# Discrete-Time Nonlinear Control of VSC-HVDC System

Authors: TianTian Qian; ShiHong Miao
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

Because VSC-HVDC is a kind of strong nonlinear, coupling, and multi-input multioutput (MIMO) system, its control problem is always attracting much attention from scholars. And a lot of papers have done research on its control strategy in the continuous-time  domain. But the control system is implemented through the computer discrete sampling in practical engineering. It is necessary to study the mathematical model and control algorithm in the discrete-time domain. The discrete mathematical model based on output feedback linearization and discrete sliding mode control algorithm is proposed in this paper. And to ensure the effectiveness of the control system in the quasi sliding mode state, the fast output sampling method is used in the output feedback. The results from simulation experiment in MATLAB/SIMULINK prove that the proposed discrete control algorithm can make the VSC-HVDC system have good static, dynamic, and robust characteristics in discrete-time domain.


---

## Body

## 1. Introduction

Voltage source converter based high voltage direct current (VSC-HVDC) technology which uses advanced (insulated gate bipolar transistor) IGBT device and pulse width modulation (PWM) method overcomes the disadvantage of traditional HVDC. It can not only adjust the active and reactive power independently, but also supply power to passive network system. Therefore the VSC-HVDC technology has a broad application prospect in the fields of distributed generation, asynchronous AC network interconnection, remote-area power supply, and so forth [1–5].Because VSC-HVDC is a multiple input multiple output, strong coupling nonlinear system, its control problem is always attracting much attention from the scholars. In chronological order, its research process can be classified into the following two phases.(1)
The steady state mathematical model and basic control strategy of VSC-HVDC system: [6] developed the steady state model and proposed control strategy which combined an inverse model controller with a PI controller. Reference [7] presented an equivalent continuous-time state space model of VSC-HVDC in the synchronous d-q reference frame and proposed a decoupled PI control strategy using the feed-forward compensation method. Reference [8] presented the elements of VSC-HVDC and proposed a feed-forward decoupled current control strategy. In this phase, the study is mainly based on the traditional PI controllers under the linear decoupling control strategy. The design method of this kind of control strategy is easy. And it shows good adjustment ability. But the parameters of PI controllers are fixed, and their adjustment ability is limited. When system suffers from large disturbance, they show weak robustness and dynamic characteristics. Some severe cases can lead to sustained oscillation of the system. Therefore some scholars started the second phase study.(2)
Optimizing and nonlinear control strategy of the VSC-HVDC system: [9] proposed an adaptive control design to improve dynamic performances of VSC-HVDC systems. The adaptive controllers designed for nonlinear characteristics of VSC-HVDC systems, which were based on back stepping method, considered parameters uncertainties. Reference [10] presented a robust nonlinear controller for VSC-HVDC transmission link using input-output linearization and sliding mode control strategy. The feedback linearization was used to cancel nonlinearity and the sliding mode control offered invariant stability to modeling uncertainties. Reference [11] proposed the controller design based on H∞ control methodologies to deal with the nonlinearities introduced by requirements to power flow and line voltage. Reference [12] addressed two different robust nonlinear control methodologies based on sliding mode control for the VSC-HVDC transmission system’s performance enhancement and stability improvement. In this phase, the studies are based on the nonlinear control, robust control, and optimizing control of the VSC-HVDC system. The experimental results from research papers show these control strategies can improve static, dynamic, and robust performance of VSC-HVDC system.However, the referred literatures are based on the continuous-time state. In practice, nowadays most controllers are implemented in discrete time. It is known that the realization of a controller using digital elements and complex programmable logic devices can achieve maximum reproducibility at minimum cost. So it is necessary to do research on the discrete model and control strategy. In [13], the discrete PI controllers of VSC-HVDC system were established. Reference [14] studied the discrete PI control algorithm of VSC-HVDC system which supplied power to the passive network.But so far, the discrete-time mathematic model and control strategy based on nonlinear control methods are seldom studied by scholars. Through proper feedback linearization control strategy, complex nonlinear system synthesis problems can be transformed to linear system synthesis problems. As a kind of robust control method, sliding mode control is versatile to linear and nonlinear system. It is easy to be designed and carried out. Because of its complete robustness to the parameters change and external disturbances which meet the match condition, it gets extensive attention from scholars and engineers [15].Therefore, the discrete mathematical model of the VSC-HVDC system based on nonlinear feedback linearization and discrete sliding mode control algorithm are proposed in this paper. And the fast output sampling (FOS) technique is adopted in the output feedback, which ensures the stability of the closed loop system in the condition of quasi sliding mode control. The simulation results performed in MATLAB/SIMULINK show that the proposed discrete control strategy can make the VSC-HVDC system have good operation performance.


## 2. The Mathematical Model of VSC-HVDC

### 2.1. Continuous-Time State Space Model

The structure diagram of VSC-HVDC is shown in Figure1. The more widely used continuous-time mathematical models based on the synchronous reference coordinates are employed in this paper, which are shown as (1) and (7).Figure 1
Structure diagram of VSC-HVDC system.


