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Sba: Controle & Automação Sociedade Brasileira de Automatica
Print version ISSN 01031759
Sba Controle & Automação vol.20 no.2 Natal Apr./June 2009
http://dx.doi.org/10.1590/S010317592009000200008
SISTEMAS DE POTÊNCIA
Security constrained optimal active power flow via network model and interior point method
Anibal T. de Azevedo^{I}; Carlos A. Castro^{II}; Aurelio R.L. Oliveira^{III}; Secundino Soares^{II}
^{I}Faculdade de Engenharia de Guaratinguetá Universidade Estadual de São Paulo Av. Dr. Ariberto Pereira da Cunha, 333, CEP 12516410, Guaratinguetá, SP, Brasil. anibal@feg.unesp.br
^{II}Faculdade de Engenharia Elétrica Universidade Estadual de Campinas Rua Albert Einstein, 400, CEP 13083852, Campinas, SP, Brasil. ccastro@ieee.org, aurelio@ime.unicamp.br
^{III}Instituto de Matemática, Estatística e Computação Científica Universidade Estadual de Campinas Rua Sérgio Buarque de Holanda, 651, CEP 13083859, Campinas, SP, Brasil. dino@cose.fee.unicamp.br
ABSTRACT
This paper presents a new formulation for the security constrained optimal active power flow problem which enables the representation of three basic constraints: branch outage, generator outage and multiple equipment congestion. It consists of a network model with additional linear equality and inequality constraints and quadratic separable objective function, which is efficiently solved by a predictorcorrector interior point method. Sparsity techniques are used to exploit the matricial structure of the problem.Case studies with a 3,535bus and a 4,238branch Brazilian power system are presented and discussed, to demonstrate that the proposed model can be efficiently solved by an interior point method, providing security constrained solutions in a reasonable time.
Keywords: Security, active power dispatch, optimal power flow, network model, power flow controls, interior point method
RESUMO
Este trabalho apresenta uma nova formulação do problema de fluxo de potência ótimo corrente contínua com restrições de segurança de três tipos: perda de ramo, perda de gerador e múltiplas perdas. A formulação emprega um modelo de fluxo em redes com restrições adicionais de igualdade e desigualdade e função objetivo quadrática separável que é eficientemente resolvido por um método de pontos interiores preditorcorretor. Técnicas de esparsidade são utilizadas para explorar a estrutura matricial do problema. Estudos de caso para o Sistema Interligado Nacional com 3535 barras e 4238 ramos são apresentados e discutidos, a fim de demonstrar que o modelo proposto pode ser resolvido através de um método de pontos interiores de maneira eficiente, fornecendo soluções que respeitam as restrições de segurança em um tempo computacional razoável.
Palavraschave: Segurança, despacho de potência ativa, fluxo de potência ótimo, fluxo em redes, controle de fluxo de potência, método de pontos interiores
1 INTRODUCTION
The optimal power flow (OPF) problem consists of obtaining the optimal settings for control variables in a power system so that certain operational goals can be achieved; these are represented by a predefined objective function f, which is subject to a set of constraints. The operating state of a power system provided by an OPF is one that guarantees affordability, reliability, security, and dependability (Momoh, 2001). Generally, the OPF problem can be expressed as
where x is the vector of dependent variables (bus voltage magnitudes and phase angles), u is a vector of control variables (as active power generation and active power flow), g (x, u) is the set of nonlinear equality constraints (power flow equations), and h (x, u) is the set of inequality constraints of the vector arguments x and u.
Minimization of active power losses, generation cost and reactive power generation of the system are possible objective functions. They may be achieved by setting control variables u, such as dispatching generating units, adjusting bus voltages and setting transformer taps. This set of constraints involves those conditions necessary to guarantee Kirchhoff's laws, bus voltage ranges, and the rated limits of equipment. The problem is usually solved for a normal operating condition of a power system (n  0 case).
A security constrained optimal power flow (SCOPF) is a special type of OPF where the optimum value of the objective function is computed while respecting the constraints, both under normal operating conditions and for specified disturbances, such as outages or equipment failures. These security constraints allow the OPF to determine the operation of the power system in a defensive manner (Wood e Wollenberg, 1996); i.e., the OPF will force the system to be operated in such a way that if a contingency is encountered, the resulting voltages and power flows will still be within the limits established. As for outages, SCOPF usually considers only single outages (n  1 case), although in some cases certain critical double outages (n  2 case) could be evolved.
As stated in (Biskas e Bakirtzis, 2004), a complete security analysis implies a specific constraint for each branch and unit outage for each monitored branch. The number of security constraints would thus be m × (m  1) for branches and n × (n  1) for generators (see Sec. for notation). In order to constitute each constraint the computation of at least one load flow is necessary. Two mains approaches to contingency selection (Stott et al., 1987) are available: direct or indirect methods.
Examples of direct methods are those that use contingency filters (Harsan et al., 1997) and distribution factors (Wai, 1981). The Inverse Matrix Modification Lemma (IMML) is used either explicitly or implicitly for most contingency analysis studies. Specific versions of these approaches have been denominated compensation methods (Alsac et al., 1983). Indirect methods involve the consideration of contingency quantities without explicitly computing them.
Various optimization methods have been used to solve the OPF problem, including linear programming, nonlinear programming and integer programming (Dommel e Tinney, 1968; Happ, 1977; Huneault e Galiana, 1991; Momoh, 2001). Nonlinear programing, such as that used in this paper, can exploit a number of techniques, such as sequential quadratic programming, augmented Lagrangian methods, generalized reduced gradient methods, projected augmented Lagrangian functions, successive linear programming, and interior point methods. The choice of interior point methods (IPMs) was based on the robustness and efficiency reported for their use in OPF problems (Granville, 1994; Wu et al., 1994; Wei, 1996; Torres e Quintana, 1998; Quintana et al., 2000; Yan et al., 2006), specially those involving SCOPF problems (Lu e Unum, 1993;Vargas et al., 1993;Yan e Quintana, 1997; Jabr, 2002).
The main contribution of the present study is to develop an efficient model for a security constrained optimal active power flow (SCOAPF) that considers three types of contingency situations: branch outage, generator outage and multiple equipment congestion. The solution is obtained by using an IPM formulated as a network model with additional linear constraints and also considers a general quadratic separable objective function that may:

