Open access peer-reviewed chapter

Sliding-Mode Perturbation Observer-Based Sliding-Mode Control for VSC-HVDC Systems

Written By

Bo Yang, Tao Yu, Hongchun Shu and Pulin Cao

Submitted: November 15th, 2017 Reviewed: January 31st, 2018 Published: October 17th, 2018

DOI: 10.5772/intechopen.74717

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Abstract

This chapter develops a sliding-mode perturbation observer-based sliding-mode control (POSMC) scheme for voltage source converter-based high voltage direct current (VSC-HVDC) systems. The combinatorial effect of nonlinearities, parameter uncertainties, unmodeled dynamics, and time-varying external disturbances is aggregated into a perturbation, which is estimated online by a sliding-mode state and perturbation observer (SMSPO). POSMC does not require an accurate VSC-HVDC system model and only the reactive power and DC voltage at the rectifier side while reactive and active powers at the inverter side need to be measured. Additionally, a considerable robustness can be provided through the real-time compensation of the perturbation, in which the upper bound of perturbation is replaced by the real-time estimation of the perturbation, such that the over-conservativeness of conventional sliding-mode control (SMC) can be effectively reduced. Four case studies are carried out on the VSC-HVDC system, such as active and reactive power tracking, AC bus fault, system parameter uncertainties, and weak AC gird connection. Simulation results verify its advantages over vector control and feedback linearization sliding-mode control. Then, a dSPACE-based hardware-in-the-loop (HIL) test is undertaken to validate the implementation feasibility of the proposed approach.

Keywords

  • sliding-mode control
  • sliding-mode perturbation observer
  • VSC-HVDC systems
  • HIL test

1. Introduction

In the past decades, the ever-increasing penetration of renewable energy (wind, solar, wave, hydro, and biomass) requires an extraordinarily reliable and effective transmission of electrical power from these new sources to the main power grid [1], in which hydropower has already been fully exploited in many grids, such that a sustainable development can be achieved in future [2]. The problems and perspectives of converting present energy systems (mainly thermal and nuclear) into a 100% renewable energy system have been discussed with a conclusion that such idea is possible, which, however, raises that advanced transmission technologies are needed to realize this goal [3].

The need for more secure power grids and ever-increasing environmental concerns continue to drive the worldwide deployment of high voltage direct current (HVDC) transmission technology, which enables a more reliable and stable asynchronous interconnection of power networks that operate on different frequencies [4]. HVDC systems use power electronic devices to convert alternative current (AC) into direct current (DC), they are an economical way of transmitting bulk electrical power in DC over long distance overhead line or short submarine cable, while advanced extruded DC cable technologies have been used to increase power transmissions by at least 50%, which is also an important onshore solution. HVDC enables secure and stable asynchronous interconnection of power networks that operate on different frequencies. Different technologies have been used to design two-terminal HVDC systems for the purpose of a point-to-point power transfer, such as line-commutated converter (LCC)-based HVDC (LCC-HVDC) systems using grid-controlled mercury-arc valves or thyristors, capacitor-commutated converter (CCC)-based HVDC (CCC-HVDC) systems, or controlled series commutated converter (CSCC)-based HVDC (CSCC-HVDC) systems [5].

Voltage source converter-based high voltage direct current (VSC-HVDC) systems using insulated gate bipolar transistor (IGBT) technology have attracted increasing attentions due to the interconnection between the mainland and offshore wind farms, power flow regulation in alternating current (AC) power systems, long distance transmission [6], and introduction of the supergrid, which is a large-scale power grid interconnected between national power grids [7]. The main feature of the VSC-HVDC system is that no external voltage source is needed for communication, while active and reactive powers at each AC grid can be independently controlled [8, 9].

Traditionally, control of the VSC-HVDC system utilizes a nested-loop d-q vector control (VC) approach based on linear proportional-integral (PI) methods [10], whose control performance may be degraded with the change of operation conditions as its control parameters are tuned from one-point linearization model [11]. As VSC-HVDC systems are highly nonlinear resulting from converters and also operate in power systems with modeling uncertainties, many advanced control approaches are developed to provide a consistent control performance under various operation conditions, such as feedback linearization control (FLC) [12], which fully compensated the nonlinearities with the requirement of an accurate system model. Linear matrix inequality (LMI)-based robust control was developed in [13] to maximize the size of the uncertainty region within which closed-loop stability is maintained. In addition, adaptive backstepping control was designed to estimate the uncertain parameters by [14]. In [8, 9], power-synchronization control was employed to greatly increase the short-circuit capacity to the AC system. However, the aforementioned methods may not be adequate to simultaneously handle perturbations such as modeling uncertainties and time-varying external disturbances.

