Model Predictive Direct Power Control of Grid-connected Converters Considering Unbalanced Filter Inductance and Grid Conditions

Weichen Yang; Shihong Miao; Zhiwei Liu; Ji Han; Yulong Xiong; Qingyu Tu

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Model Predictive Direct Power Control of Grid-connected Converters Considering Unbalanced Filter Inductance and Grid Conditions PDF

- ORCID：
Weichen Yang
✉
- ORCID：
Shihong Miao
✉
- ORCID：
Zhiwei Liu
✉
- ORCID：
Ji Han
✉
- ORCID：
Yulong Xiong
✉
- ORCID：
Qingyu Tu
✉

State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Hubei Electric Power Security and High Efficiency Key Laboratory, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Updated：2021-11-23

DOI：10.35833/MPCE.2021.000355

OUTLINE

Abstract

The grid-connected converter (GCC) is widely used as the interface between various distributed generations and the utility grid. To achieve precise power control for GCC, this paper presents a model predictive direct power control (MPDPC) with consideration of the unbalanced filter inductance and grid conditions. First, the characteristics of GCC with unbalanced filter inductance are analyzed and a modified voltage control function is derived. On this basis, to compensate for the power oscillation caused by unbalanced filter inductance, a novel power compensation method is proposed for MPDPC to eliminate the DC-side current ripple while maintaining sinusoidal grid current. Besides, to improve the control robustness against mismatched filter inductance, a filter inductance identification scheme is proposed. Through this scheme, the estimated value of filter inductance is updated in each control period and applied in the proposed MPDPC. Finally, simulation results in PSCAD/EMTDC confirm the validity of the proposed MPDPC and the filter inductance identification scheme.

Keywords

Inductance identification; grid-connected converter; model predictive control; power oscillation elimination; unbalanced grid conditions

I. Introduction

WITH the rising number of distributed renewable generations and electric vehicles, grid-connected converters (GCCs) have been widely used as the interface between various distributed generation and the utility grid, as they can realize the conversion of DC electric power to AC electric power [

1], [2].

To achieve precise power control of GCCs, various control strategies such as voltage-oriented control (VOC) [

3], direct power control (DPC) [4], model predictive control (MPC) [5] have been proposed. In VOC, the three-phase components are decoupled in the dq-frame by using a phase-locked loop (PLL), and then regulated by proportional-integral (PI) controllers [6]. Those PI controllers give good dynamic/steady-state performance. However, their linear nature is the main drawback [7]. Besides, the performance and stability of VOC scheme are highly dependent on the performance of the PLL [8] and the tuning of the PI gains [9]. In DPC, the instantaneous active power and reactive power are set as control variables. It calculates the desired voltage, and directly selects the best-fitted switching vector for the power regulation [10], [11]. However, the switching table based DPC will cause large power ripples and variable switching frequencies, and a high sampling frequency is required to ensure the control performance. Recently, space vector modulation (SVM) based DPC has been introduced for its simple algorithm, constant switching frequency [12], and lower requirement for sampling frequency. With the development of microprocessors, MPC has become a promising way to obtain a faster dynamic response and realize complex control objectives [13]. Generally, in each control period, the MPC uses the prediction information to select the best voltage vector to minimize a cost function [14], [15]. By combining the advantages of DPC and MPC, accurate power control and various control objectives can be realized based on model predictive direct power control (MPDPC) [16], [17].

In practical applications, weak grid conditions are often encountered in the distribution network and microgrid, and GCCs face unbalanced grid voltages due to the integration of single-phase generators and unbalanced loads [

18]. The unbalanced grid conditions will produce negative-sequence current, and further cause the output power oscillation and DC-side current ripple of GCC. Many DPC methods have been proposed to control GCC under unbalanced grid conditions. A new definition of instantaneous reactive power is proposed in [19] for DPC strategies, which can eliminate active power oscillation while maintaining sinusoidal grid currents. In [20], a power compensation block is added to the original DPC scheme to eliminate the negative-sequence current. The principle of power compensation can be extended to achieve other control objectives such as eliminating grid-side active and reactive power oscillations [21]. However, [22] indicates that the filter inductance will cause the variation of oscillation power. The active power oscillation at the valve side will produce DC-side current ripple and cannot be eliminated with the conventional DPC method. To address this problem, a power compensation calculation method with consideration of the DC-side voltage ripple reduction is introduced in [23] and [24]. MPC methods have also been proposed to achieve proper control performance under unbalanced grid conditions. In [25], an MPDPC method is proposed to obtain sinusoidal grid currents and eliminate the grid-side power oscillation. To achieve robust control without measured grid voltage, a sliding-mode-observer-based voltage-sensorless MPC method is proposed in [2] and [26]. To obtain a better performance comparing with table-based MPC [27], a multi-vector MPC method is proposed in [28] and a vector optimization method is proposed in [29]. The above control methods can achieve their objectives with accurate knowledge of the filter inductance. However, the actual filter inductance may vary due to the temperature and conducting current [30]. The control performance will be significantly degraded with the mismatched value of inductance applied in the control scheme.

