Gain Scheduled Torque Compensation of PMSG-Based Wind Turbine for Frequency Regulation in an Isolated Grid

Abstract

Frequency stability in an isolated grid can be easily impacted by sudden load or wind speed changes. Many frequency regulation techniques are utilized to solve this problem. However, there are only few studies designing torque compensation controllers based on power performances in different Speed Parts. It is a major challenge for a wind turbine generator (WTG) to achieve the satisfactory compensation performance in different Speed Parts. To tackle this challenge, this paper proposes a gain scheduled torque compensation strategy for permanent magnet synchronous generator (PMSG) based wind turbines. Our main idea is to improve the anti-disturbance ability for frequency regulation by compensating torque based on WTG speed Parts. To achieve higher power reserve in each Speed Part, an enhanced deloading method of WTG is proposed. We develop a new small-signal dynamic model through analyzing the steady-state performances of deloaded WTG in the whole range of wind speed. Subsequently, H_∞ theory is leveraged in designing the gain scheduled torque compensation controller to effectively suppress frequency fluctuation. Moreover, since torque compensation brings about untimely power adjustment in over-rated wind speed condition, the conventional speed reference of pitch control system is improved. Our simulation and experimental results demonstrate that the proposed strategy can significantly improve frequency stability and smoothen power fluctuation resulting from wind speed variations. The minimum of frequency deviation with the proposed strategy is improved by up to 0.16 Hz at over-rated wind speed. Our technique can also improve anti-disturbance ability in frequency domain and achieve power balance.

1. Introduction

At present, wind turbine generators (WTGs) are widely applied. However, its variability and fluctuation increase the difficulty of grid frequency stability and power balance. Especially in an isolated grid, WTGs bring a serious challenge to frequency stability. In addition, sudden load change may lead to power imbalance. In an isolated grid with a high proportion of wind power penetration, the frequency stability of isolated grid needs to be improved by adjusting WTGs output power [1,2,3,4].

WTG needs some power reserves to regulate frequency and balance power [5,6,7,8,9]. WTGs can obtain power reserves mainly in three ways: external energy storage system (ESS), pitch system and converter system [10,11,12,13,14]. Although utilizing many ESS devices can improve frequency stability, there are a few limitations such as the significant increase in economic cost [15]. Therefore, researchers developed model predictive control (MPC) to adjust pitch system for frequency regulation [16]. Some power reserve can be obtained by adjusting the pitch angle to non-zero. However, due to the delay characteristics and a long control period of pitch actuator, pitch system may lead to untimely power adjustment. Therefore, WTGs cannot fully contribute to frequency regulation when relying only on pitch system.

In contrast to ESS and pitch system, converter system has advantages of simple application and fast response to frequency variations [17,18]. Two schemes are commonly used to regulate frequency with converter control. One scheme is that WTG runs with maximum power point tracking (MPPT) control mode, and the power response to frequency variation is achieved by inertia and kinetic energy for a short time [19,20]. The other scheme is that WTG does not track the maximum power point, and the power reserve of WTG is achieved by under-speeding or over-speeding control methods. The work in [21] shows that over-speeding control method is more appropriate to gain power reserves for deloaded WTG. However, at a low wind speed, if frequency regulation is entirely achieved by power reserves of deloaded WTG, the regulation ability is weak, and severe instability may be caused by rapid change of WTG speed. If only inertial characteristic is considered, severe instability may be caused [22]. In terms of frequency regulation method applied to WTG, the most commonly used control strategy is the variable droop control [23]. Wu et al. [24] investigated which droop factor of individual deloaded WTGs can be controlled to improve the primary frequency contribution. Together with [21,25,26], they reach the following consensus. (1) At a low wind speed, there is a possibility of instability when deloaded WTG contributes more energy. Although this could be avoided by limiting the WTG power contribution with a relatively high droop value, the dynamic inertia response capability of WTG cannot be fully utilized; (2) Since utilizing wind speed as a control variable is not appropriate for actual application, variable droop parameters are computed by deloading power and effective droop factor [27]. Despite its effectiveness, the different power compensation capabilities of deloaded WTG in the whole speed range are not considered, which is important.

In summary, the above approaches cannot handle issues such as enhancing power reserve and different power compensation capabilities in each Speed Part. This paper aims to tackle them, through leveraging a new strategy in converter and pitch system for frequency regulation without external ESS. The contributions with respect to the proposed control algorithms are as follows.

• To optimize the primary frequency contribution of WTG, this paper presents a gain scheduled control strategy based on H_∞ theory which combines the advantages of steady-state power reserves and inertia and enhances the torque compensation gain at whole range of wind speed.
• In the deloading method, different deloading factors are designed to smoothen WTG output power and to ensure more power reserve.
• The WTG speed reference of pitch control system is changed to avoid decay or even power oscillation between different Speed Parts caused by torque compensation.
• Comparing with the state-of-the-art MPPT and Droop methods, the frequency deviation with the proposed strategy is reduced by 0.16 Hz and 0.04 Hz, respectively, at over-rated wind speed, and 0.38 Hz and 0.06 Hz, respectively, when wind speed changes.

