Research On The Solvability And Multiplicity Of Solutions To The System Of Equations For Electric Power Network

The system of equations for electric power network(also called “power flow equations”)is a system of high-dimensional nonlinear algebraic equations which is used to describe and reflect the electrical operating states.When the network topology and parameters as well as the nodal power injections(or the nodal voltage magnitudes)are known,the nodal voltage and angle at each node can be obtained by solving the corresponding system of equations,and then the transferred power on each line can be obtained.Solving the system of equations for electric power network provides the necessary information used for the planning,operation,control and optimization of the electric power network,hence,it is one of the most fundamental research subjects in the field of power systems.The research on solving the system of equations for electric power network dates from the 1920 s.Since then,researchers have done a lot of research regarding“improving the convergence capability of the solving methods” and “improving the computation speed of the solving methods”,which aims at obtaining the “high-voltage solution”(i.e.,the solution which is near the rated operating state)more precisely and faster.Consequently,many mature softwares for power sytems computation(such as,PSASP,BPA,MATPOWER,and PSAT)have been developed,where solving the system of equations for electric power network is the most basic functional module.These softwares have met the requirements of practical engineering well.However,at present,there still exist several key and difficult problems which need to be solved in the aspects of solvability and multiplicity of solutions to the system of equations for electrical power network,especially on the theoretical level,including: 1)When there exists a high-voltage solution,is the iterative method inevitably convergent?2)What kind of bifurcation phenomenons exist at the boundary of the solvability region? 3)How many solutions does the system of equations have? 4)How can we compute all the solutions reliably and efficiently?On the basis of the fixed-point theory,the discrete dynamical system theory,the dual principle and the computational algebraic geometry theory and methods,this dissertation investigates these key and difficult problems,and the research contents include:1.The fixed-point iteration method is one of the most common methods to solve the system of equations for electrical power network,among which the implicit Z-Bus(IZB)method is a typical representative.This dissertation investigates the convergence properties of the IZB method in the solvability region.First,the expression of IZB method defined in real vector space is derived according to the basic circuit laws.Second,we have proved that the IZB method is convergent everywhere in the solvability region(in other words,the IZB method is inevitably convergent as long as there exists a high-voltage solution);and then,we have demonstrated that the IZB method has widely convergent propery while the Newton method is locally convergent.Finally,an accelerating strategy is proposed to improve the computation speed of IZB method.2.The convergence characteristics and the bifurcation properties are two important topics in the discrete dynamical system theory,and they are closely related.Since the process of solving the parameterized system of equations for electric power network by means of the fixed-point iteration(FPI)method can also be regarded as a discrete dynamical system,it is necessary to study the bifurcation properties of FPI method in the solvability region.This dissertation has conducted bifurcation analysis for the FPI method,and the conclusion is that both the saddle-node bifurcation and the period-doubling bifurcation occur at the maximum loading point,i.e.,the FPI method becomes divergent when the total loads exceed the maximum loading point,moreover,the FPI method will eventually encounter a permanent oscillation between two periodic points of period two.3.On the basis of dual principle,this dissertation establishes a pair of inverse mappings through investigating the fixed-point iteration method,and then proposes a new method to obtain the low-voltage solution to the system of equations for electric power network.The proposed method takes full advantage of the dual property of the inverse mappings,and is able to obtain the low-voltage solution branch which is inverse to the high-voltage solution branch efficiently.4.As a system of high-dimensional nonlinear algebraic equations,the system of equations for electric power network usually has one high-voltage solution and many low-voltage solutions(which are closely related to the voltage stability phenomenon).On the basis of computational algebraic geometry theory,this dissertation proposes a new method to compute all the “real solutions”(the real solutions are composed of the high-voltage solution and all the low-voltage solutions)to the system of equations,which is called efficient homotopy continuation(EHC)method,by investigating the algebraic topology properties of the solution space.First,a special system of homotopy equations is constructed,where all the start solutions are easy to be obtained.Second,the numerical continuation method with adaptive step length control strategy is used to trace all the solution paths originating from the start solutions.Finally,all the real solutions to the system of equations for electric power network can be obtained.The case studies show that(to the author’s knowledge)the proposed EHC method in this dissertation is superior to any other existing method in the literature,in the aspect of finding all the real solutions to the system of equations for electric power network.5.Among all the low-voltage solutions,only the “type-1 low-voltage solutions”(where the Jacobian matrix in polar coordinates has exactly one positive real-part eigenvalue)are critical to voltage stability assessment.The key procedure of the energy function method for voltage stability assessmet is to obtain all the type-1 low-voltage solutions.This dissertation has presented several propositions which can be used to locate all the type-1 low-voltage solution branches,and then,the efficient homotopy continuation method is utilized to trace all the solution paths which originate from all the type-1 start solutions.Finally,all the type-1 low-voltage solutions to the system of equations for electric power network can be obtained.