To improve the control performance against the variable parameters, various control schemes have been proposed by combining parameter estimation methods with DPC or MPC such as least-squares method [

31]-[33], extended Kalman filter based method [34], gradient correction method [30], observer-based estimation method [35]-[37], and model reference adaptive method [38]. However, most of existing parameter identification methods used in converters can only work under ideal grid conditions and cannot realize the identification of unbalanced filter inductance. Another method to solve parameter mismatch is the model-free predictive control. Generally, the model-free predictive control is based on the analysis of current increment, and a state observer or neural network is designed to estimate the relationship between the reference voltage and grid current [39]-[41]. The unknown filter inductance and unmodeled dynamic characteristics are regarded as a black box, and the reference voltage can be directly obtained by using the observation results. However, due to the lack of model knowledge, the model-free predictive control has limited robustness under complex conditions.

This paper aims to propose an MPDPC with unbalanced filter inductance identification to achieve precise power control and active power oscillation elimination for GCCs under unbalanced grid conditions. The main contributions of this paper can be drawn as follows.

1) The performance characteristics of GCC with unbalanced filter inductance are analyzed, and a novel power compensation method is proposed to eliminate the valve-side active power oscillation and DC-side current ripple.

2) An unbalanced filter inductance identification scheme is proposed, which has not been fully investigated in the existing literature, especially under unbalanced grid conditions.

3) By implementing power compensation and inductance identification, the proposed MPDPC has better robustness against mismatched inductance and unbalanced grid conditions.

The remainder of this paper is organized as follows. Section II introduces the mathematical model of GCCs under unbalanced grid conditions. Then, the detailed design of the proposed MPDPC is elaborated in Section III. In Section IV, the identification scheme for unbalanced filter inductance is designed. In Section V, the validity of the proposed MPDPC is verified by simulations in PSCAD/EMTDC. Finally, conclusions are drawn in Section VI.

II. Model of GCCs Under Unbalanced Grid Conditions

Figure 1 shows the topology of a GCC, which is connected to the DC supply through the line resistance and to the AC grid through the filter inductance.

Fig. 1 Topology of a GCC.

The mathematical model of the GCC in the stationary reference frame can be expressed as:

\frac{d i}{d t} L_{α β 0} = u - e

(1)

where $u = [u_{α}, u_{β}, u_{0}]^{T}$ , $e = [e_{α}, e_{β}, e_{0}]^{T}$ , and $i = [i_{α}, i_{β}, i_{0}]^{T}$ are the converter output voltage, grid voltage, and grid current in the stationary reference frame $α β 0$ , respectively; and $L_{α β 0}$ is the inductance matrix in the stationary reference frame, which can be depicted by the three-phase filter inductance and Clarke transformation:

\begin{array}{l} L_{α β 0} = C_{3 / 2} [\begin{matrix} L_{a} & 0 & 0 \\ 0 & L_{b} & 0 \\ 0 & 0 & L_{c} \end{matrix}] C_{3 / 2}^{- 1} = \\ \frac{1}{6} [\begin{matrix} 4 L_{a} + L_{b} + L_{c} & - \sqrt[]{3} L_{b} + \sqrt[]{3} L_{c} & 4 L_{a} - 2 L_{b} - 2 L_{c} \\ - \sqrt[]{3} L_{b} + \sqrt[]{3} L_{c} & 3 L_{b} + 3 L_{c} & 2 \sqrt[]{3} L_{b} - 2 \sqrt[]{3} L_{c} \\ 2 L_{a} - L_{b} - L_{c} & \sqrt[]{3} L_{b} - \sqrt[]{3} L_{c} & 2 L_{a} + 2 L_{b} + 2 L_{c} \end{matrix}] \end{array}

(2)

where $C_{3 / 2}$ is the Clarke transformation matrix; and $L_{a}$ , $L_{b}$ , and $L_{c}$ are the three-phase filter inductances.

Considering that the converter will not generate zero-sequence current under normal conditions, the inductance matrix can be simplified as:

L_{α β} = \frac{1}{6} [\begin{matrix} 4 L_{a} + L_{b} + L_{c} & - \sqrt[]{3} L_{b} + \sqrt[]{3} L_{c} \\ - \sqrt[]{3} L_{b} + \sqrt[]{3} L_{c} & 3 L_{b} + 3 L_{c} \end{matrix}] = [\begin{matrix} L_{11} & L_{12} \\ L_{12} & L_{22} \end{matrix}]

(3)

where $L_{α β}$ is the simplified inductance matrix consisting of $L_{11}$ , $L_{12}$ , and $L_{12}$ .

Then, the control function of GCC can be written as:

[\begin{matrix} u_{α} \\ u_{β} \end{matrix}] = [\begin{matrix} e_{α} \\ e_{β} \end{matrix}] + [\begin{matrix} L_{m} & 0 \\ 0 & L_{m} \end{matrix}] [\begin{matrix} d i_{α} / d t \\ d i_{β} / d t \end{matrix}] + \underset{Δ u_{L}}{\underset{︸}{[\begin{matrix} L_{11} - L_{m} & L_{12} \\ L_{12} & L_{22} - L_{m} \end{matrix}] [\begin{matrix} d i_{α} / d t \\ d i_{β} / d t \end{matrix}]}}

(4)

where $L_{m} = (L_{a} + L_{b} + L_{c}) / 3$ is the average value of three-phase filter inductances, which represents the balanced part of filter inductance; and $Δ u_{L}$ is a disturbance voltage caused by the unbalanced part of filter inductance.

