Parameter identification of interconnected power system frequency after trip-out of high voltage transmission line

Abstract

Accurate modeling and parameters of high voltage (HV) grid are critical for stability research of system frequency. In this paper, simulation modeling of the system frequency was conducted of an interconnected power system with HV transmission lines in China. Based on recorded tripping data of the HV transmission lines, system parameters were identified by using genetic algorithm (GA). The favorable agreement between simulation results and recorded data verifies the validity of gird models and the accuracy of system parameters. The results of this paper can provide reference for the stability research of HV power grid.

Keywords

high voltage (HV) grid / frequency stability parameter identification / primary frequency regulation (PFR)

Introduction

With the development of Chinese power industry and accumulation of experiences in running large-scale regional power grids that cover several provinces, the time for interconnecting regional power networks was coming. Electricity transmission from West to East China, across North-west Grid and Central China, made a great contribution to achieving energy resources optimization configuration. However, the weak relation following long distance brought about some new problems [1]. When an abrupt disturbance occurs, the difference of frequency regulation in multi-areas will lead to the uneven distribution of power response. The tie-line transmission may overload or fluctuate, which threatens the safety of the power system [2].

As one of the most critical parameters in electric system, frequency directly indicates system stability and the consistence of interconnected grid. System frequency should be maintained within a safe range under load disturbance [3]. Frequency oscillation has become an important factor that determines the power transmission ability of the high voltage (HV) tie line. Primary frequency control (PFC) is one of the important means to ensure power system security and stability. It can regulate short period random variation and response to emergency rapidly.

Many frequency analyses of HV interconnected grid were conducted, in which some models were built based on the transfer functions of one order models to replace the PFC of generating units [4,5] while some influencing parameters were based on the typical value instead of the actual one [6,7]. These analyses did not provide detailed models or parameters, making the conclusions lack of credibility.

This paper focused on the theoretical analysis and mathematic modeling of an HV interconnected power system. Based on the measured data after transmission line trip-out, it completed the parameter identification of the frequency model by using genetic algorithm (GA). The models and parameters could provide data support for frequency simulation, which would be significant for power system frequency control and stability analysis.

Accident description and HV connected areas modeling

The interconnected power system is shown in Fig. 1 which consists of three areas. Grid A, B, and C are connected by HV transmission lines, of which, Grid B and Grid C contain other transmission lines with 500 kV.

Recorded accident data

Large power deviation occurred in the HV line in 2012. The maximum fluctuation reached by 5 percent of the capacity of Grid B. Finally the HV transmission line tripped out and the HV line between B and C lost approximately 3 percent of the active power. The recorded data by the phasor measurement unit (PMU) is depicted in Fig. 2.

As shown in Fig. 2, the frequency of Grid B and Grid C dropped approximately 0.1 Hz after the trip-out and the maximum deviation was approximately 0.2 Hz within 5 s. The frequency of Grid A started rising and the steady deviation arrived at approximately 0.05 Hz.

Frequency analysis modeling of HV grid

The model of system frequency for the HV grid is developed with the method of multi-area system. In view of multi-area system, the large power system is formed with several control areas and coupled with transmission lines. Assuming that all the generating units are equipped with speed droop, the frequency of disturbance area will change immediately when the trip-out of transmission line or load disturbance occurs. Other areas provide appropriate active power through the transmission lines to support frequency regulation. At the ending of primary frequency regulation (PFR), the system will reach a new equilibrium according to load regulation effect.

Based on the above assumption, the mathematic model for the interconnected HV grid is composed of the frequency model for single control area and the formulation of power oscillation flow on the transmission line.

Frequency model for single control area

Each regional power grid is a complex network composed of many parallel generating units. To simplify the model for frequency analysis, an equivalent generating unit can be used to represent the dynamic characteristic of the power system. Since most units in China are large reheat turbine units, the turbine unit with a speed governing system is chosen to be the equivalent unit to support the PFC in system. The block diagram of the speed governing system is illustrated in Fig. 3.

The typical parameters in this model are listed in Table 1.

