基于一类暂态稳定约束最优潮流的广义半无限优化（GSIP）及其应用研究
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摘要
广义半无限优化问题(Generalized Semi-Infinite Programming,简称GSIP)是一类包含有限多个变量无穷多个约束的复杂非线性优化问题,其约束集相关于决策变量,在动力系统控制等许多工程技术领域中都有着广泛的应用.本文以市场运营模式下的电力系统为应用背景,基于电力系统故障切除时间为变量的带暂态稳定约束最优潮流(Optimal power flow with transient stability constraints and variable clearing time of faults,简称OPF-TSCC)建立了一类GSIP模型,并对该类GSIP进行了理论分析和模型计算方法的研究.首先基于OPF-TSCC建立了GSIP模型,研究了最优性条件,进而提出了一类求解该GSIP问题的数值计算方法,理论上可证明算法的全局收敛性;作为应用,将该类GSIP用于计算OPF-TSCC以及临界故障切除时间(Critical Clearing Time,简称CCT ),数值仿真验证了该模型和求解算法的有效性.另一方面,由于电力工业的市场化改革,电价成为市场营运的关键经济指标,如何合理地建立电价预测模型是电力管理者乃至数学研究者感兴趣的问题之一.本文提出了一类基于面板数据的电价预测模型,预测结果体现了该预测模型的特点.本文主要内容如下:
第一章主要介绍了广义半无限优化问题的概念和研究现状,描述了其KKT条件,概括了求解GSIP已有方法的基本思想,并提出了一类求解GSIP的算法,阐述了电力系统最优潮流和以临界故障切除时间为变量的暂态稳定约束等基本问题,介绍了预测模型的面板数据及其相关估计模型.
第二章基于电力系统OPF-TSCC研究了一类GSIP问题.首先运用函数转换技术,等价变换OPF-TSCC为一类复杂的广义半无限优化问题,继而将其转换为双层优化模型,提出了求解转换后GSIP的一类SQP算法.理论上可证明算法的全局收敛性.电力系统的两个实例验证了该算法的有效性.
第三章基于价量面板数据建立电价预测模型.首先应用差分方法进行数据处理,进而采用面板数据单位根检验和协整检验,验证选取数据是否具有协整关系;继而应用E-G普通最小二乘法(Pooled EGLS(Cross-section weightings))对具有协整关系的面板数据进行协整建模,该模型改善了小样本问题,并提高了检验效果,及变量内生性和序列相关所导致的伪回归问题.预测结果表明该预测模型具有较高的预测精度.
Generalized semi-infinite programming (GSIP) is an important branch in the field of nonlinear programming, which comprises finite variables with infinite constraints. GSIP is wide applied in control system of engineering technique field and so on. In this paper, the power system that under the market operation model become application background. We found a kind of GSIP based on OPF-TSCC of power system, study the theory and numerical methods of GSIP. Firstly, the GSIP is found based on OPF-TSCC of power system, the KKT condition is studied, we propose an effective method the method has global convergence in theory. Then GSIP is applied to OPF-TSCC in power system. Numerical results are shown that the new method in this paper is effective. On the other hand, with the development of power industry and the gradual opening of electricity market, the electrovalence become a important index, how to build electrovalence forecast model is a crucial issue concerned by all market participants and mathematician. The electrovalence forecast model based on panel data is proposed, the forecast results indicate that this model has high predication precision. The main contents are as follows:
In the first section, we mainly introduce both the concept and current situation of generalized semi-infinite programming and the main ideas of its existing methods, propose a different theory of GSIP algorithm with its KKT system in this paper and briefly describe OPF-TSCC. In addition, the panel date and estimate model of forecast model have been introduced in this paper.
In the second section, research GSIP based on OPF-TSCC of power system. Firstly, the functional transformation technology is used, the OPF-TSCC is converted into a generalized semi-infinite programming problem, and the GSIP problem is converted into bi-level optimization. Then the SQP method is presented for the reformulated GSIP problem. The algorithm has global convergence in theory. Two practical examples are presented. Both are shown that the new method in this paper is effective.
