A Second-Order Sliding Mode Voltage Controller with Fast Convergence for a Permanent Magnet Synchronous Generator System

Abstract

This paper studies an improved super-twisting sliding mode controller (IST-SMC) for the permanent magnet synchronous generator (PMSG) voltage loop to improve the anti-disturbance capability of the system. Compared to conventional voltage controllers, the control algorithm provides advantages in terms of system resistance to load disturbances. Conventional voltage controllers have significant voltage fluctuations and long recovery times during sudden load changes. To solve this problem, a voltage loop controller based on a super-twisting sliding mode (ST-SMC) is designed to enhance the immunity of the system. Also, the ST-SMC was improved to further increase the convergence rate of the system and enhance the dynamic performance. The convergence of the system away from the balance point is accelerated by introducing an exponential term, which in turn provides an improvement in the dynamic performance of the system. The effectiveness of the proposed control scheme was verified on a PMSG.

1. Introduction

Permanent magnet synchronous generators (PMSGs) are widely used in modern industrial production. Due to the high power density, small size, and lightweight nature of PMSGs, they are used in multi-powered aircraft [1,2], micro gas turbines [3], vessels [4] flywheel energy storage [5], etc. A typical PMSG-PWM rectifier power generation system includes a PMSG, PWM rectifier, dc bus and load, etc. The structure is shown in Figure 1. The dc-side loads of PMSG-PWM rectifier power generation systems are diverse and complex. The load variations can result in voltage fluctuations on the dc-side, which can seriously affect the voltage quality.

The PMSG is a strongly coupled, non-linear system and it is difficult for the conventional control strategy of proportional-integral (PI) to meet the high precision control requirements. To suppress voltage fluctuations on the dc-side due to sudden load changes and to improve the voltage quality, different advanced control strategies have been put forward for the voltage loop in recent years, such as active disturbance rejection control (ADRC) [6,7], model predictive control (MPC) [8,9], adaptive control [10], and sliding mode control (SMC) [11,12], and so on. Among the above voltage control strategies, the ADRC has achieved better control results but the introduction of the observer has led to a complexity in the system. Voltage control strategies based on MPC require a high level of model accuracy.

It is difficult to build an accurate PMSG model, which limits the application of MPCs. Compared to ADRC and MPC, the insensitivity of sliding mode control to parameter changes and the simplicity of the design show advantages in different applications [11,12]. However, the major drawback of the sliding mode is the chattering problem caused by the high-frequency switching control.

To overcome the chattering problem, the boundary layer method and improved convergence law for a first-order sliding mode control were proposed [13,14]. The above methods are effective in suppressing the chattering problem but at the cost of sacrificing the dynamic performance of the system. In [15], a sliding mode control method based on an improved exponential convergence law was proposed for PMSG to suppress system chattering. The method effectively improves the transient performance of the dc voltage.

In addition to the aforementioned methods, the other important means of overcoming the chattering problem is the deployment of high-order sliding mode control (HO-SMC) [16,17]. By hiding the high-frequency switching function in its derivative, it can eliminate the chattering problem in theory. The super-twisting SMC (ST-SMC) is a typical high-order sliding mode controller with a relative order of one, and therefore suitable for the first-order control systems [18]. In [19], a new ST-SMC direct torque controller was proposed for induction motors to improve the robustness of the system. In [20], a current loop controller based on the ST-SMC was designed to improve the dynamic and steady-state performance of induction motors. There are numerous research results showing that ST-SMC is effective in suppressing chatter and improving system robustness. However, the significant drawback of ST-SMC is that the control parameters are dependent on the external disturbance boundary [21]. In practice, the boundaries of disturbances are normally difficult to measure precisely. The existing methods for solving perturbation boundary problems mainly include adaptive gain [4,22] and disturbance observer [23,24]. In [4], an adaptive gain adjustment technique for ST-SMC was proposed to overcome the deficiency of unknown upper bound on the perturbation; however, the adaptive law design is complex, and the control parameters are numerous. In [23,24], a disturbance observer was designed to estimate the disturbance variation in real time and feed it back into the ST-SMC control law, which in turn reduces the gain of the controller. However, the introduction of disturbance observers increases the complexity of the system, and the real-time nature of the disturbance observer is limited by the sampling frequency of the system.