Minimize the deviation from a specified generation obtained from a market pool or a dispatch model that does not consider transmission network constraints.

Minimize the transmission losses.

Realize both.
The outline of the paper is as follows. Nomenclature is presented in Sec. . The SCOAPF model is presented in detail in Sec. . An efficient interior point method for its solution is described in Sec. . In Sec. some numerical results for the Brazilian power system are reported and commented on, and in Sec. the conclusions are stated.
2 NOTATION
m number of branches.
n number of buses.
l number of independent circuit loops.
g number of generators.
c number of contingencies.
A network incidence matrix (n × m).
L network loop matrix (l × m).
X reactance diagonal matrix (m × m).
R resistance diagonal matrix (m × m).
E matrix (n × g) formed by elements E_{ij} that are equal to one if generator j is connected to bus i, otherwise it is zero.
N active power flow contingency matrix (c × m).
M active power generation contingency matrix (c × g).
p active power generation vector (g × 1).
d active power load vector (n × 1).
f active power flow vector (m × 1).
θ bus phase angle vector (n × 1).
f_{min }lower bound for active power flow f.
f_{max }upper bound for active power flow f.
p_{min }lower bound for active power generation p.
p_{max }upper bound for active power generation p
α, β weights.
ø_{1 } function associated with power flow vector.
ø_{2 } function associated with power generation vector.
* symbol for fixed or target value.
3 PROBLEM FORMULATION
The SCOAPF model is formulated as the following network model with additional linear equality and inequality constraints and quadratic separable objective function.
subject to
The objective function (2) corresponds to the association of two different criteria, the first depending on power flow, ø_{1}(f), and the second on power generation, ø_{2}(p). Both criteria are represented by quadratic and separable functions, and can be combined using scalar weights α and β within a simple biobjective optimization framework.
ø_{1}(f) is a quadratic separable function expressed by
where M_{1}, m_{2}, m_{3} are a diagonal matrix, a vector and a scalar, respectively. By setting M_{1} = R, m_{2} = 0 and m_{3} = 0, function ø_{1}(f) represents an approximation of the transmission power losses.
ø_{2}(p) is a quadratic separable function expressed by
where N_{1}, n_{2} and n_{3} are a diagonal matrix, a vector and a scalar, respectively. By setting adequate values for N_{1}, n_{2} and n_{3}, function ø_{2}(p) will represent quadratic generation costs.
A quite useful objective function ø_{2}(p) is the quadratic deviation from a desirable generation dispatch. Such a dispatch can arise from a pool auction in an electricity market or from a dispatch model which does not take into consideration transmission network constraints. In such a case ø_{2}(p) can be represented by
where W is a diagonal matrix with the component w_{i} as the penalty term associated with deviation from the desired generation . The equivalence between Eqs. (9) and (10) shows that N_{1} = W, n_{2} = p^{*t} W and n_{3} = p^{*t} W p^{*}.
Eq. (3) corresponds to the nodal balance according to the Kirchhoff's Current Law (KCL), while Eq. (4) represents the independent circuit loop equations, in accordance with Kirchhoff's Voltage Law (KVL). Efficient procedures for finding the loop matrix L from the incidence matrix A are discussed in (Oliveira et al., 2003; Expósito et al., 2006).
Eq. (5) represents a set of generic relationships between pieces of equipment in the power grid. This equation corresponds to the constraints applied to three basic situations:

Branch outage: the most important branch outages have been selected, and for these, the line outage distribution factors (LODFs) are computed to formulate postcontingency constraints in the form of f_{min} < N f < f_{max}. For instance, for the outage of branch km, an overload on branch ij is avoided by including a constraint of the type (S_{1} < P_{km} + α P_{ij} < S_{2}).

Generator outage: for certain selected generator outages, the generalized generation distribution factors (GGDFs) are computed, leading to postcontingency constraints having the form of f_{min} < N f + M p < f_{max}. For instance, for the outage of generator k, an overload on branch ij is avoided by including a constraint of the type (S_{3} < P_{k} + β P_{ij} < S_{4}).

Multiple equipment congestion: this involves limits on the interchange between areas and bottlenecks already identified by experience of the grid operator. These constraints have the form of f_{min} < N f + M p < f_{max}. For instance, the power flow between two areas can be limited by adding the constraint (S_{5} < P_{k} + γ P_{ij} + η P_{lm} < S_{6}).
The first two sets of constraints can be seen as to represent security constraints generated after the analysis of contingency cases, as discussed in (Stott, 1974; Stott e Hobson, 1978; Stott e Marinho, 1979). The automatic generation of sets of security constraints is not the main focus of this paper and the theme will not be discussed further. Although the definition and how to calculate LODFS and GGDFS using contingency cases can be seen in (Sauer, 1981) and (Ng, 1981), respectively. The contribution of this work is the use of LODFS and GGDFS to form a set that considers important contingency constraints and then efficiently solve the resulting problem. Then, the focus is how to efficiently find an OPF solution for which the three situations mentioned above can be handled by the inclusion of Eq. (5).
Eqs. (6) and (7) represent the bounds for active power flow and generation, respectively. Note that transmission limits are imposed directly on the power flow variables, which constitutes one of the main advantages of approaches based on a network model.
The SCOAPF model (2)(7) corresponds to a DC model where transmission losses are not considered in the active power balance equations. In order to compute more realistic solutions, the following procedure P1 can be adopted for the computation of transmission loss:

Solve the SCOAPF model for the original load vector d. Let (p^{0}, f^{0}) be the optimal solution.

For the solution (p^{0}, f^{0}) calculate the power loss for each branch using the equation (x_{km}f_{km})^{2}.

Compute a new load vector by including the branch power loss as an incremental load equally distributed between the terminal buses k and m.

Solve the SCOAPF model for the new load vector . Let (p^{1}, f^{1}) be the optimal solution.

