Disturbance Observer-Based Model Predictive Super-Twisting Control for Soft Open Point

Abstract

This paper presents a disturbance observer-based model predictive of super-twisting control for Soft Open Point (SOP). First, with the consideration of the disturbances caused by parameter mismatches and unmodelled dynamics, a super-twisting sliding-mode observer (STO) is proposed to observe the disturbances, and the observed disturbances are introduced into the inner-loop as the compensation to improve the anti-disturbance of SOP system. Second, the outer-loop controller is designed by applying the super-twisting sliding-mode control (STC) approach to improve the dynamic performance and robustness. Third, to deal with large current harmonics by traditional model predictive control (MPC), a Three-Vector-based MPC (TV-MPC) is proposed to increase the number of voltage vectors in a sampling time. Finally, it is verified by simulations that the proposed method can reduce current harmonics, DC-side voltage setting time and improve the dynamic performance of SOP system effectively. In case of parameter mismatches, the proposed observer can observe the disturbances correctly to enhance the robustness of the SOP system.

1. Introduction

In recent years, with the popularization and development of renewable energies, increasingly renewable energy sources are connected to electricity distribution networks. However, with the development of renewable energy sources, such problems as voltage instability, bi-directional current and unbalanced power arise in electric energy systems [1,2]. As a new power electronic device, Soft Open Point (SOP) has the advantages of flexible power management of the distribution network, fast response time and various control methods, among others [3]. Therefore, SOP has been widely used in the electricity distribution energy systems.

Two-port SOP can be seen as a back-to-back converter, which consists of two voltage source converters (VSCs) and a DC-side capacitor [4]. The two-port SOP system is depicted in Figure 1. The control proposed for SOP includes traditional proportional-integral (PI) control, droop control and model predictive control (MPC) [5,6,7]. These methods are widely applied in a SOP system. However, traditional PI control requires many parameters to be designed; when disturbances occur in the SOP system, PI control and droop control are less robust as well. In signal-vector-based MPC, only one voltage vector is applied in the system in a sampling time, which causes large current harmonics and power fluctuations. Therefore, it is necessary to design improved inner and outer loop control to improve the steady state and dynamic state of the SOP system.

In order to solve the problems above, various improved methods are proposed for the inner-loop MPC, such as increasing the number of voltage vectors in a sampling time, dual-vector MPC [8] and three-vector-based MPC [9]; combining other methods, such as adaptive MPC [10] and fuzzy MPC [11], improving the cost function [12] and delay compensation, and so on [13,14].

Since the MPC of the inner loop is created by accurate model parameters, when disturbances of parameter mismatches and unmodelled dynamics occur in the SOP system in actual operation, predictive errors may occur in the system, due to MPC being sensitive to the parameters, which leads to bad performance of the system [15,16,17]. Thus, observers are applied in the system to observe the disturbance of the system and considering the disturbance as the compensation of the system to improve the robustness of SOP. Adaptive observers are used in [18] to estimate the capacitors’ voltages, which have aa good performance during disturbances. In [19], a Kalman filter is designed to predict the current with reduced noises and estimate the disturbance due to the parameter variation, combined with extended state modeling (ESM). In [20], an extended state observer is applied to enhance the robustness of inductance to cater for the divergence caused by the stator inductance mismatch.

Sliding mode theory is widely used in the design of controllers and observers due to its fast response time and robustness [21,22,23,24]. In [25], a sliding-mode controller is designed for a grid-connected NPC inverter to inject a controlled active power while minimizing supply current harmonics, which has the advantages of fast dynamic response and low voltage ripple. Sliding-mode observer is designed for disturbances’ estimation in [26], while sliding-mode controllers are designed to solve the consensus problem in second-order nonlinear multi-agent systems. However, due to its own discontinuity, first-order sliding-mode theory has obvious chattering. Super-twisting algorithm (STA), as a second-order sliding-mode theory, can reduce the chattering effectively, while maintaining the advantages of traditional sliding mode [27,28,29,30,31,32]. In [33], a super-twisting controller is used in permanent linear synchronous motor (PMLSM) to ensure the system convergence and eliminate the chattering phenomenon, thus improving the system’s robustness. In [34], a super-twisting nonlinear disturbance observer is proposed, which can estimate the uncertainties and external disturbances. In [35], a super-twisting observer is designed to observe the lump disturbance in the permanent magnet synchronous motor (PMSM) caused by parameter mismatches and consider the disturbances as the compensation to the inner-loop, which improves the robustness of the system.