Therefore, to improve the anti-disturbance capability of PMSGs and suppress the system chattering, an improved ST-SMC voltage control strategy is proposed in this paper. The innovations of the control scheme studied are as follows: (1) to reduce the chattering problem of sliding mode control, a voltage external loop controller based on the ST-SMC is designed to improve the anti-disturbance capability of the PMSG; (2) to improve the convergence rate of the system and further enhance the immunity to disturbance, the ST-SMC is improved by introducing an exponential term in the control law, which accelerates the convergence rate when the system is away from the sliding mode surface. This improves the dynamic performance of the system. The improved ST-SMC can accelerate the convergence of the system when disturbed and improve the dynamic performance of the system.

The remainder of this article is organized as follows. In Section 2, the mathematical model of the PMSG is established and the control structure of the PMSG-PWM power generation system is described. In Section 3, the voltage outer loop controller for the PMSG based on the IST-SMC is designed and its convergence time is analyzed. Simulation and experimental results with the control scheme studied in this paper are provided in Section 4. This paper is concluded in Section 5.

2. Modelling and PMSG Rectification Systems

2.1. Modelling of PMSG

In the d-q synchronous rotating coordinate system, the mathematical model of PMSG is:

$$\left\{\begin{array}{l}{u}_d=-R_{\mathrm{s}}{i}_d-L_d\frac{\mathrm{d}{i}_d}{\mathrm{d}t}+{\omega }_{\mathrm{e}}{L}_q{i}_q\\ u_q=-R_{\mathrm{s}}{i}_q-L_q\frac{\mathrm{d}{i}_q}{\mathrm{d}t}-{\omega }_{\mathrm{e}}{L}_d{i}_d+{\omega }_{\mathrm{e}}{\psi }_f\end{array}\right.$$

(1)

where $i_d$, $i_q$ are the d-q axis components of the stator current; $u_d$, $u_q$ are the d-q axis components of the stator voltage; $R_{\mathrm{s}}$ is the stator resistance; and $L_d$, $L_q$ are the d-q axis inductances. W_e is the electrical angular speed and ${\psi }_f$ is permanent magnet flux linkage.

The PMSG current balance relationship is:

$$C\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}=i_{\mathrm{dc}}-i_{\mathrm{L}}$$

(2)

where u_dc is the dc-side voltage; C is the is the voltage regulator capacitor; i_dc is the dc-side current; and i_L is the load current.

The equation for the power balance of the PMSG is given by:

$$P_{\mathrm{e}}=u_{\mathrm{dc}}^{*}{i}_{\mathrm{dc}}=1.5(u_{\mathrm{d}}{i}_{\mathrm{d}}+u_{\mathrm{q}}{i}_{\mathrm{q}})$$

(3)

where P_e is the output electromagnetic power of the PMSG and $u_{\mathrm{dc}}^{*}$ is the reference of u_dc.

Substituting (1) into (3), (3) can be rewritten as:

$$P_{\mathrm{e}}=u_{\mathrm{dc}}^{*}{i}_{\mathrm{dc}}=1.5{\omega }_{\mathrm{e}}[{\psi }_{\mathrm{mf}}+(L_{\mathrm{q}}-L_{\mathrm{d}})i_{\mathrm{d}}]i_{\mathrm{q}}$$

(4)

For surface-mount PMSGs, L_d = L_q = L, thus (4) can be simplified as follows:

$$P_{\mathrm{e}}=u_{\mathrm{dc}}^{*}{i}_{\mathrm{dc}}=1.5{\omega }_{\mathrm{e}}{\psi }_f{i}_{\mathrm{q}}$$

(5)

Then, substituting (2) into (5), we can obtain:

$$u_{\mathrm{dc}}^{*}\ast (C\frac{d{u}_{dc}}{dt}+i_L)=1.5{\omega }_{\mathrm{e}}{\psi }_f{i}_{\mathrm{q}}$$

(6)

Further, (6) can be simplified as:

$$i_q=K(\frac{d{u}_{dc}}{dt}+\frac{i_L}{C})$$

(7)

where $K=C{u}_{dc}^{*}/1.5{\omega }_e{\psi }_f$. i_q is the output of the voltage controller.

