## 1. Introduction

With the rapid development of economy and society, people’s demand for electricity is increasing day by day. In order to meet the needs of production and life, regional grids have become increasingly interdependent and interactive and the power quality of power supply and distribution systems has become increasingly important. A large-scale interconnected power system consists of many interconnected subsystems (so called control areas), which are connected to each other by tie lines. Each area has its own generator or generator sets to meet its own load demand and power interchange needs with neighbours [

1]. If the power of an area fluctuates due to load fluctuations, communication link delays and failures, the frequency stability of the entire system will be destroyed. In order to effectively control the stability of the grid frequency and thus improve power quality, load frequency control (LFC) systems are widely used in interconnected power grids to make the deviation of the system close to zero, including area control error (ACE), frequency deviation, and tie line power deviation.

In recent years, due to abundant solar energy resources and no environmental pollution issues, distributed power generation technology based on solar energy has developed rapidly around the world. photovoltaic generation (PVG) is widely used in multi-area interconnection power systems. However, the solar-based distributed generation system has poor controllability and is easily affected by changes in the external environment (e.g., voltage and weather), which makes it difficult for PVG systems to output stable power. Moreover, as the permeability of PVG increases, it occupies part of the space of conventional generators and reduces the reserve capacity of the grid’s primary frequency modulation resources [

2,

3]. It reduces the frequency modulation capability of the grid. At the same time, due to the lack of synchronous torque, the increased penetration of PVG will continue to lead to the reduction of system inertia, which will also affect the frequency regulation capacity [

4,

5,

6,

7,

8]. Therefore, it is necessary to solve the problem of LFC of PVG integrated multi-area interconnected power systems.

Domestic and foreign scholars have carried out relevant research on the LFC of PVG integrated power systems. Abd-Elazim and Ali [

9] proposed a firefly algorithm to optimize the PI (FA-PI) controller and optimize the LFC strategy of a hybrid system composed of PV subsystems and thermal generators. Sa-ngawong and Ngamroo [

10] have proposed a Sugeno fuzzy logic controller for intelligent PV power plants based on particle swarm optimization algorithm to suppress frequency fluctuations in multi-area interconnected power systems. In order to reduce the frequency deviation caused by mismatched parameters such as PV and different load disturbances, Yang et al. [

11] proposed a sliding mode load frequency controller based on disturbance observer. Different from the above research, by considering the nonlinear characteristics of the governor’s dead zone and the turbine’s power generation rate constraints, Zeng et al. [

12] proposed an adaptive model predictive LFC method for a PVG integrated multi-area interconnected power system.

The LFC methods involved in the above studies include: PI/PID control [

13,

14,

15,

16,

17,

18], fuzzy control [

19,

20,

21,

22,

23,

24], sliding mode control [

25,

26,

27,

28,

29,

30], and model predictive control [

31,

32,

33,

34,

35,

36]. Although conventional PI/PID control, which is not combined with other algorithms, is widely used in LFC because of its simple structure, it cannot adjust parameters in real time. As a result, the system cannot response to different disturbances with good dynamic performance. Once it is combined with other algorithms to adjust the parameters in real time, its structure will be complicated. The advantage of fuzzy control is that it does not require an accurate mathematical model and is robust, but its design lacks generality and highly rely on experience. Although sliding mode control can overcome the uncertainty of the system and is robust to disturbances and unmodeled dynamics, it has a serious drawback: jitter, and the larger switching range of the control variable, the more significant the jitter. MPC uses rolling optimization strategy to compensate for the impact of uncertainty on frequency in a timely manner. However, due to the large amount of online computing, it requests a high-performance computer environment. Overall, these methods are either computationally intensive or have complex algorithm structures, and are not suitable for controlling complex PVG integrated multi-area interconnected power systems.

In fact, the LFC problems is mainly concentrated on small load disturbances. Given this, the LFC problem of PVG integrated multi-area interconnected power systems is regarded as a disturbance rejection problem. The equivalent-input-disturbance (EID) method is a very effective method for disturbance suppression. The core of the EID method is to introduce the reverse estimated equivalent external disturbance into the input channel to compensate for the actual disturbance. The EID method with a simple structure can simultaneously suppress multiple arbitrary external disturbances with equivalent disturbances. The EID method has low computation cost because the control parameters of the feedback controller, state observer, and low-pass filter in the control system can be designed independently. EID has been successfully applied in vehicle steering control [

37] and power system with wind farms [

38] and has shown excellent disturbance rejection performance.

