7《大系统优化》最优潮流

大系统优化理论在电力系统中的应用严正电信群楼1-347室上海交通大学电气工程系闵行区东川路800号上海200240Applications of Large Scale System Optimization Theory to PowerSystems7 Optimal Power Flow7.1 IntroductionIn economic dispatch, we mainly consider the economic generation output， in which the economic efficiency of system operation is emphasized. Three main aspects of constraints, relating to system security, are ignored in classical economic dispatch formulation:1）Network constraints of the system;Applications of Optimization Theory to Power Systems 2）The upper and lower limit constraints of bus voltage magnitudes in the system;3）Reactive power output constraints of generators.Since the solution of economic dispatch does not consider these constraints, when carrying out the result of economic dispatch, these constraints will possibly be violated, leading to an insecure operation state. To conquer this shortcoming, people put forward the economic dispatch considering branch security constraints, to make up theconstraints unconsidered. Aiming at the secure constraints ofvoltages, people put forward the reactive power optimizationmodel. When the effect of active and reactive power isconsidered together, the optimal power flow model isestablished. OPF can be seemed as an extension of powerflow and is an extremely important and valuable tool in poweroptimal operation of power systems and power markets.Applications of Optimization Theory to Power Systems 7.2 The mathematical model of Optimal Power Flow—OPFThe general mathematical model of optimal power flow can be expressed as:min f (u, y)s.t. h (u, y)=0g (u, y)≤0u—control variables, such as ,,P V t(transformer tap) etc;G GQ Vθ etc;y—state variables, such as ,,,G DApplications of Optimization Theory to Power Systemsh —power flow balance equations;g —inequality constraints, such as line power flow power, voltages, and reactive power output.1）Economic dispatch (ED)1min ()G ni Gi i f c P = =∑s.t. 1G nGi loss D i P P P ==+∑minmaxi=1,,Gi Gi Gi G P P P n ≤≤"In economic dispatch models, we do not consider theApplications of Optimization Theory to Power Systemseffect of system security and reactive power/voltages and there is only one equality constraint.2）Security constrained economic dispatch1min ()G ni Gi i f c P = =∑s.t. (,)0G P P Δ=θmaxij ij P P ij L ≤∈minmaxi=1,,Gi Gi Gi G P P P n ≤≤"Where, active power flow equations reflect the systemApplications of Optimization Theory to Power Systems network structure. ij P represents the limit of branch capacity. L is the branch set of this network. The problem above has N equations and L newly-added inequalities. But the system reactive powers are neglected and bus voltages are supposed to keep fixed.3）Reactive power optimizationmin loss Ps.t. (,)0P V Δ=θApplications of Optimization Theory to Power Systemsminmax min max (,)0GG G Q V Q Q Q V V VΔ=≤≤≤≤θ In this model, the active power of generators is supposedto be given, so the objective function is not the generator cost but the output of the generator in slack bus (the only unit whose output can be adjusted), equivalent to the transmission loss. Power flow equations are equality constraints, while line flow constraints do not appear because the active power is not adjustable.Applications of Optimization Theory to Power Systems4） Optimal power flow()()1max max max min max min max min max minmax min =()s.t. ,0,0,, Gn i Gi i ij ij ij ij ij ij G G G G G G f c P P V Q V P PI I S S PP PQQ Q t t t V V V =Δ=Δ=≤≤≤≤≤≤≤≤≤≤≤∑θθApplications of Optimization Theory to Power SystemsThe economic dispatch model and reactive poweroptimization model above are both special forms of the optimal power flow model.5）Security-Constrained OPFHow to consider the security in OPF, such as N-1 or N-2security?Suppose the current base case of the system is expressedby 000000(,)0(,)0h u y g u y =⎧⎨≤⎩, and the severe contingency list is k=1,2,…,k, which are severe expected contingencies screenedApplications of Optimization Theory to Power Systems by security analysis.Similarly, the contingency case corresponding to thek-th expected contingency is(,)0(,)0k k kk k kh u yg u y=⎧⎨≤⎩, where ,k kh gare different from00,h g because N-1 or N-2 expected contingency will result in the change of systematical structure.So we can list the model of SCOPF as follows:Applications of Optimization Theory to Power Systems0000000max0min ().. (,)0(,)0(,)0(,)01,2,,k k k k k k k k f u s t h u y g u y h u y g u y u u u k k=≤=≤−≤Δ=" where there are four key points:1） The objective function is only relevant to the base case,and has nothing to do with k u of the expectedcontingency. This is because what we consider here is theApplications of Optimization Theory to Power Systemsoptimal operation