According to (7), there is a non-linear relationship between the voltage controller output i_q and the dc-side voltage u_dc, whereas conventional PI controllers are linear controllers. Therefore, it is difficult to obtain a satisfactory control effect with the PI controller.

2.2. Description of the Control System for PMSG

In order to enhance the voltage control performance of the PMSG system and enhance the anti-disturbance capability, this paper proposes a voltage external loop controller based on an improved super-twisting sliding mode control (IST-SMC), the structure block diagram of which is shown in Figure 2. As can be seen from Figure 2, the overall scheme adopts the conventional double closed-loop structure, consisting of a current loop and a voltage loop. The current loop uses the conventional PI controller, and the reference value of the q-axis current is output by the voltage controller. For the voltage outer loop, the IST-STW control strategy proposed in this paper is used.

3. Proposed Control Scheme

In this section, a voltage loop controller based on IST-SMC is designed and its stability and convergence process are analyzed to demonstrate the superiority of the control strategy proposed in this paper.

3.1. Existing Voltage Controller Based on SMC

According to the existing research [25,26], for the voltage controller, the sliding surface is usually designed in the following form:

$$s=e_u+g{\displaystyle {\int }_0^t{e}_u}dt$$

(8)

where $e_u=u_{dc}^{*}-u_{dc}$, $u_{dc}^{*}$ is the reference value of the dc-side voltage. g ≥ 0 is the design parameter for the sliding-mode surface. The value of g determines the convergence rate of the system, the larger the value of g, the faster the system is to converge, and vice versa.

According to the sliding mode control theory of equivalent control [27], the output control law $i_q^{*}$ of the voltage controller can be divided into two parts, as follows:

$$i_q^{*}=u_{eq\_i_q}+u_{n\_i_q}$$

(9)

where $i_q^{*}$ is the output value of the voltage controller; $u_{eq\_i_q}$ is the equivalent control term; and $u_{n\_i_q}$ is the sliding mode control term.

The equivalent control term $u_{eq\_i_q}$ can be obtained by solving $\dot{s}=0$. Assuming i_L = 0, combining (7) and (8), $u_{eq\_i_q}$ can be obtained as:

$$u_{eq\_i_q}=Kg{e}_u$$

(10)

Then, $u_{n\_i_q}$ can be designed as:

$$u_{n\_i_q}=-gksgn(s)$$

(11)

where k is the control gain.

Thus, the $i_q^{*}$ can be represented as:

$$i_q^{*}=u_{eq\_i_q}+u_{n\_i_q}=Kg{e}_u-gksgn(s)$$

(12)

According to Lyapunov’s stability theory, to satisfy the stability condition $\dot{s}s<0$, the k needs to ensure that $k>\left|i_L/C\right|$.

From (12), it is clear that the control law contains the switching function term sgn(s), and the switching control law can result in severe chattering problems. Thus, in order to improve the system chattering problem, Gao et al. proposed the reaching law [28]. The typical exponential reaching law can be expressed as:

$$\dot{s}=-\epsilon \mathrm{sgn}(s)-qs$$

(13)

where e and q are controller parameters, e > 0, q > 0.

Then, the control law i_q can be designed as:

$$i_q^{*}=K(g{e}_u-\epsilon \mathrm{sgn}(s)-qs)$$

(14)

3.2. Voltage Controllers Based on High-Order Sliding Modes

In addition to the above-mentioned methods, the high-order sliding mode controller is also an important method for solving chattering. In this paper, a voltage controller is designed based on the high-order sliding mode.