In the third section, building the forecast model of power tariff based on panel co-integration of power tariff and demand. Firstly, the panel unit root test and panel co-integration test on the power tariff and demand are carried out. Secondly, Based on the co-integration analysis of panel data, the long-term equilibrium model of the power tariff and demand is established by Pooled EGLS method. the model is able to ameliorate the small sample problem in order to improve the test effect and solve the spurious regression. The forecast results indicate that this model has high predication precision.
第一章主要介绍了广义半无限优化问题的概念和研究现状,描述了其KKT条件,概括了求解GSIP已有方法的基本思想,并提出了一类求解GSIP的算法,阐述了电力系统最优潮流和以临界故障切除时间为变量的暂态稳定约束等基本问题,介绍了预测模型的面板数据及其相关估计模型.
第二章基于电力系统OPF-TSCC研究了一类GSIP问题.首先运用函数转换技术,等价变换OPF-TSCC为一类复杂的广义半无限优化问题,继而将其转换为双层优化模型,提出了求解转换后GSIP的一类SQP算法.理论上可证明算法的全局收敛性.电力系统的两个实例验证了该算法的有效性.
第三章基于价量面板数据建立电价预测模型.首先应用差分方法进行数据处理,进而采用面板数据单位根检验和协整检验,验证选取数据是否具有协整关系;继而应用E-G普通最小二乘法(Pooled EGLS(Cross-section weightings))对具有协整关系的面板数据进行协整建模,该模型改善了小样本问题,并提高了检验效果,及变量内生性和序列相关所导致的伪回归问题.预测结果表明该预测模型具有较高的预测精度.
Generalized semi-infinite programming (GSIP) is an important branch in the field of nonlinear programming, which comprises finite variables with infinite constraints. GSIP is wide applied in control system of engineering technique field and so on. In this paper, the power system that under the market operation model become application background. We found a kind of GSIP based on OPF-TSCC of power system, study the theory and numerical methods of GSIP. Firstly, the GSIP is found based on OPF-TSCC of power system, the KKT condition is studied, we propose an effective method the method has global convergence in theory. Then GSIP is applied to OPF-TSCC in power system. Numerical results are shown that the new method in this paper is effective. On the other hand, with the development of power industry and the gradual opening of electricity market, the electrovalence become a important index, how to build electrovalence forecast model is a crucial issue concerned by all market participants and mathematician. The electrovalence forecast model based on panel data is proposed, the forecast results indicate that this model has high predication precision. The main contents are as follows:
In the first section, we mainly introduce both the concept and current situation of generalized semi-infinite programming and the main ideas of its existing methods, propose a different theory of GSIP algorithm with its KKT system in this paper and briefly describe OPF-TSCC. In addition, the panel date and estimate model of forecast model have been introduced in this paper.
In the second section, research GSIP based on OPF-TSCC of power system. Firstly, the functional transformation technology is used, the OPF-TSCC is converted into a generalized semi-infinite programming problem, and the GSIP problem is converted into bi-level optimization. Then the SQP method is presented for the reformulated GSIP problem. The algorithm has global convergence in theory. Two practical examples are presented. Both are shown that the new method in this paper is effective.
In the third section, building the forecast model of power tariff based on panel co-integration of power tariff and demand. Firstly, the panel unit root test and panel co-integration test on the power tariff and demand are carried out. Secondly, Based on the co-integration analysis of panel data, the long-term equilibrium model of the power tariff and demand is established by Pooled EGLS method. the model is able to ameliorate the small sample problem in order to improve the test effect and solve the spurious regression. The forecast results indicate that this model has high predication precision.
引文
[1] R.Hettich, G.Still, Second order optimality conditions for generalized semi-infinite programming problems, Optimization 34(1995) 195–211.
[2] G. Still. Generalized semi-infinite programming: Theory and methods, European Journal of Operational Research 119 (1999)301–313.
