一种求解最优潮流的过滤器—信赖域内点方法
摘要
电力系统最优潮流(Optimal Power Flow,OPF)是电力系统运行、分析、控制和规划的不可或缺的网络分析和优化工具,它关系到系统运行的经济性、安全性及电能质量。至今求解OPF的方法很多,如非线性规划法、二次规划法、线性规划法及内点算法等。
在OPF算法的众多研究成果中,现代内点法具有收敛性好、多项式时间复杂性等优点,是极具潜力的优秀算法之一。但理论上,内点法求解非凸优化问题时的收敛性受到质疑,且当约束条件变得更为严格时,可能导致不收敛。因此寻找能够快速有效地求解大规模OPF问题的方法,已成为研究的一个热点。
近年来,世界著名数学家R.Fletcher提出的过滤器(Filter)法为解决以上问题提供了新的途径,他将优于(Dominate)的概念和滤点的思想引入非线性规划中。此后,结合过滤器法与其他优化方法求解非线性规划问题成为新的研究热点。
本文提出一种过滤器—信赖域内点算法(Filter Trust-RegionInterior Point Method,FTRIPM),试图解决用内点法求解含约束越界的OPF时不收敛的问题。其中,用现代内点法求解二次规划子模型(Sequential Quadratic Program,SQP)得到试探步,通过过滤器法来判断试探步是否可接受,最后采用信赖域方法来决定步长。对五个IEEE标准系统的测试表明:当OPF的约束条件变得更为严格时,原始对偶内点法(Primal Dual Interior Point Method,PDIPM)不收敛;但FTRIPM算法依然可以收敛,且拥有良好的收敛特性,因此具有良好的应用前景。
Optimal power flow (OPF) is an indispensable network analysis and optimization tool for electric power system's operation. It's vital to the system's economy, safety and power quality. There are several ways to solve OPF, such as nonlinear programming, quadratic programming, linear programming, and interior point method and so on.
Modern interior point method is one of important algorithms for the OPF, which has some advantages like good convergence and polynomial time complexity. It is extremely be an outstanding algorithm. But when to solve non-convex optimization or when the restrictions of OPF beyond the bound, the convergence of interior point method is challenged. So looking for ways to solve large-scale OPF in quickly and effectively has become a hot research.
In recent years, there is a new way called filter method can solve above problems. World-famous mathematician R. Fletcher introduced the concept of dominate and the ideological about filtrate of the filter. Since then, combining the filter method and other methods to solve nonlinear programming problems has become a new hotspot.
This paper presents a filter-trust region interior point algorithm, solving the problem that interior point method can't convergence when restrictions beyond the bound in OPF. In this algorithm, the SQP sub model is solved by the modern interior point method, and the trying step is controlled by the trust region, whether to accept it or not is decided by the filter. Numerical tests on five standard IEEE systems are very encouraging. Compare with the primal-dual interior point method, this new algorithm convergence when some restriction beyond the bound. This result is suitable to practical applications.
在OPF算法的众多研究成果中,现代内点法具有收敛性好、多项式时间复杂性等优点,是极具潜力的优秀算法之一。但理论上,内点法求解非凸优化问题时的收敛性受到质疑,且当约束条件变得更为严格时,可能导致不收敛。因此寻找能够快速有效地求解大规模OPF问题的方法,已成为研究的一个热点。
近年来,世界著名数学家R.Fletcher提出的过滤器(Filter)法为解决以上问题提供了新的途径,他将优于(Dominate)的概念和滤点的思想引入非线性规划中。此后,结合过滤器法与其他优化方法求解非线性规划问题成为新的研究热点。
本文提出一种过滤器—信赖域内点算法(Filter Trust-RegionInterior Point Method,FTRIPM),试图解决用内点法求解含约束越界的OPF时不收敛的问题。其中,用现代内点法求解二次规划子模型(Sequential Quadratic Program,SQP)得到试探步,通过过滤器法来判断试探步是否可接受,最后采用信赖域方法来决定步长。对五个IEEE标准系统的测试表明:当OPF的约束条件变得更为严格时,原始对偶内点法(Primal Dual Interior Point Method,PDIPM)不收敛;但FTRIPM算法依然可以收敛,且拥有良好的收敛特性,因此具有良好的应用前景。
Optimal power flow (OPF) is an indispensable network analysis and optimization tool for electric power system's operation. It's vital to the system's economy, safety and power quality. There are several ways to solve OPF, such as nonlinear programming, quadratic programming, linear programming, and interior point method and so on.
Modern interior point method is one of important algorithms for the OPF, which has some advantages like good convergence and polynomial time complexity. It is extremely be an outstanding algorithm. But when to solve non-convex optimization or when the restrictions of OPF beyond the bound, the convergence of interior point method is challenged. So looking for ways to solve large-scale OPF in quickly and effectively has become a hot research.
In recent years, there is a new way called filter method can solve above problems. World-famous mathematician R. Fletcher introduced the concept of dominate and the ideological about filtrate of the filter. Since then, combining the filter method and other methods to solve nonlinear programming problems has become a new hotspot.
This paper presents a filter-trust region interior point algorithm, solving the problem that interior point method can't convergence when restrictions beyond the bound in OPF. In this algorithm, the SQP sub model is solved by the modern interior point method, and the trying step is controlled by the trust region, whether to accept it or not is decided by the filter. Numerical tests on five standard IEEE systems are very encouraging. Compare with the primal-dual interior point method, this new algorithm convergence when some restriction beyond the bound. This result is suitable to practical applications.