It can be observed that when the filter inductance is symmetrical, $L_{α β}$ will be a scalar matrix. However, with the unbalanced filter inductance, $Δ u_{L}$ will be generated by the coupling of $i_{α}$ and $i_{β}$ . Considering the unbalanced grid conditions, the grid current can be written as:

\{\begin{array}{l} i_{α} = I_{p} c o s (ω t + θ_{p}) + I_{n} c o s (ω t + θ_{n}) \\ i_{β} = I_{p} s i n (ω t + θ_{p}) - I_{n} s i n (ω t + θ_{n}) \end{array}

(5)

where $ω$ is the fundamental angular frequency; $I_{p}$ and $θ_{p}$ are the amplitude and phase angle of the positive-sequence current, respectively; and $I_{n}$ and $θ_{n}$ are the amplitude and phase angle of the negative-sequence current, respectively.

Combining (4) and (5), $Δ u_{L} = [Δ u_{L α}, Δ u_{L β}]^{T}$ can be expressed as:

\{\begin{array}{l} Δ u_{L α} = \underset{Δ u_{L α 1}}{\underset{︸}{- ω (L_{11} - L_{m}) I_{p} s i n (ω t + θ_{p}) + ω L_{12} I_{p} c o s (ω t + θ_{p})}} \underset{Δ u_{L α 2}}{\underset{︸}{- ω (L_{11} - L_{m}) I_{n} s i n (ω t + θ_{n}) - ω L_{12} I_{n} c o s (ω t + θ_{n})}} \\ Δ u_{L β} = \underset{Δ u_{L β 1}}{\underset{︸}{ω (L_{22} - L_{m}) I_{p} c o s (ω t + θ_{p}) - ω L_{12} I_{p} s i n (ω t + θ_{p})}} \underset{Δ u_{L β 2}}{\underset{︸}{- ω (L_{22} - L_{m}) I_{n} c o s (ω t + θ_{n}) - ω L_{12} I_{n} s i n (ω t + θ_{n})}} \end{array}

(6)

where $[Δ u_{L α 1}, Δ u_{L β 1}]^{T} = Δ u_{L 1}$ and $[Δ u_{L α 2}, Δ u_{L β 2}]^{T} = Δ u_{L 2}$ are the disturbance voltages generated by the positive-sequence current and negative-sequence current, respectively.

Considering $L_{11} - L_{m} = - (L_{22} - L_{m})$ , it can be observed that $Δ u_{L 1}$ is a negative-sequence voltage, while $Δ u_{L 2}$ is a positive-sequence voltage. According to instantaneous power theory and the extended reactive power proposed in [

19], the active power and reactive power at the grid side can be expressed as:

\{\begin{array}{l} P = 1.5 (e_{α} i_{α} + e_{β} i_{β}) \\ Q_{n o v} = 1.5 (e_{α}^{'} i_{α} + e_{β}^{'} i_{β}) \end{array}

(7)

where $[e_{α}^{'}, e_{β}^{'}]$ lags $[e_{α}, e_{β}]$ by 90 electrical degrees.

By using $Q_{n o v}$ instead of traditional reactive power $Q$ in the control scheme, active power $P$ can be kept constant while maintaining sinusoidal grid currents when the grid voltage is unbalanced [

19].

Under unbalanced grid voltage, the active and reactive power losses $Δ P$ and $Δ Q$ on the filter inductance can be written as:

\begin{array}{l} Δ P = 1.5 [- 2 ω L_{m} I_{p} I_{n} s i n (2 ω t + θ_{p} + θ_{n}) - \\ 0.5 ω (L_{11} - L_{22}) (I_{p}^{2} s i n (2 ω t + 2 θ_{p}) + I_{n}^{2} s i n (2 ω t + 2 θ_{n})) + \\ ω L_{12} (I_{p}^{2} c o s (2 ω t + 2 θ_{p}) - I_{n}^{2} c o s (2 ω t + 2 θ_{n}))] \end{array}

(8)

\begin{array}{l} Δ Q = 1.5 [2 ω L_{m} I_{p} I_{n} c o s (2 ω t + θ_{p} + θ_{n}) + \\ 0.5 ω (L_{11} - L_{22}) (I_{p}^{2} c o s (2 ω t + 2 θ_{p}) + I_{n}^{2} c o s (2 ω t + 2 θ_{n})) + \\ ω L_{12} (I_{p}^{2} s i n (2 ω t + 2 θ_{p}) - I_{n}^{2} s i n (2 ω t + 2 θ_{n})) + ω L_{m} (I_{p}^{2} + I_{n}^{2}) + ω (L_{11} - L_{22}) I_{p} I_{n} c o s (θ_{p} - θ_{n}) + 2 ω L_{12} I_{p} I_{n} s i n (θ_{p} - θ_{n})] \end{array}

(9)

According to (4), (8), and (9), it can be observed that the unbalanced filter inductance will degrade the performance of current control and cause the double fundamental frequency power oscillation.

III. MPDPC of GCC Under Unbalanced Grid Conditions

A. Principle of MPDPC Without Output Power Oscillations

The unbalanced grid voltage and filter inductance will generate a negative-sequence current and further cause the double-frequency power oscillation. A typical control objective is to eliminate the active power oscillation at the grid side and control the active power and reactive power to follow their references. By using $Q_{n o v}$ instead of traditional reactive power $Q$ in the control scheme, the reactive power oscillation can also be eliminated while maintaining sinusoidal grid current [

19].