In this paper, by introducing the theory of the super-twisting algorithm and three-vector-based MPC, a super-twisting sliding-mode controller (STC) and TV-MPC with super-twisting sliding-mode observer (STO) method is proposed to improve the performance of the steady state and dynamic state, and to observe the disturbance correctly under parameter mismatches, the disturbances observed by STO are applied to the TV-MPC as the compensations of the inner-loop. The contributions of this article are listed as follows:

(1)

Compared with traditional outer-loop PI controller, the start-up time and dynamic response of the DC-side voltage are improved by adopting the proposed method;

(2)

Compared with signal-vector method, the current harmonics are improved by applying the TV-MPC method in SOP system;

(3)

Parameter mismatches in the SOP system are considered and compensated for by STO, which improves the adaptation of the SOP system to the disturbances.

The organization of this paper is as follows: Section 2 introduces the model of the two-port SOP system. In Section 3, STO is designed to observe the disturbances. Section 4 designs the outer-loop DC-side voltage STC. Section 5 proposes the TV-MPC method of the SOP system. Section 6 gives the simulation results. Future works are discussed in Section 7. Finally, the conclusions are summarized in Section 8.

2. Mathematical Model of SOP

Since the structure of each port of the SOP is the same, in this paper, one port of the voltage source converter (VSC) is selected to build the mathematical model. Taking VSC1 as an example, the model of one port of the SOP in synchronous rotating frame can be expressed as [36]:

$$\{\begin{array}{l}{L}_1\frac{\mathrm{d}{i}_{d1}}{\mathrm{d}t}=-R_1{i}_{d1}+\omega L_1{i}_{q1}+u_{1d}-S_{d1}{u}_{dc}\\ L_1\frac{\mathrm{d}{i}_{q1}}{\mathrm{d}t}=-R_1{i}_{q1}-\omega L_1{i}_{d1}+u_{1q}-S_{q1}{u}_{dc}\end{array}$$

(1)

where i_d1 and i_q1 are the d- and q-axis currents of VSC1, respectively. u_1d and u_1q are the d- and q-axis grid voltages of VSC1, respectively. u_dc is DC-side voltage; L₁, R₁ and ω are inductance, resistance of VSC1 and voltage angle frequency, respectively. S_d1 and S_q1 are the d- and q-axis modulation switch functions of VSC1, respectively. Similarly, VSC2 has the same mathematical model in synchronous rotating frame.

According to the current direction shown in Figure 1, the relationship between DC-side currents from VSC and from capacitance can be expressed as follows:

$$i_c=C\frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}=i_{dc1}+i_{dc2}=\frac{3}{2}(-S_{d1}{i}_{d1}+S_{q1}{i}_{q1})+\frac{3}{2}(-S_{d2}{i}_{d2}+S_{q2}{i}_{q2})$$

(2)

where C is the capacitance of DC side, i_dc1 and i_dc2 are the DC-side currents from VSC1 and VSC2, respectively. i_c is the DC-side current from capacitance; S_d2 and S_q2 are the d- and q-axis modulation switch functions of VSC2, respectively. i_d2 and i_q2 are the d- and q-axis currents of VSC2, respectively.

By discretizing (1) [37], the discrete model of VSC1 can be obtained as:

$$\{\begin{array}{l}{i}_{d1}(\mathrm{k}+1)=(1-\frac{T_s{R}_1}{L_1})i_{d1}(\mathrm{k})+T_s\omega i_{q1}(\mathrm{k})+\frac{T_s}{L_1}(u_{1d}(\mathrm{k})-u_{1Nd}(\mathrm{k}))\\ i_{q1}(\mathrm{k}+1)=(1-\frac{T_s{R}_1}{L_1})i_{q1}(\mathrm{k})-T_s\omega i_{d1}(\mathrm{k})+\frac{T_s}{L_1}(u_{1q}(\mathrm{k})-u_{1Nq}(\mathrm{k}))\end{array}$$

(3)

where i_d1(k + 1) and i_q1(k + 1) are the predicted d- and q-axis currents of VSC1 at (k + 1)th instant, respectively. u_1Nd(k) and u_1Nq(k) are the d- and q-axis voltages of VSC1, respectively. i_d1(k) and i_q1(k) are the d- and q-axis currents of VSC1 at (k)th instant, respectively. u_1d(k) and u_1q(k) are the d- and q-axis grid voltages of VSC1 at (k)th instant, respectively. T_s is the sampling time.

Each port of SOP can work in a different mode. DC-side voltage and reactive power are taken as the reference parameters in U_dcQ mode, in order to ensure the steadiness of the DC-side voltage, at least one port should work in U_dcQ mode in the SOP system. In PQ mode, the active power and reactive power are taken as reference parameters and are most used in the SOP system. In addition, taking the DC-side voltage and frequency as reference parameters the U_dcf mode can also be applied in the SOP system when system failure occurs in the system. In this paper, the U_dcQ mode is applied in port1 and PQ mode is applied in port2.