According to the control goal of the PMSG, the voltage tracking error is selected as the sliding mode surface:

$$s=e_u=u_{dc}^{*}-u_{dc}$$

(15)

$$\dot{s}={\dot{e}}_u={\dot{u}}_{dc}^{*}-1/C\ast (1.5{\omega }_e{\psi }_f{i}_q-i_L)$$

(16)

The dynamic equations for the ST-SMC control can be expressed as:

$$\left\{\begin{array}{l}ds/dt=-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)+v+f\\ dv/dt=-k_2\mathrm{sgn}(s)\end{array}\right.$$

(17)

where k₁ and k₂ are the controller parameters and both k₁ and k₂ are positive numbers, 0 < α < 1. F is the system disturbance, which includes internal parameter changes and external disturbances, and $\left|df/dt\right|\le \xi$.

In the first line of (17), two main sections are included: the first part is the power convergence law $-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)$, and the second part is the integral part ${\displaystyle \int -k_2\mathrm{sgn}(s)}dt$. From (17) it can be seen that the switching control law sgn(s) is hidden in the integral term and, hence, reduces the chattering of the system. When k₂ = 0 and α = 0, (17) can be rewritten as $-k_1\mathrm{sgn}(s)$, which is the same as the conventional sliding mode control law.

In order for the system to converge along (17), the control output $i_q^{*}$ needs to be designed as:

$$\left\{\begin{array}{l}{i}_q^{*}=u_{eq}+u_{\mathrm{n}}\\ u_{eq}=K({\dot{u}}_{dc}^{*}+i_L/C)\\ u_n=-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)+v\\ dv/dt=-k_2\mathrm{sgn}(s)\end{array}\right.$$

(18)

According to (18), the convergence rate of the system mainly depends on the control parameter k1, and the larger k1 is, the faster the system converges. However, a value of k₁ that is too large will result in the system having a large velocity when reaching the stabilization point, which leads to chattering.

In this paper, the exponential reaching law is introduced to improve the ST-SMC for increasing the convergence rate of the sliding mode control. The improved dynamic equation is as follows:

$$\left\{\begin{array}{l}ds/dt=-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)-\lambda s+v+f\\ dv/dt=-k_2\mathrm{sgn}(s)\end{array}\right.$$

(19)

where λ > 0 is the parameter to be designed. Comparing (19) and (17), it can be seen that the IST-SMC control law adds an exponential convergence law $-\lambda s$. When the system trajectory is far from the equilibrium point, the convergence rate mainly depends on the $-\lambda s$, while when the system trajectory is connected to the equilibrium point, the convergence rate mainly depends on the $-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)$ term. Thus, the convergence rate of the system trajectory away from the equilibrium point is mainly determined by λ, which avoids the problem of chattering caused by the large value of k₁.

The stability of the studied IST-SMC is proved as follows. Considering a Lyapunov function $V=s^2/2$, its derivative with respect to time is $\dot{V}=s\dot{s}$. Combined with (19), the expression for $\dot{V}$ is:

$$\begin{array}{ll}\dot{V}& =s\dot{s}=s(-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)-\lambda s-{\displaystyle \int k_2\mathrm{sgn}(s)dt}+f)\\ & =-k_1{\left|s\right|}^{\alpha +1}-\lambda s^2-s({\displaystyle \int k_2\mathrm{sgn}(s)dt}-f)\end{array}$$

(20)

According to Lyapunov stability theory, the system is stabilized when k₂ can ensure:

$$k_2>\xi$$

(21)

In this paper, the ST-SMC is improved to increase the convergence rate of the system, which in turn improves the dynamic performance of the system. The overall convergence time of the ST-SMC will be analyzed below. Considering $k_2>\xi$, the convergence time of the system is mainly determined by k₁ and k₃. Thus, (19) can be rewritten as:

$$ds/dt=-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)-\lambda s$$

(22)

According to [28], multiplying $e^{\lambda t}$ on both sides of (22) simultaneously, we can obtain:

$$\frac{d(s{e}^{\lambda t})}{dt}=-k_1{\left|s{e}^{\lambda t}\right|}^{\alpha }\ast e^{(1-\alpha )\lambda t}$$

(23)

The solution of (22) can be obtained by integrating both sides of (23) as:

$$s(t)=e^{-\lambda t}{\left(s{(0)}^{1-\alpha }+\frac{k_1}{\lambda }\left(1-e^{(1-\alpha )k_{3}t}\right)\right)}^{\frac{1}{1-\alpha }}$$

