1. Introduction
With the integration of large-scale wind power into the power grid, the internal structure of the power grid is becoming more and more complex. The new energy grid-connected equipment with power electronics as the interface significantly changes the operation characteristics of the power system dominated by the synchronous machine and reduces the transient stability of the power system, especially the small disturbance stability problem represented by low-frequency oscillation [
1,
2]. The large-scale grid connection of wind power makes the power oscillation characteristics of the system more complex, and the system damping changes significantly compared with the traditional power grid [
3,
4,
5]. Therefore, it is necessary to deeply analyze the effect of wind power grid-connected characteristics on system damping characteristics, reveal the influence mechanism of wind power grid-connected scale and wind turbine control characteristics on system oscillation stability, and then select the corresponding low-frequency oscillation suppression method.
At present, many scholars have carried out related research on the small signal stability of the wind power grid based on the damping characteristics of the system. The influence of wind power grid connection on the low-frequency oscillation of the system is studied by using the transient simulation model outlined in References [
4,
6,
7,
8,
9]. The doubly fed induction generator (DFIG) is converted into an equivalent traditional synchronous motor in Reference [
6], and the small disturbance stability of the system with wind power permeability is analyzed under the condition of high and low wind speed. Reference [
7] models wind turbines and analyzes the stability of the system based on the damping ratio from the perspectives of wind farm grid connection points, grid capacity, and transmission power of interconnection lines. In Reference [
8], research on small disturbance stability of wind turbines connected to the grid is classified into two categories: one is to directly replace synchronous units and the other is to add a wind turbine dynamic model in system simulation, and the mechanisms of these two kinds of research results are compared and analyzed. Reference [
9] used a 16-machine equivalent simulation system for feature analysis, which analyzed the impact of the depth of the penetration of wind generation on the low-frequency infrastructure modes of the interconnected power system. The mathematical model of dynamic interaction between a synchronous generator and wind turbine is established in Reference [
4]. The influence of wind turbine supplementary active power control (SAPC) on low-frequency oscillation is analyzed, and the analysis conclusion is verified by nonlinear simulation of simplified two-region system and IEEE 39-bus system. Most of these research establishes small signal stability analysis models with wind power and then builds specific electromagnetic transient simulation models to analyze the impact of wind power grid connection on the system but cannot directly quantitatively analyze the influence mechanism of wind turbine grid connection on system damping characteristics.
Therefore, the whole mathematical model of the wind turbine incorporated into the power system is constructed in References [
5,
10,
11,
12,
13,
14] to analyze the oscillation characteristics. The dynamic energy model of a DFIG with PLL is derived in References [
5,
10]. From a dissipative energy perspective, the influence of the interaction between the DFIG and power grid on the low-frequency oscillation of the system is revealed. Reference [
11] analyzes the influence of the virtual inertia of the DFIG participating in the frequency regulation of the system on the power grid dissipative energy perspective. The results show that the interaction between the virtual inertia of the wind turbine and the grid side may induce system oscillation and divergence. Reference [
12] established a closed-loop dynamic model of a power system with DFIG grid connection and found that when the open-loop oscillation modes of the DFIG subsystem and the remaining subsystems have similar frequencies, it may cause mutual repulsion of the closed-loop system oscillation modes, leading to a significant reduction in system damping. The degree of its impact is positively correlated with wind power permeability. Reference [
13] quantified the dynamic interaction introduced by wind turbines using damping torque analysis, finding that the dynamic interaction between new energy and power system is usually very weak and that the influence of this dynamic interaction on low-frequency oscillation damping is much smaller than that of power flow factors. Reference [
14] performed single-machine infinite equivalence on interconnected systems and analyzed the impact of wind power inertia control on system damping by obtaining the characteristic root variation in the inertia time constant of the wind power integration into the system. Although the above studies have modeled the whole grid-connected system including wind power, most of them consider incorporating wind power into the infinite grid for analysis and do not consider the influence of wind turbines on the low-frequency oscillation of the inter-regional power grid within the system.
