基于虚拟节点变压器模型的极坐标最优潮流
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摘要
电力系统最优潮流(Optimal Power Flow,OPF)是一个复杂的非线性规划问题,是对系统稳定运行状态的优化分析。它要求在满足一定的电力系统运行及安全约束条件下,通过调整系统中的控制变量来实现预定目标函数的最优。随着电力系统的飞速发展,越来越多的控制设备被用于电网中以调节系统的运行。这些设备的投入使用在优化系统运行方式的同时,也使得本来就十分庞大的最优潮流问题更加复杂,增加了人们对其分析和计算的难度。
变压器是电力系统中最重要的可控设备之一,广泛应用于电网中需要改变电压的各种场合。以往OPF模型多是基于传统的变压器Π型等值电路建立的,该等值电路不能直观地反映出变压器对电压的调节作用,且对应的OPF模型较为复杂,不利于计算。本文基于虚拟节点的变压器模型,建立了极坐标下以系统运行成本最小为目标函数的最优潮流模型。使用世界上广泛使用的极坐标表示方式,不但能够反映电力参数的物理意义,还可以使模型更加简洁,便于理解、分析和记忆;虚拟节点的引入,其电压幅值代替了变压器变比对功率的影响,使得表达式降低了两次。同时,等式约束的Jacobian和Hessian矩阵非零元素个数的减少量分别至少是有载调压变压器支路个数的2倍和6倍,从而减少了计算量。
用Matlab编程,在极坐标系下应用基于扰动KKT条件的原始-对偶内点法对基于传统变压器Π型等值电路的OPF模型和基于虚拟节点变压器的OPF模型分别求解。在相同的环境下,对TEST 4、IEEE 14～300节点和S-1047节点等7个测试系统进行仿真实验,仿真结果证实两种变压器模型的等价性及基于虚拟节点变压器的OPF模型的优越性。
Optimal power flow(OPF)is a complex non-linear programming problem.It is the optimization analysis of stable running.OPF must meet the requirements of the rules of the power system operation and security constraints.By adjusting control variables of the system, it can optimize the parameters of the intended target function.With the rapid development of the power system,more and more control equipment to be used for power grids in mediating system's operation.These devices put into use in optimizing the running way of the system. But,at the same time,it makes the large optimal flow problems more complicated and an increase difficult of their analysis and calculation.
Transformer is one of the most important control equipment in power system,widely used in every corner which current and voltage need to change.More than the previous model, OPF is based on the traditional transformer-type equivalent circuit∏established.This model can not be directly reflected in the surge voltage transformer's regulatory role and is complex to the analysis and calculation.Based on the virtual node transformer model,in polar coordinates system,we established a new OPF model which the objective function is to minimize operating costs.The use of the widely used polar coordinates that not only to reflect the power of the physical parameters of significance but also to make the model more concise, easy to understand,analyse and memory.The import of virtual nodes,which put the transformer voltage amplitude replace variable ratio,reduced the number of variables of expressions.At the same time,for equality constraints of the Jacobian and Hessian matrix,the reduction in the number of non-zero elements were reduced at least in the number of transformers 2 times and 6 times,thus reducing the amount of computation.
Using Matlab programming,in polar coordinates system,we applied the primal-dual interior point method to solve the OPF model based on the transformer model of traditional∏type equivalent circuit and based on virtual nodes transformer model respectively.In the same circumstances,we did such as TEST4,IEEE14～300 and S-1047 nodes seven test systems simulation.Simulation results confirm the equivalence of two transformal model and the superiority of this OPF based on a transformer model with a virtual nodes.