#### 2.1.1. Rectifier Side

Choose state variablesx1,2=irdirqT, controlled variables u1,2=UcrdUcrqT, and output variables y1,2=irdirqT. The mathematical model of the rectifier side is shown as(1)x˙1=-RrLrx1+ωrx2+UsrdLr-1Lru1,x˙2=-RrLrx2-ωrx1+UsrqLr-1Lru2,y1=x1,y2=x2,where ird,irq and Usrd,Usrq are the d-q axis currents and voltages on the rectifier side, respectively. Ucrd and Ucrq are, respectively, the d-q axis control inputs on the rectifier side. Rr and Lr are the corresponding equivalent resistance and the inductance on the rectifier side. ωr is the AC system frequency on the rectifier side.Here, the output variables on the rectifier side should bePrQrT. Pr and Qr are, respectively, the output values of active and reactive power on the rectifier side. For convenience, define Usr as the line voltage effective value of the rectifier power supply. Define the d axis in the synchronization reference frame coincidence with a axis in the three-phase reference coordinate, so Usrd=Usr, Usrq=0, PrQr=UsrdUsrq-UsrqUsrdirdirq=UsrirdUsrirq. Therefore the output variables can be chosen as y1,2=irdirqT.Define(2)f1=-RrLrx1+ωrx2+UsrdLr,f2=-RrLrx2-ωrx1+UsrqLr,fr=f1,f2T,g1=-1Lr,0T,g2=0,-1LrT.Therefore (1) can be reorganized by(3)x˙1,2=fr+g1u1+g2u2,y1,2=h1xh2x=x1x2.Then, do exact feedback linearization on system equation (3). By calculation, the relative degree of y1 is denoted by γ1=1. The relative degree of y2 is denoted by γ2=1. Based on the feedback linearization theory [16], the exact feedback linearization state equations of the rectifier side are obtained as(4)y˙1y˙2=x˙1x˙2=υ1υ2=Lfrγ1h1Lfrγ2h2+Lg1Lfrγ1-1h1Lg2Lfrγ1-1h1Lg1Lfrγ2-1h2Lg2Lfrγ2-1h2u1u2,x˙1x˙2=υ1υ2=f1f2+-1Lr00-1Lru1u2.Because -1/Lr00-1/Lr=1/Lr2≠0,  -1/Lr00-1/Lr is nonsingular.The controlled variablesu1 and u2 can be denoted by virtual control variables υ1 and υ2: (5)u1u2=Lrf1-υ1Lif2-υ2.To make it convenient for using FOS (fast output sampling) technique in Section 3.1 [17], substitute (5) into (3); then the system equation of rectifier side can be denoted by(6)x˙1=υ1,x˙2=υ2,y1=x1,y2=x2.


#### 2.1.2. Inverter Side

Choose state variablesx3,4,5=iidiiqUdc2T, controlled variables u3,4=UcidUciqT, and output variables y3,4=Udc2iiqT. The mathematical model of inverter side is shown as (7)x˙3=-RiLix3+ωix4+UsidLi-1Liu3,x˙4=-RiLix4-ωix3+UsiqLi-1Liu4,x˙5=Usidx3+Usiqx4-Rix32+x42Cx5+Udc1-x5CRdc,y3=x5,y4=x4,where iid,iiq and Usid,Usiq are the d-q axis currents and voltages on the inverter side, respectively. Ucid and Uciq are, respectively, the control inputs on the inverter side. Ri and Li are the corresponding equivalent resistance and the inductance on the inverter side. Udc1 and Udc2 are, respectively, the DC voltages on the rectifier side and inverter side. C and Rdc are, respectively, the converter station capacitance and DC line resistance. ωi is the AC system frequency on the inverter side.Here, the output variables on the inverter side should beUdc2Qi. Qi is the output value of reactive power on the inverter side. For convenience, define Usi as the line voltage effective value of the inverter side power source. Define the d axis in the synchronization reference frame that coincides with a axis in the three-phase reference coordinate, so Usid=Usi, Usiq=0, and Qi=-Usiqiid+Usidiiq=Usiiiq. Therefore the output variables can be chosen as y3,4=Udc2iiqT.Define(8)f3=-RiLix3+ωix4+UsidLi,f4=-RiLix4-ωix3+UsiqLi,f5=Usidx3+Usiqx4-Rix32+x42Cx5+Udc1-x5CRdc,fi=f3,f4,f5T,g3=-1Li,0,0T,g4=0,-1Li,0T.Therefore (7) can be reorganized by(9)x˙3,4,5=fi+g3u3+g4u4,y3,4=h3xh4x=x5x4.Then, do exact feedback linearization on system (9). By calculation, the relative degree of y3 is denoted by γ3=2. The relative degree of y4 is denoted by γ4=1. Based upon the feedback linearization theory [16], the exact feedback linearization state equations of inverter side are obtained as(10)y¨3y˙4=x¨5x˙4=υ3υ4=Lfiγ3h3Lfiγ4h4+Lg3Lfiγ3-1h3Lg4Lfiγ3-1h3Lg3Lfiγ4-1h4Lg4Lfiγ4-1h4u3u4,x¨5x˙4=υ3υ4=a1f3+a2f4+a3f5f4+-a1Li-a2Li0-1Liu3u4,where(11)a1=∂f5∂x3=Usid-2Rix3Cx5,a2=∂f5∂x4=Usiq-2Rix4Cx5,a3=∂f5∂x5=-Usidx3+Usiqx4-Rix32+x42Cx52-1CRdc.Because-a1/Li-a2/Li0-1/Li=a1/Li2≠0, -a1/Li-a2/Li0-1/Li is nonsingular.The controlled variablesu3 and u4 can be denoted by virtual control variables υ3 and υ4:(12)u3u4=a1f1+a3f3-υ3+a2υ4·Lia1Lif4-υ4.To make it convenient for using FOS (fast output sampling) technology in Section 3.1 [17], substitute a1, a2, a3, and (12) into (7); then the system equation of inverter side can be denoted by(13)x˙3=a32a1+a3a1CRdcx5+1a1υ3-a2a1υ4,x˙4=υ4,x˙5=-a3-1CRdcx5,y3=x5,y4=x4.Here, x˙5 should be x˙5=(-a3-1/CRdc)x5+(Udc1-x5)/CRdc. Since, during normal operation, Udc1 is approximately equal to x5, for convenient calculation, x˙5 is simplified into x˙5=(-a3-1/CRdc)x5.


