Discrete-Time Nonlinear Control of VSC-HVDC System

. Because VSC-HVDC is a kind of strong nonlinear, coupling, and multi-input multioutput (MIMO) system, its control problem is alwaysattractingmuchattentionfromscholars.Andalotofpapershavedoneresearchonitscontrolstrategyinthecontinuous-time domain.Butthecontrolsystemisimplementedthroughthecomputerdiscretesamplinginpracticalengineering.Itisnecessaryto studythemathematicalmodelandcontrolalgorithminthediscrete-timedomain.Thediscretemathematicalmodelbasedonoutput feedbacklinearizationanddiscreteslidingmodecontrolalgorithmisproposedinthispaper.Andtoensuretheeffectivenessofthe controlsysteminthequasislidingmodestate,thefastoutputsamplingmethodisusedintheoutputfeedback.Theresultsfrom simulationexperimentinMATLAB/SIMULINKprovethattheproposeddiscretecontrolalgorithmcanmaketheVSC-HVDC systemhavegoodstatic,dynamic,androbustcharacteristicsindiscrete-timedomain.


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, remotearea power supply, and so forth [1][2][3][4][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 - 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 However, the referred literatures are based on the continuoustime 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.

Continuous-Time State Space
Model.The structure diagram of VSC-HVDC is shown in Figure 1.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).
where   ,   and   ,   are the - axis currents and voltages on the rectifier side, respectively.  and   are, respectively, the - axis control inputs on the rectifier side.  and   are the corresponding equivalent resistance and the inductance on the rectifier side.  is the AC system frequency on the rectifier side.
Here, the output variables on the rectifier side should be (    ) .  and   are, respectively, the output values of active and reactive power on the rectifier side.For convenience, define   as the line voltage effective value of the rectifier power supply.Define the  axis in the synchronization reference frame coincidence with  axis in the three-phase reference coordinate, so   =   ,   = 0, Therefore the output variables can be chosen as y 1,2 = (    ) . Define Therefore (1) can be reorganized by Then, do exact feedback linearization on system equation (3).By calculation, the relative degree of  1 is denoted by  1 = 1.The relative degree of  2 is denoted by  2 = 1.Based on the feedback linearization theory [16], the exact feedback linearization state equations of the rectifier side are obtained as The controlled variables  1 and  2 can be denoted by virtual control variables  1 and  2 : 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 where   ,   and   ,   are the - axis currents and voltages on the inverter side, respectively.  and   are, respectively, the control inputs on the inverter side.  and   are the corresponding equivalent resistance and the inductance on the inverter side. 1 and  2 are, respectively, the DC voltages on the rectifier side and inverter side. and   are, respectively, the converter station capacitance and DC line resistance.  is the AC system frequency on the inverter side.
Here, the output variables on the inverter side should be ( 2   ).  is the output value of reactive power on the inverter side.For convenience, define   as the line voltage effective value of the inverter side power source.Define the  axis in the synchronization reference frame that coincides with  axis in the three-phase reference coordinate, so   =   ,   = 0, and   = −    +     =     .Therefore the output variables can be chosen as y 3,4 = ( 2   ) . Define Therefore ( 7) can be reorganized by Then, do exact feedback linearization on system (9).By calculation, the relative degree of  3 is denoted by  3 = 2.The relative degree of  4 is denoted by  4 = 1.Based upon the feedback linearization theory [16], the exact feedback linearization state equations of inverter side are obtained as where Because The controlled variables  3 and  4 can be denoted by virtual control variables  3 and  4 : To make it convenient for using FOS (fast output sampling) technology in Section 3.1 [17], substitute  1 ,  2 ,  3 , and ( 12) into (7); then the system equation of inverter side can be denoted by Here, ẋ 5 should be ẋ 5 = (− Mathematical Problems in Engineering 5 Discretize system equations ( 6), (13) with sampling time , shown as Rearrange ( 16), and then get where ) , Assume that the pairs (A r , B r ), (A i , B i ) are controllable and the pairs (A r , C r ), (A i , C i ) are observable through properly sampling output variables.