[3] W. Weber. Generalized semi-in?nite optimization: On some foundations, Preprint, University of Darmstadt, 1996.
[4] H.Th. Jongen, J.-J. Ruckmann, O. Stein. Generalized semi-infinite optimization: A first order optimality condition and examples, Mathematical Programming 83 (1998) 145–158.
[5] A. Hoffmann, R. Reinhardt. On reverse Chebyshev approximation problems, Technical University of Illmenau, Preprint No.M08/94, 1994.
[6] R. Hettich, G. Still. Semi-infinite programming models in Robotics, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics II, Akademie Verlag, Berlin, 1991.
[7] A. Kaplan, R. Tichatschke. On a class of terminal variational problems, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics IV, Peter Lang, Frankfurt a.M, 1997.
[8] G.-W.Weber. Generalized semi-infinite optimization and related topics, Habilitation Thesis, Darmstadt University of Technology, 1999.
[9] G.Still. Discretization in semi-infinite programming : the rate of convergence [ J ] . Math.Program. 2001,91:53-69
[10] G. Still. Generalized semi-infinite programming: Numercical aspects,1998
[11] Jan-J, Ruckmann, Juan Alfedo Gomez. On Generalized Semi-Infinite Program- ming. Sociedad de Estadistiva e Invstigacion OPerativa.2006(14),1:1-59
[12] Oliver Stein, Georg Still. On generalized semi-Infinite optimization and bilevel optimization. European Journal of Operational Research 142 (2002) 444–462
[13] Wan Zhongping, Wang Xiaojiao. He lin. Asymptotiv approximation method and its convergence on semi-infinite programming [J],Acta Mathematica Scientia,2004
[14] X.J. Tong, C. Ling, L. Qi.A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints, Journal of Computational and Applied Mathematics, 217(2008), 432-447.
[15] Yan Xia, Ka Wing, Mingbo Liu. Improved BFGS Method for Optimal Power Flow Calculation with Transient Stability Constraints, 2005, IEEE
[16] Kita H, Nishiya K, Hasegawa J. On-line preventive control for power systems based on energy function method [J]. Engineering in Japan, 1991,111(7):30-39
[17]孙景强,房大中,锺德成.暂态稳定约束下的最优潮流[J].中国电机工程学报, 2001, 25(13): 17-21
[18] A K Davaid, Lin Xu-jun. Dynamic Security Enhancment in Power Market Systems[J]. IEEE Trans on Power Systems, 2002, 17(2): 431-438
[19] Y Yuan, J.Kubokawa. Sasaki H. A solution of optimal power flow with multicontingency transient stability constraints[J], IEEE Trans. Power Systems,2003, 18(3): 1094-1102
[20] D.Gan, R.J.Thomas. Zimmerman R.D. Stability constrained optimal power flow [J], IEEE Trans. on Power Systems, 2000, 15(2): 535-540
[21] L.Chen, Y.Tada, H. Okamoto, R. Tanabe, A. Ono. Optimal operation solutions of power systems with transient stability constraints, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 48 (2001) 327-339.
[22] F. Guerra Vázquez, J. -J. Rückmann, O. Stein, G. Still. Generalized semi-infinite programming: A tutorial, Journal of Computational and Applied Mathematics, 2008, 217(3):394-419.
[23] X.J. Tong, S.Y. Wu, R. Zhou. New approach for the nonlinear programming with transient stability constraints arising from power systems, to appear in Computational Optimization and Applications, 2008.
[24] J.Gauvin, F.Dubeau. Differential properties of the marginal function in mathematical programming, Mathematical Programming, 19 (1982), 101-119.
[25] F.H. Clarke. Optimization and nonsmooth analysis, John Wiley, New York, 1983.
[26] O. Stein, G. Still. Solving semi-infinite optimization programs with interior point techniques, SIAM J. Control Optim, 2003, 24(3): 769-788.
[27] T.F. Coleman, Y. Li. An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optim., 6(1996), 418-445.
