Power System Reactive Power Optimization Based on Fuzzy Formulation and Interior Point Filter Algorithm

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1 Energy and Power Engineering, 203, 5, doi:0.4236/ee b34 Published Online July 203 (htt:// Power System Reactive Power Otimization Based on Fuzzy Formulation and Interior Point Filter Algorithm Zheng Fan, Wei Wang 2, ian-jiao Pu, Guang-yi Liu, Zhi Cai, Ning Yang 3 China Electric Power Research Institute, Beijing, China 2 Gan Electric Power Cororation, Lanzhou, China 3 Northeast China Grid Comany, Shenyang, China fzsxqs@63.com Received March, 203 ABSRAC Considering the soft constraint characteristics of voltage constraints, the Interior-Point Filter Algorithm is alied to solve the formulation of fuzzy model for the ower system reactive ower otimization with a large number of equality and inequality constraints. Based on the rimal-dual interior-oint algorithm, the algorithm maintains an udating filter at each iteration in order to decide whether to admit correction of iteration oint which can avoid effectively oscillation due to the conflict between the decrease of objective function and the satisfaction of constraints and enre the global convergence. Moreover, the filter imroves comutational efficiency because it filters the unnecessary iteration oints. he calculation relts of a ractical ower system indicate that the algorithm can effectively deal with the large number of inequality constraints of the fuzzy model of reactive ower otimization and satisfy the requirement of online calculation which realizes to decrease the network loss and maintain secified margins of voltage. Keywords: Power System; Reactive Power Otimization; Fuzzy; Filter; Interior-oint Algorithm; Online Calculation. Introduction On the remise of safe and stable oeration, the reactive ower otimization of ower system realizes hierarchical and regional balance of reactive ower, imroves voltage quality and reduces network loss by means of adjustment of the reactive ower controllers ch as teral voltages of generators, ta ositions of on load ta changers (OLCs and switchable shunt caacitor/reactors. he model of traditiona reactive ower otimization is ually exressed as imization of the active network loss under rigid voltage constraints which almost not considers the security margin of voltage and the characteristics of soft constraints when dealing with some voltage constraints. It makes the otimized voltage of some buses too close to their high limits, which becomes the threat of the system because the ability of enduring the variation is decreased remarkably [, 2]. Actually, the exected value of oerating voltage is a fuzzy concet. he voltage constraints of load nodes for reactive ower otimization are soft constraints. In [-4], a mathematical model that is closer to the reality is established by introducing the fuzzy theory for dealing with soft constraints; the formulation of fuzzy model for reactive ower otimization can decrease the network loss and maintain secified margins of voltage. Now, the rimal-dual interior-oint algorithm is widely used in the field of reactive ower otimization [5-8], because it has the advantages of raid convergence, strong robustness and insensitity to the initial value [5]. he interior-oint filter algorithm [9, 0] (IPFA is the latest achievement in nonlinear otimization research. Based on the rimal-dual interior-oint algorithm, it maintains an udating filter at each iteration in order to decide whether to admit correction of iteration oint which can avoid effectively oscillation due to the conflict between the decrease of objective function and the satisfaction of constraints and enre the global convergence, Moreover, the filter imroves comutational efficiency because it filters the unnecessary iteration oints [, 2]. In this aer, the IPFA is alied to solve the formulation of fuzzy model for the ower system reactive ower otimization with a large number of equality and inequality constraints. he examle of an actual ower system indicates that the algorithm can effectively deal with the large number of inequality constraints of the fuzzy model of reactive ower otimization and satisfy the requirement of online calculation which realizes to decrease the transmission loss and maintain secified margins of voltage. he aer is organized as follows. In section 2 the formulation of fuzzy model for the ower system reac- Coyright 203 SciRes.

