. Additionally, the variance constraint is added to the problem in such a way that the matrix entries corresponding to this equation appear near the bottom of the matrix; this reduces unnecessary fills during matrix operations and maintains computational efficiency. 1) Complex-Step Method: The complex-step method is an approximation based on the evaluation of the Taylor series [21] when exposed to a purely imaginary perturbation. An overview of the technique based on [21] is presented here. The Taylor series is written as follows [22]: (23)
V. IMPLEMENTATION DETAILS This section presents details related to the problem formulation and the precise details of the solution.
If the perturbation in (23) is replaced with a purely complex perturbation of size , the following equation can be written:
A. Problem Details This paper proposes an S-NLP maximum loading problem, including a constraint requiring that the variance of the loading
(24)
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By expanding the exponents where possible and grouping into real and imaginary parts, (24) can be split into two equations as follows: (25a) (25b) corresponding to the real and imaginary parts, respectively. If is less than mathe perturbation size is chosen such that chine precision in the language used, the equations in (25) can be further reduced to the following: (26a) (26b) since all of the other terms are multiplied by a zero resulting from the exponentials of . Therefore, based on (26b), the derivative can be computed based on the following equation: (27) Although (27) can be viewed as an approximation, results obtained using this equation in a finite precision computing environment are highly accurate. Extending the result in (27) into the gradient is accomplished by perturbing each element in individually in the complex plane and applying (27). 2) Solution Diagram: A flowchart for the solution of the proposed problem is given in Fig. 1. Details regarding SNOPT’s numerical optimization technique are well documented in [20]. The problem is first loaded into Matlab, including bus and branch information as well as cumulants associated with the random bus loads. This information is used to setup problem for the constraints as well as compute the initial vector problem variables. The problem information is then passed into SNOPT, which implements an iterative sequential quadratic solver. SNOPT requires that information regarding the nonlinear constraints be updated, and Matlab functions are called to provide these evaluations of the nonlinear functions as well as the associated gradient information. Gradient information is computed using the complex-step method presented in Section V-B1. It is during these Matlab function calls that the Cumulant Method proposed in this paper is used to compute , the variance of the loading parameter, which is included in a nonlinear constraint. The procedure continues to iterate until either a solution to the problem has been reached or the optimization process has failed and no solution can be found. VI. NUMERICAL RESULTS Numerical results are presented for a system based on the IEEE 30-bus system. Monte Carlo simulations, consisting of 10 000 samples, are used as a reference for evaluation of the proposed Cumulant Method. The Monte Carlo simulations are performed based on the control settings found using the S-NLP procedure. All real and reactive loads in the problem are treated as Gaussian random variables. The loads are modeled as independent random variables with mean values at the nominal bus
Fig. 1.
Program flow showing interaction between SNOPT and Matlab.
loading value from the original system, and the variance of each load is chosen such that the 99% confidence interval is 10% of the nominal loading value. The system tested contains a high amount of reactive compensation, and there are no limitations placed on reactive power generation or bus voltage levels. In addition, all loads in the problem are of the PQ type. Therefore, the maximum loading point will be a saddle node point, thus validating the assumption made earlier in the derivation of the proposed method. Results obtained using the procedures described in this paper are shown in Table I and show that constraining the variance in the optimization problem yields results that are at the pre-specified upper bound. In addition, the mean value for the loading parameter is reduced as the maximum value for the variance is reduced. This is a logical result since reducing the solution space should generally result in a poorer optimal solution from an objective function perspective. Therefore, there exists a trade-off between minimizing the variance, or uncertainty, of the loading parameter and the maximum possible loading level of the system. , based on Fig. 2 compares PDFs of the loading level three different values for the variance constraint. Monte Carlo simulation results presented use the control settings found using the S-NLP and 10 000 samples. The dashed curve on the left , the center curve results when the maximum variance is (dash-dot) corresponds to a maximum variance of , and the right-hand curve (solid) results when there is no constraint on the variance. As the maximum variance is reduced and the problem becomes increasingly constrained, the mean value of the optimal result is reduced. This behavior is evident in Fig. 2 is reduced. as the peak of the curve shifts to the left as Also, the variance of at the optimal solution becomes smaller as the maximum in the constraint is reduced. This can be seen as