The above studies all indicate that with the integration of large-scale wind power, the power system will exhibit weak damping and low inertia characteristics, and the risk of low-frequency oscillation will increase. Therefore, some studies have further proposed corresponding optimization strategies for wind turbine control parameters based on the analysis of the impact mechanism of wind power integration on low-frequency oscillations, in order to improve the system damping characteristics resulting from wind turbine integration. For example, Reference [
15] proposes a virtual synchronous generator (VSG) parameter optimization method based on a data-driven depth deterministic gradient strategy. With node voltage, branch current and other simulation data as input, VSG control parameters are adjusted online to achieve the target of low-frequency oscillation suppression in a model-free manner. By simplifying and reconstructing the full-order linearization model of a type 4 wind turbine, Reference [
16] established a current source damping torque model suitable for analyzing the stability of a DC voltage loop and phase-locked loop, and realized the low-frequency oscillation suppression of a type 4 wind turbine by optimizing the phase and amplitude parameters of the damping transfer loop. References [
17,
18] analyzed the influence of a power oscillation damper (POD) on low-frequency oscillation of a power system containing a DFIG, and realized the low-frequency oscillation suppression of power system by constructing an optimization model of POD parameters. At present, this kind of research is mainly carried out for a specific or a single additional controller and cannot take into account the suppression effect of multiple control parameter adjustment in the wind turbine on the low-frequency oscillation between power grid regions.
In light of this, the present paper constructs a quantitative analysis model for system damping, specifically targeting interconnected power systems with large-scale wind power integration. Based on this model, we have achieved the suppression of low-frequency oscillations in interconnected systems through the optimization of wind turbine control parameters. The primary contributions of the proposed method encompass:
- (1)
Derivation of the state-space equations for the interconnected system incorporating wind power, mapping wind turbine control parameters into the characteristic polynomial, and thereby enabling the quantitative analysis of wind turbine control characteristics within the system damping analysis model;
- (2)
Utilization of the quantitative damping analysis model to intuitively reveal the mechanism by which wind power penetration influences low-frequency oscillations in interconnected power grids;
- (3)
Introduction of the CE-PSO (Cultural Evolutionary Particle Swarm Optimization) algorithm to optimize wind turbine control parameters, achieving optimal control of low-frequency oscillations.
The remainder of this paper is organized as follows.
Section 2 derives the state-space equations for the interconnected system with wind power integration.
Section 3 establishes the analysis model for damping characteristics in regionally interconnected power grids.
Section 4 presents the damping control strategy based on the CE-PSO optimization algorithm.
Section 5 provides simulation analysis of a case study. Finally,
Section 6 concludes the paper.
2. State Equation of the Regionally Interconnected Power Grid with Wind Farms
Figure 1 shows the simplified model of the two-region interconnection system. Assuming that the transmission direction from Region 1 to Region 2 is positive, the equivalent units of the two power grids adopt the second-order generator model; the generator’s transient reactance
, the transient electromotive force
and the mechanical power
are constant; and the load adopts the constant impedance model.
In the absence of wind power generation access, the state equation of the interconnected system can be obtained through the generator swing equation [
19] as follows:
where
and
represent the power angle increment of generators G1 and G2;
and
are speed increments;
and
are mechanical power increments;
and
are electromagnetic power increments;
and
are inertial time constants;
and
are damping torque coefficients.
According to the power flow calculation, the electromagnetic power expression of G1 and G2 output in
Figure 1 without wind power can be obtained, and the synchronous torque coefficient of the generator is
. Assuming the mechanical power is constant, i.e.,
, Equation (1) can be converted to:
The eigenvalue of the characteristic polynomial
in the system state matrix can effectively reflect the low-frequency oscillation characteristics of the system, so the matrix change in Equation (2) can be obtained:
The system shown in
Figure 1 can then be simplified to the equivalent circuit shown in
Figure 2.
Based on the above equation, considering only the active power
injected into the node, we can derive:
where
represents the active power injected by node 1. The voltage phasor at node 1 is
, and the voltage phasor at node 3 is
.
Considering the incremental equation of the active power injected into the node for the above equation, we can obtain the following:
where
represents the active power injected by node 1. The voltage phasor at node 1 is
, and the voltage phasor at node 3 is
.
