Research Article | Open Access

Volume 2015 |Article ID 929467 | https://doi.org/10.1155/2015/929467

TianTian Qian, ShiHong Miao, "Discrete-Time Nonlinear Control of VSC-HVDC System", Mathematical Problems in Engineering, vol. 2015, Article ID 929467, 11 pages, 2015. https://doi.org/10.1155/2015/929467

# Discrete-Time Nonlinear Control of VSC-HVDC System

Academic Editor: Miguel A. F. Sanjuan
Revised19 May 2015
Accepted28 May 2015
Published25 Jun 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.

#### 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 .

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 .

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 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).

###### 2.1.1. Rectifier Side

Choose state variables , controlled variables , and output variables . The mathematical model of the rectifier side is shown aswhere 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 , , . Therefore the output variables can be chosen as .

DefineTherefore (1) can be reorganized byThen, do exact feedback linearization on system equation (3). By calculation, the relative degree of is denoted by . The relative degree of is denoted by . Based on the feedback linearization theory , the exact feedback linearization state equations of the rectifier side are obtained asBecause ,   is nonsingular.

The controlled variables and can be denoted by virtual control variables and : To make it convenient for using FOS (fast output sampling) technique in Section 3.1 , substitute (5) into (3); then the system equation of rectifier side can be denoted by

###### 2.1.2. Inverter Side

Choose state variables , controlled variables , and output variables . The mathematical model of inverter side is shown as 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. and 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 . 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 , , and . Therefore the output variables can be chosen as .

DefineTherefore (7) can be reorganized byThen, do exact feedback linearization on system (9). By calculation, the relative degree of is denoted by . The relative degree of is denoted by . Based upon the feedback linearization theory , the exact feedback linearization state equations of inverter side are obtained aswhere

Because , is nonsingular.

The controlled variables and can be denoted by virtual control variables and :To make it convenient for using FOS (fast output sampling) technology in Section 3.1 , substitute , , , and (12) into (7); then the system equation of inverter side can be denoted byHere, should be . Since, during normal operation, is approximately equal to , for convenient calculation, is simplified into .

##### 2.2. Discrete-Time State Space Model

Discretize the virtual control variables , , , and with sampling time :Discretize system equations (6), (13) with sampling time , shown as Rearrange (16), and then get whereAssume that the pairs , are controllable and the pairs , 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 . 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 period is divided into subintervals. Here and is equal to or greater than the observable index of system . The output variables are measured at time instants , . 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 whereAnd assume that and are invertible through appropriate choice. is the system parameter matrix with sampling rate , .

Define . This means is sampled once and is sampled once in each sampling period .

The inverter side can be expressed as whereAnd assume that and are invertible through appropriate choice. is the system parameter matrix with sampling rate , .

Define . This means is sampled twice and is sampled once in each sampling period .

According to (20) and (22), state vectors and 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 and are, respectively, equal to 1, 1, the sliding mode surfaces are defined as (27). Considerwhere and are the reference values. Their calculation processes are shown in Figure 2. , , , and are, respectively, the output value and reference value of active and reactive power. ConsiderDesired state trajectories of the discrete variable structure system shown as in (28) can be obtained by control law based on reaching law method:where , , , and are all greater than zero. And , .

Now replacing (26) with (27) and then with (28), the virtual control variables can be denoted byThe control variables shown as in (30) can be deduced by (14), (24), and (29):where and are the first and second row of in Section 3.1. , can be expressed by through FOS.

##### 3.3. The Inverter Side Control

The differences between output and reference variables are denoted by (31). Because the relative degrees of and are, respectively, equal to 2, 1, the sliding mode surfaces are defined as (32). Considerwhere and are the reference values. Usually is known. The calculation process of is shown in Figure 3. and are, respectively, the output and reference value of reactive power: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 , , , and are all greater than zero. And , .

Now replacing (31) with (32) and then with (33), the virtual control variables can be denoted byThe control variables shown as in (35) can be deduced by (15), (25), and (34):where , , and are the first, second, and third row of in Section 3.1. and can be expressed by 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 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 . Also and . The discrete SMC controllers parameters are , , , , and .

 Parameters Value AC voltages , 230 kv rad/s Transformer ratio 230 kv/100 kv 0.325 Ω 0.072 H Rated voltage in DC side ±100 kv 70 F Unit resistance of DC line Ω/km DC line length km Switching frequency 1350 HZ
##### 4.1. The Steady State Operation

In steady state operation, , , , and 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.

##### 4.2. Reversion and Step Changes

As shown in Figure 5, the change process of is as follows: the 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 is as follows: the 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 is as follows: the 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.

##### 4.3. Robustness Test

The equivalent resistance and inductance at the converter stations both reduce 20%. The experimental results are shown in Figure 7. 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.