Verify if the relative difference between and is less then a specified tolerance. If so, the procedure is finished. Otherwise recalculate the branch power loss for the new solution (p^{1}, f^{1}), and repeat the procedure.
In general, this procedure requires only two iterations of OAPF or SCOAPF to achieve convergence within a tolerance of 10^{2}.
4 SOLUTION TECHNIQUE
For the sake of simplicity, assume that the lower bounds in Eqs. (6) and (7) are all zero and that α = β = 1 in Eq. (2). The dual problem for the security constrained optimal active power flow model (2)(7) is given by
where z_{f}, z_{p}, and z_{s} are slack variables, B = ,
= and = , with 0 being a (l × g) zero matrix.
The optimality conditions for the primal and dual problems are given by primal and dual feasibility and complementarity conditions
where s_{p}, s_{f}, and s_{s} are slack variables for the bound constraints on active power generation, active power flow and security constraints, respectively. Moreover, the notation F = diag(f) for diagonal matrices formed by vectors is introduced.
4.1 PrimalDual Interior Point Methods
Most primaldual interior point methods can be seen as variants of the application of Newton's method to the first order optimality conditions. The following outlines a framework for such methods, where x = (f, p, s_{f}, s_{p}) and t = (z_{f}, z_{p}, w_{f}, w_{p}) are used.
Assume y^{0} and (x^{0}, t^{0}) > 0. For k = 0,1,2,..., do
1. Choose σ^{k} ∈ [0,1) and set µ^{k} = σ^{k} (γ^{k} /n), where n is the dimension of x and γ^{k} = (x^{k})'t^{k}.
2. Compute Newton search directions (Δx^{k}, Δy^{k}, Δt^{k}).
3. Choose an appropriate step size so that the point remains interior: α^{k} = min(1, τ^{k} , τ^{k} ), where τ^{k} ∈ (0, 1), = (1/min_{i}(Δ)), and = (1/min_{i}(Δ)).
4. Compute (x^{k}^{+1}, y^{k}^{+1}, t^{k}^{+1} ) = (x^{k}, y^{k}, t^{k}) + a^{k} (Δx^{k}, Δy^{k}, Δt^{k}).
The step size for both primal and dual variables is the same, since for quadratic problems, primal variables appear in the dual problem constraint set. Parameters σ and τ and the starting point will be discussed later. Newton search directions are defined by the following linear system^{1}.
where e is the column vector consisting exclusively of ones, r_{y} ≡ m_{2} B^{t}y  z_{f} + w_{f} + M_{1}f  N^{t} y_{2}, r_{g} ≡ n_{2} + y(p)  z_{p} + w_{p} + N_{1}p M^{t} y_{2}, r_{y}_{2} = y_{2} + w_{s}  z_{s}, and r_{ss} ≡ Nf  Mp + s + s.
4.2 The PredictorCorrector Method
For the predictorcorrector (PC) approach (Mehrotra, 1992), two linear systems must be solved. First, affine directions (Δ, Δ, Δ) are computed by solving Eqs.(11) and (12) for µ = 0. The search directions are then given by solving Eq. (11) and
4.3 Implementation Issues
Parameters τ = 0.9995 and σ = are fixed. For the predictorcorrector approach, the barrier parameter is given by µ = (/γ)^{2}(/n^{2}), where = (x + Δ)'(t + Δ). In both versions, however, if γ < 1 then µ = (γ/n)^{2}. The following starting point is suggested: y^{0} = 0, f^{0} = = f_{max}/2, p^{0} = = p_{max}/2, = (R + I)e, = e, = Ie, su = s_{max } s_{min}, s^{0} = 2su/3, = su/3.
5 LINEAR SYSTEM SOLUTION
Since the two linear systems introduced in Sec. 4.2 share the same matrix, the following discussion will consider only the system involving Eqs. (11) and (12). The dimension of this linear system can be reduced by substitutions involving various sets of variables without changing the sparse pattern of the matrix. First, slack variables are eliminated:
reducing Eq. (11) to
where D_{f} = F^{1}Z_{f} + W_{f} + M_{1}, D_{p} = P^{1}Z_{p} + W_{p} + N_{1}, D_{s} = S^{1}Z_{s} + W_{s}, r_{a} = r_{y}  F^{1}r_{zf }+ (r_{wf}  W_{f}r_{f}), r_{b} = r_{g}  P^{1}r_{zp} + (r_{wp} W_{p}r_{p}), and r_{sy}_{2} = r_{y}_{2} + (r_{ws}  W_{s}r_{s})  S^{1}r_{zs}. Note that only the inverse of diagonal matrices are involved. Now the active power generation and transmission variables in (13) can be eliminated with Δf =  (r_{a}  B^{t} Δy  N^{t} Δy_{2}), Δp =  (r_{b} + ^{t} Δy  M^{t} Δy_{2}), and Δs = (r_{sy}_{2} + Δy_{2}), resulting in
where = and
In order to solve Eq. (14), it is necessary to solve a system with dimension constituted exclusively by the number of security constraints, which in practice is much smaller than that of branches or even buses. The most intensive computational work is involved in solving Eq. (15).
6 NUMERICAL RESULTS
The proposed SCOAPF model was implemented in Matlab 7.0, running on an Intel Pentium 2.0 GHz personal computer with 2 GB of RAM in a Windows XP Professional environment. The predictorcorrector IPM approach was tested for the Brazilian Power System (BPS), a predominantly hydro system (90%) with 3,535 buses, 4,238 branches, 300 generators and 157 security constraints involving all three of the types described in Sec. 3. Three load levels (light, medium, and heavy) involving slightly different configurations were considered, as shown in Table 1. Table 2 describe the number of contingencies of each type (Branch, Generator or Multiple). All data were provided by the Brazilian Independent System Operator (ISO).
Two case studies with different objective functions were considered:

CS_{1}: Minimization of transmission power losses, effected by setting α = 1, β = 0 and ø_{1}(f) = f^{t} R f.

CS_{2}: Minimization of quadratic deviation from a predefined generation dispatch p^{*}. This involves setting α = 0, β = 1 and ø_{2}(p) = (p  p^{*})^{t} I (p  p^{*}), where I is the identity matrix. Generation dispatch p^{*} corresponds, in this case, to the economic dispatch that minimizes thermal fuel cost in the BPS and is calculated by the ISO without considering transmission network constraints.
6.1 OAPF model
An initial set of results is presented in which the proposed model is evaluated without consideration of security constraints, but considering transmission losses by the procedure P1 described in Section 4 which demands the use of only two successive OAPFs. Table 3 summarizes the performance of the OAPF model in three load scenarios by exposing the total time spent and the number of iterations for each OAPF.
The proposed predictorcorrector IPM approach presented an effective performance in terms of number of iterations and CPU time, that decreases slightly with the increase in load. Table 4 shows the power generation for each case study and load level.
As expected, the solution for CS_{1} provides lower transmission losses. The transmission power losses for light, medium and heavy loads were reduced from 3.7%, 4.6% and 5.5% for CS_{2} to 1.0%, 1.6% and 0.98% for CS_{1}, respectively.
Table 5 shows the number of security constraints violated by the optimal solutions of both case studies. It is interesting to observe that a smaller number of violations occurred for all load levels in CS_{1}. This can be explained by the network topology of the BPS, as shown in Figure 1. Most of the load is located along the east coast (Atlantic ocean), and the most important hydro plants are located in rural areas, such as the Itaipu and Tucuruí hydro plants. When economic dispatch assigns more generation to these distant hydro plants, (CS_{2}), the long transmission lines that bring this generation to the load centers operate closer to their capacities and more security constraints become active. When dispatch is designed to minimize transmission loss, (CS_{1}), more generation is assigned to conventional and nuclear thermal plants which are closer to the load centers, such as the Angra, Fluminense and Fortaleza thermal plants, thus reducing the power flow along the long transmission lines, resulting in less active security constraints. These shifts in generation are presented in Table 6 related to the generating units presented in Figure 1. These two case studies illustrate the tradeoff in the BPS between optimal dispatch from an "economic" point of view (which maximizes hydro generation and minimizes thermal fuel costs) and optimal dispatch from an "electric" point of view (which minimizes transmission loss).
Table 7 shows the values for the seven largest of the 22 security constraints violated in case study CS_{2} (heavy load).
Constraint 79 (Region 5 in Figure 1) is violated in Heavy level; it consists of maintaining the transformer shown in Fig. 2 in secure operation. This constraint is responsible for preventing the contingency that one of the three lines will cause a fault in the transformer that connects the 500 kV area and 230 kV area. This situation corresponds to the branch outage case mentioned in Section 3; it can be stated mathematically by the following expression 3100 < 1.00 (f_{23592} + f_{23593} + f_{23594} ) + 1.00 f_{3965230} + 1.00 f_{235230} < 3100 . (matricial form s_{min} < Nf < s_{max}). Constraints 135 and 155 represent similar situations.