3. Design of Super-Twisting Observer under Parameter Mismatch

In the actual operation of the SOP system, both sides work under large current and voltage fluctuations frequently, which makes the parameter changes of resistance and inductances. In this paper, a super-twisting observer (STO) is designed in the inner-loop to observe the disturbances caused by parameter mismatches and unmodelled dynamics, which maintains the strong robustness of the traditional sliding mode observer while suppressinges the high frequency vibration effectively.

Considering the parameter mismatches and unmodelled dynamics, the model of VSC1 can be rewritten as follows:

$$\{\begin{array}{l}{L}_1\frac{\mathrm{d}{i}_{d1}}{\mathrm{dt}}=-R_1{i}_{d1}+\omega L_1{i}_{q1}+u_{1d}-(u_{1Nd}-f_{d1})\\ L_1\frac{\mathrm{d}{i}_{q1}}{\mathrm{d}t}=-R_1{i}_{q1}-\omega L_1{i}_{d1}+u_{1q}-(u_{1Nq}-f_{q1})\end{array}$$

(4)

where u_1Nd and u_1Nq are the d- and q-axis voltages of VSC1, respectively. f_d1 and f_q1 are the d- and q-axis disturbances of VSC1 caused by parameter mismatches and unmodelled dynamics, respectively. Due to disturbances’ change slowly in steady-state, f_d1 and f_q1 can be written as:

$$\{\begin{array}{l}{f}_{d1}=-\frac{L_1(L_{10}\mathsf{\Delta }{R}_1-R_{10}\mathsf{\Delta }{L}_1)}{(L_{10}(L_{10}+\mathsf{\Delta }{L}_1))}{i}_{d1}-\frac{L_1\mathsf{\Delta }{L}_1}{(L_{10}(L_{10}+\mathsf{\Delta }{L}_1))}(u_{1Nd}-u_{1d})+{\epsilon }_{d1}\\ f_{q1}=-\frac{L_1(L_{10}\mathsf{\Delta }{R}_1-R_{10}\mathsf{\Delta }{L}_1)}{(L_{10}(L_{10}+\mathsf{\Delta }{L}_1))}{i}_{q1}-\frac{L_1\mathsf{\Delta }{L}_1}{(L_{10}(L_{10}+\mathsf{\Delta }{L}_1))}(u_{1Nq}-u_{1q})+{\epsilon }_{q1}\end{array}$$

(5)

where ε_d1 and ε_q1 are the d- and q-axis unmodelled dynamics disturbances of VSC1; L₁₀ and R₁₀ are offline parameters of inductance and resistance of VSC1, respectively; variable matrix x₁ is defined as:

$$\frac{{\mathit{f}}_1}{L_1}={\mathit{x}}_1$$

(6)

where f₁ = [f_d1, f_q1]^T, x₁ = [x_d1, x_q1]^T.

Errors of inductance parameters and resistance parameters ΔL₁ and ΔR₁ are uncertain and bounded, the unmodelled dynamics disturbance ε_d1 and ε_d1 are bounded, which can be written as:

$$\{\begin{array}{l}\left|\mathsf{\Delta }{L}_1\right|\le {\phi }_L\\ \left|\mathsf{\Delta }{R}_1\right|\le {\phi }_R\\ \left|{\epsilon }_{d1}\right|\le {\phi }_{\epsilon d1}\\ \left|{\epsilon }_{q1}\right|\le {\phi }_{\epsilon q1}\end{array}$$

(7)

where φ_L, φ_R and φ_εd1 and φ_εq1 are the known positive constants.

Due to the assumption above, the disturbance f_d1 and f_q1 are bounded and can be assumed as:

$$\{\begin{array}{l}\left|f_{d1}\right|\le {\phi }_{d1}\\ \left|f_{q1}\right|\le {\phi }_{q1}\\ {\phi }_{d1}\le {\phi }_1\\ {\phi }_{q1}\le {\phi }_1\end{array}$$

(8)

where φ_d1, φ_q1 and φ₁ are known positive constants.

The sliding manifold is designed as:

$$\{\begin{array}{l}{s}_{d1}={\widehat{i}}_{d1}-i_{d1}\\ s_{q1}={\widehat{i}}_{q1}-i_{q1}\end{array}$$

(9)

where ${\widehat{i}}_{d1}$ and ${\widehat{i}}_{q1}$ are the d- and q-axis estimated current of VSC1.

Proposition 1.