The rest of the paper is organized as follows. The isolated grid configuration is analyzed in Section 2. An improved deloading torque method of WTG is developed in Section 3. Characteristics of the deloaded WTG with torque compensation are studied in Section 4. Section 5 builds linearized model and designs H_∞ controller. Simulation and experiment results are shown in Section 6. Finally, the conclusion is given in Section 7.

2. Isolated Grid Configuration

In an isolated grid, load is supplied by a permanent magnet synchronous generator (PMSG) based wind turbine and a diesel generator. The PMSG is connected to the isolated grid by a converter. This work is based on the following wind turbine model in [28]. Some details of this model are included for completeness.

$$\{\begin{array}{l}{P}_m=0.5\rho C_p(\lambda ,\beta )A_r{V}_w^3,\\ C_p(\lambda ,\beta )=c_1(c_2/{\lambda }_i-c_3\beta -c_4)e^{-c_5/{\lambda }_i}+c_6\lambda ,\\ \frac{1}{{\lambda }_i}=\frac{1}{\lambda +0.08\beta }-\frac{0.035}{{\beta }^3+1},\end{array}$$

(1)

where P_m is mechanical power, ρ is air density, C_p is the power coefficient, λ is the tip speed ratio, β is the pitch angle, A_r is effective area covered by turbine blades, V_w is wind speed, and c₁, c₂, c₃, c₄, c₅ and c₆ are coefficients. The values of relevant parameters in equations and block diagrams of this paper can be seen in Appendix A. The rotor motion equation of PMSG is expressed as:

$$\frac{\mathrm{d}}{\mathrm{d}t}{\omega }_r=(T_m-T_e)\frac{1}{2H},$$

(2)

where ω_r is the generator angular speed, T_m is mechanical torque, T_e is electric magnetic torque, and H is inertia coefficient. To regulate WTG output power when isolated frequency is unstable, the torque compensation is adopted by Equation (3).

$$T_e=T_{e,del}+T_{e,com},$$

(3)

where T_e is electric magnetic torque, T_e,del is deloading torque, and T_e,com is torque compensation.

The block diagram of the isolated grid is shown in Figure 1. In the pitch system, τ_β is mechanical time constant of pitch actuator, the classical PI controller is to stabilize the WTG speed at the reference ${\omega }_r^{*}$, and the speed reference is changed. P_e is output power of WTG, P_d is diesel generator output power, P_l is load power, and P_dev is power deviation. Frequency deviation f_dev is obtained by an average aggregate frequency response model H_g(s), while M is inertia coefficient and D is damping coefficient. Diesel generator is controlled to balance power flow [29]. The governor and engine dynamics of diesel generator are modeled as in [23]. T_d and T_g are mechanical time constants of engine and governor, respectively. K_I is an integral coefficient and R is a droop factor. The proposed deloading torque method can be described in Equation (4). T_e,com is obtained by the proposed gain scheduled torque compensation strategy.

Frequency regulation strategy depends on load characteristics, and power–frequency regulation of generators can be set by adjusting control systems [30]. Thus, to reduce frequency regulation difficulty and enhance anti-disturbance ability of the isolated grid, WTG is deloaded based on over-speeding control. On the one hand, WTG could adjust output timely according to power demand of isolated grid and reduce pressure of balancing power for diesel generator. On the other hand, WTG output power could be smoothed when wind speed changes. Thus, deloaded WTG has evident advantages for enhancing grid frequency stability.

3. Deloading Method for WTG

Deloading method enables WTG to possess affluent power reserves. Two deloading factors k_f1 and k_f2 (0 < k_f1 < k_f2 < 1) are applied. Corresponding relationships between torque and speed of WTG with deloading factors are shown in Figure 2. The power reserve of WTG can be obtained with deloading factor. The power reserve of WTG increases with decreasing deloading factor. It can be seen that the power reserve with k_f2 is between those with k_f1 and MPPT. The capacity of power reserve of WTG depends on the selection of deloading factors. When wind speed is higher than its rated value, k_f2 is selected as 0.9 to obtain 0.1 pu power reserve of WTG. If k_f1 is selected, higher power reserves can be presented but causes more wind energy loss. When wind speed is lower than its rated value, k_f2 may not be able to provide WTG sufficient power reserve to regulate frequency. Therefore, k_f1 may be selected as 0.8 to possess more power reserve for WTG to regulate frequency. The deloading factors may vary theoretically from 1 (no power reserve) to 0 (total curtailment). The selection of deloading factors is related to capacity of a WTG and power requirement of grid [31].

To obtain appropriate power reserve in each Speed Part and expand speed range in Part 2, three Speed Parts are defined as follows.

Part 1:${\omega }_0<{\omega }_r\le {\omega }_1$. In this Part, to provide WTG more power reserves, k_f1 is selected. ω₀ is the cut-in angular speed and ω₁ is the initial angular speed.

Part 2:${\omega }_1<{\omega }_r<{\omega }_2,({\omega }_2>{\omega }_{r,\mathrm{nom}})$. This Part is the transition of Parts 1 and 3. ω_r,nom is the rated speed of WTG. The electromagnetic torque reference is proportional to WTG speed, and the red transition curve as shown in Figure 2 is applied for smoothening its output power. The purpose of expanding speed range is to reduce power variation when wind speed changes.