### 2.2. Discrete-Time State Space Model

Discretize the virtual control variablesυ1, υ2, υ3, and υ4 with sampling time τ:(14)υ1kυ2k=y1k+1-y1kτy2k+1-y2kτ=x1k+1-x1kτx2k+1-x2kτ=f1kf2k+-1Lr00-1Lru1ku2k,(15)υ3kυ4k=y3k+2-2y3k+1+y3kτ2y4k+1-y4kτ=x5k+2-2x5k+1+x5kτ2x4k+1-x4kτ=a1f3k+a2f4k+a3f5kf4k+-a1Li-a2Li0-1Liu3ku4k.Discretize system equations (6), (13) with sampling time τ, shown as (16)x1k+1=x1k+τυ1k,x2k+1=x2k+τυ2k,y1k=x1k,y2k=x2k,x3k+1=x3k+τa32a1+τa3a1CRdcx5k+τa1υ3k-τa2a1υ4k,x4k+1=x4k+τυ4k,x5k+1=1-τa3-τCRdcx5k,y3k=x5k,y4k=x4k.Rearrange (16), and then get (17)x1,2k+1=Arτx1,2k+Brτυ1,2k,y1,2k=Crx1,2k,x3,4,5k+1=Aiτx3,4,5k+Biτυ3,4k,y3,4k=Cix3,4,5k,where(18)Arτ=1001,Brτ=τ00τ,Cr=1001,Aiτ=10τa32a1+τa3a1CRdc010001-τa3-τCRdc,Biτ=τa1-τa2a10τ00,Ci=001010.Assume that the pairs (Arτ,Brτ), (Aiτ,Biτ) are controllable and the pairs (Arτ,Cr), (Aiτ,Ci) are observable through properly sampling output variables.


## 2.1. Continuous-Time State Space Model

The structure diagram of VSC-HVDC is shown in Figure1. The more widely used continuous-time mathematical models based on the synchronous reference coordinates are employed in this paper, which are shown as (1) and (7).Figure 1
Structure diagram of VSC-HVDC system.


### 2.1.1. Rectifier Side

Choose state variablesx1,2=irdirqT, controlled variables u1,2=UcrdUcrqT, and output variables y1,2=irdirqT. The mathematical model of the rectifier side is shown as(1)x˙1=-RrLrx1+ωrx2+UsrdLr-1Lru1,x˙2=-RrLrx2-ωrx1+UsrqLr-1Lru2,y1=x1,y2=x2,where ird,irq and Usrd,Usrq are the d-q axis currents and voltages on the rectifier side, respectively. Ucrd and Ucrq are, respectively, the d-q axis control inputs on the rectifier side. Rr and Lr are the corresponding equivalent resistance and the inductance on the rectifier side. ωr is the AC system frequency on the rectifier side.Here, the output variables on the rectifier side should bePrQrT. Pr and Qr are, respectively, the output values of active and reactive power on the rectifier side. For convenience, define Usr as the line voltage effective value of the rectifier power supply. Define the d axis in the synchronization reference frame coincidence with a axis in the three-phase reference coordinate, so Usrd=Usr, Usrq=0, PrQr=UsrdUsrq-UsrqUsrdirdirq=UsrirdUsrirq. Therefore the output variables can be chosen as y1,2=irdirqT.Define(2)f1=-RrLrx1+ωrx2+UsrdLr,f2=-RrLrx2-ωrx1+UsrqLr,fr=f1,f2T,g1=-1Lr,0T,g2=0,-1LrT.Therefore (1) can be reorganized by(3)x˙1,2=fr+g1u1+g2u2,y1,2=h1xh2x=x1x2.Then, do exact feedback linearization on system equation (3). By calculation, the relative degree of y1 is denoted by γ1=1. The relative degree of y2 is denoted by γ2=1. Based on the feedback linearization theory [16], the exact feedback linearization state equations of the rectifier side are obtained as(4)y˙1y˙2=x˙1x˙2=υ1υ2=Lfrγ1h1Lfrγ2h2+Lg1Lfrγ1-1h1Lg2Lfrγ1-1h1Lg1Lfrγ2-1h2Lg2Lfrγ2-1h2u1u2,x˙1x˙2=υ1υ2=f1f2+-1Lr00-1Lru1u2.Because -1/Lr00-1/Lr=1/Lr2≠0,  -1/Lr00-1/Lr is nonsingular.The controlled variablesu1 and u2 can be denoted by virtual control variables υ1 and υ2: (5)u1u2=Lrf1-υ1Lif2-υ2.To make it convenient for using FOS (fast output sampling) technique in Section 3.1 [17], substitute (5) into (3); then the system equation of rectifier side can be denoted by(6)x˙1=υ1,x˙2=υ2,y1=x1,y2=x2.