变压器是电力系统中最重要的可控设备之一,广泛应用于电网中需要改变电压的各种场合。以往OPF模型多是基于传统的变压器Π型等值电路建立的,该等值电路不能直观地反映出变压器对电压的调节作用,且对应的OPF模型较为复杂,不利于计算。本文基于虚拟节点的变压器模型,建立了极坐标下以系统运行成本最小为目标函数的最优潮流模型。使用世界上广泛使用的极坐标表示方式,不但能够反映电力参数的物理意义,还可以使模型更加简洁,便于理解、分析和记忆;虚拟节点的引入,其电压幅值代替了变压器变比对功率的影响,使得表达式降低了两次。同时,等式约束的Jacobian和Hessian矩阵非零元素个数的减少量分别至少是有载调压变压器支路个数的2倍和6倍,从而减少了计算量。
用Matlab编程,在极坐标系下应用基于扰动KKT条件的原始-对偶内点法对基于传统变压器Π型等值电路的OPF模型和基于虚拟节点变压器的OPF模型分别求解。在相同的环境下,对TEST 4、IEEE 14～300节点和S-1047节点等7个测试系统进行仿真实验,仿真结果证实两种变压器模型的等价性及基于虚拟节点变压器的OPF模型的优越性。
Optimal power flow(OPF)is a complex non-linear programming problem.It is the optimization analysis of stable running.OPF must meet the requirements of the rules of the power system operation and security constraints.By adjusting control variables of the system, it can optimize the parameters of the intended target function.With the rapid development of the power system,more and more control equipment to be used for power grids in mediating system's operation.These devices put into use in optimizing the running way of the system. But,at the same time,it makes the large optimal flow problems more complicated and an increase difficult of their analysis and calculation.
Transformer is one of the most important control equipment in power system,widely used in every corner which current and voltage need to change.More than the previous model, OPF is based on the traditional transformer-type equivalent circuit∏established.This model can not be directly reflected in the surge voltage transformer's regulatory role and is complex to the analysis and calculation.Based on the virtual node transformer model,in polar coordinates system,we established a new OPF model which the objective function is to minimize operating costs.The use of the widely used polar coordinates that not only to reflect the power of the physical parameters of significance but also to make the model more concise, easy to understand,analyse and memory.The import of virtual nodes,which put the transformer voltage amplitude replace variable ratio,reduced the number of variables of expressions.At the same time,for equality constraints of the Jacobian and Hessian matrix,the reduction in the number of non-zero elements were reduced at least in the number of transformers 2 times and 6 times,thus reducing the amount of computation.
Using Matlab programming,in polar coordinates system,we applied the primal-dual interior point method to solve the OPF model based on the transformer model of traditional∏type equivalent circuit and based on virtual nodes transformer model respectively.In the same circumstances,we did such as TEST4,IEEE14～300 and S-1047 nodes seven test systems simulation.Simulation results confirm the equivalence of two transformal model and the superiority of this OPF based on a transformer model with a virtual nodes.
引文
[1]王锡凡,方万良,杜正春.现代电力系统分析[M].科学出版社,2003年3月
[2]余娟,颜伟,徐国禹,等.基于预测—校正原对偶内点法的无功优化新模型[J].中国电机工程学报,2005,26(11):146-151
Yu Juan,Yan Wei,XU Guoyu,et al.A New Model of Reactive Optimization Based on Predictor Corrector Primal-Dual Interior Point Method[J].Proceedings of the CSEE,2005,26(11):146-151
[3]Wei Yan,Juan Yu,David C.Yu et al.A New Optimal Reactive Power Flow Model in Rectangular Form and its Solution by Predictor Corrector Primal Dual Interior Point Method[J].IEEE Trans on Power System,2006,21(1):61-67.