Since the MPC works in discrete time, and the digital control system needs a certain time to calculate the references of i and u, a delay of one sampling instant is taken into consideration by making predictions at instant k+2. And the active power and reactive power at the grid side are controlled to reach their references at instant k+2. Hence, the control objective is to minimize the power tracking error which is expressed as:

\{\begin{array}{l} P^{k + 2} = 1.5 (e_{α}^{k + 2} i_{α}^{k + 2} + e_{β}^{k + 2} i_{β}^{k + 2}) \Rightarrow P_{r e f} \\ Q^{k + 2} = 1.5 (e_{α}^{^{'} k + 2} i_{α}^{k + 2} + e_{β}^{^{'} k + 2} i_{β}^{k + 2}) \Rightarrow Q_{r e f} \\ J_{m i n} = (P^{k + 2} - P_{r e f})^{2} + (Q^{k + 2} - Q_{r e f})^{2} \end{array}

(10)

where the superscript $k + 2$ represents the variable at instant $k + 2$ ; $P_{r e f}$ and $Q_{r e f}$ are the references of active power and reactive power, respectively; and $J_{m i n}$ is the cost function.

To obtain the current reference with higher accuracy, the predictive values of grid voltage need to be calculated. The unbalanced grid voltage can be expressed as a complex vector:

e = e_{p} + e_{n} = E_{p} e^{j ω t} + E_{n} e^{- j ω t}

(11)

where $e_{p}$ and $e_{n}$ are the positive-sequence voltage and negative-sequence voltage, respectively; and $E_{p}$ and $E_{n}$ are the amplitudes of the positive-sequence voltage and negative-sequence voltage, respectively.

In the steady state, the grid voltage at instant k+n can be predicted as:

e_{p r e}^{k + n} = e_{p}^{k} e^{j ω n T_{s}} + e_{n}^{k} e^{- j ω n T_{s}}

(12)

where $e_{p r e}^{k + n}$ is the predicted value of grid voltage at instant k+n; and T_s is the sampling and control period.

According to (10), the optimal current references at instant k+2 can be directly calculated as:

\{\begin{array}{l} i_{α, r e f}^{k + 2} = \frac{2}{3} \frac{P_{r e f} e_{β, p r e}^{^{'} k + 2} - Q_{r e f} e_{β, p r e}^{k + 2}}{e_{α, p r e}^{k + 2} e_{β, p r e}^{^{'} k + 2} - e_{α, p r e}^{^{'} k + 2} e_{β, p r e}^{k + 2}} \\ i_{β, r e f}^{k + 2} = \frac{2}{3} \frac{- P_{r e f} e_{α, p r e}^{^{'} k + 2} + Q_{r e f} e_{α, p r e}^{k + 2}}{e_{α, p r e}^{k + 2} e_{β, p r e}^{^{'} k + 2} - e_{α, p r e}^{^{'} k + 2} e_{β, p r e}^{k + 2}} \end{array}

(13)

where $[i_{α, r e f}^{k + 2}, i_{β, r e f}^{k + 2}] = i_{r e f}^{k + 2}$ is the current reference at instant $k + 2$ ; $e_{α, p r e}^{k + 2}$ and $e_{β, p r e}^{k + 2}$ are the predicted values of $e_{α}$ and $e_{β}$ at instant $k + 2$ , respectively; and $e_{α, p r e}^{^{'} k + 2}$ and $e_{β, p r e}^{^{'} k + 2}$ are the predicted values of $e_{α}^{'}$ and $e_{β}^{'}$ at instant $k + 2$ , respectively.

The grid current at instant k+2 is determined by the grid voltage and converter output voltage at instant k+1. To make the grid current follow the references, the control function of $u$ can be derived based on the first-order Euler discretization method:

[\begin{matrix} u_{α}^{k + 1} \\ u_{β}^{k + 1} \end{matrix}] = [\begin{matrix} e_{α, p r e}^{k + 1} \\ e_{β, p r e}^{k + 1} \end{matrix}] + \frac{1}{T_{s}} [\begin{matrix} L_{11} & L_{12} \\ L_{12} & L_{22} \end{matrix}] [\begin{matrix} i_{α, r e f}^{k + 2} - i_{α, p r e}^{k + 1} \\ i_{β, r e f}^{k + 2} - i_{β, p r e}^{k + 1} \end{matrix}]

(14)

where $i_{α, p r e}^{k + 1}$ and $i_{β, p r e}^{k + 1}$ are the predictive values of grid current at instant k+1, which can be calculated by:

[\begin{matrix} i_{α, p r e}^{k + 1} \\ i_{β, p r e}^{k + 1} \end{matrix}] = T_{s} {[\begin{matrix} L_{11} & L_{12} \\ L_{12} & L_{22} \end{matrix}]}^{- 1} [\begin{matrix} u_{α}^{k} - e_{α}^{k} \\ u_{β}^{k} - e_{β}^{k} \end{matrix}] + [\begin{matrix} i_{α}^{k} \\ i_{β}^{k} \end{matrix}]

(15)

The reference $u^{k + 1}$ will be adopted on SVM at the next control period to generate the desired converter output voltage and achieve the control objectives. By using SVM, the proposed MPDPC will have a lower requirement for the sampling frequency and computational burden.