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TABLE I NUMERICAL RESULTS FOR THE 30-BUS SYSTEM
the curves appear narrower, and the peak at the mean is higher in order to maintain a unity integral. The PDFs for the bus voltage corresponding to the critical bus (Bus 30) are shown in Fig. 3. As the variance of becomes increasingly restricted, the mean value for the voltage at the critical bus rises; this corresponds to the fact that the mean value for is reduced as the constraint is tightened. In addition, the distribution for the critical bus voltage becomes narrower as the distribution for is tightened. Fig. 4 shows the PDFs for both the proposed Cumulant Method and corresponding Monte Carlo simulation when the maximum variance is constrained to . From the figure, it can be seen that the Cumulant Method solution has a slightly lower mean value than the Monte Carlo simulation result. Additionally, the Cumulant Method result has a slightly wider distribution than the Monte Carlo solution, and this corresponds to the higher variance result in the Cumulant Method data. It should be highlighted that the results shown in Fig. 4 correspond to the highest percent error found in the simulation results. In the more typical case, the two curves are extremely close. Furthermore, it is felt that the maximum error is relatively negligible given the objective of the proposed technique, which is to bound the uncertainty in the loading parameter. In all cases, mean values computed using the proposed Cumulant Method are well within 1% of the Monte Carlo simulation results. The difference in the variance between the Cumulant Method and the Monte Carlo simulation was between 2%–11%.
Fig. 2. PDFs f () versus loading level , when variance is constrained to 2e , 3:5e , and unconstrained.
VII. CONCLUSION The Cumulant Method proposed in this paper is able to successfully solve maximum loading problems with random bus loading involving a constraint on the maximum value for the variance of the loading level. The proposed method utilizes a linear mapping based on power flow equations and the characteristic equations for SNB points. The primary application of the method proposed in this paper is to systems that are highly reactively compensated and, thus, typically limited by SNBs. However, systems that are not limited by an SNB can still benefit indirectly from the proposed methodology since the manifold at points beyond the limit will still yield SNBs, provided that PQ loads are included in the problem. One clear disadvantage of the proposed technique is the increase in computational time as system size increases. This is associated with an increase in the number of variables by at least two for each additional bus and an increase in constraints associated with incorporating variance and the uncertainty mappings.
Fig. 3. PDFs of bus voltages at the critical bus (Bus 30) when variance is constrained to 2e , 3:5e , and unconstrained.
These problems can be partly overcome using optimized numerical routines that take advantage of the structure of the problem. The derivation of the method presented in this paper assumed that bus loading variables are independent; in practice, this is not always true. There are a number of open questions regarding general application of this method to problems involving correlated variables, particularly in problems involving higher order correlation.
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Fig. 4. Comparison of Monte Carlo simulation and Cumulant Method PDFs f () versus loading level , when variance is constrained to 3:5e .
The proposed methodology is tested using a sample system based on the IEEE 30-bus system using random active and reactive bus loading. Monte Carlo simulations consisting of 10 000 samples are used as a reference solution for evaluation of the accuracy of the proposed method. There is a clear reduction in the variance of the loading level as the problem becomes increasingly constrained, and the results also show that there is a trade-off between maximum loading level and variance of the loading parameter. REFERENCES [1] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming. New York: Springer-Verlag, 1997. [2] A. Papoulis and S. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. New York: McGraw-Hill, 2002. [3] M. T. Schilling, A. L. da Silva, R. Billinton, and M. El-Kady, “Bibliography on power system probabilistic analysis (1962–1988),” IEEE Trans. Power Syst., vol. 5, no. 1, Feb. 1990. [4] M. Madrigal, K. Ponnambalam, and V. H. Quintana, “Probabilistic optimal power flow,” in Proc. IEEE Can. Conf. Electrical Computer Engineering, vol. I, May 1998. [5] G. Viviani and G. Heydt, “Stochastic optimal energy dispatch,” IEEE Tran. Power App. Syst., vol. PAS-100, no. 7, pp. 3221–3228, Jul. 1981. [6] A. Schellenberg, W. Rosehart, and J. Aguado, “Cumulant-based probabilistic optimal power flow (P-OPF) with Gaussian and gamma distributions,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 773–781, May 2005. [7] A. Schellenberg, W. Rosehart, and J. A. Aguado, “Cumulant-based probabilistic optimal power flow (P-OPF) with Gaussian distributions,” in Proc. 8th Int. Conf. Probability Methods Applied Power Systems, Sep. 2004. [8] T. Yong and R. H. Lasseter, “Stochastic optimal power flow: Formulation and solution,” in Proc. IEEE Power Engineering Society Summer Meeting, vol. I, Jul. 2000. [9] N. Hatziargyriou and T. Karakatsanis, “Probabilistic load flow for assessment of voltage instability,” Proc. Inst. Elect. Eng., Gener., Transm., Distrib., vol. 145, no. 2, pp. 196–202, Mar. 1998.