If the sliding manifold is chosen as (9), the super-twisting observer can be designed as follows:

$$\{\begin{array}{l}\frac{\mathrm{d}{\widehat{i}}_{d1}}{\mathrm{d}t}=\frac{u_{1d}}{L_1}-\frac{R_1}{L_1}{\widehat{i}}_{d1}+\omega {\widehat{i}}_{q1}-\frac{u_{1Nd}}{L_1}+d_d\\ \frac{\mathrm{d}{\widehat{i}}_{q1}}{\mathrm{d}t}=\frac{u_{1q}}{L_1}-\frac{R_1}{L_1}{\widehat{i}}_{q1}-\omega {\widehat{i}}_{d1}-\frac{u_{1Nq}}{L_1}+d_q\\ d_d=-{\alpha }_1{\left|s_{d1}\right|}^{\frac{1}{2}}\mathrm{sgn}(s_{d1})+{\widehat{x}}_{d1}\\ d_q=-{\alpha }_1{\left|s_{q1}\right|}^{\frac{1}{2}}\mathrm{sgn}(s_{q1})+{\widehat{x}}_{q1}\\ {\widehat{x}}_{d1}=-{\displaystyle \int {\beta }_1\mathrm{sgn}(s_{d1})}\mathrm{dt}\\ {\widehat{x}}_{q1}=-{\displaystyle \int {\beta }_1\mathrm{sgn}(s_{q1})\mathrm{dt}}\end{array}$$

(10)

where d_d and d_q are the d- and q-axis error dynamics of the disturbance; α₁ and β₁ are the designed gains of observer.

The estimated disturbance ${\widehat{\mathit{f}}}_1$ can be expressed as follows according to (6):

$${\widehat{\mathit{f}}}_1=L_1{\widehat{\mathit{x}}}_1$$

(11)

Proof of Proposition 1.

Taking VSC1 as analysis, the Lyapunov function is designed as [38,39]:

$$V(\gamma )={\gamma }^T\mathit{M}\gamma$$

(12)

where ${\gamma }^T=\left[\begin{array}{cc}{\left|s_1\right|}^{\frac{1}{2}}\mathrm{sgn}(s_1)& {\displaystyle \int {\beta }_1\mathrm{sgn}(s_1)\mathrm{dt}}\end{array}\right]$, $\mathit{M}=\frac{1}{2}\left[\begin{array}{cc}4{\beta }_1+{\alpha }_1^2& -{\alpha }_1\\ -{\alpha }_1& 2\end{array}\right]$, s₁ = [s_d1, s_q1]^T.

The time derivative of (12) is expressed as:

$$\dot{V}(\gamma )=-{\left|s_1\right|}^{-\frac{1}{2}}{\gamma }^T{\mathit{M}}_1\gamma +{\mathit{f}}_1{\left|s_1\right|}^{-\frac{1}{2}}{\mathit{M}}_2\gamma$$

(13)

where ${\mathit{M}}_1=\frac{{\alpha }_1}{2}\left[\begin{array}{cc}2{\beta }_1+{\alpha }_1^2& -{\alpha }_1\\ -{\alpha }_1& 1\end{array}\right]$, ${\mathit{M}}_2=\left[\begin{array}{cc}(2{\beta }_1+\frac{{\alpha }_1^2}{2})& \frac{-{\alpha }_1}{2}\end{array}\right]$.

Using the bound of disturbance f₁ (8), bound of $\dot{V}(\gamma )$ can be obtained as follows:

$$\dot{V}(\gamma )\le -{\left|S_1\right|}^{-\frac{1}{2}}{\gamma }^T{\mathit{M}}_3\gamma$$

(14)

where ${\mathit{M}}_3=\frac{{\alpha }_1}{2}\left[\begin{array}{cc}2{\alpha }_1+{\alpha }_1^2-(\frac{4{\beta }_1}{{\alpha }_1}+{\alpha }_1){\phi }_1& -({\alpha }_1+2{\phi }_1)\\ -({\alpha }_1+2{\phi }_1)& 1\end{array}\right]$.

$\dot{V}(\gamma )$ is negative, and the system is strongly globally asymptotically stable when α₁ and β₁ are satisfied as follows:

$$\{\begin{array}{l}{\alpha }_1>2{\phi }_1\\ {\beta }_1>{\alpha }_1\frac{5{\alpha }_1{\phi }_1+4{\phi }_1^2}{2({\alpha }_1-2{\phi }_1)}\end{array}$$

(15)

Proofs of VSC2 are similar to the VSC1 side. The proof of the observer stability is completed, which means the values of estimated current approach the values of system current in finite time. □

Figure 2 illustrates the block diagram of STO. The initial values of the integrators are set as 0.

