THE restructuring of the power industry has lead to

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1 GLOBALLY CONVERGENT OPTIMAL POWER FLOW USING COMPLEMENTARITY FUNCTIONS AND TRUST REGION METHODS Geraldo L. Torres Universidade Federal de Pernambuco Recife, Brazil Abstract - As power systems become heavily loaded there is an increasing need for globally convergent optimal power flow (OPF) algorithms. An algorithm is said to be globally convergent if it is able to obtain a solution, if one exists, for any choice of initial point. Such an algorithm is developed here by combining three approaches: (i) reformulation of the OPF problem as a nonlinear equations system by means of complementarity functions, (ii) reformulation of the equations system as an unconstrained minimization problem, and (iii) solution of the unconstrained minimization problem by a globally convergent trust region algorithm. The proposed algorithm is implemented in MATLAB, and its performance is tested on the IEEE 30-, 57-, 118- and 300-bus systems. Keywords - Optimal Power Flow, Trust Region Method, Nonlinear Complementarity Method, Global Convergence. 1 INTRODUCTION THE restructuring of the power industry has lead to new complex optimal power flow (OPF) problems [1], requiring robust and reliable solution techniques. Concerning robustness, global convergence is a desirable property for any nonlinear OPF solution algorithm, so as to be able to obtain a solution, if one exists, for any choice of initial point. There are two classical approaches for globalizing a locally convergent optimization algorithm: use of line searches and trust regions []. This paper concentrates on a trust region method for unconstrained optimization and nonlinear systems of equations. Trust region methods are a relatively new class of nonlinear optimization algorithms that minimize a quadratic approximation of a nonlinear objective function within a closed region called the trust region. The closed region is named so because within this region the quadratic model can be trusted to be a good approximation to the original objective function. The achieved reduction in the quadratic approximation should correspond to a reduction in the nonlinear objective function, and if that is not the case, then the size of the trust region is reduced and the approximation model is solved again. There is a broad family of trust region methods, and they differ from each other mainly in the way they model the objective function and handle the constraints, the classical approach being the use of a quadratic approximation for unconstrained minimization []. Some applications of trust region methods to power systems optimization can be found in the literature [1, 3, 4, 5]. A new application of trust region methods to OPF solution is proposed here. Unlike in [5], the trust region algorithm used in this paper is for unconstrained optimization. To allow for that, first the OPF optimality conditions are reformulated as a nonlinear equations system using the complementarity function approach described in [6]. Because Newton s method may fail to converge when solving the nonlinear equations system, the equations are thus reformulated as an unconstrained minimization problem whose objective is the sum of the equations squared residuals. Finally, to globalize the solution of the unconstrained minimization problem, a trust region method is employed. The trust region OPF algorithm proposed in this paper differs from its alike in [5] in several aspects, but the three main ones are: the use here of complementarity functions to reformulate the OPF problem as a nonlinear equations system, which in turn is reformulated as an unconstrained minimization problem; the trust region algorithm used here is for unconstrained optimization, while in [5] is for constrained optimization; the trust region subproblems are solved here by the dogleg method, while in [5] are solved by interiorpoint methods for quadratic programming. The main computational implementation issues of the proposed trust region OPF algorithm are discussed in this paper, and the performance of the algorithm is studied using the IEEE 30-, 57-, 118- and 300-bus test systems. Comparisons are made with the primal-dual interior-point algorithm for direct nonlinear OPF solution [7], bearing in mind that the main focus of the globally convergent OPF algorithm is on convergence robustness rather than on processing time. The paper is organized as follows: In Section, the first-order necessary optimality conditions for a general form nonlinear OPF problem are presented and then, by using complementarity functions, are reformulated as a nonlinear equations system. In Section 3, reformulation of the equations system as an unconstrained minimization problem and globalization of the solution by a trust region method are both described. Some implementation issues are discussed in Section 4, and numerical results for various test systems are presented in Section 5. Section 6 summarizes the main contributions of the paper.