B. Principle of MPDPC Without DC-side Current Ripple

The MPDPC shown in Section III-A is used to eliminate the power oscillation at the grid side, thereby improving the control performance and decreasing the current ripple at the DC side. However, according to (8) and (9), both the balanced part and unbalanced part of the filter inductance will cause power oscillation, which makes the active power at the valve side contain a double fundamental frequency oscillation, and further generates a ripple current at the DC side. In this subsection, the control objective is to eliminate the ripple current at the DC side and control the average active power and reactive power at the grid side to follow their references.

According to (10) and (14), the active power and reactive power at instant k+2 at the grid side and the active power and reactive power at instant k+1 at the valve side are controlled by the output voltage at instant k+1. The power oscillation caused by the filter inductance can be compensated by adding double fundamental frequency components on P_ref and Q_ref, thereby eliminating the power oscillation at the valve side. Considering the above factors, the control objective can be expressed as:

\{\begin{array}{l} P_{v}^{k + 1} = 1.5 (u_{α}^{k + 1} i_{α}^{k + 1} + u_{β}^{k + 1} i_{β}^{k + 1}) \Rightarrow P_{r e f} \\ P^{k + 2} = 1.5 (e_{α}^{k + 2} i_{α}^{k + 2} + e_{β}^{k + 2} i_{β}^{k + 2}) \Rightarrow P_{r e f} + P_{2 ω}^{k + 2} \\ Q^{k + 2} = 1.5 (e_{α}^{^{'} k + 2} i_{α}^{k + 2} + e_{β}^{^{'} k + 2} i_{β}^{k + 2}) \Rightarrow Q_{r e f} + Q_{2 ω}^{k + 2} \\ \begin{array}{l} J_{m i n} = (P_{v}^{k + 1} - P_{r e f})^{2} + (P^{k + 2} - P_{r e f} - P_{2 ω}^{k + 2})^{2} + \\ (Q^{k + 2} - Q_{r e f} - Q_{2 ω}^{k + 2})^{2} \end{array} \end{array}

(16)

where $P_{v}^{k + 1}$ is the active power at the valve side at instant k; and $P_{2 ω}^{k + 2}$ and $Q_{2 ω}^{k + 2}$ are the double fundamental frequency compensation components of active power and reactive power at instant k+2, respectively.

Based on (8) and (9), it can be found that the oscillation component of $Δ Q$ lags that of $Δ P$ by 90 electrical degrees. Therefore, $Q_{2 ω}^{k + 2}$ can be obtained as:

Q_{2 ω}^{k + 2} = P_{2 ω}^{k + 2} e^{- T_{2 ω} s / 4}

(17)

where $T_{2 ω}$ is the period of double fundamental frequency.

The optimal grid current references can be obtained as:

\{\begin{array}{l} i_{α, r e f}^{k + 2} = \frac{2}{3} \frac{(P_{r e f} + P_{2 ω}^{k + 2}) e_{β, p r e}^{^{'} k + 2} - (Q_{r e f} + Q_{2 ω}^{k + 2}) e_{β, p r e}^{k + 2}}{e_{α, p r e}^{k + 2} e_{β, p r e}^{^{'} k + 2} - e_{α, p r e}^{^{'} k + 2} e_{β, p r e}^{k + 2}} = i_{α, 0}^{k + 2} + i_{α, 2 ω}^{k + 2} \\ i_{β, r e f}^{k + 2} = \frac{2}{3} \frac{- (P_{r e f} + P_{2 ω}^{k + 2}) e_{α, p r e}^{^{'} k + 2} + (Q_{r e f} + Q_{2 ω}^{k + 2}) e_{α, p r e}^{k + 2}}{e_{α, p r e}^{k + 2} e_{β, p r e}^{^{'} k + 2} - e_{α, p r e}^{^{'} k + 2} e_{β, p r e}^{k + 2}} = i_{β, 0}^{k + 2} + i_{β, 2 ω}^{k + 2} \end{array}

(18)

where $i_{α, 0}^{k + 2}$ and $i_{β, 0}^{k + 2}$ are the current references corresponding to $P_{r e f}$ and $Q_{r e f}$ , respectively; and $i_{α, 2 ω}^{k + 2}$ and $i_{β, 2 ω}^{k + 2}$ are the compensation references corresponding to $P_{2 ω}^{k + 2}$ and $Q_{2 ω}^{k + 2}$ , respectively. According to (11) and (17), it can be derived that $i_{α, 2 ω}^{k + 2}$ and $i_{β, 2 ω}^{k + 2}$ are the fundamental frequency components composed of positive-sequence and negative-sequence currents. The grid current will maintain sinusoidal and undistorted when the compensation references are applied in the MPDPC.

Combining (14)-(18), $P_{2 ω}^{k + 2}$ can be obtained as:

P_{2 ω}^{k + 2} = \frac{K_{1} - K_{4} - K_{3} Q_{2 ω}^{k + 2}}{K_{2}}

(19)

K_{1} = \frac{2 T_{s}}{3} (P_{r e f} - P_{p r e}^{k + 1}) - 2 L_{12} i_{α, p r e}^{k + 1} i_{β, p r e}^{k + 1} - L_{11} (i_{α, p r e}^{k + 1})^{2} - L_{22} (i_{β, p r e}^{k + 1})^{2}

(20)

K_{2} = \frac{2}{3} \frac{N_{1} e_{β, p r e}^{^{'} k + 2} - N_{2} e_{α, p r e}^{^{'} k + 2}}{e_{α, p r e}^{k + 2} e_{β, p r e}^{^{'} k + 2} - e_{α, p r e}^{^{'} k + 2} e_{β, p r e}^{k + 2}}