4. Design of Outer-Loop Super-Twisting Controller Design

In this paper, port1 is operated in U_dcQ mode, which keeps the stability of the DC-side voltage. However, in the traditional outer-loop PI controller, the constant parameters of controller may lead to poor dynamic performance when the parameters of the system are changed. As a second-order sliding mode controller, the super-twisting controller (STC) can effectively suppress the chattering of the first-order sliding mode controller while having a better dynamic performance compared with the PI controller. An outer-loop super-twisting voltage controller is designed in this section.

The outer-loop system of VSC1 is composed of the DC voltage loop, the sliding manifold is designed as:

$$S=u_{dcref}-u_{dc}$$

(16)

Uncertain disturbance δ is unknown, but can be assumed to be bounded as follows:

$$\left|\delta \right|\le \rho$$

(17)

where ρ is a known positive constant.

Proposition 2.

In order to track the reference DC-side voltage, according to the theory of super-twisting algorithm (STA), when u_dc is the system-state variable, STA of u_dc can be described as:

$$\frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}=-k_1{\left|S\right|}^{\frac{1}{2}}\mathrm{sgn}(S)-{\displaystyle \int k_2\mathrm{sgn}(S)}+\delta$$

(18)

where k₁ > 0 and k₂ > 0 are the controller gains.

When the grid voltages are three-phase balanced and the system is stable, the state of the d- and q-axis currents of AC side i_d and i_q can be written as:

$$\{\begin{array}{l}{i}_q=0\\ \frac{\mathrm{d}{i}_d}{\mathrm{d}t}=0\end{array}$$

(19)

where i_q = [i_q1, i_q2]^T, i_d = [i_d1, i_d2]^T.

Substituting (19) into (2) and (1), which can be rewritten as:

$$C\frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}=-\frac{3}{2}{i}_{d1}{s}_{d1}-\frac{3}{2}{i}_{d2}{s}_{d2}$$

(20)

$$\{\begin{array}{l}{S}_{d1}=\frac{u_{1d}-R_1{i}_{d1}}{u_{dc}}\\ S_{d2}=\frac{u_{2d}-R_2{i}_{d2}}{u_{dc}}\end{array}$$

(21)

where u_2d is the d-axis grid voltage of VSC2.

Substituting (21) into (20), i_d1 can be rewritten as:

$$i_{d1}=\frac{-\frac{2}{3}C{u}_{dc}\frac{\mathrm{d}{u}_{dc}}{{\mathrm{d}}_t}-i_{d2}(u_{2d}-R_2{i}_{d2})}{u_{1d}-R_1{i}_{d1}}$$

(22)

when the system is stable, i_d1 = i_d1ref, (22) can be rewritten as:

$$i_{d1ref}=\frac{-\frac{2}{3}C{u}_{dc}\frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}-i_{d2}(u_{2d}-R_2{i}_{d2})}{u_{1d}-R_1{i}_{d1}}$$

(23)

Substituting (18) into (23), the output of the controller id1ref can be written as:

$$i_{d1ref}=\frac{\frac{2}{3}C{u}_{dc}(k_1{\left|S\right|}^{\frac{1}{2}}\mathrm{sgn}(S)+{\displaystyle \int k_2\mathrm{sgn}(S)\mathrm{d}t})-i_{d2}(u_{2d}-R_2{i}_{d2})}{u_{1d}-R_1{i}_{d1}}$$

(24)

Proof of Proposition 2.

The Lyapunov function is designed as:

$$V\left(\zeta \right)={\zeta }^T\mathit{P}\zeta$$

(25)

where ${\zeta }^T=\left[\begin{array}{cc}{\left|S\right|}^{\frac{1}{2}}\mathrm{sgn}(S)& {\displaystyle \int k_2\mathrm{sgn}(S)\mathrm{dt}}\end{array}\right]$, $\mathit{P}=\frac{1}{2}\left[\begin{array}{cc}4k_2+k_1^2& -k_1\\ -k_1& 2\end{array}\right]$.

The time derivative of (25) is expressed as:

$$\dot{V}\left(\zeta \right)=-{\left|S\right|}^{-\frac{1}{2}}{\zeta }^T{\mathit{P}}_1\zeta +\delta {\left|S\right|}^{-\frac{1}{2}}{\mathit{P}}_2\zeta$$

(26)

where ${\mathit{P}}_1=\frac{k_1}{2}\left[\begin{array}{cc}2k_2+k_1^2& -k_1\\ -k_1& 1\end{array}\right]$, ${\mathit{P}}_2=\left[\begin{array}{cc}(2k_2+\frac{k_1^2}{2})& \frac{-k_1}{2}\end{array}\right]$.

