An Equivalent Model for Frequency Dynamic Analysis of Large Power Grids Based on Regulation Performance Weighting Method

Abstract

With the construction of the UHV (Ultra High Voltage) AC/DC hybrid power grid and the large-scale access to renewable energy such as wind power, frequency dynamic fluctuation has become a prominent problem affecting the safe and stable operation of large power grids. The expansion of the scale of the power system makes it impossible to use traditional fine modeling to analyze the power system. In order to reduce the calculation scale and storage capacity of power system frequency dynamic simulation, it is necessary to make appropriate equivalent simplification of the external system, so the appropriate dynamic equivalent method is of great significance. This paper mainly studies the equivalent model suitable for frequency dynamic analysis of large power grids. Firstly, the typical models of generator set and load are simplified, and the parameters that have a great influence on frequency in the simplified model are obtained through characteristic analysis. Then, a dynamic aggregation method of generator governor and prime mover parameters and load parameters based on regulation performance weighting (the parameters of the generator or load are weighted and summed according to its regulation ability on the system) is proposed. This method is applied to the simulation example of the East China Power Grid. The simulation proves that the frequency of the East China Power Grid before and after equivalence can be consistent under four different faults, which verifies the effectiveness of the equivalent method proposed in this paper in the frequency dynamic analysis of large power grids.

1. Introduction

One of the most important parameters in the power system is frequency, which can reflect the operation quality of the power grid [1]. To make the power grid safe and stable, it is necessary to maintain the frequency within a reasonable range [2]. Because the power system has a certain particularity, it is impossible to directly test the actual power grid to study the frequency problem, and it must be studied through modeling and simulation. The dynamic equivalence method of the power system is inseparable from the physical problems that need to be studied after equivalence. According to different research problems, dynamic equivalence can be divided into three methods: the coherency equivalence method [3,4], the mode equivalence method [5] and the online identification method [6].

There are four main models in the power system [7], including the generator model, excitation system model, prime mover-governor model and load model. So far, the first two of these four models have well-studied models because their models are mainly electrical quantities that can be obtained through actual measurements [8,9,10]. This paper mainly studies the latter two models. Because the power system is developing rapidly and the scale is expanding, a large number of additional governors, loads and other components must be taken into account. Therefore, the traditional comprehensive and refined modeling of the power system [11,12,13] is no longer applicable. It is necessary to make appropriate equivalent simplification of the external system. Therefore, the appropriate dynamic equivalent method is of great significance.

With the integration of large-scale renewable energy, such as wind power, into the power system, its instability will inevitably have an unpredictable impact on the grid frequency. This paper also studies the general model of wind power generation system, including four modules: wind turbine model, pitch angle controller, converter controller and generator model [14,15], so as to further study the frequency dynamic analysis of large power grids with new energy. The simulation results of the power system can be used as the reference data for the planning and construction of the power grid, but different simulation models will have different degrees of influence on the results of the simulation calculation [16,17,18]. In particular, the accuracy of the load model will play a crucial role in the final simulation results of the power grid [19]. It can be seen from references [20,21,22] that with the change in the power grid and power supply structure and load type, the frequency characteristics of the system will also change, and the security and stability of the power grid will also face new threats.

In this paper, a method of dynamic aggregation of generator governor-primitive motor parameters and load parameters based on adjustment performance weighting is proposed. This method overcomes the uncertainty of human experience values and has fast aggregation speed and high engineering accuracy.

Table 1. The comparison of regulation performance weighting method and other methods.

	Regulation Performance Weighting Method	Other Coherency-Based Equivalence Method
Polymerization rates	Faster	Fast
Reliability	Overcome the uncertainty of human experience	Relying on human experience
Accuracy	High	Low

shows the comparison between the regulation performance weighting method and other main equivalence methods. Using this method, the actual power grid can be equivalent to a single generator and a single load, which is convenient for system frequency simulation analysis. Finally, the method is applied to the East China Power Grid. The equivalent method of this paper is used to equalize the generator and load of the East China Power Grid. The simulation proves that the frequency of the East China Power Grid can be consistent before and after the equivalence under four different faults, which verifies the effectiveness of the equivalent method in the dynamic analysis of the large power grid frequency.

2. Equivalence Method for Governor and Prime Mover Based on Regulation Performance Weighting

2.1. Aggregation Equivalence Modeling for Governor and Prime Mover of Steam Turbine

Taking the governor and prime mover models commonly used in steam turbines as an example, the governor model is analyzed in detail and simplified reasonably. The governor and prime mover models commonly used in PSD-BPA (Power System Department-Bonneville Power Administration) are shown in Figure 1 and Figure 2 [23].

In Figure 1, $T_1$ represents the time constant of rotational speed measurement, $P_{ref}$ represents the given power reference value and $T_C$ represents the opening time constant of the oil motor.