[4]李文沅.电力网络N-1安全性有功实时经济调度计算的研究[博士论文].重庆大学,1987
[5]R Mota-Palomino,V H Quintana.A Penalty Function-Linear Programming Method for Solving Power System Constrained Economic Operation Problems[J].IEEE Transactions on Power Apparatus and Systems,1984,103(6):1414-1442
[6]江长明,朱继忠等.有功调度中经济性与安全性的协调问题[J].电力系统自动化,1995,19(11):43-48
Jiang Changming,Zhu Jizhong,et al.The Coordination Of Economic Characteristic and Security in Real Power Dispatching[J].Automation of Electric Power Systems,1995,19(11):43-48
[7]G Kron.Tensorial Analysis of Integrated Transmission Systems[J].AIEE Trans.,1952,71,partⅢ:814-822;1953,72,part Ⅳ:827-838
[8]L K Kirchmayer,G W Stagg.Analysis of Total and Incremental Losses in Transmission Systems[J].AIEE Trans.,1951,70,PartⅡ:1197-1205
[9]R H Kerr,L K Kirchmayer.Theory of Economic Operation of Interconnected Areas[J].AIEE Trans.,1959,78 PartⅢ:647-653
[10]J Carpentier.Contribution a letude du Dispatching Economique[M].Bulletin de ia Societe Francaise des Electriciens,1962,3:431-447
[11]Dommel H W,Tinney W F.Optimal Power Flow Solutions[J].IEEE Trans.on Power Apparatus and Systems,1968,87(10):1866-1876
[12]李彩华,郭志忠等.电力系统优化调度概述—经济调度与最优潮流[J].电力系统及其自动化学报,2002,14(2):60-63
Li Caihua,Guo Zhizhong,et al.Summary of Power System Optimal Dispatch—Economic Dispatch and Optimal Power Flow[J].Proceeding of the EPSA,2002,14(2):60-63
[13]D W Wells.Method for Economic Secure Loading of a Power System[J].Proceedings of IEEE,1968, 115(8):606-614
[14]K R Frish.Principles of Linear Programming-the Double Gradient Form of the Logarithmic Potential Method.Memorandum,Institute of Economics,University of Oslo,Oslo,Norway,1954
[15]N Karmarkar.A New Polynomial-time Algorithm for Linear Programming[J].Combinatorica,1984,4:373-395
[16]P E Gill,W Murray,M A Saunders,et.al.On the Projected Newton Barrier Methods for Linear Programming and an Equivalence to Karmarkar's Projective Method[M].Math.Programming,1956,36:183-209
[17]Hua Wei,Sasaki H,Yokoyama J et al.An Application of Interior Point Quadratic Programming Algorithm to Power System Optimization Problems[J].IEEE Trans on Power System,1996,11(1):260-266
[18]Hua Wei,Sasaki H,Kubokawa J et al.An Interior Point Power System Weighted Nonlinear L1 Norm Static State Estimation[J].IEEE Trans on Power System,1998,13(3):617-623
[19]Hua Wei,Sasaki H,Kubokawa J et al.An Interior Point Nonlinear Programming for Optimal Power Flow Problems with A Novel Data Structure[J].IEEE Trans on Power System,1998,13(3):870-877
[20]Hua Wei,Sasaki H,Kubokawa J et al.Large Scale Hydrothermal Optimal Power Flow Problems Based on Interior Point Nonlinear Programming[J].IEEE Trans on Power System,2000,15(1):396-403
[21]韦化,李滨,杭乃善,等.大规模水火电力系统最优潮流的现代内点理论分析[J].中国电机工程学报,2003,23(4):5-8
Wei Hua,Li Bin,Hang Naishan et al.An Analysis of Interior Point Theory for Large-scale Hydrothermal Optimal Power Flow Problems[J].Proceedings of the CSEE,2003,23(4):5-8
[22]韦化,李滨,杭乃善,等.大规模水-火电力系统最优潮流的现代内点算法实现[J].中国电机工程学报,2003,23(6):13-18
Wei Hua,Li Bin,Hang Naishan et al.An Implementation of Interior Point Algorithm for Large-scale Hydro-thermal Optimal Power Flow Problems[J].Proceedings of the CSEE,2003,23(6):13-18
[23]丁晓莹,韦化.现代内点理论及其在电力系统经济调度中的应用[J].广西大学学报,2000,25(1):78-82
Wei Hua,Ding Xiaoying.Application of Modern Interior Point Theory for the Economic Load Dispatch Problem in Power Systems[J].Journal of Guangxi University,2000,25(1):78-82
[24]韦化,丁晓莺.基于现代内点法理论的电压稳定临界点算法[J].中国电机工程学报,2002,22(3):27-31
Wei Hua,Ding Xiaoying.An Algorithm of Determining Voltage Stability Critical Point Based on Interior Point Theory[J].Proceedings of the CESS,2002,22(3):27-31
[25]X Y Ding,X F Wang,Y H Song.An interior Point Cutting Plane Method for Optimal Power Flow.International Journal of Management Mathematics[J],2004,15(4):355-368
[26]Ponnambalam K,Quintana V H.Vannelli A.A Fast Algorithm for Power System Optimization Problems Using an Interior Point Method.Power Industry[C].Computer Application Conference,1991,Conference Proceedings,1991,393-400.