(24)

where $s(0)$ is the initial value of s. When s converges to 0, we can obtain the convergence time with the IST-SMC controller as:

$$t_1=\ln \left(1+\frac{\lambda }{k_1}s{(0)}^{1-\alpha }\right)/\lambda (1-\alpha )$$

(25)

Similarly, with ST-SMC, the convergence time can be expressed as:

$$t_2=s{(0)}^{1-\alpha }/k_1(1-\alpha )$$

(26)

Further, differencing (26) and (27) yields:

$$t_2-t_1=\frac{\lambda s{(0)}^{1-\alpha }-k_1\ln \left(1+\frac{\lambda }{k_1}s{(0)}^{1-\alpha }\right)}{k_1\lambda (1-\alpha )}>0$$

(27)

According to (28), the convergence time of the IST-SMC is smaller than that of the ST-SMC. Thus, the IST-SMC can increase the convergence speed of the system and improve the dynamic performance of the system. Figure 3 and Figure 4 show the convergence trajectories of the ST-SMC and the IST-SMC.

This paper focuses on improving the convergence rate of the system by introducing an exponential convergence law, and the improved output control law $i_q^{*}$ is:

$$\left\{\begin{array}{l}{i}_q^{*}=u_{eq}+u_{\mathrm{n}}=-k_1{\left|s\right|}^{\alpha }\mathrm{sgn}(s)-\lambda s+v\\ dv/dt=-k_2\mathrm{sgn}(s)\end{array}\right.$$

(28)

Figure 3 shows the convergence trajectories for different values of the control parameters k₁ and k₂ in the left and right waveforms, respectively. The initial condition is: e_u = −4. It can be seen in Figure 3 that as k₁ increases, the system converges faster.

However, excessive values of k₁ will produce large accelerations around the steady state point, which will result in chattering in the system. As k₂ increases, the convergence speed of the system decreases and leads to larger overshoots. Thus, the control parameters k₁ = 6 and k₂ = 4 are selected, taking into account the chattering and convergence speed of the system.

The waveform on the left in Figure 4 shows the convergence trajectory of the ST-SMC and the IST-SMC. With the IST-SMC, the system can converge to a steady-state more quickly. The waveform on the right shows the convergence trajectory of the system when the control parameter λ is selected for different values. It can be seen from the waveforms that the system converges faster as l increases within reasonable ranges.

4. Simulation and Experimental Verification

The control scheme studied in this paper was implemented on a 0.5 kW HSPMSG. The parameters of the simulation and experimental prototype are shown in

Table 1. Parameters of the testing on an HSPMSG.

Parameters	Value	Unit
Rated power	500	W
Rated speed	12,000	r/min
Pole pairs	1	--
dc-side voltage	60	V
Phase resistance	0.1	Ω
d-axis inductance	82.5	μH
q-axis inductance	82.5	μH
Switching frequency	20	kHz

. The overall simulation and experiment consist of two parts: (1) verification of the rationality and feasibility of the proposed control scheme; and (2) comparison with existing control schemes (conventional sliding mode and ST-SMC).

4.1. Simulation Analysis

4.1.1. Performance Verification of the Proposed Control Scheme

Based on the analysis in Section 3, balancing the chattering problem and convergence time, several simulation experiments were carried out and the control parameters k₁ = 1000, k₂ = 10 and λ = 2000 were determined. Figure 5 shows the simulation results when the system is started up with the control strategy proposed in this paper.

The waveform on the left in Figure 5 shows the dc-side voltage. The waveform on the right shows the q-axis currents. As can be seen from the dc-side voltage waveform, the dc-side voltage has a good dynamic response speed, whether it is load starting or no-load starting. The dc-side voltage u_dc is able to track the reference value rapidly and with a minor overshoot. The above simulation results demonstrate the feasibility of the control strategy proposed in this paper.

The simulation results during the sudden loading and unloading of the system are shown in Figure 6. The operating conditions of the system are the system startup at no load, then putting in the load at 0.2 s and removing the load at 0.4 s. As can be seen from the simulation results, when the system is suddenly loaded, the voltage drop on the dc-side is 0.35 V and the recovery time is 0.01 s.