(21)

K_{3} = \frac{2}{3} \frac{- N_{1} e_{β, p r e}^{k + 2} + N_{2} e_{α, p r e}^{k + 2}}{e_{α, p r e}^{k + 2} e_{β, p r e}^{^{'} k + 2} - e_{α, p r e}^{^{'} k + 2} e_{β, p r e}^{k + 2}}

(22)

K_{4} = N_{1} i_{a, 0}^{k + 2} + N_{2} i_{β, 0}^{k + 2}

(23)

N_{1} = - L_{11} i_{α, p r e}^{k + 1} - L_{12} i_{β, p r e}^{k + 1}

(24)

N_{2} = - L_{22} i_{β, p r e}^{k + 1} - L_{12} i_{α, p r e}^{k + 1}

(25)

After obtaining the current references at instant k+2 by (18), the output voltage references at instant k+1 can be calculated through (14) and (15).

IV. Identification Scheme for Unbalanced Filter Inductance

As mentioned above, the filter inductance plays an important role in obtaining the power compensation components and the converter voltage references. Once the filter inductance deviates from the initial value, it will affect the dynamic performance and fail to reach the control objectives. Hence, it is necessary to estimate the accurate value of $L_{α β}$ online for better control performance.

According to (14), the grid current can be derived as:

[\begin{matrix} i_{α}^{k} \\ i_{β}^{k} \end{matrix}] = T_{s} [\begin{matrix} B_{11} & B_{12} \\ B_{12} & B_{22} \end{matrix}] [\begin{matrix} u_{L α}^{k - 1} \\ u_{L β}^{k - 1} \end{matrix}] + [\begin{matrix} i_{α}^{k - 1} \\ i_{β}^{k - 1} \end{matrix}] = T_{s} B + i^{k - 1}

(26)

where B is the inverse of the actual $L_{α β}$ ; B₁₁, B₁₂, and B₂₂ are elements in B, which are the parameters that need to be estimated; and $[u_{L α}^{k - 1}, u_{L β}^{k - 1}]^{T} = u_{L}^{k - 1}$ is the voltage drop on the filter inductance at instant $k - 1$ .

The parameter matrix B is estimated online and updated during each period. Assuming that the estimated B at instant $k - 1$ deviates from the actual value by $Δ B$ , then (26) can be rewritten as:

[\begin{matrix} {\hat{i}}_{α}^{k} + Δ i_{α}^{k} \\ {\hat{i}}_{β}^{k} + Δ i_{β}^{k} \end{matrix}] = T_{s} ([\begin{matrix} {\hat{B}}_{11}^{k - 1} & {\hat{B}}_{12}^{k - 1} \\ {\hat{B}}_{12}^{k - 1} & {\hat{B}}_{22}^{k - 1} \end{matrix}] + [\begin{matrix} Δ B_{11} & Δ B_{12} \\ Δ B_{12} & Δ B_{22} \end{matrix}]) [\begin{matrix} u_{L α}^{k - 1} \\ u_{L β}^{k - 1} \end{matrix}] + [\begin{matrix} i_{α}^{k - 1} \\ i_{β}^{k - 1} \end{matrix}]

(27)

where the superscript ˆ represents the estimated value; and $[Δ i_{α}^{k}, Δ i_{β}^{k}]^{T} = Δ i^{k}$ is the estimated error of grid currents. Accordingly, the relationship between $Δ B$ and $Δ i^{k}$ satisfies:

[\begin{matrix} Δ B_{11} & Δ B_{12} \\ Δ B_{12} & Δ B_{22} \end{matrix}] [\begin{matrix} u_{L α}^{k - 1} \\ u_{L β}^{k - 1} \end{matrix}] = [\begin{matrix} Δ i_{α}^{k} \\ Δ i_{β}^{k} \end{matrix}]

(28)

The estimated parameters in B can be updated by calculating $Δ B$ . However, (28) is an underdetermined equation and cannot be solved to obtain a unique solution. Generally, the change frequency of the filter inductance is far less than the fundamental frequency. Therefore, B₁₁, B₁₂, and B₂₂ can be assumed to be constant within a short time, and we can use the estimated and measured information at instant $k - n$ to fill in (28). After some transformations, $Δ B$ can be obtained by:

[\begin{matrix} u_{L α}^{k - 1} & u_{L β}^{k - 1} & 0 \\ 0 & u_{L α}^{k - 1} & u_{L β}^{k - 1} \\ u_{L α}^{k - n - 1} & u_{L β}^{k - n - 1} & 0 \\ 0 & u_{L α}^{k - n - 1} & u_{L β}^{k - n - 1} \end{matrix}] [\begin{matrix} Δ B_{11} \\ Δ B_{12} \\ Δ B_{22} \end{matrix}] = \frac{1}{T_{s}} [\begin{matrix} Δ i_{α}^{k} \\ Δ i_{β}^{k} \\ Δ i_{α}^{k - n} \\ Δ i_{β}^{k - n} \end{matrix}]

(29)

where $Δ i_{α}^{k - n}$ and $Δ i_{α}^{k - n}$ are the estimated errors obtained by ${\hat{B}}^{k - 1}$ .