[27]Momoh J A,Austin R F,Adapa R.,Ogbuobiri E C.Application of interior point method to economic dispatch[C].IEEE International Conference Systems,Man and Cybernelics,1992,(2):1096-1101.
[28]WUYu-Chi,Afif S Debs,Roy E Marsten.A Direct Non-linear Predictor-corrector Primal-dual Interior Point Algorithm for Optimal Power Flow[J].IEEE Trans on Power System,1994,9(2):876-883
[29]Momoh J A,Dias LG,Guo SX,Adapa R.Economic Operation and Planning of Multi-area Interconnected Power Systems[J].IEEE Transactions On Power Systems,1995,10(2):1044-1053
[30]Granville S.Optimal Reactive Dispatch Through Interior Point Methods[J].IEEE Transactions on Power Systems,1994,9(1):136-146
[31]郝玉国,刘广一,于尔铿.一种基于Karmarkar内点法的最优潮流算法[J].中国电机工程学报,1996,16(6):409-412
Hao Yuguo,Liu Guangyi,Yu Erkeng.A New OPF Algorithm Based on Karmarkar's Interior Point Method[J].Proceeding of the CSEE,1996,16(6):409-412
[32]Momoh J A,Zhu J.Z.Improved Interior Point Method for OPF Problem.IEEE Transaction on Power Systems,1999,14(3):I114-I120
[33]史继莉,邱晓燕.人工智能在最优潮流中的应用综述[J].继电器,2005,33(16):85-95
Shi Jili,Qiu xiaoyan.Artificial Intelligence Applications in Optimal Power Flow[J].Relay,2005,33(16):85-95
[34]Mcshane K A,Monma C L,Shanno D F.An Implementation of a Primal-dual Interior Point Method for Linear Programming[J].ORSA Journal on Computing,1989(1):70-83
[35]Mehrotra S.On Finding a Vertex Solution Using Interior Point Methods[M].Linear Algebra and its Appl ications,1990,152:233-533
[36]Mehrotra S.On the Implementation of a Primal Dual Interior Point Method[J].SIAM Optimization,1992,2:575-601
[37]Lustig I J.Feasibility issues in a Primal Dual Interior Point Methods for Linear Programming[M].Mathematical Programming,1991,49:145-162
[38]Laston L,Plummer J.An Interior-point Algorithm for Solving General Nonlinear Programming Problems[A].the SIAM Conference on Programming[C],1994,64(2):123-147
[39]Vanderbei Robert J,David F Shanno.An Interior-Point Algorithm for Nonconvex Nonlinear Programming[R].Princeton NJ:Princeton University,1997
[40]I I Dikin.Iterative Solution of Problem of Linear and Quadratic Programming[J].Soviet Mathematical Doklady,1967,8:674-675
[41]Fiacco A V,McCormick G P.Nonlinear Programming Sequential Unconstrained Minimization Techniques.New York:John Wiley and Sons,1968
[42]周鹗.电机学[M].中国电力出版社,1995年5月
[43]何仰赞,温增银.电力系统分析[M].华中科技大学出版社,2005年1月
[2]余娟,颜伟,徐国禹,等.基于预测—校正原对偶内点法的无功优化新模型[J].中国电机工程学报,2005,26(11):146-151
Yu Juan,Yan Wei,XU Guoyu,et al.A New Model of Reactive Optimization Based on Predictor Corrector Primal-Dual Interior Point Method[J].Proceedings of the CSEE,2005,26(11):146-151
[3]Wei Yan,Juan Yu,David C.Yu et al.A New Optimal Reactive Power Flow Model in Rectangular Form and its Solution by Predictor Corrector Primal Dual Interior Point Method[J].IEEE Trans on Power System,2006,21(1):61-67.