For PVG-integrated multi-area interconnected power systems with grid voltage fluctuations and load fluctuations, this paper proposes a double EID strategy to control the frequency stability of the system. One of the EID controllers was used in PVG subsystems to maintain stable output power by suppressing grid voltage fluctuations and controlling the output current of the inverter. Another EID controller was used in interconnected power systems to maintain system power balance and frequency stability by suppressing PV output power fluctuations and load disturbances. This strategy generates a new control signal by estimating equivalent disturbance and compensates for the effect of actual disturbance. It has a simple structure and does not require prior information about disturbance. To the best of the authors’ knowledge, this work can be considered as the first contribution of EID to the optimal LFC issue of a PVG integrated multi-area interconnected power system. Double EID LFC strategy can suppress the power grid voltage fluctuation and load demand disturbance and ensures that PVG integrated multi-area interconnected power systems operate normally.

## 4. Results and Discussion

The allowable range of the frequency deviation of the power system is ±0.2 Hz, which was used as the standard in this study. Three simulation experiments were carried out for a PVG integrated two-area interconnected power system to verify the effectiveness and applicability of the proposed double EID method under different conditions.

The values of all the coefficients in the PVG integrated two-area interconnected power system are given in

Appendix A Table A1. The following first-order low-pass filter was selected:

The state space expression of the filter in the PVG subsystem is:

Because the PVG integrated two-area interconnected power system is a multiple-input multiple-output system, the state space expression of the filter is:

For the linearization model of the PVG subsystem, by using Equations (28) and (31) and choosing Q_{K} = diag (5 10^{9}), R_{K} = 1, ρ = 10^{6}, Q_{L} = 100 and R_{L} = 1, the state feedback gain and the state observer gain could be obtained where K_{p} = (8.2411 3.1622 × 10^{4}), L_{p} = 3.1423 × 10^{3}.

For the PVG integrated two-area interconnected power system, the state feedback gain and the state observer gain could be obtained by Equations (28) and (31) as well and choosing

Q_{K} = diag (10

^{11} 10

^{5} 10

^{6} 10

^{7} 10

^{7} 10

^{5} 10

^{4} 10

^{3} 10

^{6} 10

^{3} 10

^{4} 10

^{7} 10

^{9} 10

^{5}),

R_{K} = diag (10

^{4} 111), ρ = 10

^{6},

Q_{L} = diag (50 10

^{−4} 10

^{−3} 10

^{−4} 10

^{−3} 10

^{−4} 10

^{−2} 10

^{−4} 10

^{−4}) and

R_{L} = diag (1 1 1 1 1), where:

Three cases were simulated in MATLAB. First, the fluctuation of the output power of the PVG subsystem was studied in the case of a sudden drop in the grid voltage. Secondly, the step load response of the two-area interconnected power system was investigated and compared under the EID method, the FA-PI control method [

9] and the conventional PI control method. Finally, in the case of random fluctuations in grid voltage and load, the control performance of the proposed double EID method, the FA-PI control method [

9] and the conventional PI control method were compared.

#### 4.1. Output Power Response of PVG Subsystem with Grid Voltage Sag

We performed simulation experiments of the PVG subsystem when the grid voltage plummets. The output current and output power of the PVG subsystem are actually the output current and output power of the PV grid-connected single-phase inverter. In the experiment, the reference output active power of the inverter was set to 1100 W, the frequency of the PCC voltage is 50 Hz, the simulation time is 60 s, and only t = 4.7 s~5.4 s simulation date are displayed. The root mean square (RMS) of the point of common coupling (PCC) changed from 220 V to 132 V at the 5th second, and the voltage of PCC decreased by 40%.

In

Figure 7b, after half a cycle after the voltage dip, the inverter output current starts to follow the reference current. In the third cycle, the inverter output current can better follow the reference current, and the delay of the output current is small. The output power of the inverter fluctuates the most at 5.001 s, but it converges to 1100 W, which is the reference value, at 5.05 s. This shows that when the grid voltage dips, the controller quickly estimates the disturbance and compensates for the disturbance, which enables the inverter output current to quickly track the reference current and the output power to recover to the rated value in a short time.