### 2.1.2. Inverter Side

Choose state variablesx3,4,5=iidiiqUdc2T, controlled variables u3,4=UcidUciqT, and output variables y3,4=Udc2iiqT. The mathematical model of inverter side is shown as (7)x˙3=-RiLix3+ωix4+UsidLi-1Liu3,x˙4=-RiLix4-ωix3+UsiqLi-1Liu4,x˙5=Usidx3+Usiqx4-Rix32+x42Cx5+Udc1-x5CRdc,y3=x5,y4=x4,where iid,iiq and Usid,Usiq are the d-q axis currents and voltages on the inverter side, respectively. Ucid and Uciq are, respectively, the control inputs on the inverter side. Ri and Li are the corresponding equivalent resistance and the inductance on the inverter side. Udc1 and Udc2 are, respectively, the DC voltages on the rectifier side and inverter side. C and Rdc are, respectively, the converter station capacitance and DC line resistance. ωi is the AC system frequency on the inverter side.Here, the output variables on the inverter side should beUdc2Qi. Qi is the output value of reactive power on the inverter side. For convenience, define Usi as the line voltage effective value of the inverter side power source. Define the d axis in the synchronization reference frame that coincides with a axis in the three-phase reference coordinate, so Usid=Usi, Usiq=0, and Qi=-Usiqiid+Usidiiq=Usiiiq. Therefore the output variables can be chosen as y3,4=Udc2iiqT.Define(8)f3=-RiLix3+ωix4+UsidLi,f4=-RiLix4-ωix3+UsiqLi,f5=Usidx3+Usiqx4-Rix32+x42Cx5+Udc1-x5CRdc,fi=f3,f4,f5T,g3=-1Li,0,0T,g4=0,-1Li,0T.Therefore (7) can be reorganized by(9)x˙3,4,5=fi+g3u3+g4u4,y3,4=h3xh4x=x5x4.Then, do exact feedback linearization on system (9). By calculation, the relative degree of y3 is denoted by γ3=2. The relative degree of y4 is denoted by γ4=1. Based upon the feedback linearization theory [16], the exact feedback linearization state equations of inverter side are obtained as(10)y¨3y˙4=x¨5x˙4=υ3υ4=Lfiγ3h3Lfiγ4h4+Lg3Lfiγ3-1h3Lg4Lfiγ3-1h3Lg3Lfiγ4-1h4Lg4Lfiγ4-1h4u3u4,x¨5x˙4=υ3υ4=a1f3+a2f4+a3f5f4+-a1Li-a2Li0-1Liu3u4,where(11)a1=∂f5∂x3=Usid-2Rix3Cx5,a2=∂f5∂x4=Usiq-2Rix4Cx5,a3=∂f5∂x5=-Usidx3+Usiqx4-Rix32+x42Cx52-1CRdc.Because-a1/Li-a2/Li0-1/Li=a1/Li2≠0, -a1/Li-a2/Li0-1/Li is nonsingular.The controlled variablesu3 and u4 can be denoted by virtual control variables υ3 and υ4:(12)u3u4=a1f1+a3f3-υ3+a2υ4·Lia1Lif4-υ4.To make it convenient for using FOS (fast output sampling) technology in Section 3.1 [17], substitute a1, a2, a3, and (12) into (7); then the system equation of inverter side can be denoted by(13)x˙3=a32a1+a3a1CRdcx5+1a1υ3-a2a1υ4,x˙4=υ4,x˙5=-a3-1CRdcx5,y3=x5,y4=x4.Here, x˙5 should be x˙5=(-a3-1/CRdc)x5+(Udc1-x5)/CRdc. Since, during normal operation, Udc1 is approximately equal to x5, for convenient calculation, x˙5 is simplified into x˙5=(-a3-1/CRdc)x5.


## 2.1.1. Rectifier Side

Choose state variablesx1,2=irdirqT, controlled variables u1,2=UcrdUcrqT, and output variables y1,2=irdirqT. The mathematical model of the rectifier side is shown as(1)x˙1=-RrLrx1+ωrx2+UsrdLr-1Lru1,x˙2=-RrLrx2-ωrx1+UsrqLr-1Lru2,y1=x1,y2=x2,where ird,irq and Usrd,Usrq are the d-q axis currents and voltages on the rectifier side, respectively. Ucrd and Ucrq are, respectively, the d-q axis control inputs on the rectifier side. Rr and Lr are the corresponding equivalent resistance and the inductance on the rectifier side. ωr is the AC system frequency on the rectifier side.Here, the output variables on the rectifier side should bePrQrT. Pr and Qr are, respectively, the output values of active and reactive power on the rectifier side. For convenience, define Usr as the line voltage effective value of the rectifier power supply. Define the d axis in the synchronization reference frame coincidence with a axis in the three-phase reference coordinate, so Usrd=Usr, Usrq=0, PrQr=UsrdUsrq-UsrqUsrdirdirq=UsrirdUsrirq. Therefore the output variables can be chosen as y1,2=irdirqT.Define(2)f1=-RrLrx1+ωrx2+UsrdLr,f2=-RrLrx2-ωrx1+UsrqLr,fr=f1,f2T,g1=-1Lr,0T,g2=0,-1LrT.Therefore (1) can be reorganized by(3)x˙1,2=fr+g1u1+g2u2,y1,2=h1xh2x=x1x2.Then, do exact feedback linearization on system equation (3). By calculation, the relative degree of y1 is denoted by γ1=1. The relative degree of y2 is denoted by γ2=1. Based on the feedback linearization theory [16], the exact feedback linearization state equations of the rectifier side are obtained as(4)y˙1y˙2=x˙1x˙2=υ1υ2=Lfrγ1h1Lfrγ2h2+Lg1Lfrγ1-1h1Lg2Lfrγ1-1h1Lg1Lfrγ2-1h2Lg2Lfrγ2-1h2u1u2,x˙1x˙2=υ1υ2=f1f2+-1Lr00-1Lru1u2.Because -1/Lr00-1/Lr=1/Lr2≠0,  -1/Lr00-1/Lr is nonsingular.The controlled variablesu1 and u2 can be denoted by virtual control variables υ1 and υ2: (5)u1u2=Lrf1-υ1Lif2-υ2.To make it convenient for using FOS (fast output sampling) technique in Section 3.1 [17], substitute (5) into (3); then the system equation of rectifier side can be denoted by(6)x˙1=υ1,x˙2=υ2,y1=x1,y2=x2.