[28] Z. Drici, V. Lakshmikantham, W. Walter. Convex dependence of solution of differential equations in a Banch space relative to initial time, Nonlinear Analysis, 34(1998), 629-635.
[29] D.Ralph. Nonlinear programming advances in mathematical programming with complementarity constraints, 2007.
[30]王锡凡,现代电力系统分析,科学出版社, 2003.3
[31]王建全,陈晓峰,杨梅强,王俊,考虑暂态稳定约束下的输电阻塞管理计算,电力系统及其自动化学报,2007,19(3): 66-71.
[32]程瑜,张粒子.销售电价与用电需求的协整建模分析[J].中国电机工程学报,2006,26(7):118-122. Cheng Yu, Zhang Lizi. The co-integration analysis of power tariff and demand[J]. Proceedings of the CSEE, 2006,4,26(7):118-122(in Chinese).
[33]周明,严正,倪以信,等.含误差预测校正的ARIMA电价预测新方法[J].中国电机工程学报,2004,24(12):63-68. Zhou ming, Yan Zheng, Ni yixin, et al. A novel arima approach on electricity price forecasting with the improvement of predicted error[J]. Proceedings of the CSEE, 2004, 24(12): 63-68(in Chinese).
[34]刘广建,胡三高,戴俊良.电力系统边际电价的混沌特性及预测[J].中国电机工程学报,2003,23(5):6-8,175. Liu Guangjian, Hu Sangao, Dai Junliang. The chaotic property of system marginal price and its forecasting[J]. Proceedings of the CSEE, 2003,23(5):6-8,175 (in Chinese).
[35] J M Revaz, S.Storelli, Pepin J,et al. Dynamic tariffing of industrial consumers[C]. Europe Power Delivery, London, UK, 1998, 11-18.
[36] A P Sanghvi. Flexible strategies for load/demand management using dynamic pricing [J].IEEE Transaction on Power Systems, 1989, 4(1):83-93.
[37] A K David , Y C Lee . Dynamic tariffs: theory of utility-consumer interaction [J]. IEEE Transaction on Power Systems, 1989, 4(3): 904-911.
[38]吴杰康,任震,黄雯莹,等.在全面开放的电力市场中用户用电管理及其策略[J].电网技术,2001,25(8):50-53. Wu Jiekang, Ren Zhen,Huang Wenying, et al. Customer management of electric energy use and its strategy in fully open electricity market[J]. Power System Technology, 2001,25(8):50-53(in Chinese).
[39]程瑜,张粒子.基于ARCH模型的电价联动建模研究[J].中国电机工程学报,2006,26(9):126-130. Cheng Yu, Zhang Lizi. Electricity tariff linkage modeling research based on ARCH[J]. Proceedings of the CSEE, 2006,26(9):126-130(in Chinese).
[40] P.Pedroni. 2004. Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to PPP hypothesis: new results. Econometric Theory 20, 597–627.
[41]黄旭平.混业经营下金融发展与经济增长:基于亚洲地区的非稳定面板数据分析.国际经贸探索,2007, 23(1):66-70 (in chinese).
[42] P.Pedroni. Critical Values for Cointegration Tests in Heterogenous Panels with Multiple Regressors, Oxford bulletin of economics and Statistics, Special issue.1999, 653-669.
[43] S.Johansen. Estimation & hypothesis testing of cointegration vectors in Gaussian vectorauto-regressive models, Econometrica, 1991,59(6) :1551–80.
[44] P.Pedroni. Panel Cointegration Asymptotic & Finite Sample Properties of Pooled Time Series Tests, With an Application to the PPP Hypothesis: New Results.”Working paper, Indiana University,1997.
[45] J.Durbin, G.S.Watson. Testing for Serial Correlation in Least Squares Regression, I. Biometrika 37 (1950): 409-428
[46]萧政(美).面板数据分析,北京大学出版社,2005.09
[47]王勇领.预测计算方法,科学出版社, 1986.11.01
[2] G. Still. Generalized semi-infinite programming: Theory and methods, European Journal of Operational Research 119 (1999)301–313.