It can be observed that (29) is an overdetermined equation, through which we can obtain the least square solution of $Δ B$ . The solving process is simple and does not bring a high computational burden, hence it will not be expanded here. Since the delay step n determines the correlation between the two sets of functions in (29), when the correlation is high, the overdetermined equation will become an ill-conditioned equation, making the estimated results extremely sensitive to the measurement or calculation errors. To avoid excessive correlation between the information at instant $k - n$ and that at instant k, the instant $k - n$ should not lag the instant k by 0 or 180 electrical degrees.

The estimated values of B can be updated as:

[\begin{matrix} {\hat{B}}_{11}^{k} \\ {\hat{B}}_{12}^{k} \\ {\hat{B}}_{22}^{k} \end{matrix}] = [\begin{matrix} {\hat{B}}_{11}^{k - 1} \\ {\hat{B}}_{12}^{k - 1} \\ {\hat{B}}_{22}^{k - 1} \end{matrix}] + G [\begin{matrix} Δ {\hat{B}}_{11} \\ Δ {\hat{B}}_{12} \\ Δ {\hat{B}}_{22} \end{matrix}]

(30)

where $0 < G \leq 1$ is the update step size.

Then, the estimated inductance matrix can be expressed as:

{\hat{L}}_{α β}^{k} = {[\begin{matrix} {\hat{B}}_{11}^{k} & {\hat{B}}_{12}^{k} \\ {\hat{B}}_{12}^{k} & {\hat{B}}_{22}^{k} \end{matrix}]}^{- 1}

(31)

The ${\hat{L}}_{α β}^{k}$ and ${\hat{B}}^{k}$ are used in the proposed MPDPC and will be updated again at the next control period.

Figure 2 shows the control diagram of the proposed MPDPC, in which the control objective can be selected by setting the compensation power and the blocks “ $Z^{- 1}$ ” and “ $Z^{- n}$ ” mean the unit delay processes. The sampling and control process is shown in Appendix A. The block diagram of the filter inductance identification scheme is shown in Fig. 3, where the block “inv” means the inversion of a matrix. It can be found that the slightly off-nominal frequency mainly affects the delay process. Since the obtained delay values are close to the actual values, it will have little effect on the performance of the proposed MPDPC.

Fig. 2 Control diagram of proposed MPDPC.

Fig. 3 Block diagram of filter inductance identification scheme.

V. Simulation Results

To verify the validity of the proposed MPDPC, a simulation model of GCC is established on PSCAD/EMTDC. The system parameters are listed in Table I.

TABLE I System Parameters

Parameter	Value
DC-side voltage $u_{d c}$	850 V
Root mean square of line-line voltage	380 V
Active power reference $P_{r e f}$	50 kW
Reactive power reference $Q_{r e f}$	0 kvar
Grid angular frequency $ω$	100π rad/s
Sampling and control period $T_{s}$	100 μs
Initial value of inductance identification ${\hat{L}}_{0}$	1 mH
Update step size of inductance identification $G$	0.1
Delay step of inductance identification $n$	50

A. Validity of Proposed MPDPC Under Unbalanced Grid Conditions

In the conventional MPDPC, the unbalanced grid condition of filter inductance is ignored, and only the average value is used to obtain the voltage vector. The analysis in Section II demonstrates that the unbalanced filter inductance will degrade the control performance and generate power oscillation. To solve these problems, a new converter voltage control function (14) is used in the proposed MPDPC. The conventional MPDPC in [

25] is compared with the proposed MPDPC under the same simulation conditions, and the vector selection process is replaced with SVM to obtain better performance.

The unbalanced filter inductances are set as $L_{a} = 2$ mH, $L_{b} = 6$ mH, $L_{c} = 4$ mH, respectively. In this subsection, the initial value of the average filter inductance and the actual inductance matrix $L_{α β}$ are used in the traditional MPDPC and the proposed MPDPC, respectively. The filter inductance identification scheme will be simulated and verified in the next subsection. Before 2 s, the grid voltage is balanced, and the control objective is to eliminate the active power oscillation at the grid side. At 2 s, the grid voltage of phase A is reduced by 20%. At 2.2 s, the control objective is changed to eliminate the active power oscillation at the valve side.

Figure 4 shows the simulation results, including the grid-side active power and reactive power P and Q_nov, valve-side active power P_v, compensation active power reference $P_{2 ω}$ , DC-side current I_dc, grid voltage and current e_i and $i_{i} (i = a, b, c)$ , and the fast Fourier transform (FFT) of phase-A current with the proposed MPDPC. It can be observed from Fig. 4(a), even if the grid voltage is balanced, the unbalanced filter inductance will generate power oscillation at the grid side and valve side.

Fig. 4 Simulation results of conventional and proposed MPDPC with unbalanced filter inductance and grid conditions. (a) Simulation results of conventional MPDPC. (b) Simulation results of proposed MPDPC. (c) FFT analysis of phase-A current with proposed MPDPC.

The conventional MPDPC cannot eliminate the power oscillation and track the power references. The simulation results in Fig. 4(b) indicate that the grid-side power oscillation can be effectively eliminated by the proposed MPDPC. Under the influence of unbalanced filter inductance and grid voltage, the power oscillation at the valve side and grid side cannot be eliminated simultaneously. The valve-side active power oscillation and DC-side current ripple can be eliminated by the power compensation method. Meanwhile, the FFT analysis shown in Fig. 4(c) also indicates that the proposed MPDPC can maintain sinusoidal grid currents under unbalanced grid conditions. In addition, it should be noted that the proposed MPDPC can also work when the reactive power reference is not equal to 0.