Constraint 84 (Region 3 in Figure 1) is necessary to prevent a power flow inversion from the 500 kV area to the 440 kV area, as illustrated in Fig. 3. This situation corresponds to the multiple equipment congestion case mentioned in Section 3; it can be stated mathematically as 8050 < f_{535536} + p_{500} + p_{501} + p_{502} + p_{503} + p_{507} + p_{510} + p_{513} + p_{520} < 8050. (matricial form s_{min} < Nf + Mp < s_{max}). Similar situations are represented by constraints 111 and 157.
Constraint 86 (Region 3 in Figure 1) consists of the prevention of congestion by limiting the maximum power output of the group of generators shown in Fig. 3. This constraint can be treated as a special case of multiple equipment congestion; it can be stated mathematically as 5550 < p_{501} + p_{502} + p_{503} + p_{510} + p_{513} + p_{520} < 5550. (matricial form s_{min} < Mp < s_{max}).
6.2 SCOAPF model
A second set of numerical results is now presented to evaluate the proposed model when security constraints are considered. Table 8 summarizes the performance of the proposed approach, which corresponds in this case to the SCOAPF model, for a heavy load. The inclusion of security constraints influenced the performance of the model in relation to the situation in which these constraints were not considered (Table 3).
The difference in performance between the OAPF and SCOAPF models can be explained as follows: when no security constraints are considered, Eq. (14) is no longer necessary, and an IPM is produced in which each iteration is computationally less costly. Moreover, Matrix D_{y} presents 0.2320% of nonzero elements without security constraints, whereas the same matrix with security constraints has 0.3122% of nonzero elements (an increase of 34.57% in the number of nonzero elements). Furthermore, the inclusion of constraints results in an increase in the number of iterations.
Since only a few of the security constraints are active in the optimal solution, one alternative to reduce CPU time would be to adopt a scheme similar to that found in (Stott e Hobson, 1978), in which only the security constraints actually violated, identified after running the OAPF, are included in the SCOAPF model. Note that these constraints are here simultaneously included in the SCOAPF, although in (Stott e Hobson, 1978) they were included one at a time. The application of this procedure for CS_{2} and heavy load, where only 22 of the 157 security constraints are violated, reduced the CPU time of about 30%.
7 CONCLUSION
This paper has presented a new security constrained optimal active power flow formulation which enables the representation of three basic constraints: branch outage, generator outage and multiple equipment congestion. Quadratic separable objective functions of active power generation and flows can be considered, thus allowing the minimization of transmission losses, generation cost or quadratic deviation from a desired dispatch, or a combination of both. The model was formulated as a network flow problem with additional linear constraints and enables the efficient solution using an interior point method exploiting the specific matricial structure by sparsity techniques. The model was tested on the Brazilian power system with 3,535 buses, 4,238 branches, and 157 security constraints. The results showed that the technique is flexible, robust and efficient.
ACKNOWLEDGMENT
This research was supported in part by FAPESP, CNPq, CAPES, and FINEP, Brazilian agencies for research support.
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Artigo submetido em 04/06/2008 (Id.: 00877)
Revisado em 10/10/2008, 28/01/2009
Aceito sob recomendação do Editor Associado Prof. Eduardo N. Asada
1 From this point on, superscript k will be omitted for a cleaner notation.