Part 3:${\omega }_2\le {\omega }_r<{\omega }_{r,\max }$. k_f2 is chosen in this Part. WTG output power is the highest among all Parts. The power reserve in this Part is also the largest. ω_r,max is the speed up limit. When WTG speed decreases, more energy would be obtained by adjusting pitch angle, and WTG speed reference of pitch control system is changed to be ω₂ naturally.

Based on the above description, an improved torque control method is proposed to ensure more reserve. The red transition curve as shown in Figure 2 is adopted in Part 2. The proposed deloading torque in this paper is expressed by Equation (4).

$$T_{e,del}=\{\begin{array}{ll}{k}_{f1}{k}_{opt}{\omega }_r^2,& {\omega }_0<{\omega }_r\le {\omega }_1\\ \frac{k_{f2}\frac{P_{e,\mathrm{nom}}}{{\omega }_2}-k_{f1}{k}_{opt}{\omega }_1^2}{{\omega }_2-{\omega }_1}({\omega }_r-{\omega }_2)+k_{f2}\frac{P_{e,\mathrm{nom}}}{{\omega }_2},& {\omega }_1<{\omega }_r<{\omega }_2\\ k_{f2}\frac{P_{e,\mathrm{nom}}}{{\omega }_r},& {\omega }_2\le {\omega }_r<{\omega }_{r,\max }\end{array}$$

(4)

where k_opt is the optimal factor, and P_e,nom is the rated power. Furthermore, steady-state simulation is conducted with k_f1 and k_f2. The relationship between power and speed is obtained as shown in Figure 3. The shaded part is the area of adjustable power of deloaded WTG for frequency regulation.

Control laws of the proposed deloading method are further analyzed by steady-state simulation. Simulation results in the whole range of wind speed are shown in Figure 4. When wind speed approaches the rated value, WTG with deloading method would reach the rated angular speed. Thus, the action of pitch angle with deloading method is earlier than that with MPPT control. Correspondingly, C_p with deloading method is also smaller than the value with MPPT control. In other words, the rated wind speed of deloaded WTG is derated by keeping certain power reserve.

4. Analysis of Deloaded WTG

At under-rated wind speed, the deloaded WTG operates at Point A in Figure 3, and certain power reserve of WTG is obtained. When frequency drops, WTG increases output power to isolated grid. Due to rapid change of increasing output power, WTG’s operation point moves to Point B. Furthermore, increasing output power of WTG leads to overload; thus, its speed decreases, and WTG’s operation point moves to Point C. When frequency deviation becomes stable at almost zero or within the dead band, the WTG will return to Point A. However, at over-rated wind speed, the deloaded WTG operates at speed ω₂. When isolated grid requires power, the deloaded WTG would increase torque to supply power by torque compensation controller. Since adjustment of pitch angle is not timely, the WTG speed would be decreased from speed ω₂, which leads to decrease of T_e,del of deloading torque controller. At this point, WTG speed is influenced by torque compensation controller, pitch controller and deloading torque controller. Speed decreasing leads to deloading torque reduction and untimely power adjustment. Since untimely power adjustment can be induced, angular speed reference ω₂ of pitch control system is not suitable when WTG participates in frequency regulation. However, when torque compensation component of WTG is added to support frequency, with a reasonable and higher speed reference ω₃ of pitch control system, pitch angle would begin operation timely, and more power can be absorbed by wind turbine. Thus, this paper adopts ω₃ (ω₃ > ω₂) as a new speed reference of pitch control system to avoid decay or even power oscillation caused by torque compensation. ω₃ is designed in pitch control system. This new speed reference increases adjustment range of WTG speed and regulation time for pitch system. The selection of ω₂ and ω₃ is related to capacity and inertia of WTG. The selection purpose of ω₂ is to expand speed range for reducing power variation when wind speed changes. The selection purpose of ω₃ is to ensure regulating power effectively and extend regulation time for pitch control system response.

Simulation above rated wind speed is performed, and results are shown in Figure 5. ω₂ and ω₃ are set at 1.02 pu and 1.05 pu, respectively. Torque compensation demand is added by 0.06 pu. With traditional deloading method in [23,26], deloaded WTG increases output power to 0.96 pu rapidly. However, when a constant torque compensation component demand 0.06 pu is added, the rotational speed is decreased from angular speed reference ω₂, which leads to decreases of T_e and P_e. In this period, speed decreasing leads to deloading torque decreasing and power drops. When deloading method in Equation (4) with ω₂ is applied, the output power is also not adequate. When deloading method in Equation (4) with ω₃ is applied, a higher rotational speed with a reasonable torque creates a higher power. Deloaded WTG increases output power to 0.96 pu rapidly, and some speed jitter is caused by torque compensation. Simulation results show the necessity of resetting speed reference of pitch control system.

Moreover, steady-state performances of the deloaded WTG are simulated at different wind speeds. The steady-state simulation results are shown in

Table 1. Output Characteristics of the Deloaded WTG at 10 m/s.