[3] W. Weber. Generalized semi-in?nite optimization: On some foundations, Preprint, University of Darmstadt, 1996.
[4] H.Th. Jongen, J.-J. Ruckmann, O. Stein. Generalized semi-infinite optimization: A first order optimality condition and examples, Mathematical Programming 83 (1998) 145–158.
[5] A. Hoffmann, R. Reinhardt. On reverse Chebyshev approximation problems, Technical University of Illmenau, Preprint No.M08/94, 1994.
[6] R. Hettich, G. Still. Semi-infinite programming models in Robotics, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics II, Akademie Verlag, Berlin, 1991.
[7] A. Kaplan, R. Tichatschke. On a class of terminal variational problems, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics IV, Peter Lang, Frankfurt a.M, 1997.
[8] G.-W.Weber. Generalized semi-infinite optimization and related topics, Habilitation Thesis, Darmstadt University of Technology, 1999.
[9] G.Still. Discretization in semi-infinite programming : the rate of convergence [ J ] . Math.Program. 2001,91:53-69
[10] G. Still. Generalized semi-infinite programming: Numercical aspects,1998
[11] Jan-J, Ruckmann, Juan Alfedo Gomez. On Generalized Semi-Infinite Program- ming. Sociedad de Estadistiva e Invstigacion OPerativa.2006(14),1:1-59
[12] Oliver Stein, Georg Still. On generalized semi-Infinite optimization and bilevel optimization. European Journal of Operational Research 142 (2002) 444–462
[13] Wan Zhongping, Wang Xiaojiao. He lin. Asymptotiv approximation method and its convergence on semi-infinite programming [J],Acta Mathematica Scientia,2004
[14] X.J. Tong, C. Ling, L. Qi.A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints, Journal of Computational and Applied Mathematics, 217(2008), 432-447.
[15] Yan Xia, Ka Wing, Mingbo Liu. Improved BFGS Method for Optimal Power Flow Calculation with Transient Stability Constraints, 2005, IEEE
[16] Kita H, Nishiya K, Hasegawa J. On-line preventive control for power systems based on energy function method [J]. Engineering in Japan, 1991,111(7):30-39
[17]孙景强,房大中,锺德成.暂态稳定约束下的最优潮流[J].中国电机工程学报, 2001, 25(13): 17-21
[18] A K Davaid, Lin Xu-jun. Dynamic Security Enhancment in Power Market Systems[J]. IEEE Trans on Power Systems, 2002, 17(2): 431-438
[19] Y Yuan, J.Kubokawa. Sasaki H. A solution of optimal power flow with multicontingency transient stability constraints[J], IEEE Trans. Power Systems,2003, 18(3): 1094-1102
[20] D.Gan, R.J.Thomas. Zimmerman R.D. Stability constrained optimal power flow [J], IEEE Trans. on Power Systems, 2000, 15(2): 535-540
[21] L.Chen, Y.Tada, H. Okamoto, R. Tanabe, A. Ono. Optimal operation solutions of power systems with transient stability constraints, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 48 (2001) 327-339.
[22] F. Guerra Vázquez, J. -J. Rückmann, O. Stein, G. Still. Generalized semi-infinite programming: A tutorial, Journal of Computational and Applied Mathematics, 2008, 217(3):394-419.
[23] X.J. Tong, S.Y. Wu, R. Zhou. New approach for the nonlinear programming with transient stability constraints arising from power systems, to appear in Computational Optimization and Applications, 2008.
[24] J.Gauvin, F.Dubeau. Differential properties of the marginal function in mathematical programming, Mathematical Programming, 19 (1982), 101-119.
[25] F.H. Clarke. Optimization and nonsmooth analysis, John Wiley, New York, 1983.
[26] O. Stein, G. Still. Solving semi-infinite optimization programs with interior point techniques, SIAM J. Control Optim, 2003, 24(3): 769-788.
[27] T.F. Coleman, Y. Li. An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optim., 6(1996), 418-445.