Based on the variable structure control strategy, sliding-mode control (SMC) is an effective and high-frequency switching control for nonlinear systems with modeling uncertainties and time-varying external disturbances. The main idea of SMC is to maintain the system sliding on a surface in the state space via an appropriate switching logic; it features the simple implementation, disturbance rejection, fast response, and strong robustness [15]. While the malignant effect of chattering phenomenon can be reduced by predictive variable structure [16] and self-tuning sliding mode [17], SMC has been applied on electrical vehicles [18], power converters [19], induction machines [20], wind turbines [21], etc. Moreover, a feedback linearization sliding-mode control (FLSMC) has been developed for the VSC-HVDC system to offer invariant stability to modeling uncertainties by [22]. Basically, SMC assumes perturbations to be bounded and the prior knowledge of these upper bounds is required. However, it may be difficult or sometimes impossible to obtain these upper bounds, thus the supreme upper bound is chosen to cover the whole range of perturbations. As a consequence, SMC based on this knowledge becomes over-conservative which may cause a poor tracking performance and undesirable control oscillations [23].

During the past decades, several elegant approaches based on observers have been proposed to estimate perturbations, including the unknown input observer (UIO) [24], the disturbance observer (DOB) [25], the equivalent input disturbance (EID)-based estimation [26], enhanced decentralized PI control via advanced disturbance observer [27], the extended state observer (ESO)-based active disturbance rejection control (ADRC) [28], and practical multivariable control based on inverted decoupling and decentralized ADRC [29]. Among the above listed approaches, ESO requires the least amount of system information, in fact, only the system order needs to be known [30]. Due to such promising features, ESO-based control schemes have become more and more popular. Recently, ESO-based SMC has been developed to remedy the over-conservativeness of SMC via an online perturbation estimation. It observes both system states and perturbations by defining an extended state to represent the lumped perturbation, which can be then compensated online to improve the performance of system. Related applications can be referred to mechanical systems [31], missile systems [32], spherical robots [33], and DC-DC buck power converters [34].

This chapter uses an ESO called sliding-mode state and perturbation observer (SMSPO) [35, 36] to estimate the combinatorial effect of nonlinearities, parameter uncertainties, unmodeled dynamics, and time-varying external disturbances existed in VSC-HVDC systems, which is then compensated by the perturbation observer-based sliding-mode control (POSMC). The motivation to use POSMC, in this chapter, rather than SMC and our previous work [35, 36, 37] can be summarized as follows:

  • The robustness of POSMC to the perturbation mostly depends on the perturbation compensation, while the ground of the robustness in SMC [18, 19, 20, 21, 22] is the discrete switching input. Furthermore, the upper bound of perturbation is replaced by the smaller bound of its estimation error, thus an over-conservative control input is avoided and the tracking accuracy is improved.

  • POSMC can provide greater robustness than that of nonlinear adaptive control (NAC) [35, 36] and perturbation observer-based adaptive passive control (POAPC) [37] due to its inherent property of disturbance rejection.

Compared to VC [11], POSMC can provide a consistent control performance under various operation condition of the VSC-HVDC system and improve the power tracking by eliminating the power overshoot. Compared to FLSMC [22], POSMC only requires the measurement of active and reactive power and DC voltage, which can provide a significant robustness and avoid an over-conservative control input as the real perturbation is estimated and compensated online. Four case studies are carried out to evaluate the control performance of POSMC through simulation, such as active and reactive power tracking, AC bus fault, system parameter uncertainties, and weak AC gird connection. Compared to the author’s previous work on SMSPO [35, 36], a dSPACE simulator-based hardware-in-the-loop (HIL) test is undertaken to validate its implementation feasibility.

The rest of the chapter is organized as follows. In Section 2, the model of the two-terminal VSC-HVDC system is presented. In Section 3, POSMC design for the VSC-HVDC system is developed and discussed. Sections 4 and 5 present the simulation and HIL results, respectively. Finally, conclusions are drawn in Section 6.

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2. VSC-HVDC system modeling

There are two VSCs in the VSC-HVDC system shown in Figure 1, in which the rectifier regulates the DC voltage and reactive power, while the inverter regulates the active and reactive power. Only the balanced condition is considered, e.g., the three phases have identical parameters and their voltages and currents have the same amplitude while each phase shifts 120 between themselves. The rectifier dynamics can be written at the angular frequency ω as [14].

did1dt=R1L1id1+ωiq1+ud1diq1dt=R1L1iq1ωid1+uq1dVdc1dt=3usq1iq12C1Vdc1iLC1E1

where the rectifier is connected with the AC grid via the equivalent resistance and inductance R1 and L1, respectively. C1 is the DC bus capacitor, ud1=usd1urdL1 and uq1=usq1urqL1.