The transfer function of the electro-hydraulic servo actuator is as follows:

$$G_2={\displaystyle \frac{P_{GV}}{P_{CV}}}={\displaystyle \frac{\left(K_P+K_{D}s+\frac{K_I}{s}\right).\frac{1}{T_o}.\frac{1}{s}}{1+\left(K_P+K_{D}s+\frac{K_I}{s}\right).\frac{1}{T_o}.\frac{1}{s}.\frac{1}{1+T_{2}s}}}$$

(1)

$T_2$ represents the time of the oil motor travel feedback link, the typical value is 0.02 s; $T_O$ represents the opening time constant of the oil motor, the typical value is 1–4 s; $K_P$, $K_D$ and $K_I$ are the proportional amplification link, differential link and integral link multiples of the PID module, respectively. In general, $K_D$ and $K_I$ are all 0, and $K_P$ is about 10.

Under the typical parameters of the governor model, because P_CV is a step signal, take $P_{CV}=1/s$ and the governor model is simplified to obtain the transfer function as shown in Equation (2).

$$P_{GV}={\displaystyle \frac{K_P.\frac{1}{T_o}.\frac{1}{s^2}}{1+K_P.\frac{1}{T_o}.\frac{1}{s}.\frac{1}{1+T_{2}s}}}$$

(2)

$P_{GV}$ is a step signal, so when $P_{GV}=1/s$, the transfer function of PM can be expressed as Equation (3).

$$P_M(s)={\displaystyle \frac{1}{s}}\left[{\displaystyle \frac{1}{T_{CH}s+1}}{F}_{HP}+{\displaystyle \frac{1}{T_{CH}s+1}}{\displaystyle \frac{1}{T_{RH}s+1}}(1-F_{HP})\right]$$

(3)

Therefore, the steam turbine governor-primitive engine model has seven main parameters: speed deviation magnification K, oil motor opening time constant $T_O$, PID module proportional magnification multiple $K_P$, oil motor travel feedback link time T₂, steam volume time constant T_CH, reheater time constant T_RH and high-pressure cylinder mechanical power ratio coefficient F_HP. Therefore, the frequency characteristics of these parameters are analyzed to find out the parameters that have a greater impact on the frequency.

By analyzing Equation (1), taking $K_P=7$, $T_2=0.02$, $T_O$ = 1.15 and 4, the P_GV(t) curve is obtained, as shown in Figure 3.

From Figure 3, when $T_O$ increases from 1.15 to 4, the P_GV(t) curve changes greatly. It can be seen that the oil motor opening time constant $T_O$ has a great influence on the characteristics of the steam turbine, and this parameter needs to be considered in the equivalent modeling.

The frequency characteristics of the remaining parameters are analyzed according to the same method, which is no longer shown here. Finally, it is concluded that the speed deviation amplification factor K, the oil motor opening time constant $T_O$, the reheater time constant T_RH and the high-pressure cylinder mechanical power proportional coefficient F_HP have a great influence on the frequency characteristics of the steam turbine. These four parameters are mainly considered when equivalent.

The steps of the dynamic aggregation method of turbine governor-prime mover parameters based on regulation performance weighting are shown in Figure 4:

(1)

The system to be analyzed is divided into an internal system that should remain unchanged and an external system to be equivalent. The power flow calculation of the power grid is carried out through BPA, and the parameter variables (rated output of the generator prime mover $P_{ei}$) and state variables (actual active power output of the generator $P_{GENi}$) of each generator in the external system that needs to be equivalent are obtained.

(2)

Through the calculation of the parameters of the M turbines to be aggregated, the spinning reserve of each turbine generator and the equivalent turbine generator are obtained.

According to (4), the spinning reserve of each turbine generator is obtained:

$$P_{SRi}=P_{ei}{K}_{MAXi}-P_{GENi}$$

(4)

In Equation (4), $P_{ei}$ represents the rated output of the i-th turbine generator prime mover, $K_{MAXi}$ represents the maximum regulating valve opening of the i-th turbine generator prime mover and $P_{GENi}$ represents the actual active output of the i-th turbine generator.

According to (5), the spinning reserve of the equivalent turbine generator $P_{SReq}$ is obtained:

$$P_{SReq}={\displaystyle \sum _{i=1}^M{P}_{SRi}}$$

(5)

(3)

Organize the body model and parameters of M governors to be aggregated in the external system and classify the governors of M steam turbine units according to the commonly used n governor models (G1/G2/G3/.../Gn). According to the different types of governors, the generators are classified, and the sum of the spins of turbo-generators under different types of governors is obtained.

$$\left\{\begin{array}{c}{P}_{G1}={\displaystyle \sum _{i=1}^{\mathrm{a}}{P}_{SRi}}\\ P_{G2}={\displaystyle \sum _{i=1}^{\mathrm{b}}{P}_{SRi}}\\ P_{G3}={\displaystyle \sum _{i=1}^{\mathrm{c}}{P}_{SRi}}\\ \dots \\ P_{G\mathrm{n}}={\displaystyle \sum _{i=1}^{\mathrm{n}}{P}_{SRi}}\end{array}\right.$$

(6)

Comparing the sum of the spins of n governor models, if the proportional of the sum of the spins of a governor model to the sum of the spins of all models is less than 5%, the governor is ignored, and the others are equivalent according to different governor types.