2 NONLINEAR EQUATIONS REFORMULATION OF OPTIMAL POWER FLOW PROBLEMS In this paper one deals with the solution of nonlinear OPF problems of the general form: min f(x) (1a) s. t.: g(x) = 0 (1b) x x x (1c) where x R n is a vector of decision variables, including the control and state variables; g : R n R m is a nonlinear vector function including conventional power flow equations and other equality constraints such as power balance across boundaries in a pool operation; and x and x are lower and upper bounds on the variables x, corresponding to physical and operating limits on the system. If x is a local minimizer to (1) then there exist vectors of Lagrange multipliers, say, (λ, π, υ ), that satisfy the Karush-Kuhn-Tucker (KKT) optimality conditions [6]: Sπ = 0, s 0, π 0, Zυ = 0, z 0, υ 0, x + s x = 0, x + z x = 0, g(x) = 0, f(x) + g(x)λ π + υ = 0, (a) (b) (c) (d) (e) (f) where s and z are slack vectors that transform the inequalities in (1) into the equalities (c) and (d); S and Z are diagonal matrices with S ii = s i and Z ii = z i ; f(x) is the gradient vector of f(x), and g(x) is the gradient matrix of g(x). The major difficulty in solving () is mainly related to the complementarity conditions (a) and (b). A rootfinding Newton s method applied to () cannot automatically assure (s, z, π, υ) 0, and the numerical solution of Sπ = 0 and Zυ = 0 is intricate. For instance, the Newton equation for the i-th complementarity equation s i π i = 0 is s k i π i + π k i s i = s k i π k i. (3) If a variable becomes zero, say, πi k = 0, then (3) becomes s k i π i = 0, leading to a zero update, π i = 0. Thus, πi k will remain zero all the time once it becomes zero during the iterations, which is fatal because the algorithm will never be able to recover from such a situation. In [6] a nonlinear complementarity (NC) approach is proposed to handle (a) and (b). Unlike interior-point methods, the NC approach does not require that the strict positivity conditions (s, z, π, υ) > 0 be verified at each iteration. Derived from techniques for solving complementarity problems [8], the most attractive feature of the NC approach is that it reformulates the KKT conditions () as a nonlinear equations system, allowing for a Newton-type method to be used. The sign conditions (s, z, π, υ) 0 are automatically satisfied at the limit point, without imposing additional conditions during the iterations. The reformulation of the OPF as nonlinear equations is obtained by handling each complementarity condition in () by a function ψ : R R that holds the property ψ(a, b) = 0 ab = 0, a 0, b 0. (4) Thus, the i-th complementarity conditions in (a) and (b), respectively, are equivalent to the equations ψ(s i, π i ) = 0, (5) ψ(z i, υ i ) = 0. (6) Any function that holds the property (4), such as ψ(a, b) = a + b a + b, (7) ψ(a, b) = 1 min{a, b}, (8) ψ(a, b) = 1 ((ab) + min{0, a} + min{0, b}), (9) is said to be a NC-function. The function (7) was proposed by Fischer [8] in 199, and has attracted a lot of attention. By exploiting the property (4), the KKT conditions () can be reformulated as the nonlinear equations system ψ(s, π) ψ(z, υ) h(y) = x + s x x + z x = 0, (10) g(x) f(x) + g(x)λ π + υ where y = (s, z, π, υ, λ, x). The advantage of equation reformulation (10) is that, unlike KKT conditions (), it can be solved iteratively by well-known methods [9]. The conditions (s, z, π, υ) 0 are automatically assured by the function ψ at the limit point of the iterations. Newton s method is the most popular method for solving systems of nonlinear equations. Because with a poor initial estimate it may diverge, step size control can be used to improve the convergence, i.e., given the point y k, solve the Newton system for the correction term y, h(y k ) T y = h(y k ), (11) and then compute a new solution estimate from y k+1 = y k + α k y. (1) The matrix h(y k ) is the gradient of h(y) evaluated at y k, and the scalar α k (0, 1] is the step size parameter to enhance the convergence. 3 GLOBALIZATION BY A TRUST REGION METHOD Newton s method can be made more robust by using line searches or trust region techniques. A trust region method is considered in this paper due to its significant success in globalizing algorithms for unconstrained optimization and systems of nonlinear equations []. Note that solving the equations system (10) is equivalent to solving the unconstrained minimization problem min y φ(y) = 1 h(y)t h(y). (13)