2 694 Z. FAN E AL. tive ower otimization is introduced. In section 3 the solution of the model based on an interior oint filter algorithm is resented. est relts of a ractical system are reorted in section 4 and conclusions are made in section Formulation of Fuzzy Model for Reactive Power Otimization he traditional mathematical model of reactive ower otimization is established under a certain given active ower disatching mode by using the voltage amlitudes, hase angles, oututs of reactive ower comensation dece and transformation ratios of OLC as decision variables and reresenting other variables in the form of function of decision variables. he roblem can be described as follows: f st.. g 0 0 ( h h h h h 2 3 where x [ x, x, x ] ; x is the vector comosed of teral voltages of generators, oututs of reactive ower comensation deces and transformation ratios of OLCs; x2 is the vector comosed of the voltages of PQ buses, it has the characteristics of soft constraint; x3 is the vector comosed of voltage hase angles of all buses excet slack bus; f ( x is active network loss of ower system; g 0 are active balance equations( N -dimensional, N is the number of nodes in the system; g 0 2 are reactive balance equations( N NG -dimensional, NG is the number of generators in the system; hx ( are the inequality constraints of teral voltages and reactive ower oututs of generators, oututs of reactive ower comensation deces and transformation ratios of OLCs, which are regarded as hard constraints; h are the inequality constraints of voltages of PQ buses, which are regarded as soft constraints; h and h are the vectors of lower limits and uer limits of hard constraints resectively( -dimensional, N H is the m of 2 N G, N C and N, N C and N are the number of reactive ower comensation deces and OLCs resectively; h and h are the vectors of lower limits and uer limits of soft constraints resectively ( - dimensional, N S is the number of PQ buses. Fuzzy theory is introduced for the justfication of the roblem, and it is the key to select membershi functions. In this aer, the membershi functions for objective function and soft constraint variables are exressed as iecewise linear functions []. he membershi function of objective function is reresented as follows: f fm fm f ( f fm f fm (2 0 f( f x m where is the ideal imum reduction of network loss; f m is imum value of network loss. he membershi function of soft constraint variables is defined as follows: hi hi hi hi hi( x hi hi( x hi ( hi (3 hi( x hi hi hi( hi x 0 hi hi orhi h i where is allowable imum offset of h i. On the basis of fuzzy set theory [], the roblem ( is converted to imum satisfaction degree roblem [2], namely S where S st.. g 0 g 0 2 f S f εv S h εv S h h h h 0 S is satisfaction degree. 3. Fuzzy Reactive Power Otimization Based on Interior Point Filter Algorithm he interior-oint filter algorithm [9, 0] is based on the rimal-dual interior-oint algorithm and maintains an udating filter at each iteration in order to avoid oscillation when contradiction exists between the decrease of objective function and the satisfaction of constraints and enre the global convergence. he algorithm also imroves calculation efficiency due to the filter filters unnecessary iteration oints. o solve (4 using IPFA, inequality constraints are transformed into equality constraints by adding to slack variables and new objective function is built by introducing barrier function. Meanwhile, the roblem (4 is converted to nonlinear imizing roblem, that is m (4 Coyright 203 SciRes.

3 Z. FAN E AL. 695 ( xx,, S S ( ln sil ln siu i i ln sil i i ln s iu ln s ln s ln s sl f st.. g 0 0 f S fm sf 0 sf 0 εv S h s l 0 sl 0 εv S h 0 0 hx ( h sl 0 sl 0 hx ( h 0 0 S ssl 0 ssl 0 S s 0 s 0 l, u, l, u, ssl, s where x, s s s s and s f are the vectors comosed of original variables, x [ s, s, s, s, s, s, s ]. l u l u sl f In IPFA, the decrease of objective function (, xx, S is equivalent to the satisfaction of constraints which is denoted as: g f S fm sf εv S h sl ( xx,, S h εv S h (6 hx ( h sl hx ( h S ssl S s A set called filter is maintained at each iteration k, which is defined as: k k k k k k k F ( ( x, x, S, ( x, x, S R ( ( x, x, S, ( x, x, S F is called if the condition k k k k k k ( x, x, S ( x, x, S k k k k k k ( xk, x k, k ( xk, x k, k or S S is satisfied. he iteration oint ( xk, x k, Sk is acceted only if objective function ( xx,, S and infinite norm of the constraints set ( xx,, S meet (8. At iteration k, the filter is udated according to 2 k (5 (7 (8 (9. 2 (, R : ( xk, x k, Sk Fk Fk (9 and ( xk, x k, Sk In order to solve (5, the Lagrangian function is reresented as: L S ( lns lns il i i ln sil ln siu i i ln ssl ln s ln sf y g y2 y l ( εv S h sl y u ( εv S h s u yl ( h h sl yu ( h h y sl ( S ssl y ( S s y ( f S f s f m f l u l u sl and iu (0 where y, y2, y, y, y, y,y,y y f are the vectors comosed of dual variables. According to Karush-Kuhn-ucker condition, it is the necessary condition that all the artial derivatives of the Lagrangian function are equal to zero if the imum of the roblem (5 is existed. Derivation rocess and solution rocedure for using the IPFA to solve the nonlinear rogramg model can be referred to []. 4. Case Study est case of a ractical 244-bus ower system is emloyed to validate the solution of fuzzy model for reactive ower otimization based on interior-oint filter algorithm. he system contains 76 generation units, 2 on-load ta changers and 56 shunt caacitor/reactors. he current controllable deces include 36 generation units, 2 on-load ta changers and 52 shunt caacitor/reactors. he comuter configuration used for case study is Intel Core i GHz and 2 GB memory. In this case, the reference ower is 00 MVA; the lower and uer limits of bus voltages are set at 0.9 and. (.u.; there are 2 buses olated voltage limits in the initial state; the initial network loss is.256 (.u.. ε and the elements of vector v are set to and resectively. After otimization calculation for fuzzy model, the satisfaction degree is , all bus voltages are within limits and the network loss is reduced to Comarisons of otimal relts and the initial values are in able. Moreover, comarisons of solng relts of fuzzy model and traditiona model for reactive Coyright 203 SciRes.