(4)

Through the superposition calculation of the rated output power of the prime mover of the M number of turbine generator sets to be aggregated, the rated output power of the equivalent turbine generator is obtained. Through the total rated output power, total actual active power and total spinning reserve of the unit to be aggregated, the maximum valve opening of the prime mover of the equivalent governor is obtained.

The rated output power of the equivalent turbine generator is as follows:

$$P_{eM}={\displaystyle \sum _{i=1}^M{P}_{ei}}$$

(7)

The actual active power of the equivalent turbine generator is as follows:

$$P_{GENM}={\displaystyle \sum _{i=1}^M{P}_{GENi}}$$

(8)

Maximum gate opening of prime mover of equivalent governor is as follows:

$$K_{MAXi}={\displaystyle \frac{P_{S\mathrm{Re}q}+P_{GENi}}{P_{eM}}}$$

(9)

(5)

The parameters of the equivalent steam turbine governor-primitive engine model are obtained by the method of dynamic aggregation of turbine governor-primitive engine parameters based on the weighted adjustment performance.

Define the proportional coefficient of turbine speed regulation $K_{PMi1}$:

$$K_{PMi1}={\displaystyle \frac{F_{HP}K}{T_{RH}{T}_O}}$$

(10)

From the above analysis, it can be seen that the speed deviation amplification factor K, the oil motor opening time constant $T_O$, the reheater time constant T_RH and the high-pressure cylinder mechanical power proportional coefficient F_HP in the turbine governor-prime mover have a great influence on the characteristics of the turbine. Therefore, Equation (10) is expressed as the product of these four parameters, where $T_O$ and T_RH takes the reciprocal form.

Define the turbine governor weight $R_i$:

$$R_i=P_{SRi}{K}_{PMi1}$$

(11)

The parameters in the turbine equivalent governor are aggregated as follows:

$$\epsilon ={\displaystyle \frac{{\displaystyle \sum _{i=1}^M{\epsilon }_i{R}_i}}{{\displaystyle \sum _{i=1}^M{R}_i}}}$$

(12)

In Equation (12), ε can be the amplification factor K of the equivalent machine speed deviation, the opening time constant $T_O$ of the oil motor, the reheater time constant T_RH and the mechanical power proportional coefficient F_HP of the high-pressure cylinder, and ${\epsilon }_i$ are the corresponding parameters of the i-th generator governor.

2.2. Aggregation Equivalence Modeling for Governor and Prime Mover of Hydro-Turbine

The parameter aggregation method of the hydro-turbine governor and prime mover model is similar to that of the steam turbine. However, due to the large difference between the governor and prime mover model of hydro-turbine and steam turbine, the parameters are also different, so the aggregation steps are also different. The following describes the different steps of hydro-turbine parameter aggregation and steam turbine parameter aggregation.

In step 2 of the parameter aggregation method for steam turbine governors, the maximum governor valve opening $K_{MAXi}$ of the steam turbine generator prime mover is required to obtain the spinning $P_{SRi}$ of each generator. However, this parameter is not available in the typical governor model of the hydro-turbine. Therefore, the spinning $P_{SRi}$ of each hydro-generator is obtained according to Equation (13):

$$P_{SRi}=P_{ei}-P_{GENi}$$

(13)

In Equation (13), $P_{ei}$ represents the rated output of the prime mover of the i-th hydro-generator and $P_{GENi}$ represents the actual active output of the i-th hydro-generator.

In step 4, only the rated output power of the equivalent hydro-generator and the actual active power output of the equivalent hydro-generator are required.

In step 5, because the parameters of the turbine governor are different from those of the steam turbine, the proportional coefficient $K_{PMi2}$ of the turbine governor is defined:

$$K_{PMi2}={\displaystyle \frac{D_d}{R{T}_d{T}_G(T_W/2)}}$$

(14)

According to the same method as the steam turbine, the frequency characteristics of hydro-turbine parameters are analyzed. It is found that the adjustment coefficient R, the soft feedback link coefficient D_d, the soft feedback time constant T_d, the governor response time T_G and the water hammer effect time constant T_W/2 have a great influence on the characteristics of the hydro-turbine. Therefore, $K_{PMi2}$ in Equation (14) is expressed as the product of these five parameters, where R, T_d, T_G and T_W/2 are in the reciprocal form.