3 Any root y of h(y) has φ(y ) = 0, and since φ(y) 0 for all y each root of h(y) is a minimizer of φ(y). The converse is not always true; local minimizers of φ(y) with φ(y ) > 0 are not roots of h(y). However, local minima like these which are not roots of h(y) satisfy an interesting property []. Since φ(y ) = h(y )h(y ) = 0 (14) one can have h(y ) 0 only if h(y ) is singular. Note that h(y ) is the coefficient matrix in the Newton system (11) to solve the KKT conditions and under a constraint qualification (regularity condition) it is nonsingular. The trust region method to solve the unconstrained minimization problem (13) uses information gathered about φ(y) to construct a model function m k whose behavior near the approximation point y k should be similar to φ(y). The model m k is usually a quadratic function derived from the truncated Taylor series expansion of φ(y) around y k, m k (y k +d) = φ(y k )+ φ(y k ) T d+ 1 dt φ(y k )d, (15) where φ(y k ) is the gradient and φ(y k ) is the Hessian of φ(y), both evaluated at y k. If the point y = y k + d is far from y k, the model function m k (y k + d) may not be a good approximation to φ(y); thus, the solution d = arg min m k (y k + d) does not always make sense as a minimization step for φ(y). For instance, the Hessian φ(y k ) may be indefinite, and there are directions along which m k (y k +d) is unbounded from below; in this case, d is infinite. To globalize the algorithm the search for a minimizer of m k (y k + d) is restricted to some region around y k, called the trust region. This trust region is usually a ball defined by the Euclidean norm d k, where the scalar k is called the trust region radius. The trust region subproblem around the point y k for the unconstrained minimization problem (13) is min φ(y k ) + φ(y k ) T d + 1 dt φ(y k )d s. t.: d k. (16) If the candidate solution y k+1 = y k +d k does not produce a sufficient decrease in φ(y), then the trust region as given by the radius k is too large. To proceed, the radius k is reduced and the trust region subproblem is solved again. 3.1 Solving the Trust Region Subproblems As shown in [10], a vector d is a global solution to problem (16) if and only if it is feasible (i.e., d k ) and there is a Lagrange multiplier σ 0 such that ( φ(y k ) + σ I)d = φ(y k ), (17a) σ ( k d ) = 0, (17b) ( φ(y k ) + σ I) is positive semidefinite. (17c) The condition (17b) is a complementarity condition that states that at least one of the nonnegative quantities σ 0 and ( k d ) 0 must be zero. Thus, when d < k one must have σ = 0 and, from (17a), φ(y k )d = φ(y k ). When σ > 0 the solution d is collinear with the negative gradient of m k and normal to its contours. Although in principle one seeks the optimal solution of (16), it is enough for purposes of global convergence to find an approximate solution d k that lies within the trust region and gives a sufficient reduction in model m k []. This approximate solution can be obtained, e.g., using Powell s dogleg method [11]. Another approach is the two-dimensional subspace minimization. While the former can be used only when φ(y k ) is positive definite, the later is more general and can handle directions of negative curvature. The dogleg method is used here due to its lower computational cost. Before presenting it, one need to discuss the Cauchy step and the Newton step The Cauchy Step The Cauchy point for problem (16) is the constrained minimizer in the direction of steepest descent at d = 0. Thus, it is given by d c = α k φ(y k ) (18) where α k is the optimal step length found by solving min α>0 φ T k ( α φ k ) + 1 (α φ k) T φ k (α φ k ) (19a) s. t.: α φ k k (19b) whose solution is given by φ T k φ k if δ k > 0 and ( φt k φ k )3/ k δ α k = k δ k k otherwise φ k (0) where, for compactness, δ k = φ T k φ k φ k The Newton Step The Newton step is the unconstrained minimizer to problem (16), which is given by d n = φ(y k ) 1 φ(y k ). (1) Because the Newton step is an unconstrained minimizer, it may not satisfy the trust region constraint The Dogleg Method If the Newton step d n lies inside the trust region (i.e., if d n k ), then it is an approximate solution to (16). If it lies outside, then dogleg method finds the constrained minimizer along the dogleg path subject to d k. The dogleg path consists of the two line segments from d = 0 to d = d c and from d = d c to d = d n, denoted by { τd c for τ [0, 1] d(τ) = () d c + (τ 1)(d n d c ) for τ (1, ].