## 2.1.2. Inverter Side

Choose state variablesx3,4,5=iidiiqUdc2T, controlled variables u3,4=UcidUciqT, and output variables y3,4=Udc2iiqT. The mathematical model of inverter side is shown as (7)x˙3=-RiLix3+ωix4+UsidLi-1Liu3,x˙4=-RiLix4-ωix3+UsiqLi-1Liu4,x˙5=Usidx3+Usiqx4-Rix32+x42Cx5+Udc1-x5CRdc,y3=x5,y4=x4,where iid,iiq and Usid,Usiq are the d-q axis currents and voltages on the inverter side, respectively. Ucid and Uciq are, respectively, the control inputs on the inverter side. Ri and Li are the corresponding equivalent resistance and the inductance on the inverter side. Udc1 and Udc2 are, respectively, the DC voltages on the rectifier side and inverter side. C and Rdc are, respectively, the converter station capacitance and DC line resistance. ωi is the AC system frequency on the inverter side.Here, the output variables on the inverter side should beUdc2Qi. Qi is the output value of reactive power on the inverter side. For convenience, define Usi as the line voltage effective value of the inverter side power source. Define the d axis in the synchronization reference frame that coincides with a axis in the three-phase reference coordinate, so Usid=Usi, Usiq=0, and Qi=-Usiqiid+Usidiiq=Usiiiq. Therefore the output variables can be chosen as y3,4=Udc2iiqT.Define(8)f3=-RiLix3+ωix4+UsidLi,f4=-RiLix4-ωix3+UsiqLi,f5=Usidx3+Usiqx4-Rix32+x42Cx5+Udc1-x5CRdc,fi=f3,f4,f5T,g3=-1Li,0,0T,g4=0,-1Li,0T.Therefore (7) can be reorganized by(9)x˙3,4,5=fi+g3u3+g4u4,y3,4=h3xh4x=x5x4.Then, do exact feedback linearization on system (9). By calculation, the relative degree of y3 is denoted by γ3=2. The relative degree of y4 is denoted by γ4=1. Based upon the feedback linearization theory [16], the exact feedback linearization state equations of inverter side are obtained as(10)y¨3y˙4=x¨5x˙4=υ3υ4=Lfiγ3h3Lfiγ4h4+Lg3Lfiγ3-1h3Lg4Lfiγ3-1h3Lg3Lfiγ4-1h4Lg4Lfiγ4-1h4u3u4,x¨5x˙4=υ3υ4=a1f3+a2f4+a3f5f4+-a1Li-a2Li0-1Liu3u4,where(11)a1=∂f5∂x3=Usid-2Rix3Cx5,a2=∂f5∂x4=Usiq-2Rix4Cx5,a3=∂f5∂x5=-Usidx3+Usiqx4-Rix32+x42Cx52-1CRdc.Because-a1/Li-a2/Li0-1/Li=a1/Li2≠0, -a1/Li-a2/Li0-1/Li is nonsingular.The controlled variablesu3 and u4 can be denoted by virtual control variables υ3 and υ4:(12)u3u4=a1f1+a3f3-υ3+a2υ4·Lia1Lif4-υ4.To make it convenient for using FOS (fast output sampling) technology in Section 3.1 [17], substitute a1, a2, a3, and (12) into (7); then the system equation of inverter side can be denoted by(13)x˙3=a32a1+a3a1CRdcx5+1a1υ3-a2a1υ4,x˙4=υ4,x˙5=-a3-1CRdcx5,y3=x5,y4=x4.Here, x˙5 should be x˙5=(-a3-1/CRdc)x5+(Udc1-x5)/CRdc. Since, during normal operation, Udc1 is approximately equal to x5, for convenient calculation, x˙5 is simplified into x˙5=(-a3-1/CRdc)x5.


## 2.2. Discrete-Time State Space Model

Discretize the virtual control variablesυ1, υ2, υ3, and υ4 with sampling time τ:(14)υ1kυ2k=y1k+1-y1kτy2k+1-y2kτ=x1k+1-x1kτx2k+1-x2kτ=f1kf2k+-1Lr00-1Lru1ku2k,(15)υ3kυ4k=y3k+2-2y3k+1+y3kτ2y4k+1-y4kτ=x5k+2-2x5k+1+x5kτ2x4k+1-x4kτ=a1f3k+a2f4k+a3f5kf4k+-a1Li-a2Li0-1Liu3ku4k.Discretize system equations (6), (13) with sampling time τ, shown as (16)x1k+1=x1k+τυ1k,x2k+1=x2k+τυ2k,y1k=x1k,y2k=x2k,x3k+1=x3k+τa32a1+τa3a1CRdcx5k+τa1υ3k-τa2a1υ4k,x4k+1=x4k+τυ4k,x5k+1=1-τa3-τCRdcx5k,y3k=x5k,y4k=x4k.Rearrange (16), and then get (17)x1,2k+1=Arτx1,2k+Brτυ1,2k,y1,2k=Crx1,2k,x3,4,5k+1=Aiτx3,4,5k+Biτυ3,4k,y3,4k=Cix3,4,5k,where(18)Arτ=1001,Brτ=τ00τ,Cr=1001,Aiτ=10τa32a1+τa3a1CRdc010001-τa3-τCRdc,Biτ=τa1-τa2a10τ00,Ci=001010.Assume that the pairs (Arτ,Brτ), (Aiτ,Biτ) are controllable and the pairs (Arτ,Cr), (Aiτ,Ci) are observable through properly sampling output variables.


## 3. Discrete Sliding Mode Control of VSC-HVDC System

In ideal continuous-time case, the SMC (sliding mode control) switches at infinite frequency and forces the states to slide on the so-called switching hyperplane. In practical applications, direct implementation of continuous-time SMC schemes using digital elements, which are considered as the device for imperfect switching, will inevitably induce chattering phenomenon and deteriorate performance or even induce instability. Chattering will cause serious harmonics which is undesirable in VSC-HVDC systems. Hence, the controller design using the discrete-time SMC (DSMC) algorithm is desirable for a successful implementation of the VSC-HVDC control systems. And due to the finite sampling frequency, the controller inputs are calculated once per sampling period and held constant during that interval. Under such a circumstance, the trajectories of the system states of interest are unable to precisely move along the sliding surface, which will lead to a quasi sliding mode motion only [18]. Therefore, only using static output feedback technology has not effectively ensured the control effect of discrete sliding mode control. The fast output sampling technology should be used [17, 19].