The other steps of the equivalent method of the hydro-turbine governor are the same as those of the steam turbine. Finally, the adjustment coefficient R, the soft feedback link coefficient D_d, the soft feedback time constant T_d, the governor response time T_G and the water hammer effect time constant T_W/2 in the hydraulic turbine equivalent governor can be obtained by Equation (12).

2.3. Aggregation Equivalence Modeling for Prime Mover and Controller of Wind Turbine

The general model of wind turbines mainly includes four modules: a wind turbine model, pitch angle controller, converter controller and generator model. The mechanical power captured by a wind turbine can be expressed as follows [14]:

$$P_{\mathrm{m}}={\displaystyle \frac{1}{2}}\rho \pi R^2{C}_p(\lambda ,\beta )v^3$$

(15)

In the formula, $\rho$ is the air density, $R$ is the radius of the wind wheel, $v$ is the wind speed, $\lambda$ is the tip speed ratio and $C_p$ is the wind energy utilization coefficient.

In the general model, the wind turbine adopts the linear aerodynamic model, and the wind speed is assumed to be constant [14]:

$$P_m=P_{m0}-K_a\beta (\beta -{\beta }_0)$$

(16)

In the formula, $\beta$ is the slurry pitch angle, ${\beta }_0$ is the initial pitch angle, $P_{m0}$ is the initial mechanical power and $K_a$ is the aerodynamic power coefficient.

The wind turbine drive shaft system model adopts a single mass model, and a rigid body is used to simulate the wind turbine blade, the drive shaft and the rotor shaft of the generator. Ignoring the internal differences of the shaft system, that is, the first-order inertia link is used to simulate the transmission process of the shaft torque, as shown in Equations (17) and (18):

$$T_J{\displaystyle \frac{d\omega }{dt}}=T_m-T_e$$

(17)

$${\displaystyle \frac{d{T}_m}{dt}}={\displaystyle \frac{1}{t_h}}(T_{ae}-T_m)$$

(18)

In the formulas, $T_J$ is the total inertia time constant of the wind turbine and the generator; $T_m$, $T_e$ and $T_{ae}$ denote the mechanical torque of the rotor, electromagnetic torque of the generator and the torque of the shaft.

The pitch angle controller model and the converter-level controller model are shown in Figure 5 and Figure 6.

Suppose the intermediate variables are $x_1$, $x_2$, $x_3$ and $x_4$. The transfer function expressions are shown as Equations (19)–(21):

$$\beta ={\displaystyle \frac{1}{1+s{T}_{\beta }}}[(K_{pc}+{\displaystyle \frac{K_{ic}}{s}})\Delta P+(\Delta PKcc+\Delta \omega )(K_{pw}+{\displaystyle \frac{K_{iw}}{s}})]$$

(19)

$$u_{qr}=K_{p2}(K_{p1}\Delta P+K_{i1}{x}_1-i_{qr})+K_{i2}{x}_2+s{\omega }_s{L}_m{i}_{ds}+s{\omega }_s{L}_{rr}{i}_{qr}$$

(20)

$$u_{dr}=K_{p2}(K_{p3}\Delta u+K_{i3}{x}_3-i_{dr})+K_{i2}{x}_4-s{\omega }_s{L}_m{i}_{qs}-s{\omega }_s{L}_{rr}{i}_{dr}$$

(21)

In the equation, $T_{\beta }$ is the reaction time constant of the blade; $L_{rr}$ is the inductance of the rotor winding; $L_m$ is the mutual inductance between the stator and the rotor; ${\omega }_s$ is the electrical angular velocity of the stator. $K_{p1}$ and $K_{i1}$ are the proportional and integral coefficients of active power control; $K_{p2}$ and $K_{i2}$ are the proportional and integral coefficients of rotor-side current control;$K_{p3}$ and $K_{i3}$ are the proportional and integral coefficients of reactive power control.

Similarly, through the analysis of frequency characteristics, it is found that the inertia time constant $T_J$, the blade reaction time constant $T_{\beta }$, the active power control magnification $K_1$, the current control magnification $K_2$ and the reactive power control magnification $K_3$ have a great influence on the frequency. Therefore, these five parameters are mainly considered in the equivalence.

The parameter aggregation method of the wind turbine is similar to that of the hydro-turbine, but the parameters are different due to the large difference between the wind turbine and the hydro-turbine. The following describes the different steps of wind turbine parameter aggregation and water turbine parameter aggregation.

In step 2, the wind turbine is similar to the hydro-turbine in obtaining the spinning reserve of each unit, but the wind turbine must be grouped before aggregation. The working characteristics of the wind turbine are different under different wind speeds. The wind turbine is grouped according to the method based on critical wind speed and wind speed similarity.