Using the bound of disturbance δ (17), the bound of $\dot{V}(\zeta )$ can be obtained as follows:

$$\dot{V}(\zeta )\le -{\left|S\right|}^{-\frac{1}{2}}{\zeta }^T{\mathit{P}}_3\zeta$$

(27)

where ${\mathit{P}}_3=\frac{k_1}{2}\left[\begin{array}{cc}2k_1+k_1^2-(\frac{4k_2}{k_1}+k_1)\rho & -(k_1+2\rho )\\ -(k_1+2\rho )& 1\end{array}\right]$.

$\dot{V}(\zeta )$ is negative and the system is strongly globally asymptotically stable when k₁ and k₂ are satisfied as follows:

$$\{\begin{array}{l}{k}_1>2\rho \\ k_2>k_1\frac{5k_1\rho +4{\rho }^2}{2(k_1-2\rho )}\end{array}$$

(28)

When k₁ and k₂ are satisfied in the equation above, the error dynamic of DC-side voltage approaches the ideal sliding manifold S = 0 in finite time. The proof is completed. □

Figure 3 shows the block diagram of outer-loop STC.

5. Inner-Loop TV-MPC for SOP

5.1. Traditional MPC Method

In the SOP system consisting of two-level VSC, the switch states can be divided into six active voltage vectors (V1, V2, V3, V4, V5, V6) and two zero vectors (V0, V7), as shown in Figure 4. As shown in Figure 1, ‘1’ in Figure 4 means the switch S_x is on and S′_x is off, while ‘0’ means the switch S_x is off and S′_x is on (x = 1, 2, 3, 4, 5, 6).

Port1 is operated in U_dcQ mode to ensure the stability of the DC-side voltage. In the traditional MPC method, the PI controller is used in the outer-loop subsystem to output d-axis reference current i_d1ref, which can be expressed as:

$$i_{d1ref}=k_p(u_{dcref}-u_{dc})+k_i{\displaystyle \int (u_{dcref}-u_{dc})\mathrm{d}t}$$

(29)

where k_p and k_i are the proportional and integral coefficients of PI controller, respectively. u_dcref is the reference voltage of the DC side.

The PQ mode is used in Port2. With a stable grid voltage, a given reference current is equivalent to a given reference power, which can transform the PQ mode control into a problem of tracking the reference current.

Taking VSC1 for analysis, since two zero vectors V0 and V7 obtain the same voltage vector, in traditional MPC, seven voltage vectors are substituted into (3) to calculate the corresponding predicted currents, and each predicted current is substituted into the cost function, the voltage vector with the smallest cost function value can be selected as the optimal voltage vector and applied to the switch in the next sampling time. The cost function can be designed as:

$$f=\left|i_{d1ref}-i_{d1}(\mathrm{k}+1)\right|+\left|i_{q1ref}-i_{q1}(\mathrm{k}+1)\right|$$

(30)

where i_q1ref is the q-axis current reference value of VSC1.

Similarly, VSC2 with given reference current, cost function values can be calculated and the optimal voltage vector will be applied to the switch in the next sampling time.

5.2. TV-MPC Method

In the traditional MPC method, only one voltage vector can be used in a sampling time and seven times of cost function calculations are required at both port, which leads to large current harmonics and control delays. Therefore, using the principle of deadbeat control, a low-complexity TV-MPC method is proposed. By calculating the predicted value of the reference voltage of VSC at the next sampling time, using

Table 1. Voltage vector combination selection.

Sector	Voltage Vector Combination
I	V1, V2, V0 or V7
II	V2, V3, V0 or V7
III	V3, V4, V0 or V7
IV	V4, V5, V0 or V7
V	V5, V6, V0 or V7
VI	V6, V1, V0 or V7

to judge the sector and selecting a zero vector and two active voltage vectors in this sector as the vector combination. The vector combination is applied to the VSC at the next sampling time.

Taking VSC1 as the example, in the principle of deadbeat control, assuming the predicted currents at (k + 1)th instant as follows:

$$\{\begin{array}{l}{i}_{d1ref}=i_{d1}(\mathrm{k}+1)\\ i_{q1ref}=i_{q1}(\mathrm{k}+1)\end{array}$$

(31)

Substituting (31) into (3), the reference voltages of VSC1 u_1Ndref and u_1Nqref can be written as:

$$\{\begin{array}{l}{u}_{1Ndref}=\frac{L_1}{T_s}(i_{d1}(\mathrm{k})-i_{d1ref})-R_1{i}_{d1}(\mathrm{k})+L_1\omega i_{q1}(\mathrm{k})+u_{1d}(\mathrm{k})\\ u_{1Nqref}=\frac{L_1}{T_s}(i_{q1}(\mathrm{k})-i_{q1ref})-R_1{i}_{q1}(\mathrm{k})-L_1\omega i_{d1}(\mathrm{k})+u_{1q}(\mathrm{k})\end{array}$$