4 696 Z. FAN E AL. able. Comarisons of otimal relts and the initial values. Initial values Otimal relts network loss(.u number of buses olated voltage limits 2 0 imum voltage of PQ bus(.u imum voltage of PQ bus (.u able 2. Comarisons of solng relts of fuzzy model and traditional model. fuzzy model traditiona model network loss(.u number of buses olated voltage limits 0 0 imum voltage of PQ bus(.u imum voltage of PQ bus (.u comuting time(s ower otimization are shown in able 2. From the able and able 2, it can be seen that the algorithm can effectively deal with the large number of inequality constraints of the fuzzy model of reactive ower otimization and has high comute efficiency. After otimization calculation, there is no bus olated voltage limit through the adjustment of the reactive ower controllers. he imum voltage of PQ bus is its uer limit when traditiona model is used, however, the voltages of PQ buses maintain secified margins when fuzzy model is used, which imroves the safety level of the system. Meanwhile, the network loss is decreased to.083 from.256 and the decreasing range is 8.63% when traditiona model is used, whereas the network loss is decreased to.0258 from.256 and the decreasing range is 8.04% when fuzzy model is used. So the decreasing range of network loss obtained by solng the fuzzy model is less than that obtained by solng the traditiona model, but the level of voltage security is imroved when fuzzy model is used. he examle of the ractical ower system indicates that the algorithm can effectively solve the fuzzy model of reactive ower otimization and satisfy the requirement of online calculation. After otimization calculation, the olated voltages are corrected; the voltage soft constraint buses are ket secified margins; at the same time the network loss is reduced. 5. Conclusions his aer uses the interior oint filter algorithm to solve the formulation of fuzzy model for the ower system reactive ower otimization considering the soft constraint characteristics of voltage constraints. Otimization relts show that the algorithm can effectively deal with the large number of equality and inequality constraints of the fuzzy model for the ractical ower system and satisfy the requirement of online calculation which realizes to decrease the network loss and maintain security margins of voltage. REFERENCES [] H. Yuan, G. G. Xu and J. Y. Zhou, A Reactive Power/voltage Otimization Based on Fuzzy Linear Programg, Power System echnology, Vol. 27, No. 2, 2003, [2] F. R. u and X. R. Wang, Fuzzy Modeling for Power System Reactive Power Otimization, Power System Protection and Control, Vol. 38, No. 3, 200, [3] Y. N. Li, L. Z. Zhang and Y. H. Yang, Reactive Power Otimization Under Voltage Constraints Margin, Proceedings of the CSEE, Vol. 2, No. 9, 200,. -4. [4] K. omsoc, A Fuzzy Linear Programg Aroach to the Reactive Power/Voltage Control Problem, IEEE rans on Power Systems, 992, Vol. 7, No., doi:0.09/ [5] M. B. Liu, Y. Cheng and S. H. Lin, Comarative Studies of Interior-oint Linear and Nonlinear Programg Algorithms for Reactive Power Otimization, Automation of Electric Power Systems, Vol. 26, No., 2002, [6] M. B. Liu, S. K. so, Y. Cheng, An Extended Nonlinear Primal-dual Interior-oint Algorithm for Reactive-ower Otimization of Large-scale Power Systems with Discrete Control Variables, IEEE rans on Power Systems, Vol. 7, No. 4, 2002, doi:0.09/pwrs [7] J. D. Xu, X. Q. Ding, Z. C. Qin Zhencheng, et a., A Nonlinear Predictor-Corrector Interior Point Method for Reactive Power Otimization in Power System, Power System echnology, Vol. 29, No. 9, 2005, [8] K. Pan, X. S. Han and X. X. Meng, Solution Princiles Study of Nonlinear Correction Equations in Primal-dual Interior Point Method for Reactive Power Otimization, Power System echnology, Vol. 30, No. 9, 2006, [9] U. Michael, S. Ulbrich, L. N. Vicente, A Globally Convergent Primal-dual Interior-oint Filter Method for Nonlinear Programg, Mathematical Programg, Vol. 00, No. 2, 2004, doi:007/s [0] W. Andreas and L.. Bieglery, Line Search Filter Methods for Nonlinear Programg: Motivation and Global Convergence, Yorktown Heights, USA: IBM. J. Watson Research Center, 200. Coyright 203 SciRes.

5 Z. FAN E AL. 697 [] S. Yang, J. Y. Zhou, Q. Li, et a., An Interior-oint Reactive Power Otimization Based on Filter Set, Power System Protection and Control, 20, Vol. 39, No. 8, [2] Y. Y. Sun, G. Y. He and S. W. Mei, A New Otimal Power Flow Algorithm Based on Filter Interior Point Method, Advanced echnology of Electrical Engineering and Energy, Vol. 26, No. 2, 2007, Coyright 203 SciRes.