In step 5, because the parameters of the wind turbine are different from those of the hydro-turbine, the proportional coefficient $K_{PMi3}$ of wind turbine speed regulation is defined:

$$K_{PMi3}={\displaystyle \frac{K_1{K}_2{K}_3}{T_J{T}_{\beta }}}$$

(22)

The other steps of the wind turbine equivalent method are the same as those of the hydro-turbine. Finally, the parameters of the wind turbine can be obtained by Equation (12).

3. Equivalence Method for Load Based on Regulation Performance Weighting

3.1. Static Load Model

The static load model is mainly represented by the polynomial model, and the LB (Load Balance) model is mainly used in BPA. The expression of the static load model is shown in Equation (23):

$$\left\{\begin{array}{l}P=P_0\left[P_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2+P_2\left({\displaystyle \frac{V}{V_0}}\right)+P_3\right]\left(1+\Delta f\cdot L_{DP}\right)\\ Q=Q_0\left[Q_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2+Q_2\left({\displaystyle \frac{V}{V_0}}\right)+Q_3\right]\left(1+\Delta f\cdot L_{DQ}\right)\end{array}\right.$$

(23)

where the second order term represents the constant impedance term, the first order term represents the constant current term, and the zeroth order term represents the constant power term. $P_1$, $P_2$ and $P_3$ are the active power proportional coefficients of each term, $Q_1$, $Q_2$ and $Q_3$ are the reactive power proportional coefficients of each term and $P_1+P_2+P_3=1$, $Q_1+Q_2+Q_3=1$. $L_{DP}$ is the frequency response factor of active load, and $L_{DQ}$ is the frequency response factor of reactive load.

The frequency characteristics of the load are characterized by term $\left(1+\Delta f\cdot L_{DP}\right)$ and $\left(1+\Delta f\cdot L_{DQ}\right)$, so in the static load model, the main factors affecting the frequency are the frequency response factor L_DP of the active load and the frequency response factor L_DQ of the reactive load.

Assuming that there are M static loads in the system when the static load is equivalent, the parameter aggregation steps of the equivalent model are as follows.

The capacity of the equivalent static load is the sum of the static load capacity (MVA) of each node:

$$S_M={\displaystyle \sum _{i=1}^M{S}_i}$$

(24)

The parameter aggregation method based on the adjustment performance weighting method is also used to obtain the parameters of the equivalent static load.

Define the static load ratio coefficient:

$$K_{PMi3}=L_{DP}{L}_{DQ}$$

(25)

Considering the frequency characteristics of the static load model, the main influencing factors are the frequency response factor L_DP of the active load and the frequency response factor L_DQ of the reactive load. Therefore, $K_{PMi3}$ in Equation (25) is expressed as the product of these two parameters.

Define the static load weight:

$$R_i=S_i{K}_{PMi3}$$

(26)

Then, the parameters in the equivalent static load model can also be aggregated according to Equation (12), where ε is the parameter of the equivalent static load model, which can be the frequency response factor L_DP of the active load and the frequency response factor L_DQ of the reactive load and ${\epsilon }_i$ is the corresponding parameters of the static load model of the i-th node.

3.2. Dynamic Load Model

The induction motor model is mainly considered in the dynamic load model, and its expressions are shown in Equations (27)–(29):

$$\left\{\begin{array}{c}{\displaystyle \frac{d{E^{\prime }}_d}{dt}}=-{\displaystyle \frac{1}{T^{\prime }}}\left[{E^{\prime }}_d+(X-X^{\prime })I_q\right]+{\omega }_{0}s{E^{\prime }}_q\\ {\displaystyle \frac{d{E^{\prime }}_q}{dt}}=-{\displaystyle \frac{1}{T^{\prime }}}\left[{E^{\prime }}_q-(X-X^{\prime })I_d\right]-{\omega }_{0}s{E^{\prime }}_d\end{array}\right.$$

(27)

$$\left\{\begin{array}{l}{I}_d={\displaystyle \frac{1}{R_s^2+X^{\prime 2}}}\left[R_s(U_d-E_d^{\prime })+X^{\prime }(U_q-E_q^{\prime })\right]\\ I_q={\displaystyle \frac{1}{R_s^2+X^{\prime 2}}}\left[R_s(U_q-E_q^{\prime })+X^{\prime }(U_d-E_d^{\prime })\right]\end{array}\right.$$

(28)

where $s$ is the rotor slip; $T^{\prime }$ is the transient open circuit time constant; $X$ is open-circuit impedance for the rotor and $X^{\prime }$ is short-circuit reactance for rotor stalling. $I_d$ and $I_q$ represent the d- and q-axis currents of the stator, respectively. $R_s$ represent the stator resistance.