Figure 1.

A standard two-terminal VSC-HVDC system.

The inverter dynamics is written as

did2dt=R2L2id2+ωiq2+ud2diq2dt=R2L2iq2ωid2+uq2dVdc2dt=3usq2iq22C2Vdc2+iLC2E2

where the inverter is connected with the AC grid via the equivalent resistance and inductance R2 and L2, respectively. C2 is the DC bus capacitor, ud2=usd2uidL2 and uq2=usq2uiqL2.

The interconnection between the rectifier and inverter through DC cable is given as

Vdc1iL=Vdc2iL+2R0iL2E3

where R0 represents the equivalent DC cable resistance.

The phase-locked loop (PLL) [38] is used during the transformation of the abc frame to the dq frame. In the synchronous frame, usd1, usd2, usq1, and usq2 are the d, q axes components of the respective AC grid voltages; id1, id2, iq1, and iq2 are that of the line currents; urd, uid, urq, and uiq are that of the converter input voltages. P1, P2, Q1, and Q2 are the active and reactive powers transmitted from the AC grid to the VSC; Vdc1 and Vdc2 are the DC voltages; and iL is the DC cable current.

At the rectifier side, the q-axis is set to be in phase with the AC grid voltage us1. Correspondingly, the q-axis is set to be in phase of the AC grid voltage us2 at the inverter side. Hence, usd1 and usd2 are equal to 0, while usq1 and usq2 are equal to the magnitude of us1 and us2. Note that this chapter adopts such framework from [12, 14, 22] to provide a consistent control design procedure and an easy control performance comparison, other framework can also be used as shown in [8, 9, 11]. The only difference of these two alternatives is the derived system equations, while the control design is totally the same. In addition, it is assumed that the VSC-HVDC system is connected to sufficiently strong AC grids, such that the AC grid voltage remains as an ideal constant. The power flows from the AC grid can be given as

P1=32usq1iq1+usd1id1=32usq1iq1Q1=32usq1id1usd1iq1=32usq1id1P2=32usq2iq2+usd2id2=32usq2iq2Q2=32usq2id2usd2iq2=32usq2id2E4
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3. POSMC design for the VSC-HVDC system

3.1. Perturbation observer-based sliding-mode control

Consider an uncertain nonlinear system which has the following canonical form:

ẋ=Ax+Bax+bxu+dty=x1E5

where x=x1x2xnTRn is the state variable vector, uR and yR are the control input and system output, respectively. ax: RnR and bx: RnR are unknown smooth functions, and dt: R+R represents the time-varying external disturbance. The n×n matrix A and the n×1 matrix B are of the canonical form as follows:

A=0100001000010000n×n,B=0001n×1E6

The perturbation of system (5) is defined as [35, 36, 37]

Ψxut=ax+bxb0u+dtE7

From the original system (5), the last state xn can be rewritten in the presence of perturbation (6) as follows:

ẋn=ax+bxb0u+dt+b0u=Ψxut+b0uE8

Define a fictitious state xn+1=Ψxut. Then, system (5) can be extended as

y=x1ẋ1=x2ẋn=xn+1+b0uẋn+1=Ψ̇E9

The new state vector becomes xe=x1x2xnxn+1T, and following assumptions are made [35]:

  • A.1 b0 is chosen to satisfy: bx/b01θ<1, where θ is a positive constant.

  • A.2 The functions Ψxut:Rn×R×R+R and Ψ̇xut:Rn×R×R+R are bounded over the domain of interest: Ψxutγ1, Ψ̇xutγ2 with Ψ0,0,0=0 and Ψ̇0,0,0=0, where γ1 and γ2 are positive constants.

  • A.3 The desired trajectory yd and its up to n th-order derivative are continuous and bounded.