[28] Z. Drici, V. Lakshmikantham, W. Walter. Convex dependence of solution of differential equations in a Banch space relative to initial time, Nonlinear Analysis, 34(1998), 629-635.
[29] D.Ralph. Nonlinear programming advances in mathematical programming with complementarity constraints, 2007.
[30]王锡凡,现代电力系统分析,科学出版社, 2003.3
[31]王建全,陈晓峰,杨梅强,王俊,考虑暂态稳定约束下的输电阻塞管理计算,电力系统及其自动化学报,2007,19(3): 66-71.
[32]程瑜,张粒子.销售电价与用电需求的协整建模分析[J].中国电机工程学报,2006,26(7):118-122. Cheng Yu, Zhang Lizi. The co-integration analysis of power tariff and demand[J]. Proceedings of the CSEE, 2006,4,26(7):118-122(in Chinese).
[33]周明,严正,倪以信,等.含误差预测校正的ARIMA电价预测新方法[J].中国电机工程学报,2004,24(12):63-68. Zhou ming, Yan Zheng, Ni yixin, et al. A novel arima approach on electricity price forecasting with the improvement of predicted error[J]. Proceedings of the CSEE, 2004, 24(12): 63-68(in Chinese).
[34]刘广建,胡三高,戴俊良.电力系统边际电价的混沌特性及预测[J].中国电机工程学报,2003,23(5):6-8,175. Liu Guangjian, Hu Sangao, Dai Junliang. The chaotic property of system marginal price and its forecasting[J]. Proceedings of the CSEE, 2003,23(5):6-8,175 (in Chinese).
[35] J M Revaz, S.Storelli, Pepin J,et al. Dynamic tariffing of industrial consumers[C]. Europe Power Delivery, London, UK, 1998, 11-18.
[36] A P Sanghvi. Flexible strategies for load/demand management using dynamic pricing [J].IEEE Transaction on Power Systems, 1989, 4(1):83-93.
[37] A K David , Y C Lee . Dynamic tariffs: theory of utility-consumer interaction [J]. IEEE Transaction on Power Systems, 1989, 4(3): 904-911.
[38]吴杰康,任震,黄雯莹,等.在全面开放的电力市场中用户用电管理及其策略[J].电网技术,2001,25(8):50-53. Wu Jiekang, Ren Zhen,Huang Wenying, et al. Customer management of electric energy use and its strategy in fully open electricity market[J]. Power System Technology, 2001,25(8):50-53(in Chinese).
[39]程瑜,张粒子.基于ARCH模型的电价联动建模研究[J].中国电机工程学报,2006,26(9):126-130. Cheng Yu, Zhang Lizi. Electricity tariff linkage modeling research based on ARCH[J]. Proceedings of the CSEE, 2006,26(9):126-130(in Chinese).
[40] P.Pedroni. 2004. Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to PPP hypothesis: new results. Econometric Theory 20, 597–627.
[41]黄旭平.混业经营下金融发展与经济增长:基于亚洲地区的非稳定面板数据分析.国际经贸探索,2007, 23(1):66-70 (in chinese).
[42] P.Pedroni. Critical Values for Cointegration Tests in Heterogenous Panels with Multiple Regressors, Oxford bulletin of economics and Statistics, Special issue.1999, 653-669.
[43] S.Johansen. Estimation & hypothesis testing of cointegration vectors in Gaussian vectorauto-regressive models, Econometrica, 1991,59(6) :1551–80.
[44] P.Pedroni. Panel Cointegration Asymptotic & Finite Sample Properties of Pooled Time Series Tests, With an Application to the PPP Hypothesis: New Results.”Working paper, Indiana University,1997.
[45] J.Durbin, G.S.Watson. Testing for Serial Correlation in Least Squares Regression, I. Biometrika 37 (1950): 409-428
[46]萧政(美).面板数据分析,北京大学出版社,2005.09
[47]王勇领.预测计算方法,科学出版社, 1986.11.01