The rotor equation is as follows:

$$\left\{\begin{array}{l}{T}_J{\displaystyle \frac{ds}{dt}}=T_m-T_e\\ T_m=(A{\omega }_r^2+B{\omega }_r+C)T_0\end{array}\right.$$

(29)

where $T_J$ is the inertia time constant of the rotor, ${\omega }_r$ is the angular velocity of the rotor, $T_m$ is the mechanical torque of the motor and A, B and C are the torque coefficients of the mechanical load.

The induction motor model in BPA software (version number: FSDEdit 2.8) mainly adopts the ML model [24]. Among the main parameters, the inertia time constant T_J, the proportion of motor power to bus power P_per, the load rate K_L, the stator reactance X_s, the rotor reactance X_r and the torque equation constant A have a great influence on the frequency, which should be considered emphatically.

Assuming that there are M induction motors in the system, the following is the process of obtaining the parameters of the equivalent machine.

The capacity of the equivalent machine can be obtained by summing the capacity (MVA) of each induction motor:

$$S_M={\displaystyle \sum _{i=1}^M{S}_i}$$

(30)

The parameters of the equivalent induction motor are also obtained by using the parameter aggregation method based on the adjustment performance weighting method.

Define the dynamic load ratio coefficient $K_{PMi4}$:

$$K_{PMi4}=T_J\cdot P_{per}\cdot K_L\cdot X_s\cdot X_r\cdot A$$

(31)

When studying the frequency characteristics of induction motors, the main influencing factors are the inertia time constant T_J, the ratio of motor power to bus power P_per, the load rate K_L, the stator reactance X_s, the rotor reactance X_r and the torque equation constant A. Therefore, $K_{PMi4}$ in Equation (31) is expressed as the product of these six parameters.

Define the dynamic load weight $R_i$:

$$R_i=S_i{K}_{PMi4}$$

(32)

Then, the parameters in the equivalent induction motor model are also aggregated according to Equation (12), where ε is the parameter of the equivalent induction motor, which can be respectively the inertia time constant T_J, the ratio of motor power to bus power P_per, the load rate K_L, the stator reactance X_s, the rotor reactance X_r and the torque equation constant A. ${\epsilon }_i$ is the corresponding parameters of the i-th induction motor.

4. Case Analysis and Application of East China Power Grid

Using the equivalent method in this paper, Anhui, Shanghai and Fujian power grids are equivalent to a single generator (equivalent if there is a turbine and a steam turbine) and a single load (equivalent if there is a static load and a dynamic load). The Jiangsu and Zhejiang power grids remain unchanged to see whether the system frequency is consistent before and after the equivalence.

Among them, there are 31 generators in the Anhui power grid, all of which are turbine generators, 120 static load nodes and 102 induction motors. There are 20 generators in the Shanghai power grid, all of which are turbine generators, 64 static load nodes and 66 induction motors. Fujian power grid has 29 turbo-generators and 11 hydro-generators, 120 static load nodes and 125 induction motors. By using the governor-primitive motor and load equivalent method based on the weighted adjustment performance proposed in this paper, the generator and load models in the three power grids are equivalent. Each power grid is finally equivalent to a single generator (the Fujian power grid is a steam turbine and a turbine), a static load node and an induction motor. The following table shows some governors and load parameters of the Anhui power grid, Shanghai power grid and Fujian power grid. The following

Table 2. Governor parameters of Anhui power grid (part).

Unit Name/Governor Parameters	Pei/MW	KMAXi	PGENi/MW	PSRi/MW	TO	K	TRH	FHP
Wan Anqing_4	336.2	1.06	228.9	127.472	1.15	16.7	10	0.3
Wan Huaier_2	336.2	1.06	212.8	143.572	1.15	16.7	10	0.3
Wan Jiuhua_1	336.2	1.06	183.4	172.972	1.3	20	12	0.4
Wan Qiandian_2	151.2	1.06	87.25	73.022	1.3	20	12	0.4
Wan Tianmen_1	698.4	1.06	426.9	313.404	1.5	23	14	0.5
Wan Luohe_1	358.6	1.06	198.4	181.716	1.5	23	14	0.5
Wan Yongfeng_2	635.2	1.06	316.9	356.412	1.7	25	16	0.6
Wan Mayi_1	635.2	1.06	442.4	230.912	1.7	25	16	0.6
Wan Linhuan_3	336.2	1.06	164	192.372	1.9	28	18	0.7
Wan Hualiu_3	685.2	1.06	467.5	258.812	1.9	28	18	0.7
…	…	…	…	…	…	…	…	…
Equivalent generator	15,442.8	1.06	8755.21	7211.36	1.55	23.06	14.43	0.52

,

Table 3. Anhui power grid induction motor parameters (part).