The above three assumptions ensure the effectiveness of such perturbation estimation-based approach. In particular, assumptions A.1 and A.2 guarantee the closed-loop system stability with perturbation estimation, while assumption A.3 ensures POSMC can drive the system state x to track a desired state xd=ydyd1ydn1T [39]. In the consideration of the worst case, e.g., y=x1 is the only measurable state, an (n+1)th-order SMSPO [35, 36] for the extended system (8) is designed to estimate the system states and perturbation, shown as follows:

x̂̇1=x̂2+α1x˜1+k1satx˜1x̂̇n=Ψ̂+αnx˜1+knsatx˜1+b0uΨ̂̇=αn+1x˜1+kn+1satx˜1E10

where x˜1=x1x̂1, ki and αi, i=1,2,,n+1, are positive coefficients, function satx˜1 is defined as satx˜1=x˜1/x˜1 when x˜1>ϵ and satx˜1=x˜1/ϵ when x˜1ϵ. The effect and setting of the SMSPO parameters are provided as follows:

  • The Luenberger observer constants αi, which are chosen to place the observer poles at the desired locations in the open left-half complex plane. In other words, αi are chosen such that the root of sn+1+α1sn+α2sn1++αn+1=s+λαn+1=0 is in the open left-half complex plane. A larger value of αi not only will accelerate the estimation rate of SMSPO, but also will result in a more significant effect of peaking phenomenon. Thus, a trade-off between the estimation rate and effect of peaking phenomenon must be made through trial-and-error. Normally, they are set to be much larger than the root of the closed-loop system to ensure a fast online estimation [37].

  • The sliding surface constants ki. k1x˜2max must be chosen to guarantee the estimation error of SMSPO (9) will enter into the sliding surface Sspox˜=x˜1=0 at t>ts and thereafter remain Sspo=0, tts [35, 39]. While the poles of the sliding surface λk are determined by choosing the ratio ki/k1i=23n+1 to put the root of pn+k2/k1pn1++kn/k1p+kn+1/k1=p+λkn=0 to be in the open left-half complex plane. Under Assumption A.2, SMSPO converges to a neighborhood of the origin if gains ki are properly selected, which has been proved in [35, 40]. For a given k1, a larger ki not only will accelerate the estimation rate of SMSPO, but also will result in a degraded observer stability. Thus, a trade-off between the estimation rate and observer stability must be made through trial-and-error [39].

  • The layer thickness constant of saturation function ϵ, which is a positive small scaler to replace the sign function by the saturation function, such that the chattering effect can be reduced. A larger ϵ will result in a smoother chattering, but a larger steady-state estimation error. Consequently, a trade-off between the chattering effect and steady-state estimation error must be made through trial-and-error. In practice, a value closes to 0 is recommended.

Moreover, the reduced estimation error dynamics on the sliding mode can be written as [35]

x˜̇2=k2k1x˜2+x˜3x˜̇3=k3k1x˜2+x˜4x˜̇n=knk1x˜2+Ψ˜Ψ˜̇=kn+1k1x˜2+Ψ̇E11

Lemma 1 [39]. Consider extended system (8), design an SMSPO (9). If assumption A.2 holds for some value γ2, then given any constant δ, the gains ki can be chosen such that, from an initial estimation error x˜e0, the estimation error x˜e converges exponentially into the neighborhood

x˜eδE12

In particular,

x˜iδλkn+1i,i=2,,n+1,t>t1.E13

where t1 is the time constant which definition can be found in [39].

Remark 1. When SMSPO is used to estimate the perturbation, the upper bound of the derivative of perturbation γ2 is required to guarantee the estimation accuracy, and such upper bound will result in a conservative observer gain. However, the conservative gain is only included in the observer loop, not in the controller loop.

Define an estimated sliding surface as

Ŝxt=i=1nρix̂iydi1E14

where the estimated sliding surface gains ρi=Cn1i1λcni, i=1,,n, place all poles of the estimated sliding surface at λc, where λc>0.

Besides, the actual sliding surface is written by

S=i=1nρixiydi1E15

Hence, the estimation error of the sliding surface can be directly calculated as

S˜=SŜ=i=1nρix˜iE16

Construct a Lyapunov function as follows:

V=12Ŝ2E17

The attractiveness of sliding surface is achieved if V̇<0 for all x˜Ŝ, that is, the control u needs to be designed to enforce ŜŜ̇<0 outside a prescribed manifold Ŝ<εc.

The POSMC for system (5) is designed as

u=1b0ydni=1n1ρix̂i+1ydiζŜφsatŜΨ̂E18

where ζ and φ are control gains which are chosen to fulfill the attractiveness of the estimated sliding surface Ŝ.