### 3.1. Fast Output Sampling (FOS) Technology

Compared with the static output feedback technology, FOS not only keeps its advantage, but also can randomly configure the system poles and always make the closed loop system stable. And then FOS can ensure the effectiveness of the discrete sliding mode control. In the FOS, every sampling periodτ is divided into N subintervals. Here Δ=τ/N and N is equal to or greater than the observable index of system A,B. The output variables are measured at time instants t=lΔ, l=0,1,…,N-1. Consider the discrete-time system having be at time t=kτ; the fast output samples are obtained as [17, 19](19)yk=yk-1τyk-1τ+Δ⋮yk-1τ+N-1Δ.Then the rectifier side can be expressed as (20)x1,2k=Arτx1,2k-1+Brτυ1,2k-1,y1,2k=Cr0x1,2k-1+Dr0υ1,2k-1,where(21)Cr0=CrCrAr⋮CrArN1-1,Dr0=0CrBr⋮Cr∑j=0N1-2ArjBr.And assume that Cr0 and Dr0 are invertible through appropriate choice. ArBrCr is the system parameter matrix with sampling rate 1/Δ1, Δ1=τ/N1.Definey1,2k=y1((k-1)τ)y2((k-1)τ). This means y1 is sampled once and y2 is sampled once in each sampling period τ.The inverter side can be expressed as(22)x3,4,5k=Aiτx3,4,5k-1+Biτυ3,4k-1,y3,4k=Ci0x3,4,5k-1+Di0υ3,4k-1,where(23)Ci0=CiCiAi⋮CiAiN2-1,Di0=0CiBi⋮Ci∑j=0N2-2AijBi.And assume that Ci0 and Di0 are invertible through appropriate choice. AiBiCi is the system parameter matrix with sampling rate 1/Δ2, Δ2=τ/N2.Definey3,4k=y3((k-1)τ)y3((k-1)τ+Δ2)y4((k-1)τ). This means y3 is sampled twice and y4 is sampled once in each sampling period τ.According to (20) and (22), state vectors x1,2(k) and x3,4,5(k) can be deduced as (24)x1,2k=ArτCr0-1y1,2k+Brτ-ArτCr0-1Dr0υ1,2k-1,(25)x3,4,5k=AiτCi0-1y3,4k+Biτ-AiτCi0-1Di0υ3,4k-1.


### 3.2. The Rectifier Side Control

The differences between output and reference variables are denoted by (26). Because the relative degrees of y1 and y2 are, respectively, equal to 1, 1, the sliding mode surfaces are defined as (27). Consider(26)e1,2=e1e2=y1k-irdrefky2k-irqrefk,where irdref(k) and irqref(k) are the reference values. Their calculation processes are shown in Figure 2. Pr, Prref, Qr, and Qrref are, respectively, the output value and reference value of active and reactive power. Consider(27)s1,2=s1s2=e1ke2k.Desired state trajectories of the discrete variable structure system shown as in (28) can be obtained by control law based on reaching law method:(28)s1k+1-s1k=-ρ1τs1k-ε1τsgn⁡s1k,s2k+1-s2k=-ρ2τs2k-ε2τsgn⁡s2k,where ρ1, ρ2, ε1, and ε2 are all greater than zero. And 1-ρ1τ>0, 1-ρ2τ>0.Figure 2
Control block diagrams of output references value at the rectifier station.
(a)
The reference value ofd axis current
(b)
The reference value ofq axis currentNow replacing (26) with (27) and then with (28), the virtual control variables can be denoted by(29)υ1k=-ρ1s1k-ε1sgn⁡s1k,υ2k=-ρ2s2k-ε2sgn⁡s2k.The control variables shown as in (30) can be deduced by (14), (24), and (29):(30)u1k=-RrQr1+ωLrQr2-Lrυ1+Usrd,u2k=-ωLrQr1-RrQr2-Lrυ2+Usrq,where Qr1 and Qr2 are the first and second row of x1,2(k) in Section 3.1. u1(k), u2(k) can be expressed by y1,2(k) through FOS.


### 3.3. The Inverter Side Control

The differences between output and reference variables are denoted by (31). Because the relative degrees of y3 and y4 are, respectively, equal to 2, 1, the sliding mode surfaces are defined as (32). Consider(31)e3,4=e3e4=y3k-Udcrefky4k-iiqrefk,where Udcref(k) and iiqref(k) are the reference values. Usually Udcref(k) is known. The calculation process of iiqref(k) is shown in Figure 3. Qi and Qiref are, respectively, the output and reference value of reactive power:(32)s3,4=s3s4=c3e3k+e3k+1-e3kτe4k.Similar to the rectifier side, desired state trajectories of the discrete variable structure system shown as in (33) can be obtained by control law based on reaching law method:(33)s3k+1-s3k=-ρ3τs3k-ε3τsgn⁡s3k,s4k+1-s4k=-ρ4τs4k-ε4τsgn⁡s4k,where ρ3, ρ4, ε3, and ε4 are all greater than zero. And 1-ρ3τ>0, 1-ρ4τ>0.Figure 3
Control block diagram of output reference value at the inverter station.Now replacing (31) with (32) and then with (33), the virtual control variables can be denoted by(34)υ3k=-ρ3s3k-ε3sgn⁡s3k-c3y3k+1-y3kτ,υ4k=-ρ4s4k-ε4sgn⁡s4k.The control variables shown as in (35) can be deduced by (15), (25), and (34):(35)u3k=-RiQi3+ωiLiQi4-Lia3a1a3+2CRdcQi5-Lia1υ3+Lia2a1υ4+Lia3Udc1a1CRdc+Usid,u4k=-ωiLiQi3-RiQi4-Liυ4+Usiq,where Qi3, Qi4, and Qi5 are the first, second, and third row of x3,4,5(k) in Section 3.1. u3(k) and u4(k) can be expressed by y3,4(k) through FOS.