state of current base case.2） The constraint max 0k k u u u −≤Δ connects controlvariables of expected contingencies with that of the basecase. It is called coupled constraint.3） If the control variable demand is kept the same for thebase case and the contingency k , then max 0k uΔ=. The dispatch here is called pre-dispatch.4） If max0k u Δ≠, the dispatch is called corrective dispatch,i.e., the control variables can be conditionallyApplications of Optimization Theory to Power Systemsre-dispatched after contingency k .5） If maxk u Δ=∞, the corrective dispatch is totallydecoupled from the base case, i.e., the control variablescan be freely re-dispatched after contingency k .7.3 Algorithms commonly used in optimal power flow computationThe major challenge of solving an OPF problem is that theproblem size is huge — a great deal of variables andApplications of Optimization Theory to Power Systems constraints. Most of the work is around identifying which inequality constraints will become equality constraints, which are not known before optimal solution is reached.Until now, most reliable and successful methods are successive quadratic programming (SQP) method and Newton method. Technically, all these methods require flexible utilization of Lagrangian multipliers, penalty functions (exterior or interior) and check of Kuhn-tucker conditions. The general idea is to form an Augmented Lagrangianobjective function by Lagrangian multipliers, and inequalityconstraints are included into the objective function by penaltyfunctions. The most successful penalty methods are thequadratic penalty function (exterior point method) andlogarithmic barrier function (interior function method). Wewill introduce these techniques in another Chapter.Applications of Optimization Theory to Power Systems 7.4 Modern interior point method based optimal power flow7.4.1 IntroductionIn 1949, Dantzig brought forward simplex method for linear programming problems. Since then, this method and its variants have been widely used in optimization problems. In practice it is an efficient computation method. But there exist some shortcomings in this method, such as:(1) The iteration number increases quickly when constraintApplications of Optimization Theory to Power Systems and variable number of problems increases. In the worst case, iteration number of simplex method will go up in complexity of exponential order, resulting in slow convergence.(2) Simplex method stops at the optimal primal and dual basis. In degenerate condition, it needs large number of basis exchanges to reach the optimal solution,.In 1979, Leonid Khachian, a Soviet Union scholar, brought forward the first polynomial-time algorithm — ellipsoid method, which is a breakthrough in optimizationApplications of Optimization Theory to Power Systems theory. But due to the restriction of limited accuracy of calculations, the application of this method is not optimistic, for realistic problems, not as successful as simplex method.In 1984, Karmarkar put forward a new algorithm for linear programming—interior point method. This method is also a polynomial time algorithm, showing great potential in practical computation. Different from simplex method, which seeks the optimal solution along the boundary of feasible region, interior point method is based on the structure of theApplications of Optimization Theory to Power Systems simplex. It starts from the initial interior point, goes along the steepest descent direction, and walks up to the optimal solution directly from the inside of the feasible region. Because it seeks the optimal solution within feasible region, for large-scale linear programming problems, when constraints and variable number increase, iteration number of this method will not change significantly. Moreover, the convergence and computation speed are better than that of simplex method.Applications of Optimization Theory to Power Systems In 1985, P. Gill and his colleagues proved that interior point method and classical interior penalty function method (or barrier penalty function method) are equivalent in form. Since classical barrier method can be applied to nonlinear programming problems, interior point method can also be used in solving other optimization problems, such as quadratic or nonlinear programming, etc. Thus optimization problems with continuous variables can be expressed in a unified form, and solved by the same algorithm.Applications of Optimization Theory to Power Systems 7.4.2 Classification of modern interior point methodsFor many years, people have carried out much research on Karmarkar’s algorithm, and many variants have been devised. As to its practical application, it is classified as the following