Te,com (pu)	ωr (pu)	Te (pu)	Pe (pu)	Power Reserve (pu)
−0.2	0.9737	0.5443	0.5299	0~0.0488
−0.1	0.9591	0.5623	0.5393	−0.0094~0.0394
0	0.9195	0.6088	0.5598	−0.0299~0.0189
0.1	0.8772	0.654	0.5737	−0.0438~0.005
0.2	0.8306	0.6967	0.5787	−0.0488~0

and

Table 2. Output Characteristics of the Deloaded WTG at 13.5 m/s.

Te,com (pu)	ωr (pu)	Te (pu)	Pe (pu)	Power Reserve (pu)
−0.2	1.05	0.6571	0.7	0~0.3
−0.1	1.05	0.7571	0.795	−0.095~0.205
0	1.05	0.8571	0.9	−0.2~0.1
0.1	1.05	0.9571	1	−0.3~0

. In Table 1, wind speed is set at 10 m/s, and WTG is deloaded by Equation (4). Torque and power of WTG increase with the increasing torque compensation component T_e,com, and then corresponding speed of WTG decreases. In Table 2, wind speed is set at 13.5 m/s, and WTG speed is adjusted at 1.05 pu of its rated speed by pitch control. It can be seen that the greater the absolute value of T_e,com is, the more output power P_e can be compensated. The steady-state power reserve in Part 1 is less than that in Part 3, and the inertial characteristic would be utilized to possess more available power reserve in Part 1 for short-time frequency regulation. The gap of power reserve between different Speed Parts is narrowed, which states the rationality of using different reasonable deloading factors based on Speed Parts.

5. Design of Torque Compensation Controller

5.1. Linearized Models of the Deloaded WTG

According to the motion equation and mechanical power of WTG in [28], linearized equation can be expressed by Equation (5).

$$\{\begin{array}{l}\Delta P_m-\Delta P_e=H{\omega }_{r0}\frac{\mathrm{d}\Delta {\omega }_r}{\mathrm{d}t},\\ \Delta P_m=a_1\Delta \beta +a_2\Delta {\omega }_r+a_3\Delta V_w,\end{array}$$

(5)

where “Δ” indicates a small-signal value around the operating point, subscript “0” indicates steady-state value around the operating point, and a₁, a₂ and a₃ depending on the variables at stable operating point are calculated by Equation (6) [32].

$$\{\begin{array}{l}{a}_1=\frac{\partial P_m}{\partial \beta }=0.5\rho A_r{V}_{w0}^3\frac{\partial C_p}{\partial \beta },\\ a_2=\frac{\partial P_m}{\partial {\omega }_r}=0.5\rho A_r{V}_{w0}^3\frac{\partial C_p}{\partial \lambda }\frac{\partial \lambda }{\partial {\omega }_r},\\ a_3=\frac{\partial P_m}{\partial V_w}=0.5\rho A_r(3C_{p0}{V}_{w0}^2+V_{w0}^3\frac{\partial C_p}{\partial \lambda }\frac{\partial \lambda }{\partial V_w}).\end{array}$$

(6)

According to Equations (4)–(6), linearized models of deloaded WTG can be written as

$$\Delta {\dot{\omega }}_r=\{\begin{array}{ll}\frac{a_1\Delta \beta +(a_2-3k_{f1}{k}_{opt}{\omega }_{r0}^2-T_{e,com0})\Delta {\omega }_r+a_3\Delta V_w-{\omega }_{r0}\Delta T_{e,com}}{H{\omega }_{r0}},& {\omega }_0<{\omega }_r\le {\omega }_1\\ \frac{a_1\Delta \beta +[a_2-\frac{k_{f2}\frac{P_{e,\mathrm{nom}}}{{\omega }_2}-k_{f1}{k}_{opt}{\omega }_1^2}{{\omega }_2-{\omega }_1}(2{\omega }_{r0}-{\omega }_2)-k_{f2}\frac{P_{e,\mathrm{nom}}}{{\omega }_2}-T_{e,com0}]\Delta {\omega }_r+a_3\Delta V_w-{\omega }_{r0}\Delta T_{e,com}}{H{\omega }_{r0}},& {\omega }_1<{\omega }_r<{\omega }_2\\ \frac{a_1\Delta \beta +(a_2-T_{e,com0})\Delta {\omega }_r+a_3\Delta V_w-{\omega }_{r0}\Delta T_{e,com}}{H{\omega }_{r0}},& {\omega }_r\ge {\omega }_2.\end{array}$$

(7)

The linearized models take deloading factors and torque compensation value into account. In Parts 1 and 2, k_f1 and k_f2 are introduced, WTG speed is mainly affected by wind speed and torque compensation. In Part 3, the output power is constant without power compensation, and a₁, a₂, a₃ and relevant variables are related to k_f2. Pitch angle needs to be considered, and the linearized model of pitch control system as [33]:

$$\{\begin{array}{l}\Delta {\beta }^{*}=(K_{\beta p}+K_{\beta i}/s)\Delta {\omega }_r,\\ \Delta \beta =\frac{1}{{\tau }_{\beta }s+1}\Delta {\beta }^{*},\end{array}$$

(8)

where K_βp and K_βi are proportional and integral coefficients.

