外文翻译--基于优化的牛顿—拉夫逊法和牛顿法的潮流计算

外文翻译--基于优化的牛顿—拉夫逊法和牛顿法的潮流计算英文文献Power Flow Calculation by Combination of Newton-Raphson Method and Newton’s Method in Optimization.Andrey Pazderin, Sergey YuferevURAL STATE TECHNICAL UNIVERSITY ? UPIE-mail: pav@//0>., usv@//.Abstract--In this paper, the application of the Newton’s method in optimization for power flow calculation is considered. Convergence conditions of the suggested method using an example of a three-machine system are investigated. It is shown, that the method allows to calculate non-existent state points and automatically pulls them onto the boundary of power flow existence domain. A combined method which is composed of Newton-Raphson method and Newton’s method in optimization is presented in the paper.Index Terms?Newton method, Hessian matrix, convergence of numerical methods, steady state stabilityⅠ.INTRODUCTIONThe solution of the power flow problem is the basis on which otherproblems of managing the operation and development of electrical power systems EPS are solved. The complexity of the problem of power flow calculation is attributed to nonlinearity of steady-state equations system and its high dimensionality, which involves iterative methods. The basic problem of the power flow calculation is that of the solution feasibility and iterative process convergence [1].The desire to find a solution which would be on the boundary of the existence domain when the given nodal capacities are outside the existence domain of the solution, and it is required to pull the state point back onto the feasibility boundary, motivates to develop methods and algorithms for power flow calculation, providing reliable convergence to the solution.The algorithm for the power flow calculation based on the Newton's method in optimization allows to find a solution for the situation when initial data are outside the existence domain and to pull the operation point onto the feasibility boundary by an optimal path. Also it is possible to estimate a static stability margin by utilizing Newton's method in optimization.As the algorithm based on the Newton’s method in optimization has considerable computational cost and power control cannot be realized in all nodes, the algorithm based on the combination of the Newton-Raphson met hods and the Newton’s method in optimization is offered to be utilizedfor calculating speed, enhancing the power flow calculation.II. THEORETICAL BACKGROUNDA.Steady-state equationsThe system of steady-state equations, in general, can be expressed as follows: where is the vector of parameters given for power flow calculation. In power flow calculation, real and reactive powers are set in each bus except for the slack bus. Ingeneration buses, the modulus of voltage can be fixed. WX,Y is the nonlinear vector function of steady-state equations. Variables Y define the quasi-constant parameters associated with an equivalent circuit of an electrical network. X is a required state vector, it defines steady state of EPS. The dimension of the state vector coincides with the number of nonlinear equations of the system 1. There are various known forms of notation of the steady-state equations. Normally, they are nodal-voltage equations in the form of power balance or in the form of current balance. Complex quantities in these equations can be presented in polar or rectangular coordinates, which leads to a sufficiently large variety forms of the steady-state equations notation. There are variable methods of a nonlinear system of steady-state equations solution. They are united by the incremental vector of independent variables ΔX being searched and the condition of convergence being assessed at each iteration.B. The Newton's method in optimizationAnother way of solving the problem of power flow calculation is related to defining a zero minimum of objective function of squares sum of discrepancies of steady-stateequations:2?The function minimum 2 is reached at the point where derivatives on all required variables are equal to zero: 3It is necessary to solve a nonlinear set of equations 3 to find the solution for the problem. Calculating the power flow, which is made by the system of the linear equations with a Hessian matrix at each iteration, is referred to as the Newton'smethod in optimization [4]: 4The Hessian matrix contains two items: 5During the power flow calculation, the determinant of Hessian matrix is positive round zero and negative value of a determinant of Jacobian .This allows to find the state point during the power flow calculation, when initial point has been outside of the existence domain.The convergence domain of the solution of the Newton's optimization method is limited by a positive value of the Hessian matrix determinant. The iterative process even for a solvable operating point can converge to an incorrectsolution if initial approximation has been outside convergencedomain. This allows to estimate a static stability margin of the state and to