three kinds:1. Projective scaling method, i.e., Karmarkar’s originalmethod. This method is built on a constructed canonical form of linear programming. Namely, it requires that problems should have special simplex structure and theApplications of Optimization Theory to Power Systems optimal objective value is zero. In calculation process, practical problems need to be transformed to this canonical form through complicated transformation.Therefore, the application field of projective scaling method is quite limited.2. Affine-scaling method. This method is technicallymature and has been widely applied. What is used more at present is primal affine-scaling method and dual affine-scaling method. But their polynomial timeApplications of Optimization Theory to Power Systems complexity has not been proved in theory.3. Path-following method, also named centertrajectory-following method. This method combines logarithmic barrier function and Newton method for linear programming problems. Its polynomial time complexity has been proved in theory. In addition, it hasfast convergence, powerful robustness, and is insensitiveto the selection of initial points. As a result, it has greatdeveloping potential.Applications of Optimization Theory to Power Systems 7.4.3 Application of modern interior point methods in power systemsModern interior point methods have gained people’s recognition with the advantages of fast convergence and powerful robustness. They are widely used in programming problems, leading to a revolution in optimization area. The application of modern interior point methods to power system optimization problems started from 1990s. Among them affine-scaling and path-following methods behave better.Applications of Optimization Theory to Power Systems In 1991, Clements and other people proposed a nonlinear programming interior point method for power system state estimation problems. This is the first report on the applicationof interior point theory to power systems.In 1992, Pbnnambalam transformed reservoir optimization dispatch problem to a linear programming modelin canonical form, and then used dual affine-scaling method to solve it. Test result of multiple systems showed, for large systems, the iteration number of this method was far less thanApplications of Optimization Theory to Power Systems that of simplex method.In 1994, Singh adopted primal and dual affine-scaling methods respectively to solve weighted minimal absolute value state estimation problems. The former was used for solving primal model, while the latter was used for dual model. Simulation result told us that dual model was better than primal model.In the same year, Granville used path-following method to solve reactive power dispatch problem in nonlinear model.Applications of Optimization Theory to Power Systems Test result of 3467-node system showed, this kind of interior method has good numerical stability, whose capacity in dealing with inequality constraints was better than that of Newton method for nonlinear programming models.In 1995, Clements used path following method for state estimation, mainly using it to deal with all kinds of inequality constraints.In 1996, Granville used path following method to determine the least load curtailment of heave-load systems in severeApplications of Optimization Theory to Power Systems contingency condition in order to recover the system solvability. Furthermore, he used the curtailment value to weigh the insolvable extent of tested systems.In 1997, Li Jingbo and others used primal affine-scaling method to solve the reactive power optimization model of large power systems.In 1998, Liu Mingbo used primal-dual interior point method to solve power system reactive power optimization model in linear programming form. In the same year, they broughtbetter.From 1998 to 2000, Wei Hua proposed a new interior pointnonlinear programming algorithm-perturbed KKT conditionof the primal problem based primal-dual interior point method,and used it to solve optimal power flow of power systems,L norm state estimation problems and weighted nonlinear1hydro-thermal optimal power flow problems.Applications of Optimization Theory to Power Systems7.4.4 Perturbed KKT condition based primal-dualinterior point methodFor convenience of explanation, we consider nonlinear programming problem in the following form:min () s.t. ()0 () nmrf x x R h x h Rg g x g g R ∈=∈≤≤∈ (7-1) Firstly, by introducing slack vector ,r l u R ∈, inequalityconstraints are transformed to equality constraints, and system (7-1) is transformed to:Applications of Optimization Theory to Power Systemsmin ()s.t. ()0()0 ()0(,)0f x h xg x l g g x u g l u =−−=+−=≥(7-2) Secondly, define a Lagrangian