Based on the above analysis, the linearized models in Parts 1 and 2 could not reflect operation characteristics in Part 3. Therefore, it is necessary to study compensation ability of deloaded WTG in different Speed Parts. Considering the complexity and small speed range of the dynamic model in Part 2, one torque compensation controller is commonly applied to Parts 1 and 2, another one is adopted in Part 3.

5.2. Design of Gain Scheduled Torque Compensation Controller

The purpose of torque compensation controller is to complement power shortage and ensure stable operation of WTG. H_∞ control can overcome the influences of wind speed or sudden load changes, and achieve optimal operation of WTG [34,35]. Thus, H_∞ controller is more suitable for frequency regulation in the deloaded WTG. The overall block diagram of H_∞ control is shown in Figure 6. Specific designs of the controller are as follows.

According to linearized models in Equation (8), [x₁ x₂]^T are chosen as state variables. Small-signal of WTG speed Δω_r is added to the state variables x = [x₁ x₂ Δω_r]^T. The external input is w = [ΔV_w ΔP_loss]^T. P_loss is the sum of P_d and P_l. The input signal is u = ΔT_e,com. The output signal of plant P tracks deviation ΔP_dev = ΔP_e − ΔP_loss. The plant P is described by state equation:

$$\dot{x}=A_{1}x+B_{11}w+B_{12}u$$

(9)

with $A_1=\left[\begin{array}{ccc}-\frac{1}{{\tau }_{\beta }}& 0& 1\\ 1& 0& 0\\ \frac{a_1{q}_1}{H{\omega }_{r0}}& \frac{a_1{q}_2}{H{\omega }_{r0}}& \frac{a_2-q_3}{H{\omega }_{r0}}\end{array}\right],B_{11}=\left[\begin{array}{cc}0& 0\\ 0& 0\\ \frac{a_3}{H{\omega }_{r0}}& 0\end{array}\right],B_{12}=\left[\begin{array}{c}0\\ 0\\ -\frac{1}{H}\end{array}\right],$ and output equation:

$$y=C_{1}x+D_{11}w+D_{12}u,$$

(10)

with $C_1=\left[\begin{array}{ccc}0& 0& q_3\end{array}\right]$, $D_{11}=\left[\begin{array}{cc}0& -1\end{array}\right]$, $D_{12}={\omega }_{r0}$, where $q_1=\frac{K_{\beta p}}{{\tau }_{\beta }}$, $q_2=\frac{K_{\beta i}}{{\tau }_{\beta }}$, $q_3=\{\begin{array}{ll}3k_{f1}{k}_{opt}{\omega }_{r0}^2+T_{e,com0},& \mathrm{Parts}1,2\\ T_{e,com0},& \mathrm{Part}3\end{array}$.

Subsequently, the corresponding plant transfer function is:

$$P=\left[\begin{array}{ccc}{A}_1& B_{11}& B_{12}\\ C_1& D_{11}& D_{12}\end{array}\right].$$

(11)

In Figure 6, H_g(s) is given by Equation (12).

$$H_g=\left[\begin{array}{cc}{A}_g& B_g\\ C_g& D_g\end{array}\right],$$

(12)

where A_g = −D/M, B_g = 1, C_g = 1/M, and D_g = 0. Weighting function W₂(s) is chosen as W₂(s) = µ, and weighting function W₁(s) is chosen as:

$$W_1=\left[\begin{array}{cc}{A}_2& B_2\\ C_2& D_2\end{array}\right].$$

(13)

Thus, frequency regulation issues for the deloaded WTG are transferred into H_∞ norm minimization of the transfer function $T_{\tilde{z}\tilde{w}}$ from input $\tilde{w}={\left[\begin{array}{ccc}\Delta f_{dev}^{*}& \Delta V_w& \Delta P_{loss}\end{array}\right]}^{\mathrm{T}}$ to output signal $\tilde{z}={\left[\begin{array}{cc}{z}_1& z_2\end{array}\right]}^{\mathrm{T}}$. The closed loop system is expressed as:

$$\left[\begin{array}{c}\tilde{z}\\ \tilde{y}\end{array}\right]=\tilde{P}\left[\begin{array}{c}\tilde{w}\\ u\end{array}\right],$$

(14)

where $\tilde{P}$ of the system transfer function can be expressed as:

$$\tilde{P}=\left[\begin{array}{cccccc}{A}_1& \mathit{O}& \mathit{O}& \mathit{O}& B_{11}& B_{12}\\ B_{\mathit{g}}{C}_1& A_{\mathit{g}}& 0& 0& B_{\mathit{g}}{D}_{11}& B_{\mathit{g}}{D}_{12}\\ -B_2{D}_{\mathit{g}}{C}_1& -B_2{C}_{\mathit{g}}& A_2& B_2& -B_2{D}_{\mathit{g}}{D}_{11}& -B_2{D}_{\mathit{g}}{D}_{12}\\ -D_2{D}_{\mathit{g}}{C}_1& -D_2{C}_{\mathit{g}}& C_2& D_2& -D_2{D}_{\mathit{g}}{D}_{11}& -D_2{D}_{\mathit{g}}{D}_{12}\\ \mathit{O}& 0& 0& 0& 0& \mu \\ -D_{\mathit{g}}{C}_1& -C_{\mathit{g}}& 0& 1& -D_{\mathit{g}}{D}_{11}& -D_{\mathit{g}}{D}_{12}\end{array}\right].$$