Differentiate estimated sliding surface (13) along SMSPO (9), use the reduced estimation error dynamics (10), it yields

Ŝ̇=Ψ̂+b0u+knk1x˜2ydn+i=1n1ρix̂i+1ydi+kik1x˜2E19

Substitute control (17) into the above Eq. (18), leads to

Ŝ̇=i=1nρikik1x˜2ζŜφsatŜεcE20

Consequently, the attractiveness of sliding surface can be derived as

ζŜ+φ>i=1nρikik1x˜2E21

which will be fulfilled with the relationship of k1 if

ζŜ+φ>k1i=1nρikik1E22

The above condition can be immediately satisfied if control gain φ is chosen as

φ>k1i=1nρikik1E23

which, using gains ki, yields

φ>k1i=1nρiCni1λki1E24

This condition ensures the existence of a sliding mode on the boundary layer Ŝεc. From system (15) one can easily calculate

S˜̇=i=1n1ρix˜i+1i=1nρikik1x˜2+Ψ˜E25

As Ŝ=SS˜, the actual S-dynamics of sliding surface can be obtained with dynamics (19) as

Ṡ+ζ+φεcS=ζ+φεci=1nρix˜i+i=1n1ρix˜i+1+Ψ˜E26

It is definite that the driving term of S-dynamics is the sum of the estimation errors of states and the perturbation. The bounds of the sliding surface can be calculated by

ŜεcSS˜εcSS˜+εcSi=1nρix˜i+εcδλkn+1i=2nρiλki+εc,t>t1.E27

Based on bounds (26), together with the polynomial gains ρi, the states tracking error satisfies the following relationship [41]

xitxdit2λciεcλcn+δλkn+1j=2nλkλcjCn1j,i=0,1,,n1.E28

Note that POSMC does not require an accurate system model, and only one state measurement y=x1 is needed. As the upper bound of perturbation Ψ is replaced by the smaller bound of its estimation error Ψ˜, a smaller control gain is needed such that the over-conservativeness of SMC can be avoided [35].

Remark 2. The motivation to use SMSPO is due to the fact that the sliding-mode observer potentially offers advantages similar to those of sliding-mode controllers, in particular, inherent robustness to parameter uncertainty and external disturbances [42]. It is a high-performance state estimator with a simple structure and is well suited for uncertain nonlinear systems [31]. Moreover, it has the merits of simple structure and easy analysis of the closed-loop system stability compared to that of ADRC which uses a nonlinear observer [28], while they can provide almost the same performance of perturbation estimation.

The overall design procedure of POSMC for system (5) can be summarized as follows:

Step 1. Define perturbation (6) for the original n th-order system (5);

Step 2. Define a fictitious state xn+1=Ψ to represent perturbation (6);

Step 3. Extend the original n th-order system (5) into the extended n+1 th-order system (8);

Step 4. Design the n+1 th-order SMSPO (9) for the extended n+1 th-order system (8) to obtain the state estimate x̂ and the perturbation estimate Ψ̂ by the only measurement of x1;

Step 5. Design controller (17) for the original n th-order system (5), in which the estimated sliding surface Ŝ is calculated by (13).

3.2. Rectifier controller design

Choose the system output yr=yr1yr2T=Q1Vdc1T, let Q1 and Vdc1 be the given references of the reactive power and DC voltage, respectively. Define the tracking error er=er1er2T=Q1Q1Vdc1Vdc1T, differentiate er for rectifier (1) until the control input appears explicitly, yields

ėr1e¨r2=fr1Q̇1fr2V¨dc1+Brud1uq1E29

where

fr1=3usq12R1L1id1+ωiq1fr2=3usq12C1Vdc1ωid1R1L1iq1iq1Vdc13usq1iq12C1Vdc1iLC112R0C13usq1iq12C1Vdc1iLC13usq2iq22C2Vdc2iLC2E30

and

Br=3usq12L1003usq12C1L1Vdc1E31

The determinant of matrix Br is obtained as Br=9usq12/4C1L12Vdc1, which is nonzero within the operation range of the rectifier, thus system (28) is linearizable.

Assume all the nonlinearities are unknown, define the perturbations Ψr1 and Ψr2 as

Ψr1Ψr2=fr1fr2+BrBr0ud1uq1E32

where the constant control gain Br0 is given by

Br0=br1000br20E33

Then system (28) can be rewritten as

ėr1e¨r2=Ψr1Ψr2+Br0ud1uq1Q̇1V¨dc1E34

Define z11=Q1, a second-order sliding-mode perturbation observer (SMPO) is used to estimate Ψr1 as

ẑ̇11=Ψ̂r1+αr1Q˜1+kr1satQ˜1+br10ud1Ψ̂̇r1=αr2Q˜1+kr2satQ˜1E35

where observer gains kr1, kr2, αr1, and αr2 are all positive constants.