4 Two properties make the dogleg method well-defined: the Newton point is always farther away than the Cauchy point (i.e., d n > d c ), and the quadratic objective decreases monotonically along the dogleg path. Thus, the constrained minimizer is the intersection of the dogleg path d(τ) with the trust region boundary (shown in Fig. 1), characterized by d c + (τ 1)(d n d c ) = k. (3) Equation (3) is quadratic and thus has two solutions; since d c < k the intersection point is given by (d T τ = c d nc ) d nc ( d c k ) dt c d nc d nc + 1 (4) where d nc = d n d c. The main steps of the dogleg method are as follows: 1. Given φ(y k ) and φ(y k ), obtain d n from (1). If d n k, then return with d k = d n.. Compute d c from (18). If d c k, then return with d k = d c. 3. Given d n and d c from steps 1 and, compute d(τ) from () and (4), and return with d k = d(τ). Clearly, the most expensive task in the dogleg method is setting up and solving the linear system (1) to obtain d n. All other steps are inexpensive. Figure 1: The Cauchy point and Newton point in the dogleg path Further Developments Alternatively, one can solve problem (16) using LSTRS [1], a MATLAB software for large-scale trust region subproblems. LSTRS is based on a reformulation of the trust region subproblem as a parameterized eigenvalue problem. An iterative procedure finds the optimal value for the parameter, which is used to compute a solution for problem (16). The eigenvalue formulation is based on the fact that there exists a value of a scalar parameter α such that problem (16) (without the term φ(y k ) in the objective) is equivalent to min 1 wt B α w s. t.: w T w 1 +, e T 1 w = 1, (5) where B α is the bordered matrix [ ] α φ(y B α = k ) T φ(y k ) φ(y k ) (6) and e 1 is the first canonical vector in R n d+1. The optimal value for α is given by α = σ φ(y k ) T d, with σ, d the optimal pair in (17). If we knew α, we could compute a solution to the trust region subproblem from the algebraically smallest eigenvalue of B α and a corresponding eigenvector with a special structure. The solution would consist of the last n d components of the eigenvector and the Lagrange multiplier would be the eigenvalue. LSTRS starts with an initial guess for α and iteratively adjusts this parameter toward the optimal value. This is accomplished by solving a sequence of eigenvalue problems for B α for different α s, as clearly described in [1]. 3. Updating the Trust Region Radius Critical to the performance of the trust region algorithm is the strategy for choosing the trust region radius k at each iteration []. If k is too small, the algorithm misses an opportunity to make substantial progress to the solution. If too large, the minimizer of the model may be far from the minimizer of the objective function in the region, so one may have to reduce the size of the region and try again. The updating of k is usually based on the agreement between the model function m k and the objective function φ(y) at previous iterations. Given a step d k, a reduction ratio ρ k is defined as: ρ k = φ(y k) φ(y k + d k ) m k (0) m k (d k ). (7) The numerator is the actual reduction in the objective function, and the denominator is the predicted reduction (the reduction in φ(y) predicted by m k ). The predicted reduction is always nonnegative, since d k is obtained by minimizing the model m k over a region that includes d = 0. Thus if ρ k is negative then φ(y k +d k ) φ(y k ) and the step d k must be rejected. If ρ k is close to 1, then there is good agreement between m k and φ over this step, and it is safe to expand the trust region for the next iteration. If ρ k is positive but significantly smaller than 1, then the trust region is not altered. Based on the Algorithm 4.1 in [], the main steps of the trust region algorithm for solving (13) are as follows: 1. Set k = 0, set maximum trust region size > 0, choose 0 (0, ) and η [0, 1 4 ).. Form and approximately solve the trust region subproblem (16) for the step d k. Compute the reduction ratio ρ k from (7). 3. If ρ k < 1 4, then set k+1 = 1 4 d k and go to step 5. Otherwise, go to step If ρ k > 3 4 and d k = k, then set k+1 = min( k, ). Otherwise, set k+1 = k. 5. If ρ k > η, then x k+1 = x k + d k. Otherwise, x k+1 = x k. Set k = k + 1 and return to step.