(15)

W₁(s) is a tracking performance weighting function of frequency deviation. To improve the sensitivity of frequency deviation response, W₁(s) has the characteristics of high gain and low pass. W₂(s) is the upper bound of the uncertainty for model parameters. W₁(s) and W₂(s) are designed by testing repeatedly in this paper as follows:

$$\{\begin{array}{l}{W}_1(s)=\frac{4.2}{s+100},W_2(s)=0.03(\mathrm{Parts}1,2)\\ W_1(s)=\frac{1.4}{s+100},W_2(s)=0.03(\mathrm{Part}3).\end{array}$$

(16)

According to the augmented matrix $\tilde{P}$, K₁(s) and K₂(s) are obtained by using a MATLAB (7.13) function hinfsyn and reduced as:

$$\{\begin{array}{l}{K}_1(s)=\frac{9002s^2+1.093\times {10}^{6}s+1.931\times {10}^7}{s^3+2.021\times {10}^4{s}^2+4.119\times {10}^{6}s+2.109\times {10}^8},\\ K_2(s)=\frac{1097s^2+1.097\times {10}^{5}s+421.2}{s^3+1.424\times {10}^4{s}^2+1.414\times {10}^{6}s+5429}.\end{array}$$

(17)

The K₁(s) has a steady-state gain of 0.0916, two zeros at −21.4599 and −99.9576, and three poles at −101, −104 and −2 × 10⁴. K₁(s) is simplified to K₁(s) = 0.0916(s + 21.4599)/(s + 104). The K₂(s) has a steady-state gain of 0.0776, two zeros at −0.0038 and −100, and three poles at −0.00384, −100 and −14,140. Subsequently, K₂(s) is simplified to K₂(s) = 0.0776/(s + p), where p is a sufficiently large pole.

The purposes of gain scheduled torque compensation controller are to improve anti-disturbance ability for frequency and ensure power balance. The proposed H_∞ control has mainly two merits as follows: (1) Different power reserves in each Speed Part are considered in the piecewise controller. In this way, the proposed strategy could support power shortage based on frequency deviation and WTG Speed Part. If the same gain is applied, the generated power compensation in Part 3 is more than that in Parts 1 and 2. To improve the frequency regulation ability in Parts 1 and 2, the steady-state gain of K₁(s) is bigger than that of K₂(s); (2) The H_∞ controller has another advantage of making full use of power reserve and inertia characteristic for frequency regulation. In Parts 1 and 2, the power compensation of WTG is mainly dependent on less power reserve and inertial characteristic for a short time. When the frequency change behavior is slow, it is not suitable to use the inertia characteristic for a long time. The overload and instability of WTG are easily induced, and the gain of K₁(s) is required to be smaller. When the frequency change is more rapid, it is available to use the inertia characteristic. Here, the gain of K₁(s) is expected to be bigger. The power reserve in Part 3 is larger and constant. Thus, the gain of K₂(s) is almost constant at low frequency and attenuates rapidly for noise immunity in higher frequencies. K₁(s) and K₂(s) are consistent with the expected design. It is also clear that the anti-disturbance ability in Part 3 is stronger than that in Parts 1 and 2.

6. Simulation and Experiment

6.1. Simulation Setup

To verify the effectiveness and validity of the proposed H_∞ controller, it is compared with MPPT controller and variable droop controller (Droop for short) in simulation and experiment. H_∞ controller and Droop controller are based on the deloading method in Equation (4). Droop controller is introduced in Equation (18) [23]. Parameters of isolated grid are shown in

Table 3. Parameters of Isolated Grid.

System	Parameters	Values
Diesel generator	Rated power	10 kW
-	Rated voltage	400 V
-	Rated frequency	50 Hz
Constant load	Rated power	5.5 kW
Electronic load	Rated power	10 kW
Wind turbine	Rated speed	12 m/s
PMSG	Rated power	10 kW

.

$$T_e=T_{e,del}+K_P(f_{rated}-f),$$

(18)

where K_P is variable droop factor, and f_rated is the rated frequency of isolated grid. When K_P is small, frequency regulation ability of WTG will be weakened. When K_P is great, unreasonable torque compensation component may worsen system stability.

6.2. Simulation Results

To compare the effectiveness of the presented strategy when load changes, simulation is conducted. Simulation period is 10 s, and wind speeds are set as constants. At the instant of 4 s, load demand is increased by 1.5 kW. The simulation results are shown in Figure 7 and Figure 8.