Power oscillation flow on the transmission line

When the trip of transmission line occurs, the response of system frequency is mainly determined by the distribution of impact power deviation in the whole grid, the PFC of the power system after the disturbance happens, and the tie-line power transmission caused by frequency difference. Transfer function diagram for single control area is shown in Fig. 4.

Actually, the trip-out of transmission line or load disturbance in a place will impact the whole grid. For the disturbance area k, there exists the power balance equation

(1)P_k\left({\mathrm{0}}^{+}\right)=-{\displaystyle \sum _{i=\mathrm{1}}^n{P}_i\left({\mathrm{0}}^{+}\right)}=-P_{\mathrm{L}}\left({\mathrm{0}}^{+}\right).

Thus the impact power deviation in area i can be described by

(2)P_i\left({\mathrm{0}}^{+}\right)={\displaystyle \frac{K_i}{{\displaystyle \sum _{i=\mathrm{1}}^n{K}_i}}}{P}_{\mathrm{L}}\left({\mathrm{0}}^{+}\right).

During the period of impact power distribution, the output power of generators in area i is calculated by the synchronous torque coefficient K_{\mathrm{s}} between area i and disturbance area k. K_{\mathrm{s}} depends on the susceptance and initial phase difference {\delta }_{ik\mathrm{0}}. The shorter electric distance between area i and k means the smaller phase difference and more impact power distribution in area i.

For the typical two-area connected system, take {\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}} as the distribution coefficient impact power after transmission line trip-out. When the HV transmission line disturbance occurs, the impact power distribution of each area is calculated as

(3)P_{\mathrm{L}\mathrm{1}}\left({\mathrm{0}}^{+}\right)={\lambda }_{\mathrm{1}}{P}_{\mathrm{L}},

(4)P_{\mathrm{L}\mathrm{2}}\left({\mathrm{0}}^{+}\right)={\lambda }_{\mathrm{2}}{P}_{\mathrm{L}},

where {\lambda }_{\mathrm{1}}+{\lambda }_{\mathrm{2}}=\mathrm{1}.

The power balance equation in area i is described as

(5)\mathrm{\Delta }{P}_{\mathrm{T}i}-\mathrm{\Delta }{P}_{\mathrm{L}i}={\displaystyle \frac{\mathrm{d}{W}_{\mathrm{K}i}}{\mathrm{d}t}}+K_{\mathrm{L}i}\mathrm{\Delta }{f}_i+\mathrm{\Delta }{P}_{\mathrm{t}i},

where W_{\mathrm{K}i} is the total kinetic energy in the whole grid, K_{\mathrm{L}i} is the frequency regulation coefficient of load, \mathrm{\Delta }{f}_i is the frequency deviation of control area i, \mathrm{\Delta }{P}_{\mathrm{t}i}is the normalized incremental change tie-line power of area i, \mathrm{\Delta }{P}_{\mathrm{T}i} is the normalized incremental change fluctuation in turbine out power in area i, and \mathrm{\Delta }{P}_{\mathrm{L}i} is the normalized incremental change of electric load variations. All parameters in Eq. (5) are relative ones.

Since kinetic energy W_{\mathrm{K}i} varies with the square of frequency in area i,

W_{\mathrm{K}i}={\left({\displaystyle \frac{f_i}{f_{\mathrm{e}}}}\right)}^{\mathrm{2}}{W}_{\mathrm{K}i\mathrm{e}},

where f_e is the fundamental frequency (50 Hz in China); W_{\mathrm{K}ie} is the total kinetic energy in the whole grid i when system frequency is f_e.

The system can be linearized at the equilibrium point

(6)W_{\mathrm{K}i}=\left(\mathrm{1}+\mathrm{2}{\displaystyle \frac{\mathrm{\Delta }{f}_i}{f_{\mathrm{e}}}}\right)W_{\mathrm{K}i\mathrm{e}},

(7){\displaystyle \frac{\mathrm{d}{W}_{\mathrm{K}i}}{\mathrm{d}t}}={\displaystyle \frac{\mathrm{2}{W}_{\mathrm{K}i\mathrm{e}}}{f_{\mathrm{e}}}}\cdot {\displaystyle \frac{\mathrm{d}\mathrm{\Delta }{f}_i}{\mathrm{d}t}}=M_i{\displaystyle \frac{\mathrm{d}\mathrm{\Delta }{f}_i}{\mathrm{d}t}},

where M_i is the inertia time constant of grid.

