Initialization of Electromagnetic Transient Simulation Using Boundary Value Solution Method

Abstract

Initialization is an important preliminary work in electromagnetic transient simulation which is typically executed from the periodic steady-state. To this end, researchers have proposed so far a variety of initialization methods, among which the initialization based on fixed-point iteration or the so-called the EMTP-based approach has been widely used. However, this type of method can be very time consuming for a large power system due to its weak convergence. In this paper, the initialization is described as a two-point differential boundary value problem. On this basis, the boundary value solution technique is used to solve this problem, therefore, a new electromagnetic transient initialization method is derived. The proposed method is a strict Newton algorithm, so it has better convergence than the traditional fixed-point iteration method. Case studies conducted on two typical network have confirmed the effectiveness of the proposed method. Introduction The electromagnetic transient simulation and related programs (the EMTP-type programs) are widely used in the electromagnetic transient studies of power systems. The transient behavior of a network can be represented mathematically by ordinary differential equations in the continuous time-domain. Using numerical integration methods such as the implicit trapezoidal rule to solve the ordinary differential equations, the transient solution of the network can be obtained in the discrete time-domain. The transient study of a network normally begins from the network’s steady state. If the system is lightly damped, the time for it to reach its steady state can be significant and the simulation can be prohibitively expensive. The effective initialization for electromagnetic transient simulation or the EMTP-type programs, therefore, becomes unavoidable. Alternatively, the steady state solution for the periodic ordinary differential equations may be of interest in itself for the computer-aided design of nonlinear circuits and harmonic analysis [1]. Some methods to initialize the EMTP-type programs have been proposed [2, 3]. Based on the needs of the EMTP program development, the earliest proposed electromagnetic transient initialization method should be the phasor-solution technique [4]. In this approach all the elements of a network are represented in the phasor-domain. The system solution is obtained at the fundamental system frequency by solving the network matrix equation. The so-obtained system quantities are used to generate the initial state of the network. This solution technique is simple but limited to the linear and lumped-element networks. Another important initialization method is the so-called EMTP-based approach [5]. This approach performs a network’s initialization within the EMTP solution frame. That is, the initial steady state of a network can be directly established for a given initial state by simply integrating the system equations until the response becomes periodic. From a mathematical point of view, the solution process of this method is equivalent to the classical fixed-point iteration. It is well known that the fixed-point iteration is a general numerical method, but its convergence is not as good as the Newton method. So, the EMTP-based approach can closely predict the network’s initial state even for a nonlinear and time-variant network, but it is not practical for large systems due to its low efficiency. Since the phasor-solution approach and the EMTP-based technique may not be appropriate for the initialization of large power systems, the load-flow program-based initialization technique has been sought to initialize an EMTP-type program [2]. This technique is efficient and suitable for large AC power systems, but it is obviously not suitable for DC transmission systems when using the electromagnetic transient model. Note that the currently widely used EMTP-based approach is just to simply integrate the system differential equations step by step for a sufficient number of periods until the transient response becomes negligible. The main problem with this approach is its expensively slow convergence. To solve this problem, a natural idea is to use Newton iteration. However, a precise expression for the Jacobian matrix may be difficult to obtain in general. In fact, evaluating the Jacobian matrix by finite differences is inefficient and may cause numerical error because we must solve the nonlinear differential equations many times if nonlinearities exist. A more accurate and efficient method involves the solution of related sensitivity systems which describe the linearization of a nonlinear system [1, 3], but this improvement is still not a strict Newton method. In this paper, the electromagnetic transient initialization is described as a differential boundary value problem, thus, the boundary value solution technique can be used to solve the initialization. In this approach, theoretically we can use different boundary value methods [6], but considering that the EMTP-type programs are mainly using the trapezoidal rule, we also use the trapezoidal integration method to implement initialization. In order to solve the problem that the Jacobian matrix involved in the boundary value solution is too large, a block recursive solution method is proposed in this paper. In theory, the proposed initialization method is a rigorous Newton algorithm and thus has better convergence. Section 2 introduces the mathematical expression or the formation of electromagnetic transient initialization problem. On this basis, this section also introduces the basic fixed-point iteration method, which is currently widely used for initialization. Section 3 describes simply the solution technique for differential boundary value problem, meanwhile this section derives the initialization method based on this technique. To further elaborate the proposed method, in Section 4, the proposed method is applied to a general linear differential system with periodic input, and compared with the fixed-point iteration. This leads to an interesting result. In section 5, the proposed methods are tested and compared with the fixed-point iteration method using two simple but typical networks. The Problem Formation and EMTP-Based Methods From the mathematical point of view, the electromagnetic transient simulation is just the solution of the differential initial value problem. That is 0 ( , ) ( 0) d t t dt    x x f x x x , (1) where, x and ( , ) t f x are n vectors. For a given initial value 0 x , we can obtain discrete time-domain numerical solutions of state variables ( ) t x by numerical integration. Electromagnetic transient simulation is mainly to study the transient response of the power system under disturbance or fault conditions. Usually, this process must start from a steady state initial condition. This is the so-called initialization problem. Note that, in steady state of power systems, ( , ) t f x is periodic in t of period T . Henceforth, we can assume that the Eq. 1 has a periodic solution ( ) t x of period T . In this case, the goal of the initialization is to determine the initial state 0 x y such that integrating Eq. 1 from this initial state over the interval   0 T , we obtain the periodic solution ( ) t x of period T . This is essentially a two-point boundary value problem in which the solution to Eq. 1 in the interval   0 T , must satisfy the following boundary condition: 0 ( )= T y x x y , (2) Since 0 ( ) ( , ) T T d      x y f x (3) we can express the above problem in terms of the mapping 0 ( ) ( , ) T d      y F y y f x (4) where, ( ) F y is a function of the initial state y . For the solution of y , the most basic method is the so-called fixed-point iteration method which can be described as ( 1) +1 0 ( ) ( , ) k T k k k d       y F y y f x (5) where, the superscript k represents the number of iterations. This iteration must be repeated until the following condition is met: +1 ( ) k k k     y y F y (6) where,  is the required accuracy. It is easy to understand that numerical integration is required in the iterative process described by Eq. (5). For this, the trapezoidal rule is usually used because most EMTP-Type programs use this integration method. Obviously, we need to integrate the system step-by step for a full period to complete one iteration, and repeat this process multiple times until the transient response becomes negligible. This is precisely the method currently used with the EMTP-Type programs to arrive at the steady-state response. Therefore, this method is also commonly referred to as the EMTP-based approach. This procedure is satisfactory if the transient decays rapidly. However, for lightly damped networks typical of power systems, the transient usually decays very slowly and this will make the initialization very time-consuming. Initialization Using Boundary Value Solution Technique The main purpose of this study is to improve the convergence of initialization, so as to improve the efficiency of electromagnetic transient simulation. To this end, one of the most important ways is to use the Newton method to solve the Eq. 4. That is ( ) ( ) k k k k     F I J y F y y (7) where, I is a unit matrix and F J is the Jacobian matrix, ( )   F F y J y (8) However, it can be seen from Eq. 4 that the precise expression for the Jacobian matrix F J is difficult to obtain when ( , ) t f x is nonlinear. In fact, if we use stepwise integration, which is a serial solution to calculate the integral term involved in ( ) F y , it is really difficult to get the exact expression of F J . Since the initialization is a two-point boundary value problem, the efficient method to solve it is naturally to use the boundary value method [6], [7]. Note that, for the initialization problem, the most important is the solution technique within the boundary value methods (BVM). Different from the traditional step-by-step integration process, the in