6.2.1. Case 1: At Over-Rated Wind Speed

Wind speed is set to 13.5 m/s. In Figure 7a, due to lack of wind power contribution, P_d with MPPT control is less than those with deloading method. From 4.02 s to 4.15 s, P_d with H_∞ control is 100 W which is less than that with Droop control. The variation of P_d is smoothened by power reserve of deloaded WTG, the more WTG output power is, means that the less P_d is. As shown in Figure 7b, P_e with MPPT is constant and the power reserve with deloading method is 1 kW for regulating frequency. Compared with Droop control, the power compensation is increased up to 120 W with H_∞ control. The frequency curves are shown in Figure 7c. With MPPT control, the frequency drops to 49.31 Hz when load is increased. The frequency variation with H_∞ control is the smallest. The torque compensation responses to frequency deviation are shown in Figure 7d. In this case, controller K₂(s) is applied, the maximum possible torque compensation is 0.1 pu. When frequency drops rapidly, due to the use of inertia characteristic and power reserve, the peak torque compensation value with H_∞ control is 0.057 pu. The peak torque compensation value with Droop control is 0.045 pu. The torque compensation of H_∞ control is up to 1.27 times of that of Droop control. It can be seen that the torque compensation gain of H_∞ control is higher than Droop control. The pitch angle curves of WTG with different methods are shown in Figure 7e. At a same over-rated wind speed, since the obtained wind power with deloading method is always less than its rated value, the pitch angles with deloading method are 6.7° bigger than that (3.4°) with MPPT control. When the deloaded WTG increases output power to regulate frequency, more energy is obtained by adjustment of pitch angle.

6.2.2. Case 2: At Under-Rated Wind Speed

Wind speed is set to 10 m/s. In Figure 8a, when load demand is increased at 4 s, P_d is lowered. Meantime, the WTG with deloading method participates in balancing power. Due to the WTG’s power contribution since 4 s, compared to MPPT control, diesel generator outputs with deloading methods are reduced. In Figure 8b, the initial outputs of WTG with MPPT and deloading method are 5.79 kW and 5.66 kW, respectively, and the steady-state power reserve of deloaded WTG is 130 W. When load increases, the instantaneous power compensation of WTG with H_∞ control is about 400 W which is larger than that (340 W) with Droop control. Due to the inertia characteristic, the power compensations of deloaded WTG are more than the steady-state power reserve of 130 W. The frequency curves are shown in Figure 8c. When the frequency is disturbed by a load change at 4 s, the frequency variation with H_∞ control is the smallest. In Figure 8d, T_e,com with Droop control is weaker than that with H_∞ control. The torque compensation response gain is enhanced by H_∞ control.

6.3. Experimental Setup

Experimental platform is shown in Figure 9. The platform in laboratory contains diesel generator, constant load, electronic load, indoor WTG and RTLAB real-time simulator. The task of diesel generator is to support voltage and frequency. The constant load is utilized to imitate conventional load. The load change is realized by the electronic load. The indoor WTG is simulated by operating cabinet, grid connected cabinet, main control cabinet, motor and PMSG. The driving system of wind turbine simulation is composed of inverter and motor. The output power of motor is adjusted by inverter. The wind turbine simulation algorithm is applied to PLC. According to wind speed, pitch angle and real-time generator speed, wind turbine mechanical power is calculated by Equation (1). Wind speed is obtained by look-up table or external input. The pitch angle control system is also simulated in PLC, and its specific strategy is included in Figure 1. In this way, the mechanical torque is calculated. WTG speed reference is obtained by Equation (1) which is sent to the inverter. PMSG is driven by the motor. Subsequently, the generated power is output to the isolated grid through the converter. The simulation of indoor WTG is implemented. Voltage signals of isolated grid are transmitted to RTLAB real-time simulator for calculating frequency. Torque compensation controllers are programmed in the RTLAB. Torque compensation component is sent to the PLC through analog interfaces. The operation condition of isolated grid is monitored by HMI. The key data are recorded by measurement instrument.

The dead zone of frequency regulation is set at 0.18 Hz. When frequency deviation is outside the dead zone, the signal of torque compensation component is enabled. Once frequency deviation inside the dead zone, the present compensation component is held in the compensation controller. To verify the effectiveness of the proposed strategy, experiments for different cases were carried out.

6.4. Experimental Results

6.4.1. Case 1: At Over-Rated Wind Speed

Wind speed is set at 13.5 m/s, load demand is added by 1.5 kW at 4 s, and experimental period is 10 s. Experimental results with different control methods are shown in Figure 10a,b. When MPPT control is adopted, at 4 s, since the load is increased, frequency deviation of isolated grid dips to −0.64 Hz. WTG output power is 10 kW, and its torque is constant. With Droop control, initial output power of deloaded WTG is 9 kW. The output power of deloaded WTG is increased by Droop control according to frequency deviation. The output power of WTG is up to 9.34 kW, and the minimum frequency deviation is −0.52 Hz. With H_∞ control, the maximum output power is about 9.40 kW, the minimum frequency deviation is −0.48 Hz. It can be seen that the variation of frequency deviation with H_∞ control is smaller than the other control methods.