Together with Eq. (5), there are the transfer functions by Laplace Transform

(8)\mathrm{\Delta }{P}_{\mathrm{M}i}\left(s\right)-\mathrm{\Delta }{P}_{\mathrm{L}i}\left(s\right)=M_{i}s\mathrm{\Delta }{F}_i\left(s\right)+K_{\mathrm{L}i}\mathrm{\Delta }{F}_i\left(s\right)+\mathrm{\Delta }{P}_{\mathrm{t}i}\left(s\right),

(9)G_{\mathrm{P}i}\left(s\right)={\displaystyle \frac{\mathrm{\Delta }{F}_i\left(s\right)}{\mathrm{\Delta }{P}_{\mathrm{T}i}\left(s\right)-\mathrm{\Delta }{P}_{\mathrm{L}i}\left(s\right)-\mathrm{\Delta }{P}_{\mathrm{t}i}\left(s\right)}}={\displaystyle \frac{\mathrm{1}}{M_{i}s+K_{\mathrm{L}i}}},

where G_{\mathrm{P}i} is the transfer function of system frequency, and Tie-line power transmission of areas, P_{ij}, can be calculated by [2]

(10)P_{ij}=-P_{ji}={\displaystyle \frac{\left|U_i\right|\left|U_j\right|}{X_{ij}}}\sin {\delta }_{ij},

where U_i and U_j is the bus voltage magnitude of tie-line ends; {\delta }_{ij}={\delta }_i-{\delta }_j, {\delta }_i, {\delta }_j is the bus voltage phase of tie-line ends; {\delta }_{ij} is phase difference, and X_{ij} is the reactance between bus line i and j.

When load disturbance occurs, the variation of tie line power can be written as

(11)$$\begin{gathered} \mathrm{\Delta }{P}_{\mathrm{t}ij}={\displaystyle \frac{\mathrm{d}{P}_{\mathrm{t}ij}}{\mathrm{d}{\delta }_{ij}}}\mathrm{\Delta }{\delta }_{ij}={\displaystyle \frac{\left|U_i\right|\left|U_j\right|}{X_{ij}}}\cos {\delta }_{ij}\mathrm{\Delta }{\delta }_{ij} \\ ={\displaystyle \frac{\left|U_i\right|\left|U_j\right|}{X_{ij}}}\cos {\delta }_{ij}\left(\mathrm{\Delta }{\delta }_i-\mathrm{\Delta }{\delta }_j\right)=T_{ij}\left(\mathrm{\Delta }{\delta }_i-\mathrm{\Delta }{\delta }_j\right), \end{gathered}$$

where T_{ij}={\displaystyle \frac{\left|U_i\right|\left|U_j\right|}{X_{ij}}}\cos {\delta }_{ij} is tie line synchronous frequency coefficient, \mathrm{\Delta }{\delta }_i and \mathrm{\Delta }{\delta }_j is the phase difference of tie-line ends.

Since \mathrm{\Delta }\delta =\mathrm{2}\mathrm{\pi }{\displaystyle \int \mathrm{\Delta }f\mathrm{d}t}, Eq. (11) can be written as

(12)\mathrm{\Delta }{P}_{\mathrm{t}i}=\mathrm{2}\mathrm{\pi }{\displaystyle \sum _{i=\mathrm{1}}^n{T}_{ij}\left({\displaystyle \int \mathrm{\Delta }{f}_i\mathrm{d}t-{\displaystyle \int \mathrm{\Delta }{f}_j\mathrm{d}t}}\right)}.