Define z11=Vdc1 and z12=ż11, a third-order SMSPO is used to estimate Ψr2 as

ẑ̇11=ẑ12+αr1V˜dc1+kr1satV˜dc1ẑ̇12=Ψ̂r2+αr2V˜dc1+kr2satV˜dc1+br20uq1Ψ̂̇r2=αr3V˜dc1+kr3satV˜dc1E36

where observer gains kr1, kr2, kr3, αr1, αr2, and αr3 are all positive constants.

The above observers (31) and (32) only need the measurement of reactive power Q1 and DC voltage Vdc1 at the rectifier side, which can be directly obtained in practice.

The estimated sliding surface of system (28) is defined as

Ŝr1Ŝr2=ẑ'11Q1ρ1ẑ11Vdc1+ρ2ẑ12V̇dc1E37

where ρ1 and ρ2 are the positive sliding surface gains. The attractiveness of the estimated sliding surface (33) ensures reactive power Q1 and DC voltage Vdc1 can track to their reference.

The POSMC of system (28) is designed as

ud1uq1=Br01Ψ̂r1+Q̇1ζrŜr1φrsatŜr1Ψ̂r2+V¨dc1ρ1(ẑ12V̇dc1ζrŜr2φrsatŜr2E38

where positive control gains ζr, ζr, φr, ρ1, and φr are chosen to ensure the attractiveness of estimated sliding surface (33).

During the most severe disturbance, both the reactive power and DC voltage reduce from their initial value to around zero within a short period of time Δ. Thus, the boundary values of the system state and perturbation estimates can be obtained as ẑ11Q1, Ψ̂r1Q1/Δ, ẑ11Vdc1, ẑ12Vdc1/Δ, and Ψ̂r2Vdc1/Δ2, respectively.

3.3. Inverter controller design

Choose the system output yi=yi1yi2T=Q2P2T, let Q2 and P2 be the given references of the reactive and active power, respectively. Define the tracking error ei=ei1ei2T=Q2Q2P2P2T, differentiate ei for inverter (2) until the control input appears explicitly, yields

ėi1ėi2=fi1Q̇2fi2Ṗ2+Biud2uq2E39

where

fi1=3usq22R2L2id2+ωiq2fi2=3usq22R2L2iq2ωid2E40

and

Bi=3usq22L2003usq22L2E41

The determinant of matrix Bi is obtained as Bi=9us22/4L22, which is nonzero within the operation range of the inverter, thus system (35) is linearizable.

Assume all the nonlinearities are unknown, define the perturbations Ψi1 and Ψi2 as

Ψi1Ψi2=fi1fi2+BiBi0ud2uq2E42

where the constant control gain Bi0 is given by

Bi0=bi1000bi20E43

Then system (35) can be rewritten as

ėi1ėi2=Ψi1Ψi2+Bi0ud2uq2Q̇2Ṗ2E44

Similarly, define z21=Q2 and z21=P2, two second-order SMPOs are used to estimate Ψi1 and Ψi2, respectively, as

ẑ̇21=Ψ̂i1+αi1Q˜2+ki1satQ˜2+bi10ud2Ψ̂̇i1=αi2Q˜2+ki2satQ˜2E45

where observer gains ki1, ki2, αi1, and αi2 are all positive constants.

ẑ̇21=Ψ̂i2+αi1P˜2+ki1satP˜2+bi20uq2Ψ̂̇i2=αi2P˜2+ki2satP˜2E46

where observer gains ki1, ki2, αi1, and αi2 are all positive constants.

The above observers (38) and (39) only need the measurement of reactive power Q2 and active power P2 at the inverter side, which can be directly obtained in practice.

The estimated sliding surface of system (35) is defined as

Ŝi1Ŝi2=ẑ21Q2ẑ21P2E47

Similarly, the attractiveness of the estimated sliding surface (40) ensures the reactive power Q2 and active power P2 can track to their reference.

The POSMC of system (35) is designed as

ud2uq2=Bi01Ψ̂i1+Q̇2ζiŜi1φisatŜi1Ψ̂i2+Ṗ2ζiŜi2φisatŜi2E48

where positive control gains ζi, ζi, φi, and φi are chosen to ensure the attractiveness of estimated sliding surface (40).

Similarly, the boundary values of the system state and perturbation estimates can be obtained as ẑ21Q2, Ψ̂i1Q2/Δ, ẑ21P2, and Ψ̂i2P2/Δ, respectively.