[4]李文沅.电力网络N-1安全性有功实时经济调度计算的研究[博士论文].重庆大学,1987
[5]R Mota-Palomino,V H Quintana.A Penalty Function-Linear Programming Method for Solving Power System Constrained Economic Operation Problems[J].IEEE Transactions on Power Apparatus and Systems,1984,103(6):1414-1442
[6]江长明,朱继忠等.有功调度中经济性与安全性的协调问题[J].电力系统自动化,1995,19(11):43-48
Jiang Changming,Zhu Jizhong,et al.The Coordination Of Economic Characteristic and Security in Real Power Dispatching[J].Automation of Electric Power Systems,1995,19(11):43-48
[7]G Kron.Tensorial Analysis of Integrated Transmission Systems[J].AIEE Trans.,1952,71,partⅢ:814-822;1953,72,part Ⅳ:827-838
[8]L K Kirchmayer,G W Stagg.Analysis of Total and Incremental Losses in Transmission Systems[J].AIEE Trans.,1951,70,PartⅡ:1197-1205
[9]R H Kerr,L K Kirchmayer.Theory of Economic Operation of Interconnected Areas[J].AIEE Trans.,1959,78 PartⅢ:647-653
[10]J Carpentier.Contribution a letude du Dispatching Economique[M].Bulletin de ia Societe Francaise des Electriciens,1962,3:431-447
[11]Dommel H W,Tinney W F.Optimal Power Flow Solutions[J].IEEE Trans.on Power Apparatus and Systems,1968,87(10):1866-1876
[12]李彩华,郭志忠等.电力系统优化调度概述—经济调度与最优潮流[J].电力系统及其自动化学报,2002,14(2):60-63
Li Caihua,Guo Zhizhong,et al.Summary of Power System Optimal Dispatch—Economic Dispatch and Optimal Power Flow[J].Proceeding of the EPSA,2002,14(2):60-63
[13]D W Wells.Method for Economic Secure Loading of a Power System[J].Proceedings of IEEE,1968, 115(8):606-614
[14]K R Frish.Principles of Linear Programming-the Double Gradient Form of the Logarithmic Potential Method.Memorandum,Institute of Economics,University of Oslo,Oslo,Norway,1954
[15]N Karmarkar.A New Polynomial-time Algorithm for Linear Programming[J].Combinatorica,1984,4:373-395
[16]P E Gill,W Murray,M A Saunders,et.al.On the Projected Newton Barrier Methods for Linear Programming and an Equivalence to Karmarkar's Projective Method[M].Math.Programming,1956,36:183-209
[17]Hua Wei,Sasaki H,Yokoyama J et al.An Application of Interior Point Quadratic Programming Algorithm to Power System Optimization Problems[J].IEEE Trans on Power System,1996,11(1):260-266
[18]Hua Wei,Sasaki H,Kubokawa J et al.An Interior Point Power System Weighted Nonlinear L1 Norm Static State Estimation[J].IEEE Trans on Power System,1998,13(3):617-623
[19]Hua Wei,Sasaki H,Kubokawa J et al.An Interior Point Nonlinear Programming for Optimal Power Flow Problems with A Novel Data Structure[J].IEEE Trans on Power System,1998,13(3):870-877
[20]Hua Wei,Sasaki H,Kubokawa J et al.Large Scale Hydrothermal Optimal Power Flow Problems Based on Interior Point Nonlinear Programming[J].IEEE Trans on Power System,2000,15(1):396-403
[21]韦化,李滨,杭乃善,等.大规模水火电力系统最优潮流的现代内点理论分析[J].中国电机工程学报,2003,23(4):5-8
Wei Hua,Li Bin,Hang Naishan et al.An Analysis of Interior Point Theory for Large-scale Hydrothermal Optimal Power Flow Problems[J].Proceedings of the CSEE,2003,23(4):5-8
[22]韦化,李滨,杭乃善,等.大规模水-火电力系统最优潮流的现代内点算法实现[J].中国电机工程学报,2003,23(6):13-18
Wei Hua,Li Bin,Hang Naishan et al.An Implementation of Interior Point Algorithm for Large-scale Hydro-thermal Optimal Power Flow Problems[J].Proceedings of the CSEE,2003,23(6):13-18
[23]丁晓莹,韦化.现代内点理论及其在电力系统经济调度中的应用[J].广西大学学报,2000,25(1):78-82
Wei Hua,Ding Xiaoying.Application of Modern Interior Point Theory for the Economic Load Dispatch Problem in Power Systems[J].Journal of Guangxi University,2000,25(1):78-82
[24]韦化,丁晓莺.基于现代内点法理论的电压稳定临界点算法[J].中国电机工程学报,2002,22(3):27-31
Wei Hua,Ding Xiaoying.An Algorithm of Determining Voltage Stability Critical Point Based on Interior Point Theory[J].Proceedings of the CESS,2002,22(3):27-31
[25]X Y Ding,X F Wang,Y H Song.An interior Point Cutting Plane Method for Optimal Power Flow.International Journal of Management Mathematics[J],2004,15(4):355-368
[26]Ponnambalam K,Quintana V H.Vannelli A.A Fast Algorithm for Power System Optimization Problems Using an Interior Point Method.Power Industry[C].Computer Application Conference,1991,Conference Proceedings,1991,393-400.