Busbar Name	Capacity/MW	TJ	Pper	KL	XS	XR	A
Wan Kongdian21	−7.51652	3	1	0.8	0.067	0.17	1
Wan Huaibei22	1.456613	3	1	0.8	0.067	0.17	1
Wan Caishi22	64.64777	5	0.9	0.7	0.087	0.2	0.9
Wan Changlong22	105.7431	5	0.9	0.7	0.087	0.2	0.9
Wan Huizhou2E	87.15075	7	0.8	0.6	0.107	0.23	0.8
Wan Huigong22	96.08173	7	0.8	0.6	0.107	0.23	0.8
Wan Xishan22	84.51662	9	0.7	0.5	0.127	0.26	0.7
Wan Xiantong22	75.42928	9	0.7	0.5	0.127	0.26	0.7
…	…	…	…	…	…	…	…
Equivalent load	8489.95	6.87	0.81	0.61	0.106	0.23	0.81

,

Table 4. Static load parameters of security grid (part).

Busbar Name	Capacity/MW	LDP	LDQ
Wan Chenqiao22	40.5122	1.8	−2
Wan Linjiang22	100.8627	1.8	−2
Wan Shanmen21	26.27139	1.9	−2.1
Wan Puqing21	48.43346	1.9	−2.1
Wan Wenchang23	49.11798	2	−2.2
Wan Madian22	−0.48465	2	−2.2
Wan Qiaotou2A	63.72802	2.1	−2.5
Wan Qiaotou2_	67.3105	2.1	−2.5
…	…	…	…
Equivalent load	5365.01	1.88	−2.08

,

Table 5. Governor parameters of Shanghai power grid (part).

Unit Name/Governor Parameters	Pei/MW	KMAXi	PGENi/MW	PSRi/MW	TO	K	TRH	FHP
Hu Jinshan_5	71.4	1.06	47.95	27.734	1.15	25	10	0.3
Hu Shidong_1	369.5	1.06	164	227.67	1.15	25	10	0.3
Hu Waigao_3	311.4	1.06	136.9	193.184	1.3	20	12	0.4
Hu Wure_1	336.2	1.06	230.1	126.272	1.5	16.7	14	0.5
Hu Waisan_1	1059	1.06	616.7	505.84	1.7	28	16	0.6
Hu Shangdian_2	1059	1.06	558.4	564.14	1.7	28	16	0.6
…	…	…	…	…	…	…	…	…
Equivalent generator	8370.3	1.06	4741.91	4130.68	1.51	24.68	14.06	0.50

,

Table 6. Induction motor parameters of Shanghai power grid (part).

Busbar Name	Capacity/MW	TJ	Pper	KL	XS	XR	A
Hu Hailu23	86.72697	3	1	0.8	0.067	0.17	1
Hu Haiyang22	57.55015	3	1	0.8	0.067	0.17	1
Hu Jingyi23	155.8081	5	0.9	0.7	0.087	0.2	0.9
Hu Jingfeng22	58.73741	5	0.9	0.7	0.087	0.2	0.9
Hu Shikou24	75.46927	7	0.8	0.6	0.107	0.23	0.8
Hu Shiran22	0.432161	9	0.7	0.5	0.127	0.26	0.7
…	…	…	…	…	…	…	…
Equivalent load	7303.38	6.75	0.81	0.61	0.104	0.23	0.81

,

Table 7. Static load parameters of Shanghai power grid (part).

Busbar Name	Capacity/MW	Ldp	Ldq
Hu Zhangqiao23	155.7823	1.8	−2
Hu Dongting23	65.48597	1.9	−2.1
Hu Shenzhuan23	63.27155	2	−2.2
Hu Senlin23	71.13826	2	−2.2
Hu Hangji23	29.34815	2.1	−2.5
Hu Baobei2A	22.31632	2.1	−2.5
…	…	…	…
Equivalent load	5912.57	1.95	−2.22

,

Table 8. Governor parameters of Fujian power grid (part).

Unit Name/Governor Parameters	Pei/MW	KMAXi	PGENi/MW	PSRi/MW	K	FHP	TRH	TO
Qingchuan Hong	1217.1	1.06	1079	211.126	25	0.3	10	1.15
Qingchuan Zhu	1217.1	1.06	1066	224.126	25	0.3	10	1.15
Houshi5	672.4	1.06	338.6	374.144	20	0.4	14	1.5
Houshi6	672.4	1.06	374.4	338.344	20	0.4	14	1.5
Jiangyin2	635.2	1.06	497.2	176.112	16.7	0.5	12	1.3
Gaoyu2	336.2	1.06	237.9	118.472	23	0.7	16	1.7
Gaoyu4	336.2	1.06	217	139.372	23	0.7	16	1.7
Zhangping6	336.2	1.06	219.7	136.672	28	0.6	18	1.9
Lianhua1	180.4	1.06	90.53	100.694	28	0.6	18	1.9
…	…	…	…	…	…	…	…	…
Equivalent generator	16,654.4	1.06	11,463.83	6189.834	21.7	0.5	13.6	1.47

,

Table 9. Induction motor parameters of Fujian power grid (part).