(32)

According to u_1Ndref and u_1Nqref, the sector can be determined. The optimal vector combination u_1opt, u_2opt and u_0opt include two active voltage vectors and one zero vector. For the selected optimal vector combination, the action time of each vector in a sampling time needs to be calculated. The principle of modulation MPC is applied, assuming the voltage vector action time is inversely proportional to the cost function value. The action time of voltage vectors t₁, t₂ and t₀ can be calculated as:

$$\{\begin{array}{l}{t}_1=\frac{n}{f_1}{T}_s\\ t_2=\frac{n}{f_2}{T}_s\\ t_0=\frac{n}{f_0}{T}_s\\ n=\frac{1}{f_1{}^{-1}+f_2{}^{-1}+f_0{}^{-1}}\end{array}$$

(33)

where f₁, f₂ and f₀ are the values of cost function (30) corresponding to u_1opt, u_2opt and u_0opt, respectively. In this section, i_d1ref is calculated by STC (24).

VSC2 is similar to VSC1, the optimal voltage vectors with the corresponding action times are output to the switches at the next sampling time, which will control the whole SOP system.

5.3. TV-MPC with STO

After adding the disturbance observed by STO, (32) can be rewritten as follows:

$$\{\begin{array}{l}{u}_{1Ndref}=\frac{L_1}{T_s}(i_{d1}(\mathrm{k})-i_{d1ref})-R_1{i}_{d1}(\mathrm{k})+L_1\omega i_{q1}(\mathrm{k})+u_{1d}(\mathrm{k})+{\widehat{f}}_{d1}\\ u_{1Nqref}=\frac{L_1}{T_s}(i_{q1}(\mathrm{k})-i_{q1ref})-R_1{i}_{q1}(\mathrm{k})-L_1\omega i_{d1}(\mathrm{k})+u_{1q}(\mathrm{k})+{\widehat{f}}_{q1}\end{array}$$

(34)

The new voltage combination can be selected by the reference voltage u_d1ref and u_q1ref. By the new voltage combination, the values of cost function and action time of each vector can be recalculated.

After adding the disturbance, (3) can be rewritten as:

$$\{\begin{array}{l}{i}_{d1}(\mathrm{k}+1)=(1-\frac{T_s{R}_1}{L_1})i_{d1}(\mathrm{k})+T_s\omega i_{q1}(\mathrm{k})+\frac{T_s}{L_1}(u_{1d}(\mathrm{k})-u_{1Nd}(\mathrm{k})+{\widehat{f}}_{d1})\\ i_{q1}(\mathrm{k}+1)=(1-\frac{T_s{R}_1}{L_1})i_{q1}(\mathrm{k})-T_s\omega i_{d1}(\mathrm{k})+\frac{T_s}{L_1}(u_{1q}(\mathrm{k})-u_{1Nq}(\mathrm{k})+{\widehat{f}}_{q1})\end{array}$$

(35)

After substituting the rewritten predicted currents of (k + 1)th instant by new vector combinations chosen by (34) into (30) the values of the cost function can be calculated. Action times of each voltage vectors t₁, t₂ and t₀ can be calculated by (33).

VSC2 is similar to VSC1, new voltage combinations and action times are output to each side of switch and control the whole system of SOP.

Figure 5 is the block diagram of the STC and TV-MPC with STO for SOP.

6. Simulations

In this paper, traditional MPC, TV-MPC and TV-MPC with STC were designed to demonstrate the effectiveness of the proposed method and were simulated in MATLAB/Simulation. TV-MPC and STO with STC was designed when parameter mismatches occurred in the SOP system.

Table 2. Parameters of SOP.

Name	Symbol	Value
VSC1 grid voltage	u1	220 V
VSC2 grid voltage	u2	220 V
VSC1 resistance	R1	0.03 Ω
VSC2 resistance	R2	0.03 Ω
VSC1 inductance	L1	3 mH
VSC2 inductance	L2	3 mH
DC-side capacitance	C	5000 μF f
Sampling time	Ts	1 × 10−6 s

lists the parameters of the SOP system.

6.1. Current Pefromance of Steady-State

The proposed current of VSC2 side is set as −40 A in simulations. Outer-loop controller parameters are listed in

Table 3. Controller parameters.

Method	Outer-Loop Controller Parameters
PI and MPC	kp = 3.5, ki = 4.125
PI and TV-MPC	kp = 3.5, ki = 4.125
STC and TV-MPC	k1= 150, k2 = 3000

. Steady-state currents of PI and MPC, PI and TV-MPC and STC and TV-MPC are shown in Figure 6.