引文
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[4]Wei H,H.Sasaki,J.Kubokawa.An interior point nonlinear programming for optimal power flow problems with a novel data structure.IEEE Trans on Power System.1998,13:870-877.
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[8]孙英云,何光宇,梅生伟.基于Filter集合的内点最优潮流新算法.电工电能新技术.2007.26:29-33.
[9]J.Shaw,D.P.Bertsekas.Optimal Scheduling of large hydrothermal power system.IEEE Trans on Power System.1985,104:286-294.
[10]R.A.Duncan,G.E.Seymore,et al.Optimal hydrothermal coordination for multiple reservoir river systems.IEEE Yrans on Power System.1985,104:1154-1159.
[11]Ying X,Y.H.Song,Y.Z.Sun.Power flow control approach to power system with embedded FACTS devices.IEEE Trans on Power System.2002,17:943-950.
[12]U.P.Mhaskar,A.B.Mote,A.M.Kulkarni.A new formulation for load flow solution of power system with series FACTS devices.IEEE Trans on Power System.2003,18:1307-1315.
[13]F.Allella,D.Lauria.Fast optimal power dispatch with global transient stability constraint.IEE proc.Gener.Transm.System.2001,148:471-476.
[14]A.A.Fouad,J.Z.Tong.Stability constrained optimal rescheduling of generation.IEEE Trans on Power System.1993,8:105-112.
[15]袁亚湘.信赖域方法的收敛性.计算数学.1994,12:334-346.
[16]M.J.D.Powell.A new algorithm for unconstrained optimization.J.B.Rosen,O.L.Mangasarian and K.Ritter,et al.Nonlinear Programming.New York:Academic Press.1970,7:31-66.
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[19]M.J.D.Powell,Y.Yuan.A trust region algorithm for equality constrained optimization.Math.Programming.1991,49:189-211.
[20]A.Forsgren,P.E.Gill.Primal-dual interior point methods for nonconvex nonlinear programming.SIAM J.Optim..1998,8:1132-1152.
[21]E.M.Gertz.Combination Trust-Region Line-Search Methods for Unconstrained Optimization.PhD thesis,Department of Mathematics,University of California,San Diego.1999,4:25-31.
[22]E.M.Gertz,E.Gill.A primal-dual trust region algorithm for nonlinear optimization.Math.Programming.2004,100:49-94.
[23]Fletcher R.Numerical experiments with an l_1 exact penalty function method.In:Meyer,R.R.,Mangasarian,O.L.,Robinson,S.M.(eds.)Nonlinear Programming.Ann.Oper.Res.1981,47:303-334.
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[25]Fletcher R,Leyffer S,Toint P L.On the global convergence of SQP-filter algorithm.Numerical analysis report NA/97,University of Dundee,UK,November 2000.Submitted to SIOPT.
[26]Fletcher R,N.T.M.Could,S.Leyffer,Ph.L.Toint,A.Wachter.Global convergence of trust-region SQP-filter algorithm for general nonlinear programming.SIAM Journal on Optimization. 2002,13:635-659.
[27] Ulbrich S. On the superlinear local convergence of a filter-SQP method. Mathematical Programming. 2004,12:217-245.
[28] Ulbrich M, Ulbrich S, VicenteA L N. A globally convergent primal-dual interior-point filter method for nonlinear programming, Mathematical Programming. 2004, 100:379-410.
[29] C. M. Chin. A global convergence theory of a filter line search method for nonlinear programming. Technical report, Department of Electrical Engineering, University of Malaya, Kuala Lumpur, Malaysia. 2002, 4:12-20.
[30] C. M. Chin, R. Fletcher. On global convergence theory of an SLP-filter algorithm that takes EQP steps. Mathematical Programming. 2003, 96:161-177.
[31] Fletcher R, Leyffer S, Toint P L. On the global convergence of SLP-filter algorithm. Numerical analysis report NA/183, University of Dundee, UK. 1998, 11:23-33.
[32] Fletcher R, N. T. M. Could, S. Leyffer, Ph. L. Toint, A. Wachter. Global convergence of trust-region SQP-filter algorithm for general nonlinear programming. SIAM Journal on Optimization. 2002, 13:635-659.
[33] Ulbrich S. On the superlinear local convergence of a filter-SQP method. Mathematical Programming. 2004, 14:217-245.
[34] Y. Yuan. Conditions for convergence of trust region algorithm for nonsmooth optimization. Math. Prog.. 1985,31:220-228.
[35] Y. Yuan. An example of only linearly of trust region algorithm for nonsmooth optimization. SIAM J. Number. 1984,4:327-335.
[36] H.Yabe, H.Yamashita, T.Tanabe. A primal-dual interior point method for large-scale nonlinear optimization problems. Surikaisekikenkyusho Kokyuroku, 1998, pp.204-211.
[37] R. H. Byrd, J. C. Gilbert, J. Nocedal. A trust region method based on interior point techniques nonlinear programming. Math. Programming. 2000, 89:149-185.
[38] R. H. Byrd, M. E. Hribar, J. Nocedal. An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim.. 1999, 9:877-900.
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