Busbar Name	Capacity/MW	TJ	Pper	KL	XS	XR	A
Min Qingchuan23	1.353096	3	1	0.8	0.067	0.17	1
Min Qingchuanhe	4.723641	3	1	0.8	0.067	0.17	1
Min Fuxing23	4.958399	3	1	0.8	0.067	0.17	1
Min Panlong22	146.6867	5	0.9	0.7	0.087	0.2	0.9
Min Shanfeng22	206.7169	5	0.9	0.7	0.087	0.2	0.9
Min Shangjing22	42.68303	5	0.9	0.7	0.087	0.2	0.9
Min Fuhuo24	3.775402	7	0.8	0.6	0.107	0.23	0.8
Min Gaolong23	8.835766	7	0.8	0.6	0.107	0.23	0.8
Min Gutian1A	−31.0459	7	0.8	0.6	0.107	0.23	0.8
Min Xiangshan23	73.66411	9	0.7	0.5	0.127	0.26	0.7
Min Xindian22	8.197	9	0.7	0.5	0.127	0.26	0.7
Min Xindu22	101.8102	9	0.7	0.5	0.127	0.26	0.7
…	…	…	…	…	…	…	…
Equivalent load	15,934.64	6.95	0.80	0.60	0.106	0.23	0.80

and

Table 10. Static load parameters of Fujian power grid (part).

Busbar Name	Capacity/MW	Ldp	Ldq
Min Qingchuan23	1.353096	1.8	−2
Min Qingchuanhe	4.723641	1.8	−2
Min Fuxing 23	4.958399	1.8	−2
Min Huangli23	14.73251221	1.9	−2.1
Min Dongtai#2	−1.789323056	1.9	−2.1
Min Dongtai #1	189.0300536	1.9	−2.1
Min Fengban22	139.8962934	2	−2.2
Min Fuhuo24	3.7754017	2	−2.2
Min Gaolong23	8.835766464	2	−2.2
Min Shudou24	74.59382699	2.1	−2.3
Min Taqian22	158.8000653	2.1	−2.3
Min Tianbian23	44.59348724	2.1	−2.3
Min Tongcheng22	69.96865655	2.1	−2.3
…	…	…	…
Equivalent load	15,934.64112	1.965098481	−2.165098481

show some governor and load parameters of Anhui power grid, Shanghai power grid and Fujian power grid.

After the equivalence is completed according to the above method, the unipolar blocking fault of ‘Jin-Su DC’, the bipolar blocking fault of ‘Jin-Su DC’, the bipolar blocking fault of ‘Jin-Su DC’ and the bipolar blocking fault of ‘Long-Zheng DC’ and the load shedding fault are simulated. The frequency change of East China Power Grid with and without equivalence is monitored to verify the effectiveness of the equivalence method. The following is the graph of the frequency change of the power grid with and without the equivalence when four different faults occur in the system.

It can be seen from Figure 7, Figure 8 and Figure 9 that under the three DC blocking conditions, the frequency changes of East China Power Grid with and without equivalence are consistent, which proves that the equivalence method has a good equivalence effect. And as shown in Figure 10, under the simulated load of −7200 MW, which is a high-frequency fault, the frequency difference of the whole network of the East China Power grid with and without equivalent is no more than 0.03 Hz, which shows that the overall equivalent effect of this equivalent method is good. It can be seen that the equivalent method of the governor-prime mover and load based on regulation performance weighting proposed in this paper has good effectiveness under small power shortage, high power shortage, low-frequency fault or high-frequency fault and can adapt to the frequency dynamic analysis of large power grid.

5. Conclusions

This paper studies the equivalent model suitable for large power grid frequency dynamic analysis. From the two aspects of generator and load, a method of equivalent aggregation of governor-primitive motor model and load model parameters based on regulation performance weighting is proposed. This method overcomes the uncertainty of artificial experience value, having fast aggregation speed and high engineering accuracy. Finally, the simulation verification is carried out in the example of the East China Power Grid. The results show that the equivalent method can meet the accuracy and safety requirements of the project in the dynamic analysis of large power grid frequency.

However, due to the complexity and particularity of the governor and load model, when the parameters of the governor-primitive motor are equivalently aggregated in this paper, there are fewer types of governors in the selected examples. Although the equivalent results are ideal, the case of more types of governors is not considered. In the follow-up study, a power grid with more types of governors can be selected as an example analysis to comprehensively verify the effect of the equivalent method.