Considering the large-scale wind power grid connection, ignoring the line power loss and the line’s own resistance, the electromagnetic power increment equation in Regions 1 and 2 is as follows:
where
is the voltage increment of the wind power junction point;
is the phase increment of the wind power junction point;
and
are the reactance of generators G1 and G2 to the junction point of wind power, respectively. The variable subscript 0 represents the initial value.
At this time, without considering the load changes and network consumption, the following can be obtained:
where
is the active power output of the wind farm.
In the interconnected power system, if the frequency of the wind farm junction point slightly changes to and the dynamic frequency coefficient of the wind farm is , then the active power output of the wind farm is . In the interconnected power grid, the frequency variation in the junction point of the wind farm can be represented by the frequency variation in the generator close to it. When the wind farm is located in Region 1, ; when the wind farm is located in Region 2, .
According to the small-value oscillation micro-interference theory, when the rotor has small-value oscillation, the frequency of the node is determined by the frequency of the nearby generator, i.e.,
and
. By combining Equations (6) and (7), the non-state variables in the middle are eliminated. Subsequently, when the wind farm is located in either the power feeding region or the power accepting region, the state equation of the interconnected system can be obtained by incorporating Equation (1).
Taking the wind farm in the power feeding region as an example, the characteristic equation of Equation (8) is as follows:
As can be seen from Equation (10), when the junction point of the wind farm changes, and will change, resulting in changes in and , thus changing the system damping. When the output power of the wind farm changes, will also change, which will cause a change in system damping. According to the same analysis, when the wind farm is in the accepting region, the output power of the wind farm and the junction point will also change the damping characteristics of the interconnected power grid.
To sum up, for interconnected power systems with large-scale wind power grid connection, the damping characteristics of the system are closely related to multiple aspects such as wind farm junction points, wind farm dynamic frequency characteristic coefficients, and wind farm output power (i.e., wind power permeability).
4. Optimization Control Strategy for Wind Power System Parameters Based on CE-PSO Algorithm
According to the analysis in
Section 3, it can be seen that several groups of wind turbine control parameters have an impact on system damping. In order to improve the stability of the interconnected power grid, this paper considers optimizing the multiple control parameters of the wind turbine by building an optimization model, so as to improve the system damping characteristics.
4.1. Optimized Damping Control Model
This paper proposes an optimization control objective function by setting the desired damping ratio of the system and selecting the minimum difference between the damping ratio of the oscillation mode and the desired damping ratio under typical operating conditions.
where
represents the maximum deviation value of the desired damping ratio under the
-th operating mode;
is the expected damping ratio; and
is the actual damping ratio.
Theoretically, the solutions to Equation (10) (i.e., the eigenvalues ) can reveal the low-frequency oscillation behavior of the system. However, solving Equation (10) directly is exceedingly difficult; thus, we resort to utilizing the Lagrangian function for the solution.
Assuming the dominant low-frequency oscillation mode to be
, when this mode exhibits weak damping characteristics, the eigenvalue is considered to be proximate to the imaginary axis, signifying
. Under this condition,
, and substituting these values into Equation (31) yields
[
20].
In the equation, represents the real part of , while represents the imaginary part of .
To simplify the analysis, it is assumed that the parameters and operating conditions of the wind turbine units within the wind farm are uniform. The total output power of the wind farm is taken as the sum of the output powers of the individual units. Assuming a wind power penetration rate of
, let
By substituting Equations (47) and (48) into Equation (10) and rearranging the terms, we obtain:
The damping ratio is an indicator used to evaluate the dynamic performance of a system. When the damping ratio is larger, the number of oscillations to reach a steady state will decrease. In power systems, the damping ratio is typically required to be no less than 0.05 s−1, and in some specific systems, the minimum damping ratio requirement is not less than 0.15 s−1. Here, is taken as the expected damping ratio.
The constraint conditions for each control parameter of the wind turbine should meet the following:
where
,
,
and
are the minimum and maximum values of
and
, respectively;
,
,
and
are the minimum and maximum values of
and
, respectively; and
,
,
, and
are the minimum and maximum values of
and
. respectively.