find the most perilous path of its weighting.III. INVESTIGATIONS ON THE TEST SCHEMEConvergence of the Newton's method in optimization with a full Hessian matrix has been investigated. Calculations were made based on program MathCAD for a network comprising three buses the parameters of which are presented in Figure 1.Dependant variables were angles of vectors of bus voltage 1 and 2 ,independent variables were capacities in nodes 1 and 2, and absolute values of voltages of nodes 1, 2 and 3 were fixed.Fig. 1 ? The Test schemeIn Figure 2, the boundary of existence domain for a solution of the steady-sta te is presented in angular coordinates δ1-δ2. This boundary conforms to a positive value of the Jacobian determinant:As a result of the power flow calculation based on the Newton method in optimization, the angle values have been received, these values corresponding to the given capacities in Fig.2 generation is positive and loading is negative.For the state points which are inside the existence domain, the objective function 2 has been reduced to zero. For the state points which are on the boundary of the existence domain, objective function 2 has not been reduced to zero and the calculated values of capacities differed fromthe given capacities.Fig. 2 ? Domain of Existence for a SolutionFig.3 - Boundary of existence domain In Fig.3, the boundary of the existence domain is presented in coordinates of capacities P1-P2. State points occurring on the boundary of the existence domain 6 have been set by the capacities which were outside the existence domain. As a result of power flow calculation by minimization 2 based on the Newton's method in optimization, the iterative process converges to the nearest boundary point. It is due to the fact that surfaces of the equal level of objective function 2 in coordinates of nodal capacities are proper circles for threemachine system having the centre on the point defined by given values of nodal capacitiesThe graphic interpretation of surfaces of the equal level of objective function for operating point state with 13000 MW loading bus 1 and 15000 MW generating bus 2 is presented in Fig.3.Hessian matrix is remarkable in its being not singular on the boundary of existence domain. The determinant of a Hessian matrix 5 is positive around zero and a negative value of the Jacobian matrix determinant. This fact allows the power flow to be calculated even for the unstable points which are outside existence domain. The iterative process based on the system of the linear equations 4 solution has converged to the critical stability point within 3-5 iteration. Naturally,the iterative process based on Newton-Rapson method is divergent for such unsolvable operating points.The convergence domain of the method under consideration has been investigated. What is meant is that not all unsolvable operating points will be pulled onto theboundary of existence domain. A certain threshold having been exceeded the iterative process has begun to converge to the imaginary solution with angles exceeding 360It is necessary to note that to receive a critical stability operating point in case when initial nodal capacities are set outside the boundary of the existence domain, there is no necessity to make any additional terms as the iterative process converges naturally to the nearest boundary point.Pulling the operation point onto feasibility boundary is not always possible by the shortest and optimal path. There are a number of constraints, such as impossibility of load consumption increase at buses, constraints of generation shedding/gaining at stations. Load following capability of generator units is various, consequently for faster pulling the operation point onto the feasibility boundary it is necessary to carry out this pulling probably by longer, but faster path.The algorithm provides possibility of path correction of pulling. It is carried out by using of the weighting coefficients, which define degree of participation of eachnode in total control action. For this purpose diagonal matrix A of the weighting coefficients for each node is included into the objective function 2:All diagonal elements of the weighting coefficient matrix A should be greater-than zero:When initial approximation lies into the feasibility domain, coefficients are not influence on the computational process and on the result.In the figure 4 different paths of the pulling the same operation point onto feasibility boundary depending on the weighting coefficients are presented. Paths are presented for two different operating points.In tables I and II effect of weighting coefficients on the output computation is presented. In tables I and II k1 and k2 are weighting coefficient for buses 1 and 2, respectively.TABLE IWEIGHTING COEFFICIENT EFFECT ON OUTPUT COMPUTATION FOR INITIAL SET CAPACITIES P1 -13000 MW AND P2 15000 MWCoefficients ,MW ,MW ,deg ,deg1,1 -7800 9410 -45 555,1 -8600 8080 -69 250.005,1 -5700 10140 -1 93TABLE