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[10] P. Zhang and S. T. Lee, “Probabilistic load flow computation using the method of combined cumulants and Gram-Charlier expansion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 676–682, Feb. 2004. [11] W. Tian, D. Sutanto, Y. Lee, and H. Outhred, “Cumulant based probabilistic power system simulation using Laguerre polynomials,” IEEE Trans. Energy Convers., vol. 4, no. 4, pp. 567–574, Dec. 1989. [12] A. Schellenberg, W. Rosehart, and J. A. Aguado, “Cumulant based stochastic optimal power flow (S-OPF) for variance optimization,” in Proc. IEEE Power Engineering Society General Meeting, Jun. 2005. [13] C. Cañizares, Ed., Voltage Stability Assessment: Concepts, Practices and Tools: IEEE/PES Power System Stability Subcommittee, Aug. 2002, ch. 4. [14] T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems. Norwell, MA: Kluwer, 1998. [15] W. D. Rosehart, C. A. Cañizares, and V. H. Quintana, “Multiobjective optimal power flows to evaluate voltage security costs in power networks,” IEEE Trans. Power Syst., vol. 18, no. 2, pp. 578–587, May 2003. [16] W. Rosehart, C. Cañizares, and V. Quintana, “Optimal power flow incorporating voltage collapse constraints,” in Proc. IEEE Power Engineering Society Summer Meeting, vol. 2, Jul. 1999, pp. 820–825. [17] N. Maratos and C. Vournas, “Relationships between static bifurcations and constrained optima,” in Proc. IEEE Int. Symp. Circuits Systems, vol. 2, May 2000, pp. 477–480. [18] M. G. Kendall and A. Stuart, The Advanced Thoery of Statistics, 4th ed. New York: Macmillan, 1977. [19] C. A. Cañizares and F. L. Alvarado, “Point of collapse and continuation methods for large ac/dc systems,” IEEE Trans. Power Syst., vol. 8, no. 1, pp. 1–8, Feb. 1993. [20] SNOPT Optimizer, S. B. S. Inc.. (2004, May 4). http://www. sbsi-soloptimize.com/asp/sol.product_snopt.htm [Online] [21] J. R. R. A. Martins, P. Sturdza, and J. J. Alonso, “The complex-step derivative approximation,” ACM Trans. Math. Softw. (TOMS), vol. 29, no. 3, pp. 245–262, Sep. 2003. [22] Eric Wesstein’s World of Mathematics, E. W. Weisstein.http://malhworld.wolfram.com/ [Online]
Antony Schellenberg (S’03) received the B.Sc. degree in electrical engineering from the University of Calgary, Calgary, AB, Canada, in 2002. He is currently working toward the Ph.D. degree in electrical engineering at the University of Calgary. His research is focused on probabilistic and stochastic optimal power flow.
William Rosehart (M’01) received the B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 1996, 1997, and 2001, respectively. He is currently an Assistant Professor in the Department of Electrical and Computer Engineering at the University of Calgary, Calgary, AB, Canada. His main research interests are in the areas of numerical optimization techniques, stability issues, and modeling power systems in a deregulated environment.
José A. Aguado (M’01) received the Ingeniero Eléctrico and Ph.D. degrees from the University of Málaga, Málaga, Spain, in 1997 and 2001, respectively. Currently, he is an Associate Professor and Head of the Department of Electrical Engineering at the University of Málaga. His research interests include operation, planning, and deregulation of electric energy systems and numerical optimization techniques.