## 3.1. Fast Output Sampling (FOS) Technology

Compared with the static output feedback technology, FOS not only keeps its advantage, but also can randomly configure the system poles and always make the closed loop system stable. And then FOS can ensure the effectiveness of the discrete sliding mode control. In the FOS, every sampling periodτ is divided into N subintervals. Here Δ=τ/N and N is equal to or greater than the observable index of system A,B. The output variables are measured at time instants t=lΔ, l=0,1,…,N-1. Consider the discrete-time system having be at time t=kτ; the fast output samples are obtained as [17, 19](19)yk=yk-1τyk-1τ+Δ⋮yk-1τ+N-1Δ.Then the rectifier side can be expressed as (20)x1,2k=Arτx1,2k-1+Brτυ1,2k-1,y1,2k=Cr0x1,2k-1+Dr0υ1,2k-1,where(21)Cr0=CrCrAr⋮CrArN1-1,Dr0=0CrBr⋮Cr∑j=0N1-2ArjBr.And assume that Cr0 and Dr0 are invertible through appropriate choice. ArBrCr is the system parameter matrix with sampling rate 1/Δ1, Δ1=τ/N1.Definey1,2k=y1((k-1)τ)y2((k-1)τ). This means y1 is sampled once and y2 is sampled once in each sampling period τ.The inverter side can be expressed as(22)x3,4,5k=Aiτx3,4,5k-1+Biτυ3,4k-1,y3,4k=Ci0x3,4,5k-1+Di0υ3,4k-1,where(23)Ci0=CiCiAi⋮CiAiN2-1,Di0=0CiBi⋮Ci∑j=0N2-2AijBi.And assume that Ci0 and Di0 are invertible through appropriate choice. AiBiCi is the system parameter matrix with sampling rate 1/Δ2, Δ2=τ/N2.Definey3,4k=y3((k-1)τ)y3((k-1)τ+Δ2)y4((k-1)τ). This means y3 is sampled twice and y4 is sampled once in each sampling period τ.According to (20) and (22), state vectors x1,2(k) and x3,4,5(k) can be deduced as (24)x1,2k=ArτCr0-1y1,2k+Brτ-ArτCr0-1Dr0υ1,2k-1,(25)x3,4,5k=AiτCi0-1y3,4k+Biτ-AiτCi0-1Di0υ3,4k-1.


## 3.2. The Rectifier Side Control

The differences between output and reference variables are denoted by (26). Because the relative degrees of y1 and y2 are, respectively, equal to 1, 1, the sliding mode surfaces are defined as (27). Consider(26)e1,2=e1e2=y1k-irdrefky2k-irqrefk,where irdref(k) and irqref(k) are the reference values. Their calculation processes are shown in Figure 2. Pr, Prref, Qr, and Qrref are, respectively, the output value and reference value of active and reactive power. Consider(27)s1,2=s1s2=e1ke2k.Desired state trajectories of the discrete variable structure system shown as in (28) can be obtained by control law based on reaching law method:(28)s1k+1-s1k=-ρ1τs1k-ε1τsgn⁡s1k,s2k+1-s2k=-ρ2τs2k-ε2τsgn⁡s2k,where ρ1, ρ2, ε1, and ε2 are all greater than zero. And 1-ρ1τ>0, 1-ρ2τ>0.Figure 2
Control block diagrams of output references value at the rectifier station.
(a)
The reference value ofd axis current
(b)
The reference value ofq axis currentNow replacing (26) with (27) and then with (28), the virtual control variables can be denoted by(29)υ1k=-ρ1s1k-ε1sgn⁡s1k,υ2k=-ρ2s2k-ε2sgn⁡s2k.The control variables shown as in (30) can be deduced by (14), (24), and (29):(30)u1k=-RrQr1+ωLrQr2-Lrυ1+Usrd,u2k=-ωLrQr1-RrQr2-Lrυ2+Usrq,where Qr1 and Qr2 are the first and second row of x1,2(k) in Section 3.1. u1(k), u2(k) can be expressed by y1,2(k) through FOS.


## 3.3. The Inverter Side Control

The differences between output and reference variables are denoted by (31). Because the relative degrees of y3 and y4 are, respectively, equal to 2, 1, the sliding mode surfaces are defined as (32). Consider(31)e3,4=e3e4=y3k-Udcrefky4k-iiqrefk,where Udcref(k) and iiqref(k) are the reference values. Usually Udcref(k) is known. The calculation process of iiqref(k) is shown in Figure 3. Qi and Qiref are, respectively, the output and reference value of reactive power:(32)s3,4=s3s4=c3e3k+e3k+1-e3kτe4k.Similar to the rectifier side, desired state trajectories of the discrete variable structure system shown as in (33) can be obtained by control law based on reaching law method:(33)s3k+1-s3k=-ρ3τs3k-ε3τsgn⁡s3k,s4k+1-s4k=-ρ4τs4k-ε4τsgn⁡s4k,where ρ3, ρ4, ε3, and ε4 are all greater than zero. And 1-ρ3τ>0, 1-ρ4τ>0.Figure 3
Control block diagram of output reference value at the inverter station.Now replacing (31) with (32) and then with (33), the virtual control variables can be denoted by(34)υ3k=-ρ3s3k-ε3sgn⁡s3k-c3y3k+1-y3kτ,υ4k=-ρ4s4k-ε4sgn⁡s4k.The control variables shown as in (35) can be deduced by (15), (25), and (34):(35)u3k=-RiQi3+ωiLiQi4-Lia3a1a3+2CRdcQi5-Lia1υ3+Lia2a1υ4+Lia3Udc1a1CRdc+Usid,u4k=-ωiLiQi3-RiQi4-Liυ4+Usiq,where Qi3, Qi4, and Qi5 are the first, second, and third row of x3,4,5(k) in Section 3.1. u3(k) and u4(k) can be expressed by y3,4(k) through FOS.