5 Note that the trust region radius is increased only if d k actually reaches the boundary of the trust region (i.e., d k = k ). If the step stays strictly inside the region, then one can infer that the value of k is not interfering with the progress of the algorithm, so one leaves its value unchanged for the next iteration. 4 SOME IMPLEMENTATION ISSUES 4.1 Non-Smoothness of the Complementarity Function In this paper, the nonlinear system of equations (10) is defined using the Fischer-Burmestein function (7). Since ψ(a, b) a = 1 (8) a a + b ψ(a, b) b = 1 b a + b (9) ψ(a, b) is differentiable everywhere but in (0, 0). Some approaches to deal with the non-smoothness of ψ(a, b) are discussed in the literature. One approach is based on Clarke s generalized Jacobian [13]: ψ(a, b) a ψ(a, b) b a 1 if a + b 0 = a + b 1 α otherwise b 1 if a + b 0 = a + b 1 β otherwise (30) (31) where α and β are arbitrary nonnegative constants with α + β = 1. Another approach is to use smoothing approximations, such as: ψ(a, b) = a + b a + b + µ (3) ψ(a, b) = a + b (a b) + 4µ (33) where µ > 0 is a smoothing parameter so that property (4) becomes ψ(a, b) = 0 ab = µ, a > µ, b > µ. (34) 4. Computing Gradients and Hessians To set up the trust region subproblem (13) one has to compute the gradient and Hessian of composite function φ(h(y)). The gradient is computed from φ(y k ) = h(y k )h(y k ) (35) where D s 0 D π D z 0 D υ 0 0 h(y) = I I 0 I I (36) g(x) T 0 0 I I g(x) H(y) D s, D π, D z and D υ are diagonal matrices given by and Dii s s i = 1, s i + πi (37) Dii π π i = 1, s i + πi (38) Dii z z i = 1, z i + υi (39) Dii υ υ i = 1, z i + υi (40) H(y k ) = f(x k ) + m λ k i g i (x k ). (41) i=1 The Hessian φ(y k ) is computed from φ(y k ) = h(y k ) h(y k ) T + h j (y k ) h j (y k ). n y j=1 (4) Computing φ(y k ) can be computationally expensive and requires efficient sparse matrix structures and linear algebra kernel, as it involves the evaluation and summation of a large number of functions Hessians h j (y k ) plus the product of the matrix h(y k ) to its transpose. The gradient h(y k ) is relatively simple and inexpensive to obtain [7], and using h(y k ) one can compute the first term h(y k ) h(y k ) T in φ(y k ) without computing any Hessian h j (y k ). As remarked in [], the first term h(y k ) h(y k ) T is more important than the second summation term in (4). The first term will be dominant when the norm of each second-order term (i.e., h j (y k ) h j (y k ) ) is significantly smaller than the eigenvalues of h(y k ) h(y k ) T. Such a behavior can be seen either when the residuals h j (y) are small or are nearly affine (so that the h j (y) are small). Note that in the application in this paper ( root finding problem!) the residuals h j (y) s must be zero at the solution. 5 NUMERICAL RESULTS The proposed trust region nonlinear complementarity OPF algorithm (TRNC) is applied here to the IEEE 30-, 57-, 118- and 300-bus test systems. Its computational performance is compared with that of a widely used primaldual interior-point (PDIP) algorithm [7]. Both algorithms are implemented in the MATLAB language. In all test runs PDIP algorithm uses the parameters: µ 0 = 0.01, γ = , σ = 0. and ɛ 1 = , while TRNC algorithm uses the parameters: 0 = 1, = 5, and η = 0.5. The OPF problem solved is the classical active power losses minimization. The constraints include the active and reactive power balance equations for all buses, and lower and upper bounds on voltage magnitudes, transformer tap ratios, shunt susceptances, generators reactive power, and selected branch flows. The corresponding

6 mathematical formulation of this OPF is as follows: min g ij (Vi + Vj V i V j cos θ ij ) (i,j) B s. t.: P i (θ, V, t) P Gi + P Di = 0, i N where Q i (θ, V, t) Q Gi + Q Di = 0, i G Q i (θ, V, t) Q Gi + Q Di = 0, i F Q i (θ, V, t) + Q Di + b sh i Vi = 0, i E F ij (θ, V, t) f ij = 0, {(i, j)} B V min i t min ij b min i V i V max i, i N t ij t max ij, {(i, j)} T b sh i b max i, i E Q min i Q Gi Q max i, i G f min ij P i (θ, V, t) = V i f ij f max ij, {(i, j)} B (43) j N i V j (G ij cos θ ij + B ij sin θ ij ) (44) Q i (θ, V, t) = V i j N i V j (G ij sin θ ij B ij cos θ ij ) (45) V i and θ i are the voltage magnitude and phase angle, with θ ij = θ i θ j ; P Gi and P Di are active power generation and demand; Q Gi and Q Di are reactive power generation and demand, and b sh i is the shunt susceptance, all at bus i; g ij is the conductance