According to (20), it is clear that the voltage control law contains a term relating to the speed w_e, and the performance of the proposed control strategy should be verified at different speeds. Figure 7 shows the results of the simulation when the speed changes. The waveforms from top to bottom are: speed n, dc-side voltage u_dc, q-axis current i_q and a-phase current i_a. The initial speed of the motor is 6000 r/min, and at 0.2 s the motor starts to speed up, reaching 9000 r/min at 0.3 s, a continuous speed increase at 0.4 s and finally reaching the rated speed at 0.5 s. The dc-side voltage waveform u_dc shows that u_dc can be stabilized at the reference value when the speed changes and without fluctuation. The above simulation results can prove that the control strategy studied in this paper can achieve stable voltage control over a wide speed range.

In order to verify the performance of the studied control method when the capacitance parameters on the dc-side is changed, Figure 8 shows the simulation results when the capacitance is changed. During the simulation, the capacitance parameters were set to 50%C₀, 100%C₀, and 150%C₀, respectively, where C₀ is the initial value of the dc-side capacitance. As can be seen from Figure 7, when the capacitor parameters change, the dc-side voltage can follow the reference value well and with high accuracy. Therefore, the control strategy proposed in this paper can effectively suppress the parameter mismatch.

4.1.2. Verification of the Performance for Different Control Schemes

To verify the superiority of the control strategy proposed in this paper, a comparison is made with existing control algorithms, including the PI control, the conventional sliding mode control, and the ST-SMC. Figure 9 shows the simulation results of the system with different control methods for sudden loading and unloading, where the parameters of the PI controller are selected as: k_p = 2 and k_i = 0.01. The parameters of the sliding mode controller are selected as: g = 2000 and k = 10,000. The parameters for the high-order sliding mode control are selected as: k₁ = 1000 and k₂ = 10. The summary of the simulation results for the different control methods is shown in

Table 2. Comparison of the performance with different controllers.

Different Control Methods	Δudc/V	tv/ms
PI	1.8	45
SMC	1.4	20
ST-SMC	0.6	15
IST-SMC	0.35	10

. As can be seen from Figure 9 and Table 2, the control strategy proposed in this paper has the lowest voltage drop and the shortest recovery time when the load changes suddenly. Compared to conventional PI control, the voltage drop is reduced by 80% and the recovery time is improved by 78% with the control strategy proposed in this paper. The proposed IST-SMC can further reduce the voltage drop and the recovery time compared to the ST-SMC. It can be concluded from the above analysis that the control scheme proposed in this paper has a stronger immunity to disturbances compared to other control schemes.

4.2. Simulation Analysis

The feasibility of the studied control scheme was verified for a 500 W PMSG platform. The experimental setup is shown in Figure 10, and the parameters of the PMSG prototype are shown in Table 1. The overall experimental setup consists of the following four parts: (a) A coaxial permanent magnet synchronous motor with one side of the motor acting as the prime mover and the other side as the generator for control algorithm verification; (b) The inverter is used to drive the prime mover and the rectifier is used to convert the AC power from the PMSG into DC power. Both rectifier and inverter are based on intelligent power modules (FS100R12KT4G) from Infineon; (c) A microcontroller based on the DSP-TMS320F28377D is used as the control unit; (d) The load resistor is used to consume the electrical energy generated by the PMSG.

Firstly, in this section, the feasibility of the control scheme studied in this paper is verified. Secondly, a comparison experiment is carried out with different control schemes in order to demonstrate the superiority of the proposed scheme.

4.2.1. Experimental Validation of Feasibility

Figure 11 shows the experimental waveforms when the system is started up with the control scheme proposed in this paper. The waveforms shown are the a-phase current, the dc-side voltage, and the q-axis current. The initial value of the dc-side voltage $u_{dc}^{*}$ is 20 V. When the speed of the PMSG reaches the rated speed, the dc-side voltage reference value rises to 60 V. As can be seen in Figure 11, the system has a better dynamic response speed during the start-up process and there is almost no overshoot. In the steady-state, the dc-side voltage can be stabilized at the reference value.