## 4. Simulation Results

The typical VSC-HVDC system composed of two converter stations is taken as example and the detailed parameters are shown in Table1. The simulation experiment is performed in MATLAB/SIMULINK. In the per-unit value system, the based power is 200 MW, the based voltage at AC side is 81.65 KV, and the based voltage at DC side is 100 KV. The sampling period τ=74μs. Also N1=1 and N2=2. The discrete SMC controllers parameters are ρ1=ρ2=1100, ε1=ε2=3, ρ3=ρ4=1000, ε3=ε4=3, and c3=1200.Table 1
The system parameters of three-level VSC-HVDC.
Parameters
Value
AC voltagesUsr, Usi
230 kv
ω
r
,
ω
i
2
∗
pi
∗
50 rad/s
Transformer ratioT1,T2
230 kv/100 kv
R
r
,
R
i
0.325 Ω
L
r
,
L
i
0.072 H
Rated voltage in DC side
±100 kv
C
70μF
Unit resistance of DC line
1.39
e
-
002 Ω/km
DC line length
75
∗
2 km
Switching frequency
1350 HZ


### 4.1. The Steady State Operation

In steady state operation,Prref, Udcref, Qiref, and Qiref are, respectively, 1 pu, 0 pu, 1 pu, and 0 pu. As shown in Figure 4, the active, reactive power and DC voltage can track their reference values effectively. The proposed mathematical model and control strategy can be proved to make the system operate well in steady state condition.Figure 4
Responses of the rectifier and inverter in steady state operation.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter side
(c)
The DC voltage


### 4.2. Reversion and Step Changes

As shown in Figure5, the change process of Prref is as follows: the Prref keeps 1 pu from 0 s to 1.5 s. At 1.5 s, it steps to −1 pu, and then keeps this value until 3.0 s. At last, it steps to 1 pu at 3.0 s. From the experimental results, the referred changes least affect the reactive power on the rectifier side and reactive power on the inverter side. As shown in Figure 6, the change process of Qrref is as follows: the Qrref keeps 0 pu from 0 s to 1.5 s. And then steps to 0.1 pu at 1.5 s. The change process of Qiref is as follows: the Qiref keeps 0 pu from 0 s to 2.5 s. And then steps to −0.1 pu at 2.5 s. From the experimental results, the referred changes least affect the active power on the rectifier side and the active power on the inverter side. These can prove that the proposed discrete SMC strategy can make the active and reactive power decoupled and independent.Figure 5
Responses of the rectifier and inverter when active power reverses.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter sideFigure 6
Responses of the rectifier and inverter when the reactive power step changes.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter side


### 4.3. Robustness Test

The equivalent resistance and inductance at the converter stations both reduce 20%. The experimental results are shown in Figure7. The reference values are the same as those in Section 4.1. The active, reactive power and DC voltage can track their reference values smoothly and quickly. This can prove that the proposed control strategy has good robustness.Figure 7
Responses of the rectifier and inverter when the inner parameters change at both converter stations.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter side
(c)
The DC voltage


## 4.1. The Steady State Operation

In steady state operation,Prref, Udcref, Qiref, and Qiref are, respectively, 1 pu, 0 pu, 1 pu, and 0 pu. As shown in Figure 4, the active, reactive power and DC voltage can track their reference values effectively. The proposed mathematical model and control strategy can be proved to make the system operate well in steady state condition.Figure 4
Responses of the rectifier and inverter in steady state operation.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter side
(c)
The DC voltage


## 4.2. Reversion and Step Changes

As shown in Figure5, the change process of Prref is as follows: the Prref keeps 1 pu from 0 s to 1.5 s. At 1.5 s, it steps to −1 pu, and then keeps this value until 3.0 s. At last, it steps to 1 pu at 3.0 s. From the experimental results, the referred changes least affect the reactive power on the rectifier side and reactive power on the inverter side. As shown in Figure 6, the change process of Qrref is as follows: the Qrref keeps 0 pu from 0 s to 1.5 s. And then steps to 0.1 pu at 1.5 s. The change process of Qiref is as follows: the Qiref keeps 0 pu from 0 s to 2.5 s. And then steps to −0.1 pu at 2.5 s. From the experimental results, the referred changes least affect the active power on the rectifier side and the active power on the inverter side. These can prove that the proposed discrete SMC strategy can make the active and reactive power decoupled and independent.Figure 5
Responses of the rectifier and inverter when active power reverses.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter sideFigure 6
Responses of the rectifier and inverter when the reactive power step changes.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter side


## 4.3. Robustness Test

The equivalent resistance and inductance at the converter stations both reduce 20%. The experimental results are shown in Figure7. The reference values are the same as those in Section 4.1. The active, reactive power and DC voltage can track their reference values smoothly and quickly. This can prove that the proposed control strategy has good robustness.Figure 7
Responses of the rectifier and inverter when the inner parameters change at both converter stations.
(a)
The active and reactive power at rectifier side
(b)
The active and reactive power at inverter side
(c)
The DC voltage


## 5. Conclusions

This paper deduces the discrete mathematical model of VSC-HVDC system by nonlinear input-output feedback linearization method. Based on this, the discrete sliding mode robust controllers are designed. And to ensure the effectiveness of the controllers in quasi sliding mode condition, the FOS technology is used in output feedback. The results from simulation experiment in MATLAB/SIMULINK prove the effectiveness of proposed discrete mathematical model and control strategy. Since the actual computer control system is discrete sampling and SMC has a promising prospect, the proposed discrete mathematical model and control strategy have certain practical application prospect.



---

*Source: 929467-2015-06-25.xml*