Application of Constrained Nonlinear Minimization to Estimate Parameters of Transmission Line in Power System

Prechanon   Kumkratug

Department of Electrical Engineering, Faculty of Engineering

Kasetsart University at Sriracha Campus

Thailand, 20230

The accurate parameters of the transmission line consisting of the series impedance (Z) and the shunt admittance(Y) play are very important data in analysis, designing, planing, operation, and protection of powersystem [1]. The Zand Y are due to the effect of magnetic field and electric fields, respectively. There are two possible ways to access the parameters of the transmission line. The first method uses the commissioning data such as, transmission line type, and arrangement, etc., to calculate the parameters [2]. However, commissioning data may be changed by many factors such as conductor degradation, temperature, sag, length, etc. Then it affects on the accuracy of the parameters. The second approach is the so called estimation method. This method uses the measurable data such as voltage, power, etc., with some iterative numerical methods to estimate parameters as can be seen in many previous researches [3, 4]. Research [3] applied the state estimation to eliminate inaccuracies and errors of calculation. The effects of temperature and sag of transmission were also considered in algorithm as given in [4]. Most of previous researches required the data from PMU [5–7]. However, the PMU is generally implemented at some important power substations because it is an expensive instrument.

This paper proposes the method of estimating parameters of transmission line without the use of the PMU data. The ready data at power substations voltage magnitude (V), active (P) and reactive power (Q) is incorporated into mathematical model to estimate the parameters. The constrained nonlinear minimization available in MATLAB optimization toolbox is applied in this paper.

This section will provide the mathematical model of transmission line in power system. Figure 1 (a) shows the lumped π model of transmission line equivalent. Each π model consists of the series impedance Z and the shunt admittance Y. After some mathematical manipulation of nonlinear differential equation, the transmissionline equivalent in one π model is represented in Figure 1(b). The Z′ and Y′ are given by [8]

(1)

(2)

(a)

(b)

(c)

Figure 1. Transmission line equivalent (a) distributed π model (b) with a π model (c) two-port representation

The transmission line equivalent in Figure 1(b) can be represented in two-port network as shown in Figure 1(c). The sending end voltage V_S and current I_S are given by

(3)

(4)

Here

(5)

(6)

(7)

Expressing the sending end voltage in polar form as and the receiving end voltage as reference, the complex power at sending end S_S is written by

(8)

After some mathematical manipulations of (4) and (8), the real (P_S) and reactive power (Q_S) at sending end are given by

(9)

(10)

The complex power at recieving end S_R is written by

(11)

Similarly, some mathematical manipulations of (4) and (11), the real (P_R) and reactive power (Q_R) at recieving end are given by

(12)

(13)

It can be seen in (9), (10), (12), and (13) that they are in the nonlinear equations of the parameters of the transmission line (R, X, and B) and measurable data at power substation (V_S, δ, V_R, P_S, Q_S, P_R, and Q_R). The V_S, V_R, P_S, Q_S, P_R, and Q_R are typically available at power substation. However, the δ must utilize the PMU which is usually implemented at some important power substation because it is expensive instrument. Thus, this paper proposes the method of access the δ. The general form of four equations in four variables of the above equations written by

(14)

Here

(15)

(16)

Nonlinear function optimization is important method to solving parameter in the objective function (F(x)). The necessary condition to minimize the objective equations written by [9, 10]

(17)

Subject to the inequality constrains

(18)

Figure 2. Window of MATLAB optimization toolbox

This paper applies the MATLAB optimization toolbox to solve the above objective equations. Figure 2 shows the window of the MATLAB optimization toolbox [11]. It may be mentioned here that normalization techniques called per unit power system will reduce the effects of the variation in the scale of the solution. In the practical power system industry, the lower x_min and upper x_max bounds in the range between 0 and 1. The initial point x₀ may come from the historical data of transmission line implementation. The proposed method is tested on an example to validate the concept involved. A three-phase, 50 Hz, 220 kV transmission line 340 km long. The transmission line parameters considered as the actual parameters are

Z = 0.04+ j37.7 ΩY=j3.761×10−4siemens

The measurable data at power substation is

P_S=147.03 MW Q_S=111.68 MVAR V_S=241.14 kV

P_R=144 MWQ_R=108 MVAR V_R=220 kV

Figure 3. Comparisons of various numerical methods

There are many solver types in Optimization Toolbox such as fsolve, fmin, fmincon, ga, etc. In this paper, the fslove, fmin and fmincon are compared. It can be seen from Figure 3 that the parameter estimation using the fmincon with constrained nonlinear minimization provides the best results. With 25 iterations, the value of the objective function is less than 1×10^-5 and it is considered as global minimum. Then the estimated parameters are approximately equal to the actual parameters.

The parameters of the transmission line may be changed by many factors such as the physical properties of a transmission line, temperature, etc. This paper applied constrained nonlinear minimization available in the MATLAB optimization toolbox for estimating parameters of transmission line in power system. It was found that the fmincon with nonlinear constrained minimization provided the best results. This method allows us to utilize the historical data to faster achieve the desired accuracy. The normalization techniques called per unit power system also help to reduce the effects of the variation in the scale of the solution. The simulation results indicated that the proposed method provides the perfect agreement of estimating parameters of transmission line.

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