4.2. CE-PSO Optimization Method
In order to improve the traditional particle swarm optimization (PSO) algorithm’s tendency to fall into local optima in solving the damping control optimization model, this paper combines the cross-entropy (CE) algorithm and PSO algorithm and uses the CE-PSO algorithm to achieve control parameter optimization. This algorithm combines the discrete probability estimation of the cross-entropy algorithm and the random update strategy of the particle swarm optimization algorithm, which can greatly improve the global optimization ability and optimization effect of complex models [
21,
22].
The essence of the CE algorithm is to transform an optimization problem into a probability estimation problem. Assuming
is a real-valued function defined on a finite state set
, the CE algorithm converts the optimization problem of finding the maximum value of this function into a probability estimation problem.
Equation (51) represents the original optimization problem, which is to find the maximum value of the function within the statistical sample set , as well as the state that makes achieve the value of . Equation (52) represents the transformed estimation problem, where is a value close to , and under the parameter , achieves the maximum probability, which is the expected value corresponding to the indicator function .
To solve this problem, an unbiased estimation of
needs to be made, ultimately transforming the original optimization problem into the following maximum optimization problem:
This paper employs the CE-PSO algorithm to solve the established optimization model. Firstly, the CE algorithm is used to construct a discrete probability distribution function, which randomly generates the initial particle swarm for the PSO algorithm. Then, through the random walk and iterative optimization of the PSO algorithm, excellent samples are generated to update the discrete probability distribution function. This process is repeated continuously. The overall flowchart of the algorithm is shown in
Figure 6.
The implementation steps of the CE-PSO algorithm are as follows:
- (1)
Convert the optimization variable matrix into a discrete probability distribution matrix and obtain a probability distribution function (where is part of the context). This realizes the initialization of the discrete probability distribution matrix .
- (2)
Based on , randomly generate samples of and use them as the initial particles for the PSO algorithm.
- (3)
According to the particle position update equation of the PSO algorithm, perform multiple rounds of iterative updates on the particles.
- (4)
Calculate the evaluation function for the updated particles, sort the particles in descending order based on their values, and select the top particles as excellent samples.
- (5)
Based on the update equation of the discrete probability distribution matrix
Using the excellent samples to update the discrete probability distribution matrix ;
- (6)
Repeat steps (1) to (5) until all elements in become 0 or 1, or the set iteration count is reached. At this point, the most excellent sample can be considered as the optimized optimal control parameters for the wind turbine.
5. Case Simulation
In this paper, the PSCAD platform is used to build a two-region interconnected power grid with wind farms connected to the grid shown in
Figure 1, in which the parameters of a single fan are shown in
Table 1, the load adopts a constant impedance model, the rated power of the equivalent generator
and
in the two regions are set to 900 MW, the rated power rate of a single typhoon is 2 MW, and the wind speed is set 11 m/s. As the number of wind turbines in the wind farm depends on the wind power access capacity, the wind power permeability of the system is adjusted by adjusting the number of wind turbines and the equivalent generator capacity of
.
The speed control parameters and virtual inertia control parameters are analogous, and their value ranges can be referenced from the value range of virtual inertia.
(1) Calculation of the value range for
After the introduction of virtual inertia control, wind turbines exhibit similar inertia response and frequency regulation characteristics as equivalent synchronous generators. The unit regulating power
of a traditional synchronous motor can be expressed as:
where
represents the change in generator power,
represents the change in grid frequency, and
signifies the static characteristic of active power.
From Equation (55), we can derive the following:
Upon comparing this with the power-frequency characteristic equation of a synchronous generator, it becomes evident that the value range of can be referenced from the unit regulating power of a synchronous generator. The unit regulating power ranges for steam turbines and hydroturbines are, respectively, 20~33.3 and 25~50. Consequently, a suitable range for in this context is chosen as 20~50.
(2) Calculation of the value range for
Based on the principle of energy conservation, we can utilize the conservation of rotor kinetic energy between wind turbine generators and synchronous generator sets to convert the rotor speed of the wind turbine generator to synchronous speed. Additionally, we must consider the safety of the system frequency and the rotor operation of the wind turbine generator to arrive at the following equation:
where
represents the inertia time constant of the wind turbine generator, and
is the base value of the system’s synchronous frequency.