[27]Momoh J A,Austin R F,Adapa R.,Ogbuobiri E C.Application of interior point method to economic dispatch[C].IEEE International Conference Systems,Man and Cybernelics,1992,(2):1096-1101.
[28]WUYu-Chi,Afif S Debs,Roy E Marsten.A Direct Non-linear Predictor-corrector Primal-dual Interior Point Algorithm for Optimal Power Flow[J].IEEE Trans on Power System,1994,9(2):876-883
[29]Momoh J A,Dias LG,Guo SX,Adapa R.Economic Operation and Planning of Multi-area Interconnected Power Systems[J].IEEE Transactions On Power Systems,1995,10(2):1044-1053
[30]Granville S.Optimal Reactive Dispatch Through Interior Point Methods[J].IEEE Transactions on Power Systems,1994,9(1):136-146
[31]郝玉国,刘广一,于尔铿.一种基于Karmarkar内点法的最优潮流算法[J].中国电机工程学报,1996,16(6):409-412
Hao Yuguo,Liu Guangyi,Yu Erkeng.A New OPF Algorithm Based on Karmarkar's Interior Point Method[J].Proceeding of the CSEE,1996,16(6):409-412
[32]Momoh J A,Zhu J.Z.Improved Interior Point Method for OPF Problem.IEEE Transaction on Power Systems,1999,14(3):I114-I120
[33]史继莉,邱晓燕.人工智能在最优潮流中的应用综述[J].继电器,2005,33(16):85-95
Shi Jili,Qiu xiaoyan.Artificial Intelligence Applications in Optimal Power Flow[J].Relay,2005,33(16):85-95
[34]Mcshane K A,Monma C L,Shanno D F.An Implementation of a Primal-dual Interior Point Method for Linear Programming[J].ORSA Journal on Computing,1989(1):70-83
[35]Mehrotra S.On Finding a Vertex Solution Using Interior Point Methods[M].Linear Algebra and its Appl ications,1990,152:233-533
[36]Mehrotra S.On the Implementation of a Primal Dual Interior Point Method[J].SIAM Optimization,1992,2:575-601
[37]Lustig I J.Feasibility issues in a Primal Dual Interior Point Methods for Linear Programming[M].Mathematical Programming,1991,49:145-162
[38]Laston L,Plummer J.An Interior-point Algorithm for Solving General Nonlinear Programming Problems[A].the SIAM Conference on Programming[C],1994,64(2):123-147
[39]Vanderbei Robert J,David F Shanno.An Interior-Point Algorithm for Nonconvex Nonlinear Programming[R].Princeton NJ:Princeton University,1997
[40]I I Dikin.Iterative Solution of Problem of Linear and Quadratic Programming[J].Soviet Mathematical Doklady,1967,8:674-675
[41]Fiacco A V,McCormick G P.Nonlinear Programming Sequential Unconstrained Minimization Techniques.New York:John Wiley and Sons,1968
[42]周鹗.电机学[M].中国电力出版社,1995年5月
[43]何仰赞,温增银.电力系统分析[M].华中科技大学出版社,2005年1月