To verify that the control scheme proposed in this paper is able to maintain a stable dc-side voltage over a wide speed range, Figure 12 presents the experimental results of the dc-side voltage as the PMSG speed varies. The voltage on the dc-side of the PMSG is 60 V. The voltage on the dc-side of the PMSG is 60 V. The PMSG is first operated at 6000 r/min, then the motor speed rises to 9000 r/min and, after a period of stable operation, the motor accelerates to the rated speed of 12,000 r/min. As can be seen from Figure 12, the dc-side voltage is able to track the reference value without any difference, both in the low- and high-speed conditions. The dc-side voltage waveform does not fluctuate significantly when the speed changes, which indicates the robustness of the proposed control strategy. The above results show that the control scheme studied can achieve regulated voltage control over a wide range of speeds.

Figure 13 shows the experimental results when the capacitance parameters C are varied to verify the robustness of the proposed control scheme to the parameter variations. The PMSG is operated at rated speed, the dc-side voltage reference is set to 60 V, and the capacitance parameters C are set to 50%C₀, 100%C₀, and 150%C₀, respectively. It can be seen from Figure 13 that the dc-side voltage still tracks the reference value relatively well when C changes. When the system is suddenly loaded, the voltage fluctuations on the dc-side do not change significantly as C changes. Thus, the experimental results demonstrate the high anti-disturbance performance of the studied control scheme.

4.2.2. Contrast Experiments with Different Controllers

The feasibility of the proposed control scheme was verified in the previous subsection, and further experiments with different controllers are carried out in the following. Figure 14 illustrates the experimental results when loading with different controllers. The value of load resistance added to the dc-side in the experiment is 81 Ω. In Figure 14a, the voltage fluctuation when the system is suddenly loaded with the PI controller is 11 V and the recovery time is 80 ms. Comparing Figure 14a,b, it can be seen that using an SMC control can reduce the voltage fluctuations and recovery time to some extent but the improvement is not significant. As can be seen from Figure 14c, with the ST-SMC controller, the anti-disturbance performance of the controller can be significantly improved, with the dc-side voltage fluctuation reducing by 35.4% (from 8.2 V to 5.3 V) and the recovery time reducing to 64 ms. Comparing Figure 14c,d, the voltage control performance is further improved by using the controller proposed in this paper compared to the ST-SMC controller. The voltage fluctuation is reduced from 5.3 V to 3.5 V and the recovery time is reduced to 52 ms.

The voltage fluctuations and recovery times of all the control schemes are provided in

Table 3. Experimental results comparison of four control methods.

Different Control Methods	Δudc/V	tv/ms
PI	11	80
SMC	8.2	78
ST-SMC	5.3	64
IST-SMC	3.5	52

. In addition, for clarity, Figure 15 presents a comparison diagram of the voltage fluctuations and recovery times for the four control schemes. It is shown in Table 3 and Figure 15 that there is a significant improvement in the dynamic performance of the system when using the ST-SMC. In particular, the voltage fluctuation is reduced by 51.8% and the recovery time is improved by 20% with respect to the conventional PI control. Further, the voltage drop is improved by 33.9% and the recovery time is reduced by 18.75% with the IST-SMC control. Thus, the studied control scheme can effectively improve the anti-disturbance capability of the system.

5. Conclusions

In this paper, an IST-SMC controller for PMSG voltage loops is proposed to reduce voltage fluctuations and shorten recovery times during sudden load changes. Compared to the ST-SMC controller, this scheme further improves the anti-disturbance capability of the system by introducing an exponential term to the traditional control law. The analysis of the convergence time demonstrates that the proposed control scheme has a faster convergence rate and stronger immunity to disturbances. The proposed control scheme was validated on a PMSG system. The simulation and experimental results demonstrate that the proposed control scheme is effective in improving the resistance of the system to load disturbances and parameter variations. In addition, the dc-side voltage remains stable at the reference value when the PMSG speed changes; therefore, the control scheme studied is able to achieve regulated voltage control over a wide range of speeds.