IIWEIGHTING COEFFICIENT EFFECT ON OUTPUT COMPUTATION FOR INITIAL SET CAPACITIES P1 -8000 MW AND P2 -5000 MWCoefficients ,MW ,MW ,deg ,deg1,1 -4360 -1680 -92 -800.01,1 -1050 -4920 -76 -941,0.35 5800 0 -99 -71Fig.4 - Paths of pulling the operation point onto the feasibility boundaryIV. COMBINATION OF METHODSIf to compare the Newton’s method in optimization for power flow calculation with newton-Raphson using a Jacobian matrix, the method computational costs on eachiteration will be several times greater as the property of Hessian matrix being filled up by nonzero elements 2.5-3 times greater than with Jacobian one. Each row of Jacobian matrix corresponding to any bus contains nonzero elements corresponding to all incident buses of the scheme. Each row of Hessian matrix contains nonzero elements in the matrix corresponding not only to the neighboring buses, but also their neighbors. However, it is possible to compensate this disadvantage through the combination Newton-Rap son method with Newton’s method in optimization. It means that the part of nodes can be calculated by conventional Newtonmethod, and the remaining buses will be computed by Newton’s method in optimization. The first group of passive nodes consists of buses in which it is not possible to changenodal capacity or it is not expedient. Hence, emergency control actions are possible only in a small group of buses supplying with telecontrol. Most of the nodes includingpurely transit buses are passive. Active nodes are generating buses in which operating actions are provided. Such approach allows to fix nodal capacity for all passive buses of the scheme which have been calculated by Newton-Rap son method. In active buses which have been calculated by Newton’s method in optimization, deviations from set values of nodal capacity are possible. These deviations can be considered as control action. The power flow calculation algorithm based on combination Newton ? Ra phson method and Newton’s method in optimization can be presented as follows:1.The linear equation system with Jacobian matrix is generated for all buses of the scheme.2. The solution process of the linear equation system with Jacobian is started by utilizing the Gauss method for all passive buses. Factorization of the linear equations system is terminated when all passive buses are eliminated. Factorizedequations are kept.3.The nodal admittance matrix is generated from not factorized the part of Jacobian matrix corresponding to active buses. This admittance matrix contains parameters of the equivalent network which contains only active buses.4.The linear equation system with Hessian matrix 4 is generated for the obtained equivalent by Newton’s method in optimization.5.The linear equation system with Hessian matrix is calculated and changes of independent variables are defined for active buses.6.Factorized equations of passive buses are calculated, and changes of independent variables are defined for passive buses.7.The vector of independent variables is updated using the changes of independent variables for all buses.8. New nodal capacities in all buses of the network are defined; constraints are checked; if it necessary, the list of active buses will be corrected.9. Convergence of the iterative process is checked. If changes of variables are significant, it is necessary to return to item 1.Taking into account the number of active buses in the network aren’t large, computationa l costs of such algorithm slightly exceed computational costs of the Newton-Rapson method.V. CONCLUSION1. The power flow calculation of an electric network by minimizingthe square sum of discrepancies of nodal capacities based on the Newton's method in optimizationmaterially increases the productivity of deriving a solution for heavy in terms of conditions of stability states and the unstable states outside the existence domain of the solution.2. During the power flow calculation, the determinant of Hessian matrix is positive around zero and negative value of the Jacobian matrix determinant. The iterative process naturally converges to the nearest marginal state point during the power flow calculation, when the initial operating point has been outside of the existence domain.3. There is a possibility of control action correction for the pulling operation point onto feasibility boundary by using matrix of weighting coefficients.4. Utilization of the combined method for power flow calculation all ows to use all advantages of Newton’s method in optimization and to provide high calculating speed.5. In case when the setting nodal powers are outside the existence domain, there are discrepancies in the active buses, which can be considered as control actions for pulling the state point onto the feasibility boundary. When the initial state point is inside the existence domain, the iterative process converges with zero discrepancies for both active and passive buses.中文翻译基于优化的牛顿??拉夫逊法和牛顿法的潮流计算摘要??在本文中,考虑到了优化的牛顿法在潮流计算中的应用。