Note that control outputs (34) and (41) are modulated by the sinusoidal pulse width modulation (SPWM) technique [6] in this chapter. The overall controller structure of the VSC-HVDC system is illustrated by Figure 2, in which only reactive power Q1 and DC voltage Vdc1 need to be measured for rectifier controller (34), while active power P2 and reactive power Q2 for inverter controller (41).

Remark 3 The conventional linear PI/PID control scheme employs an inner current loop to regulate the current [11], which could employ a synchronous reference frame (SRF)-based current controller [43] to avoid overcurrent. In contrast, the proposed POSMC (34) and (41) actually contains no current in its control law, while it cannot handle the overcurrent. Hence, the overcurrent protection devices [44] will be activated to prevent the overcurrent to grow, which can be seen in Figure 2.

Figure 2.

The overall controller structure of the VSC-HVDC system.

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4. Simulation results

POSMC is applied on the VSC-HVDC system illustrated in Figure 1. The AC grid frequency is 50 Hz and VSC-HVDC system parameters are given in Table 1. POSMC parameters are provided in Table 2, in which the observer poles are allocated as λαr=100 and λαr=λαi=λαi=20, while control inputs are bounded as uqi80 kV and udi60 kV, where i=1,2. The switching frequency is 1620 Hz for both rectifier and inverter, which is taken from [22]. The control performance of POSMC is compared to that of VC [11] and FLSMC [22] by the following four cases. In addition, two identical three-level neutral-point-clamped VSCs model for each rectifier and inverter from Matlab/Simulink SimPowerSystems are employed, which structure and parameters are taken directly from [11]. The simulation is executed on Matlab/Simulink 7.10 using a personal computer with an IntelR CoreTMi7 CPU at 2.2 GHz and 8 GB of RAM.

AC system-based voltageVACbase132 kV
DC cable base voltageVDCbase150 kV
System base powerSbase100 MVA
AC system resistance (25 km)R1,R20.05 Ω/km
AC system inductance (25 km)L1,L20.026 mH/km
DC cable resistance (50 km)R00.21 Ω/km
DC bus capacitanceC1,C211.94 μ F

Table 1.

The VSC-HVDC system parameters.

Rectifier controller gains
br10=100br20=7000ρ1=800ρ2=1
ζr=20ζr=10φr=20φr=20
Rectifier observer gains
αr1=300αr1=40αr2=3×104αr2=400
αr3=106Δ=0.01ϵ=0.1kr1=100
kr1=75kr2=105kr2=3.75×104kr3=2.5×107
Inverter controller gains
bi10=50bi20=50ζi=10ζi=10
φi=10φi=10
Inverter observer gains
αi1=40αi1=40αi2=400αi2=400
ki1=75ki1=75ki2=3.75×104ki2=3.75×104

Table 2.

POSMC parameters for the VSC-HVDC system.

(1) Case 1: Active and reactive power tracking: The references of active and reactive power are set to be a series of step change occurs at t=0.2 s, t=0.4 s and restores to the original value at t=0.6 s, while DC voltage is regulated at the rated value Vdc1=150 kV. The system responses are illustrated by Figure 3. One can find that POSMC has the fastest tracking rate and maintains a consistent control performance under different operation conditions.

Figure 3.

System responses obtained under the active and reactive power tracking.

(2) Case 2: 5-cycle line-line-line-ground (LLLG) fault at AC bus 1: A five-cycle LLLG fault occurs at AC bus 1 when t=0.1 s. Due to the fault, AC voltage at the corresponding bus is decreased to a critical level. Figure 4 shows that POSMC can effectively restore the system with the smallest active power oscillations. Response of perturbation estimation is demonstrated in Figure 5, which shows that SMSPO and SMPO can estimate the perturbations with a fast tracking rate.

Figure 4.

System responses obtained under the five-cycle LLLG fault at AC bus 1.

Figure 5.

Estimation errors of the perturbations obtained under the five-cycle LLLG fault at AC bus 1.

(3) Case 3: Weak AC grid connection: The AC grids are assumed to be sufficiently strong such that AC bus voltages are ideal constants. It is worth considering a weak AC grid connected to the rectifier, e.g., offshore wind farms, which voltage us1 is no longer a constant but a time-varying function. A voltage fluctuation that occurs from 0.15 to 1.05 s caused by the wind speed variation is applied, which corresponds to us1=1+0.15sin0.2πt. System responses are presented in Figure 6, it illustrates that both DC voltage and reactive power are oscillatory, while POSMC can effectively suppress such oscillation with the smallest fluctuation of DC voltage and reactive power.

Figure 6.

System responses obtained with the weak AC grid connection.