B. Validity of Filter Inductance Identification Scheme with Variable Filter Inductance

To demonstrate the validity of the filter inductance identification scheme and further illustrate the adaptability of the proposed MPDPC to variable filter inductance, two simulation cases are formulated in this subsection.

In the first case, the identification scheme is applied at 1.9 s, with the initial value ${\hat{L}}_{11} = {\hat{L}}_{22} = 1$ mH, ${\hat{L}}_{12} = 0$ , update step size $G = 0.1$ , and delay step $n = 50$ (90 electrical degrees). The other simulation conditions are the same as Section V-A. Figure 5 shows the corresponding simulation results, including grid-side active and reactive power P and Q_nov, valve-side active power P_v, DC-side current I_dc, estimated and true inductances ${\hat{L}}_{i j}$ and $L_{i j}$ , and estimated errors of grid current $Δ i$ . It can be observed from Fig. 5 that after the identification scheme is enabled, the estimated inductance values can converge to their true values with high accuracy and speed. The estimated errors of grid current can also be eliminated. Meanwhile, the active power and reactive power can track their references to achieve different control objectives under both balanced and unbalanced grid conditions.

Fig. 5 Simulation results with inductance identification scheme.

In the second case, the filter inductances are set as variable parameters. Before 0.6 s, the filter inductances are balanced with the initial values L_a=L_b=L_c=1 mH. After 0.6 s, the inductances change every 0.3 s according to the following formula:

\{\begin{array}{l} L_{a} = 5 |s i n (π (t - 0.6))| + 1 \\ L_{b} = 5 |s i n (3 π (t - 0.6))| + 1 \\ L_{c} = 5 |s i n (2 π (t - 0.6))| + 1 \end{array}

(32)

The grid voltage is unbalanced (e_a is reduced by 20%), and the control objective is to eliminate the valve-side power oscillation and DC-side current ripple. From the simulation results shown in Fig. 6, the estimated inductances can track the time-varying values, and the valve-side power oscillation and DC-side current ripple are effectively eliminated by the proposed MPDPC.

Fig. 6 Simulation results with variable filter inductance when $n = 50$ , $G = 0.1$ .

C. Performance of Filter Inductance Identification Scheme with Different Parameters

The performance of the filter inductance identification scheme is associated with two main parameters: the update step size G and the delay step n.

It is clear that G determines the convergence speed, n determines the similarity of the functions in the overdetermined equation, and further affects the solving accuracy of (29). To show the impact of G and n, a series of simulation tests are conducted with different parameters. The simulation conditions are the same as the second case of Section V-B. Figures 7 and 8 show the identification results with different G and n, respectively.

Fig. 7 Filter inductance identification results with different update steps. (a) $n = 50, G = 0.01 .$ (b) $n = 50, G = 0.5 .$ (c) $n = 50, G = 1$ .

Fig. 8 Filter inductance identification results with different delay steps. (a) $n = 1$ , $G = 0.1$ . (b) $n = 25$ , $G = 0.1$ . (c) $n = 75$ , $G = 0.1$ .

It can be observed from Figs. 6 and 7 that when G increases, the convergence rate of the identification scheme becomes higher. However, a smoother and better transient response is supposed to be obtained when G has a smaller value. Figures 6 and 8 show that the identification accuracy is significantly degraded when the delay step is too small or close to 0 electrical degrees. This is because in these situations, the correlation of functions in (29) is very high, and a tiny measurement error will lead to large calculation deviations. When the delay step is close to 90 electrical degrees, the two sets of functions in (29) are orthogonal and will provide more information about the filter inductance, hence the identification results can be obtained with higher accuracy. Therefore, it is recommended to set the delay step to 90 electrical degrees, and the update step size should be selected with a bargain of the convergence rate and transient response of the identification process.

VI. Conclusion

In this paper, an MPDPC for the GCCs considering unbalanced filter inductance and grid conditions is proposed. The main conclusions are as follows.

1) The unbalanced filter inductance will generate negative-sequence current and double fundamental-frequency power oscillation. Besides, the mismatched inductance will significantly degrade the control performance of conventional MPDPC.

2) The proposed filter inductance identification scheme can estimate the mismatched and unbalanced inductance with high accuracy and good dynamic performance. The estimation process is simple and will not bring a high computational burden.

3) Compared with the conventional MPDPC in existing research, the proposed MPDPC can effectively eliminate the current ripple at the DC side with unbalanced grid conditions.

Appendix

Appendix A

MPC works in discrete time domination, and the sampling and control process shares one cycle. For each instant k, the sensors first obtain the real-time sampling values of grid voltage $e^{k}$ and grid current $i^{k}$ . Then, the prediction process is conducted to predict $e_{p r e}^{k + 1}$ , $e_{p r e}^{k + 2}$ , and $i_{p r e}^{k + 1}$ . The filter inductance estimation value is also updated by the inductance identification scheme. Finally, the voltage references will be applied on SVM at the next instant to trigger the valves.

The simplified diagram of the sampling and control process is shown in Fig. A1.

Fig. A1 Simplified diagram of sampling and control process.

The green line and blue line represent the sampled values and predicted values, respectively. The black line and red line represent the control references calculated in the past and instant k, respectively. The yellow dot line represents the desired sinusoidal waves of different variables.

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