of a circuit connecting the bus i to bus j; Y ij = G ij + jb ij is the ij-th element of the bus admittance matrix; N is the index set of all system buses, G of generator buses, F of load buses with fixed shunt var control, E of load buses eligible for shunt var control, B of branches (lines and transformers), and T of transformers with LTCs. Table 1: Sizes of the power systems and the nonlinear OPF problem (1) System N G E B T n m p IEEE IEEE IEEE IEEE Table : Initial loading (active and reactive) and active power losses System MW Load MVAr Load MW Loss IEEE IEEE IEEE IEEE Table 1 shows the sizes of the index sets N, G, E, B, and T for the various test systems, as well as the number of primal variables n, number of equality constraints m, and number of simple bound constraints p of problem (1). Table displays the total system active and reactive power loads, and the transmission active power losses for the base condition, i.e., prior to the application of any optimization procedure. The performance of TRNC algorithm using different complementarity functions to reformulate the OPF as a nonlinear equations system is also studied. Three commonly used complementarity functions are tested: (7), (3) and (33). Complementarity function (7) is differentiable everywhere but in (0, 0), while complementarity functions (3) and (33) are differentiable everywhere but only provide µ-approximations to (a) and (b). When using (3) or (33) the parameter µ is set to Four initialization rules are considered for variables x: (i) as given by an initial power flow solution, (ii) the middle point of bounded variables (i.e., x 0 i = (x i + x i )/), (iii) a flat start (i.e., Vi 0 = 1 and θi 0 = 0), and (iv) random points within the limits. The slack variables s and z are initialized as the middle range, i.e., s 0 = z 0 = (x x)/. The Lagrange multipliers π i and υ i are all set to 0.1. The Lagrange multipliers λ i are either set to 1 if associated with an active power balance constraint or set to 0 if associated with a reactive power constraint. Table 3: Number of PDIP and TRNC iterations for convergence and minimum active losses System Number of Iterations Minimum Losses PDIP TRNC S / F (MW) (Red. %) IEEE / IEEE / IEEE / IEEE / Table 3 displays the number of PDIP and TRNC iterations using initialization rule (i) (a power flow solution), the number of successful and failed TRNC iterations (column S/F), and the minimum active losses. In these simulations TRNC algorithm uses complementarity function (4). Both algorithms converged with all test systems. The numbers of TRNC iterations are slightly higher than the number of PDIP iterations. The computational cost per iteration of TRNC algorithm is only slightly higher than the cost of a PDIP iteration. In each iteration the two algorithms solve a large sparse linear system of same size, but the system matrix in TRNC algorithm is less sparse than the matrix in PDIP algorithm. The column S/F shows the number of successful and failed TRNC iterations. For instance, for IEEE 300 bus system 15 out of 19 iteration were successful. This means that in 4 iterations the computed step d k was discarded and the trust region radius reduced. In the failed iterations the nonlinear objective function increased while the objective in the approximation model was slightly reduced. Thus, the reduction ratio ρ k for that iterations were negative indicating unsuccessful trial steps. Table 4: Number of TRNC and PDIP iterations using three different initial points: initial power flow solution, middle point and flat start System Power Flow Middle Point Flat Start TRNC PDIP TRNC PDIP TRNC PDIP IEEE (4) 10 6 (0) 11 6 (0) 11 IEEE (4) 10 9 (3) 1 9 (3) 1 IEEE (3) 1 7 (0) 13 7 (0) 14 IEEE (4) 14 1 (4) 17 (5) 17

7 Table 4 displays the number of TRNC and PDIP iterations using three different initialization rules: (i) an initial power flow solution, (ii) the middle point within limits, and (iii) a flat start. The two algorithms were able to optimize all test systems. Except for IEEE 300 bus system, when using the middle point and the flat start initializations the performance of TRNC algorithm is better than that of PDIP algorithm. When the initialization is an initial power flow solution TRNC algorithm presents an average of 4 unsuccessful steps per test system, thus requiring a few more iterations to converge than PDIP algorithm. Table 5: Performance of TRNC and PDIP using 50 random starting points System TRNC Iterations PDIP Iterations Total Avrg Min Max Total Avrg Min Max IEEE IEEE IEEE IEEE Table 5 summarizes the performance of the algorithms when the initial points are random points within the limits. The random points are generated using the MATLAB command: x = xmin + rand(n,1).