#### 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the Natural Science Foundation of China (51377068).

1. N. Flourentzou, V. G. Agelidis, and G. D. Demetriades, “VSC-based HVDC power transmission systems: an overview,” IEEE Transactions on Power Electronics, vol. 24, no. 3, pp. 592–602, 2009. View at: Publisher Site | Google Scholar
2. Y. Wei, Q. He, Y. Sun, and C. Ji, “Improved power flow algorithm for VSC-HVDC system based on high-order newton-type method,” Mathematical Problems in Engineering, vol. 2013, Article ID 235316, 10 pages, 2013. View at: Publisher Site | Google Scholar
3. L. Zhang, L. Harnefors, and H.-P. Nee, “Interconnection of two very weak AC systems by VSC-HVDC links using power-synchronization control,” IEEE Transactions on Power Systems, vol. 26, no. 1, pp. 344–355, 2011. View at: Publisher Site | Google Scholar
4. L. Zhang, L. Harnefors, and H.-P. Nee, “Modeling and control of VSC-HVDC links connected to island systems,” IEEE Transactions on Power Systems, vol. 26, no. 2, pp. 783–793, 2011. View at: Publisher Site | Google Scholar
5. F. Xu and Z. Xu, “A modular multilevel power flow controller for meshed HVDC grids,” Science China Technological Sciences, vol. 57, no. 9, pp. 1773–1784, 2014. View at: Publisher Site | Google Scholar
6. G. Zhang and Z. Xu, “Steady-state model for VSC based HVDC and its controller design,” in Proceedings of the IEEE Power Engineering Society Winter Meeting, vol. 3, pp. 1085–1090, February 2001. View at: Google Scholar
7. M. Yin, G.-Y. Li, T.-Y. Niu, G.-K. Li, H.-F. Liang, and M. Zhou, “Continuous-time state-space model of VSC-HVDC and its control strategy,” Proceedings of the Chinese Society of Electrical Engineering, vol. 25, no. 18, pp. 34–39, 2005. View at: Google Scholar
8. R. Song, C. Zheng, R. Li, and Z. Xiaoxin, “VSCs based HVDC and its control strategy,” in Proceedings of the IEEE/PES Transmission and Distribution Conference and Exhibition: Asia and Pacific, pp. 1–6, August 2005. View at: Publisher Site | Google Scholar
9. S.-Y. Ruan, G.-J. Li, X.-H. Jiao, Y.-Z. Sun, and T. T. Lie, “Adaptive control design for VSC-HVDC systems based on backstepping method,” Electric Power Systems Research, vol. 77, no. 5-6, pp. 559–565, 2007. View at: Publisher Site | Google Scholar
10. A. Moharana and P. K. Dash, “Input-output linearization and robust sliding-mode controller for the VSC-HVDC transmission link,” IEEE Transactions on Power Delivery, vol. 25, no. 3, pp. 1952–1961, 2010. View at: Publisher Site | Google Scholar
11. N. Nayak, S. K. Routray, and P. K. Rout, “State feedback robust ${H}_{\infty }$ controller for transient stability enhancement of Vsc-Hvdc transmission systems,” Procedia Technology, vol. 4, pp. 652–660, 2012, Proceedings of the 2nd International Conference on Computer, Communication, Control and Information Technology (C3IT '12) on February 25-26, 2012. View at: Google Scholar
12. H. S. Ramadan, H. Siguerdidjane, M. Petit, and R. Kaczmarek, “Performance enhancement and robustness assessment of VSC–HVDC transmission systems controllers under uncertainties,” International Journal of Electrical Power & Energy Systems, vol. 35, no. 1, pp. 34–46, 2012. View at: Publisher Site | Google Scholar
13. X.-G. Wei, G.-F. Tang, and J.-C. Zheng, “Study of VSC-HVDC discrete model and its control strategies,” Proceedings of the Chinese Society of Electrical Engineering, vol. 27, no. 28, pp. 6–11, 2007. View at: Google Scholar
14. H. Yang, N. Zhang, and M. J. Ye, “Study of VSC-HVDC connected to passive network discrete model and its control strategies,” Power System Protection and Control, vol. 40, no. 4, pp. 37–42, 2012. View at: Google Scholar
15. V. I. Utkin and H.-C. Chang, “Sliding mode control on electro-mechanical systems,” Mathematical Problems in Engineering, vol. 8, no. 4-5, pp. 451–473, 2002. View at: Publisher Site | Google Scholar | MathSciNet
16. H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Prentice hall, Upper Saddle River, NJ, USA, 1996.
17. H. Werner, “Multimodel robust control by fast output sampling—an lmi approach,” Automatica, vol. 34, no. 12, pp. 1625–1630, 1998. View at: Publisher Site | Google Scholar
18. T.-L. Tai and J.-S. Chen, “UPS inverter design using discrete-time sliding-mode control scheme,” IEEE Transactions on Industrial Electronics, vol. 49, no. 1, pp. 67–75, 2002. View at: Publisher Site | Google Scholar
19. M. C. Saaj, B. Bandyopadhyay, and H. Unbehauen, “A new algorithm for discrete-time sliding-mode control using fast output sampling feedback,” IEEE Transactions on Industrial Electronics, vol. 49, no. 3, pp. 518–523, 2002. View at: Publisher Site | Google Scholar