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].

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 period  is divided into  subintervals.Here Δ = / and  is equal to or greater than the observable index of system (A, B).The output variables are measured at time instants  = Δ,  = 0, 1, . . .,  − 1.
Consider the discrete-time system having be at time  = ; the fast output samples are obtained as [17,19] Then the rectifier side can be expressed as where And assume that C r0 and D r0 are invertible through appropriate choice.(A r B r C r ) is the system parameter matrix with sampling rate 1/Δ 1 , ).This means  1 is sampled once and  2 is sampled once in each sampling period .
The inverter side can be expressed as where And assume that C i0 and D i0 are invertible through appropriate choice.(A i B i C i ) is the system parameter matrix with sampling rate 1/Δ 2 , Δ 2 = / 2 .
Define y 3,4k = ( ).This means  3 is sampled twice and  4 is sampled once in each sampling period .According to (20) and ( 22), state vectors  1,2 () and  3,4,5 () can be deduced as 3.2.The Rectifier Side Control.The differences between output and reference variables are denoted by (26).Because the relative degrees of  1 and  2 are, respectively, equal to 1, 1, the sliding mode surfaces are defined as (27).Consider where  ref () and  ref () are the reference values.Their calculation processes are shown in Figure 2.   ,  ref ,   , and  ref are, respectively, the output value and reference value of active and reactive power.Consider Desired state trajectories of the discrete variable structure system shown as in (28) can be obtained by control law based on reaching law method: where  1 ,  2 ,  1 , and  2 are all greater than zero.And 1 − 1  > 0, 1 −  2  > 0. Now replacing (26) with ( 27) and then with (28), the virtual control variables can be denoted by The control variables shown as in (30) can be deduced by ( 14), (24), and (29): where  1 and  2 are the first and second row of x 1,2 (k) in Section 3.1. 1 (),  2 () can be expressed by y 1,2 (k) through FOS.

The Inverter Side
Control.The differences between output and reference variables are denoted by (31).Because the relative degrees of  3 and  4 are, respectively, equal to 2, 1, the sliding mode surfaces are defined as (32).Consider where  ref () and  ref () are the reference values.Usually  ref () is known.The calculation process of  ref () is shown in Figure 3.   and  ref are, respectively, the output and reference value of reactive power: ) . (32) 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: where  3 ,  4 ,  3 , and  4 are all greater than zero.And 1 − 3  > 0, 1 −  4  > 0. Now replacing (31) with (32) and then with (33), the virtual control variables can be denoted by The control variables shown as in (35) can be deduced by ( 15), (25), and (34): where  3 ,  4 , and  5 are the first, second, and third row of x 3,4,5 (k) in Section 3.1. 3 () and  4 () can be expressed by y 3,4 (k) through FOS.

Simulation Results
The typical VSC-HVDC system composed of two converter stations is taken as example and the detailed parameters are shown in Table 1.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  = 74 s.Also

Reversion and
Step Changes.As shown in Figure 5, the change process of  ref is as follows: the  ref 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  ref is as follows: the  ref 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  ref is as follows: the  ref 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.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.

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.

Figure 2 :
Figure 2: Control block diagrams of output references value at the rectifier station.

Figure 3 :
Figure 3: Control block diagram of output reference value at the inverter station.

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 The active and reactive power at rectifier side The active and reactive power at inverter side

Figure 5 :
Figure 5: Responses of the rectifier and inverter when active power reverses.

Figure 6 :
Figure 6: Responses of the rectifier and inverter when the reactive power step changes.

Figure 7 :
Figure 7: Responses of the rectifier and inverter when the inner parameters change at both converter stations.

Table 1 :
4.1.The Steady State Operation.In steady state operation,  ref ,  ref ,  ref , and  ref are, respectively, 1 pu, 0 pu, 1 pu, and 0 pu.As shown in Figure4, 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.The system parameters of three-level VSC-HVDC.