* (xmax - xmin) A set of 50 random points is generated for each system. The same random points are used by TRNC and PDIP algorithms. The column Total shows the total number of TRNC or PDIP iterations to solve the whole set of 50 cases. The column Avrg is the average number of iterations, column Min is the lowest number of iterations observed for that test set, and column Max is the largest number of iterations taken by a single run. The two algorithms were able to converge with the 50 random points for IEEE 30, 57 and 118 bus systems, TRNC algorithm converged with the 50 random points for IEEE 300 bus system, and PDIP algorithm failed to converge with 6 out of 50 random points for IEEE 300 bus system. Overall, the total number of iterations taken by TRNC algorithm is lower than the total number of iterations taken by PDIP algorithm. The total of 90 iterations of PDIP algorithm applied to IEEE 300 bus system does not include the number of iterations in the 6 unsuccessful cases. While the average number of iterations is quite close in the two algorithms, the minimum number of iterations is observed in all instances with TRNC algorithm. For instance, 10 versus 17 iterations for IEEE 118 bus system, and 11 versus 18 iterations for IEEE 300 bus system. Thus if one is able to reduce the number of failed steps in TRNC algorithm to a minimum the computational cost can be significantly reduced. In the tests performed the unsuccessful TRNC iterations are due to increases in the original objective, and an interesting approach to handle this issue is presented in [14], combining trust region with line search iterations. Table 6: Performance of TRNC using three different complementarity functions System Function (4) Function (3) Function (33) Middle Flat Middle Flat Middle Flat IEEE IEEE IEEE IEEE The performance of TRNC algorithm using different complementarity functions to reformulate the OPF as a nonlinear equations system is also studied. The results are presented in Table 6, and each function is tested with two initial points, the middle point and the flat start ones. The performances of the complementarity functions (7) and (3) are similar and better than the performance of complementarity function (33). Although complementarity function (7) is not differentiable in the origin, it never required using Clark s generalized Jacobian as described in Section 4.1. Overall, the three complementarity functions studied performed successfully, but the choice of an appropriate complementarity function to reformulate the OPF problem requires further investigation. 6 CONCLUSIONS The paper has presented the mathematical development of a globally convergent OPF algorithm combining three major ideas: (i) using complementarity functions to transform the OPF problem into a system of nonlinear equations, (ii) transforming the nonlinear equations system into an unconstrained minimization problem, and (iii) using trust region methods for unconstrained minimization to globalize the solution convergence. The motivation is that one of the main difficulties to apply a trust region method to OPF problems is handling inequality constraints. Thus this paper uses complementarity functions to handle inequality constraints. Three complementarity functions have been implemented and tested. The performance of the proposed trust-region nonlinear complementarity algorithm is very promising. Globally convergent algorithms are inherently time consuming, but the proposed algorithm shows potential to be close to interior-point algorithms in terms of speed. The numerical results presented here are preliminary, and the performance of the algorithm can be improved in several fronts as, e.g., better choice of parameters and more efficient solutions of the trust region subproblems. ACKNOWLEDGMENTS The author gratefully acknowledges the financial support from CNPq, a Brazilian research funding agency. This work was partially developed at the EMSOL Lab, University of Waterloo, Canada. The author gratefully thanks Dr. Claudio Cañizares for providing the research conditions at EMSOL.

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