In summary, the value range for is 5~10.
The ranges for the rotor-side controller parameters can be inferred from references [
23,
24]:
The value range for
is 5~20. The value range for
is 1~10. The value range for
is 5~20. The value range for
is 1~10.
According to the damping quantitative analysis method of the regional power grid with wind farm interconnection proposed in this paper, the quantitative relationship between low-frequency oscillation frequency and wind power permeability can be obtained as shown in
Figure 7.
At the same time, the quantitative relationship between the system oscillation damping ratio and the wind power permeability shown in
Figure 8 can also be analyzed.
As evident from
Figure 7 and
Figure 8, the low-frequency oscillation between regional power grids exhibits an increasing trend with the augmentation of wind power penetration. Concurrently, the damping ratio declines as the level of penetration rises. Notably, when the penetration exceeds 30%, the diminishing effect on the damping ratio becomes less pronounced, and the system’s oscillation frequency essentially stabilizes.
By adjusting the number of wind turbines to set the wind power permeability of the interconnected power grid to 30%, according to the damping quantitative analysis method of the interconnected regional power grid with wind farms proposed in this paper, we can quantitatively analyze the relationship between different control parameters of wind turbines and inter-regional low-frequency oscillation characteristics.
Firstly, the relationship between the speed control PD parameters
,
of the DFIG and the system oscillation damping ratio is shown in
Figure 9 below.
It can be seen from
Figure 9 that when
is less than 38, the damping increases with the increase in
, and when
is greater than 38, the damping decreases with the increase in
.
Secondly, the relationship between the power outer loop control PD parameters
and
of the DFIG rotor side converter and the system oscillation damping ratio is shown in
Figure 10 below.
It can be seen from
Figure 10 that improper PD parameters of the power outer loop control can easily lead to system damping cross-boundary instability, and the proportional parameters should be larger and the integral parameters should be smaller.
Furthermore, the relationship between the current inner loop control PD parameters
and
of the DFIG rotor side converter and the system oscillation damping ratio is shown in
Figure 11 below.
It can be seen from
Figure 11 that the damping decreases with the increase in the proportional parameters of the current inner loop control, and the damping increases with the increase in the integral parameters.
Through the analysis of the above cases, the interconnected power grid will produce different inter-regional oscillation modes under different wind power permeabilities. Different control parameters of wind turbines likewise exert varying influences on the system’s oscillation characteristics. Therefore, according to the damping ratio requirements and the specific wind power permeability, the reasonable control parameters of the wind turbine can be obtained by optimizing the model to improve the oscillation characteristics and realize the system damping control.
Utilizing the CE-PSO algorithm, this paper optimizes the control parameters of the established model and ultimately obtains three optimized sets of PD control parameters, as shown in
Table 2.
To verify that the low-frequency oscillations between power grid regions can be effectively suppressed after optimizing the aforementioned wind turbine control parameters, in this study, a comparative analysis of the low-frequency oscillation characteristics under before optimization conditions (BOP), after optimization using traditional PSO, and after optimization using CE-PSO was conducted. The results, which show the comparative oscillation characteristics, are presented in
Figure 12 and
Table 3 below, which summarize the key findings.
As clearly demonstrated in
Figure 12 and
Table 3, following the optimization of wind turbine control parameters, the oscillation decay between regions of the power grid is significantly accelerated compared to its pre-optimization state. Specifically, the system oscillation damping ratio is enhanced by a factor of 2.54 after the application of the proposed optimization method, indicating the effectiveness of the proposed wind turbine parameter optimization control approach in suppressing low-frequency oscillations across system regions. Furthermore, the optimization performance of the CE-PSO algorithm proposed in this paper is significantly superior to the traditional PSO algorithm. While both methods are able to ensure that the system damping ratio remains above 0.05 s
−1 after optimizing the wind turbine control parameters, the system damping ratio achieved by the CE-PSO used in this study increased by 63.37% compared to the traditional PSO. This verifies the distinct advantages of the proposed method in this paper.