#### 4.2. Load Frequency Response of Step Load Disturbances

From this case, it should be noted that a +1% step load disturbance was set in the area 1 at the 100th second. The suppression performance of step load disturbance of EID control method, FA-PI control [

9] method and conventional PI control method were evaluated and compared. The simulation diagram of the proposed EID method in MATLAB is shown in

Figure 8.

Figure 9 shows the corresponding performance of the three methods under step load disturbance. The light blue dashed line represents the conventional PI method, the dark blue dashed line is the FA-PI method, and the red solid line is the EID method. The simulation time is 600 s.

Table 1 shows the performance indicators of the three methods under step load including the integral of absolute value of the error (IAE), the integral of time multiply absolute value of the error (ITAE), the integral of square error (ISE) and the integral of time multiply square error (ITSE). IAE, ITAE, ISE, ITSE are defined as follows [

9]:

It can be seen from

Figure 9 that at 130 s, the ACE and tie-line power deviations of the conventional PI control method, FA-PI control method and EID control method are 8.7 × 10

^{−3} p.u, 6.3 × 10

^{−3} p.u, and 6 × 10

^{−4} p.u, respectively. Therefore, the ACE and the tie line power deviation in the proposed method is significantly smaller than that of the FA-PI method and the conventional PI method. At 108 s, the conventional PI control method and FA-PI control method have the maximum Δ

f_{1}, which is 3.45 × 10

^{−4} Hz and 3.4 × 10

^{−4} Hz, respectively. At 150 s, the frequency of area 1 under these two control methods again has a large deviation, which is 2.8 × 10

^{−4} Hz and 1.3 × 10

^{−4} Hz respectively. Under the EID control method, Δ

f_{1} only has a large fluctuation of 0.4 × 10

^{−4} Hz at 101 s. After 150 s, Δ

f_{1} under the conventional PI control method and FA-PI control method is still fluctuating, and Δ

f_{1} under the EID control method has tended to 0. The curve fluctuation of Δ

f_{2} is similar to Δ

f_{1}.

Table 1 can show that the performance of the EID method is significantly better than that of the FA-PI method and the conventional PI method. Therefore, compared with the suppression performance of the FA-PI method and the conventional PI method, the EID method is more capable of suppressing step load disturbances, making the system frequency deviation smaller and the convergence speed faster.

#### 4.3. Load Frequency Response for Random Loads

In the area 1, from the 100th second to the 200th second the PVG subsystem suffers random fluctuations of grid voltage, and there are random load disturbances in the interconnected power system. The experimental simulation diagram of this case is similar to the simulation diagram of the previous case. The simulation time is 600 s. The simulation diagrams of the two random disturbances in MATLAB are shown in

Figure 10 and

Figure 11. We implemented separately simulation experiments of the double EID method, the FA-PI method [

9] and the conventional PI method. The simulation results are shown in

Figure 12.

In

Figure 12a–e, by using the EID control method, the ACE and the tie line power deviation are in the range of (−0.2 × 10

^{−2}, 0.2 × 10

^{−2}) and the frequency deviation is respectively within (−1.2 × 10

^{−4}, 1.4 × 10

^{−4}) (Δ

f_{1}) and (−0.22 × 10

^{−4}, 0.25 × 10

^{−4}) (Δ

f_{2}). Using the FA-PI control method, the ACE and the tie line power deviation are in the range of (−1.1 × 10

^{−2}, 0.15 × 10

^{−2}) and the frequency deviation is respectively within (−8 × 10

^{−4}, 6 × 10

^{−4}) (Δ

f_{1}) and (−2.65 × 10

^{−4}, 0.4 × 10

^{−4}) (Δ

f_{2}). Using the conventional PI control method, the ACE and the tie line power deviation are in the range of (−3 × 10

^{−2}, 0) and the frequency deviation is respectively within (−11 × 10

^{−4}, 8.5 × 10

^{−4}) (Δ

f_{1}) and (−2.5 × 10

^{−4}, 0) (Δ

f_{2}). Under the EID control method, the range of ACE, tie line power deviation and frequency deviation is significantly smaller than the range under the conventional PI control method and FA-PI control method. The four performance indicators of IAE, ITAE, ISE, ITSE under the double EID method in

Table 2 are at most 30% of the other two methods.

Figure 12 and

Table 2 show that the double EID LFC strategy is more effective in disturbance suppression than the FA-PI method and the conventional PI method when the system has both load disturbances and voltage fluctuations.