6.4.2. Case 2: At Under-Rated Wind Speed

Wind speed is set to 10 m/s, load demand is added by 1.5 kW at 4 s, and the experimental results with different control methods are shown in Figure 10a,b. The WTG output power with MPPT control is 5.7 kW. When load is increased, the frequency deviation dips to −0.66 Hz. With Droop control, the output power of WTG is compensated by 0.2 kW, and the minimum of frequency deviation is reduced to −0.6 Hz. With the proposed H_∞ control, frequency deviation variation is suppressed, and restores to dead zone. The maximum power compensation of WTG is nearby 0.3 kW, the minimum frequency deviation is −0.56 Hz. It can be seen that the frequency regulation capability is further enhanced by H_∞ control.

6.4.3. Summary of Results

To quantify the comparison between the MPPT, droop and H_∞ control algorithms for simulations and experiments, the important values are shown in

Table 4. Comparison with different control methods in simulation and experiment.

Control Strategy		MPPT Control	Droop Control	H∞ Control
Simulation	Case 1, Pe,commax	0	450 W	570 W
	Case 1, fdev,min	−0.69 Hz	−0.5 Hz	−0.45 Hz
	Case 2, Pe,commax	0	354 W	420 W
	Case 2, fdev,min	−0.71 Hz	−0.55 Hz	−0.52 Hz
Experiment	Case 1, Pe,commax	0	340 W	400 W
	Case 1, fdev,min	−0.64 Hz	−0.52 Hz	−0.48 Hz
	Case 2, Pe,commax	0	200 W	300 W
	Case 2, fdev,min	−0.66 Hz	−0.6 Hz	−0.56 Hz

. P_e,commax indicates the maximum of power compensation of WTG. The minimum of frequency deviation f_dev,min is also shown. It can be seen that, at different cases, P_e,commax with H_∞ control is larger than those with the other control methods, and f_dev,min with H_∞ control is the largest. The power compensations of WTG at over-rated wind speed is larger than that those at under-rated speed. With MPPT control, WTG has no power compensation. At over-rated wind speed, comparing the other methods, the minimum of frequency deviation with H_∞ control is enhanced by 0.16 Hz and 0.04 Hz, respectively. The primary frequency regulation contribution of WTG is further enhanced by H_∞ control. Compared with simulation results, there are oscillations in the experimental results. The oscillations could be caused by the weak mechanical system of diesel generator. In addition, according to the simulation and experimental curves shown in Figure 7, Figure 8, Figure 10 and Figure 11, the overall trend is consistent.

6.4.4. Case 3: Wind Speed Change

In this case, wind speed is variable, as shown in Figure 12; load capacity is constant; and experimental period is 100 s. Wind speed change leads to output power variation of WTG. To suppress frequency fluctuation caused by WTG output power, deloaded WTG provides power compensation to isolated grid when wind speed changes. Experimental results are shown in Figure 13a,b. When MPPT control is applied, WTG has no power compensation, the minimum P_e is 6.2 kW, and the minimum frequency deviation is −0.9 Hz. When Droop control is adopted, the minimum P_e is 6.0 kW, and the minimum frequency deviation is −0.58 Hz. With the proposed H_∞ control, change of P_e is smaller than those with other control methods. The minimum P_e is 6.2 kW, and the minimum frequency deviation is reduced to −0.52 Hz. The torque compensation controller provides a damping to the power variation of WTG. The capability of torque compensation with H_∞ control is stronger than that with droop control. The change of frequency with H_∞ control is the smallest. The pitch angle curves are shown in Figure 13b. At over-rated speed, the pitch angles with deloading method are greater than that with MPPT control. At under-rated speed, the pitch angles go to zero to obtain the maximum wind energy. Due to the delay characteristic of pitch control system, the pitch angle curve with H_∞ control is roughly the same as the curve with Droop control. The specific data comparison is shown in

Table 5. Comparison with different control methods when wind speed changes.

Control Strategy	MPPT Control	Droop Control	H∞ Control
Maximum of Pe	10 kW	9 kW	9 kW
Minimum of Pe	6.2 kW	6.0 kW	6.2 kW
Maximum of fdev	0.2 Hz	0.1 Hz	0.1 Hz
Minimum of fdev	−0.9 Hz	−0.58 Hz	−0.52 Hz

. Comparing the other methods, the minimum of frequency with H_∞ control is enhanced by 0.38 Hz and 0.06 Hz, respectively, which shows that the H_∞ control enhances the ability of WTG to suppress frequency variation.

7. Conclusions

The gain scheduled control strategy based on H_∞ theory strategy in this paper enhances the frequency stability of isolated grid. Compared with other control methods, the torque compensation gain is further enhanced by the proposed strategy for the whole range of wind speeds. The proposed strategy nicely integrates the advantages of steady-state power reserves and inertia for regulating frequency, which enables achieving more power reserves with different deloading factors. The speed reference of pitch system is also improved to avoid decay of power compensation at over-rated wind speed. Various H_∞ control performance metrics including inhibiting frequency fluctuation, minimization of active power change, and smoothing power are confirmed by experiments. Comparing with the state-of-the-art MPPT and Droop methods, the frequency deviation with the proposed strategy is reduced by 0.16 Hz and 0.04 Hz, respectively, at over-rated wind speed, and 0.38 Hz and 0.06 Hz, respectively, when wind speed changes.