In this paper, Eq. (12) is solved by Laplace Transform

(13)\mathrm{\Delta }{P}_{\mathrm{t}i}\left(s\right)=\mathrm{2}\mathrm{\pi }{\displaystyle \sum _{i=\mathrm{1}}^n{T}_{ij}{\displaystyle \frac{\mathrm{1}}{s}}\left[\mathrm{\Delta }{F}_i\left(s\right)-\mathrm{\Delta }{F}_j\left(s\right)\right]}.

Based on the theoretical modeling, the trip-out of the HV transmission line can be summarized as follows:

1) The transmission line between A and B tripped out. In this case, the connected relation between Grid A and Grid B was cut off while Grid B and Grid C still kept connected by the HV and 500 kV transmission lines. The mathematic model in this case is demonstrated in Fig. 5.

2) The transmission line between Grids B and C tripped out. In this case, the three areas were still connected while the tie line synchronous frequency coefficient between Grid B and Grid C was much lower than the original one. Model for interconnected HV grid is shown in Fig. 6.

Parameter identification of power system after HV line trip-out

The importance of frequency simulation is suggested by a voluminous literature. System parameters, such as equivalent speed droop, impact power coefficient and inertia time, prove to play an important role in frequency regulation and stability. To obtain the detailed value of these models, the GA-based method was proposed for parameter identification based on the recorded data.

GA identification procedure

GA is a search heuristic that mimics the process of natural selection. This heuristic is routinely used to generate useful solutions to optimization and search problems [8].

In a GA, a population of candidate solutions to an optimization problem is evolved toward better solutions. The identification procedure can be illustrated as shown in Fig. 7. The input and the output signal are measured and transformed to the identification module. The simulated system response Y' with the parameters are compared with the measured response. The estimation algorithm will update the parameters considering the corresponding errors. The procedure repeats and minimizes the errors until the convergence limit is met.

The objective function of a problem is a main source providing the mechanism to evaluate the status of each individual. Every solution evolves toward the increase of objective function. The objective of the optimization is to maximize the following objective function through continuous adaptation of the parameters.

(14)f=-{\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{i=\mathrm{1}}^N{\mathrm{e}}_i^{\mathrm{2}}}=-{\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{i=\mathrm{1}}^N}{\left[y_i-{\widehat{y}}_i\right]}^{\mathrm{2}},

where N is number of the sampling data points of the output signal, y_i and {\widehat{y}}_i are the measured and simulated values of the system output at time instant i. The value of the objective is always lower than zero, so this generational process stops when successive iterations no longer produce better results, which means the simulated results are in great agreement with the measured data.

Simulation results

The mathematic model for Grid B and C after HV trip-out is demonstrated in Fig. 5. The input signal is the change of power flow on the HV transmission line. The frequency and power oscillation of the two areas are the output signals, as displayed in Fig. 2. The identified results are exhibited in Figs. 8-10 and key parameters are listed in Table 2.

The validity of the identified parameters is proved by comparing the simulated transients using the identified parameters with respect to the recorded data. It is apparent that the recorded data and the estimated results are in favorable agreement, indicating the accuracy of the identified parameters.

When the tripping of the HV transmission line happens, Grid A is disconnected with Grid B and C. It can be regarded as a large single control area. The power lost on the HV transmission is regarded as the source system disturbance. The identified results are presented in Fig. 11 and parameters are tabulated in Table 3.

It can be observed from the comparison of simulated results and recorded data in Fig. 2 that the absolute error of steady and overshoot value of frequency is less than 0.02 Hz, which is allowed in dynamic simulation of power system. So the accuracy of these models and obtained parameters can be well proved.

Conclusions

Focusing on the dynamic frequency response of a power system after the trip-out of the HV transmission line, the mathematic formulation of the tie line power oscillation and frequency variation were derived strictly. Based on the theory, simulation models were built for different cases according to the real network. A GA based method was adopted to identify the key parameters. A great agreement between simulation results and recorded data verifies the validity of gird models and the accuracy of system parameters. The parameters obtained in this paper may lay a solid foundation for further research of frequency and tie line power after disturbance such as HV trip-out.

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