function:(,,,,,,,)()()[()][()]T T T T T L x l u y z w z w f x y h x z g x l g w g x u g z l w u =−−−−−+−−−Applications of Optimization Theory to Power Systemswhere, m y R ∈, ,,,rz w z w R ∈ are Lagrangian multipliers.Derive the KKT first-order optimality condition: ()()L z z l L w w u ∂⋅⎧=−=⎪⎪∂⎨∂⋅⎪=−−=⎪∂⎩we obtain:z zw w =⎧⎨=−⎩Moreover, we haveApplications of Optimization Theory to Power Systems()()()()0()0 ()0()0x y z w L f x h x y g x z w L h x L g x l g L g x u g =∇−∇−∇+=⎧⎪==⎪⎨=−−=⎪⎪=+−=⎩ (7-3) 00l u L LZe L UWe ==⎧⎨==⎩(7-4) where, (,,)0,0,0l u z w y ≥≤≠,()(,,,)r r L U Z W R ×∈ are diagonal matrices, x L representsL x ∂∂, other expressions are similar, and [1,1,1]T r e R ≡∈".Applications of Optimization Theory to Power SystemsThus the following perturbed complementary slackness condition (perturbed factor 0>μ) is obtained:00l u L LZe e L UWe e ⎧≡−=⎨≡+=⎩μμμμ (7-5) In the above formula, the function of 0>μ is identical with that of penalty factors in conventional interior point penalty function. The corrective equation of perturbed KKT condition derived by Newton method is:Applications of Optimization Theory to Power Systems000000()()()00()00000()0000()0000000000000x T y T z T w l u L F h x g x g x x L h x y L g x I z L g x I w L Z l L U W u μμ∇∇∇Δ⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥−∇Δ⎢⎥⎢⎥⎢⎥−⎢⎥∇−Δ⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥−∇Δ⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−Δ⎢⎥⎢⎥⎢⎥−Δ⎣⎦⎣⎦⎣⎦ (7-6) where,00000,,,,x y l u z L L L L L μμ and 0w L are residuals of perturbed KKT equations. And222()()()()F h x y g x z w f x =∇+∇+−∇ Where 2()h x ∇, 2()g x ∇ and 2()f x ∇ are Hessian matricesApplications of Optimization Theory to Power Systemsof (),()h x g x and ()f x .Solve the correction equation (7-6) for the k -th iterative correction, so a new approximation of the optimal solution is:, p d x x x y y y l l step l z z step z u u u w w w ΔΔ⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=+Δ=+Δ⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥ΔΔ⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦(7-7) where, p step and d step are primal and dual step lengths, respectively. To guarantee the feasibility of variables, the step lengths are selected according to the following formulae:Applications of Optimization Theory to Power Systems0.9995min min(:0;:0),1, 0.9995min min(:0;:0),1 1,2,i i p i i i i i i i d i i i i i l u step l u l u z w step z w i r z w ⎧⎫−−=Δ<Δ<⎨⎬ΔΔ⎩⎭⎧⎫−−=Δ<Δ>=⎨⎬ΔΔ⎩⎭" (7-8)Additionally, we give some supplementary explanation about every parameter.（1）Complementary gap 1()ri i i i i Gap l z u w =≡−∑. Gap is avery important parameter, determining the solution property.Applications of Optimization Theory to Power SystemsWhen complementary gap tends to zero, the solution also tends to zero. This method uses complementary gap as the condition to determine the optimal solution.（2）The selection of perturbed factor μ is a key point of primal-dual interior point algorithm. We should make μ change with the variance of convergence speed. In this algorithm perturbed factor is updated to be /2Gap r ≡⋅μσ in every step of iteration, to improve the solution optimality step by step.Applications of Optimization Theory to Power Systems（3）Centering parameter σ, generally (0,1]∈σ. When 1=σ, the direction decided by (,,,,,)x y l u z w ΔΔΔΔΔΔ is called the centering direction. Along this direction, the algorithm mainly improves feasibility of the solution. When 0=σ, the direction decided by (,,,,,)x y l u z w ΔΔΔΔΔΔ is known as affine-scaling direction. Along this direction, the algorithm mainly improves optimality of the solution. By experience, we can obtain good convergent effect in most situations by setting 0.1or = 0.5σ.Applications of Optimization Theory to Power SystemsAccording to the above description, the computational procedure of this method can be summarized as follows:（1）Initialization: set k =0,max 100K =, centering parameter(0,1]∈σ, tolerance 610−=ε; choose slack variables [,]0T l u ≥, dual variables [0,0,1]T z w y ><=, set initialvariable x with a proper value; compute initial complementary gap 1()ri i i i i Gap l z u w =−∑ .（2）If max or k K Gap <>=ε, repeat the following:A. Compute perturbed factor =/2μσGap r ⋅;Applications of Optimization Theory to Power SystemsB. Solve the corrective equation for[,], [,], [,]x y l u z w ΔΔΔΔΔΔ;C. Determine primal and dual step lengths p step andd step according to the formula （7-8）, and update primal and dual variables according to （7-7）;D. Set 1k k =+, and compute complementary gap1()ri i i i i Gap l z u w =≡−∑.（3）If max k K ≥, the computation does not converge andthen stops.<, the optimal solution is reached and thecomputation ends.。