It can be seen that in VSC1 side, the total harmonic distortions (THDs) of a-phase of PI and MPC, PI and TV-MPC and STC and TV-MPC are 1.07%, 0.43% and 0.58%, respectively. While on VSC2 side, the THD of a-phase of PI and MPC, PI and TV-MPC and STC and TV-MPC are 1.06%, 0.43% and 0.44%, respectively. Compared with PI and MPC, it is obvious that PI and TV-MPC has a better steady-state current performance and reduces the current ripple of each side. Current performances of STC and TV-MPC are similar to PI and TV-MPC, maintaining the low current ripple of the system.

6.2. DC Voltage Performance

Regulating the proposed DC-side voltage u_dcref at 850 V, in order to verify the effectiveness of STC+TV-MPC in start-up response performance and dynamic state performance, the reference current of VSC2 side is changed from 40 A to 80 A at 0.25 s. As is shown in Figure 7a, the time used to track the proposed DC voltage under PI and MPC, PI and TV-MPC and STC and TV-MPC are 0.108 s, 0.029 s and 0.016 s, respectively. It can be seen that STC and TV-MPC track the proposed DC voltage faster than PI and MPC and PI and TV-MPC without overshoot, which has a better start-up response performance.

Changes of the DC-side voltage in dynamic state are shown in Figure 7b. It can be seen that the time used to recover the proposed DC-side voltage and voltage drops of STC and TV-MPC is much less than PI and MPC and PI and TV-MPC.

As a result, it can be seen that STC and TV-MPC has better performance in start-up response and current change response.

6.3. STC and TV-MPC with STO under Parameter Mismatches

In order to prove the effectiveness of STC and TV-MPC with STO, the SOP system under parameter mismatches is simulated in this section. STO and STC parameters are listed in

Table 4. STO and STC parameters.

k1	k2	α1	β1	α2	β2
100	200	50,000	1,500,000	50,000	1,500,000

.

The proposed current of VSC2 side is set as 40 A, when parameter mismatches have not occurred in the SOP system, Figure 8a shows the d-axis currents of SOP system i_d1, i_d2, i_d1ref and i_d2ref. Figure 8b shows the q-axis currents of SOP system i_q1, i_q2, i_q1ref and i_q2ref. The disturbances f_d1, f_q1, f_d2 and f_q2 estimated by STO are shown in Figure 8c. It can be seen that when parameter mismatches have not occurred in SOP system, i_d1, i_q1, i_d2 and i_q2 can track the reference currents correctly and the disturbances estimated by STO are small.

When inductance mismatches have occurred in the SOP system, STO can observe the disturbances caused by inductance mismatches. The SOP system starts at 0 s and ends at 0.8 s, while the observers are applied to the system at 0.4 s. When system inductances become 5L₁ and 5L₂, as shown in Figure 9a, inductances’ mismatches influence the d-axis currents obviously, Figure 9b shows the q-axis currents, when observers are applied to the system at 0.4 s, currents can track the reference currents correctly. Figure 9c shows the disturbances estimated by STO, it can be seen that after a short time, the estimated disturbances can be stabilized. When system inductances become 0.3L₁ and 0.3L₂, Figure 10a–c present the d-axis current, q-axis current and disturbances estimated by STO, respectively, it can be seen the STO can also estimate the disturbances correctly. These results verify the effectiveness of proposed STO.

Figure 11 shows the influences of resistance mismatches, when system resistances become 3R₁ and 3R₂, Figure 11a,b show the d-axis and q-axis currents of SOP system, it can be seen that resistance mismatches have less influence than inductance mismatches. As shown in Figure 11c, disturbances caused by resistance mismatches can be observed by STO. Therefore, it can be proved that the proposed STC and TV-MPC with STO can reduce the currents error obviously when parameter mismatches have occurred in the SOP system.

7. Discussion

For the better performance of the SOP system, increasing the number of ports, considering the faults that may occur in the system, taking Vf mode into consideration, and an experimental study of the SOP system will be the future work.

8. Conclusions

In this paper, a STC and TV-MPC with STO was proposed to improve the operation of the SOP system. Considering the disturbances caused by parameter mismatches and unmodelled dynamics, the STO was designed to observe the disturbances, and steady-state currents can track the reference currents correctly with the use of STO. In addition, aiming at the problems such as long start-up time, overshoot and poor robustness caused by traditional out-loop PI controller, an outer-loop STC was designed to improve the performance of the DC-side voltage. Moreover, in order to reduce the current harmonics, an inner-loop TV-MPC method was proposed. The simulation results show that STC and TV-MPC with STO can reduce current harmonics effectively while the DC voltage has a better operation state, when parameter mismatches have occurred in the SOP system, STO can observe the disturbances correctly, which prove the effectiveness and correctness of the proposed method.