(4) Case 4: System parameter uncertainties: When there is a fault in the transmission or distribution grid, the resistance and inductance values of the grid may change significantly. Several tests are performed for plant-model mismatches of R2 and L2 with ±20% uncertainties. All tests are undertaken under the nominal grid voltage and a corresponding −120 A in the DC cable current iL at 0.1 s. The peak active power P2 is recorded, which uses per unit (p.u.) value for a clear illustration of system robustness. It can be found from Figure 7 that the peak active power P2 controlled by POSMC is almost not affected, while FLSMC has a relatively large range of variation, i.e., around 3% to R2 and 8% to L2, respectively. Responses to mismatch of R2 and L2 changing at the same time are demonstrated in Figure 8. The magnitude of changes is around 10% under FLSMC and almost does not change under POSMC. This is because POSMC estimates all uncertainties and does not need an accurate system model, thus it has better robustness than that of FLSMC which requires accurate system parameters.

Figure 7.

The peak active power P2 (in p.u.) to a −120 A in the DC cable current iL obtained at nominal grid voltage for plant-model mismatches in the range of 20% (one parameter changes and others keep constant).

Figure 8.

The peak active power P2 (in p.u.) to a −120 A in the DC cable current iL obtained at nominal grid voltage for plant-model mismatches in the range of 20% (different parameters may change at the same time).

The integral of absolute error (IAE) indices of each approach calculated in different cases are tabulated in Table 3. Here, IAEQ1=0TQ1Q1dt, IAEVdc1=0TVdc1Vdc1dt, IAEQ2=0TQ2Q2dt, and IAEP2=0TP2P2dt. The simulation time T = 3 s. Note that POSMC has a little bit higher IAE than that of FLSMC in the power tracking due to the estimation error, while it can provide much better robustness in the case of 5-cycle LLLG fault and weak AC grid connection. In particular, its IAEQ1 and IAEVdc1 are only 8.57 and 9.51% of those of VC, 16.42 and 20.36% of those of FLSMC with the weak AC grid connection. The overall control costs are illustrated in Figure 9, with IAEu=0Tud1+uq1+ud2+uq2dt. It is obvious that POSMC has the lowest control costs in all cases, which is resulted from the merits that the upper bound of perturbation is replaced by the smaller bound of its estimation error, thus an over-conservative control input can be avoided.

MethodPower tracking
Case
IAEQ1IAEVdc1IAEQ2IAEP2
VC3.83E024.44E032.13E022.71E02
FLSMC2.19E021.73E032.23E022.18E02
POSMC2.33E022.00E032.42E022.33E02
MethodFive-cycle LLLG fault Weak AC grid connection
Case
IAEQ1IAEVdc1IAEQ1IAEVdc1
VC2.62E022.15E034.53E034.13E03
FLSMC1.13E024.13E034.08E033.33E03
POSMC5.64E031.38E033.88E046.78E04

Table 3.

IAE indices (in p.u.) of different control schemes calculated in different cases.

Figure 9.

Overall control costs IAEu (in p.u.) obtained in different cases.

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5. Hardware-in-the-loop test results

HIL test is an important and powerful technique used in the development and test of complex real-time embedded systems, which provides an effective platform by adding the complexity of the plant under control to the test platform. The complexity of the plant under control is included in test and development by adding a mathematical representation of all related dynamic systems.

A dSPACE simulator-based HIL test is used to validate the implementation feasibility of POSMC, which configuration and experiment platform are given by Figures 10 and 11, respectively. The rectifier controller (34) and inverter controller (41) are implemented on one dSPACE platform (DS1104 board) with a sampling frequency fc=1 kHz, and the VSC-HVDC system is simulated on another dSPACE platform (DS1006 board) with the limit sampling frequency fs=50 kHz to make HIL simulator as close to the real plant as possible. The measurements of the reactive power Q1, DC voltage Vdc1, active power P2, and reactive power Q2 are obtained from the real-time simulation of the VSC-HVDC system on the DS1006 board, which are sent to two controllers implemented on the DS1104 board for the control inputs calculation.

Figure 10.

The configuration of the HIL test.

Figure 11.

The experiment platform of the HIL test.

It follows from [37] that an unexpected high-frequency oscillation in control inputs may emerge as the large observer poles would result in high gains, which lead to highly sensitive observer dynamics to the measurement disturbances in the HIL test. Note that this phenomenon does not exist in the simulation. One effective way to alleviate such malignant effect is to reduce the observer poles. Through trial-and-error, an observer pole in the range of λαr1525 and λαr=λαi=λαi310 can avoid such oscillation but with almost similar transient responses, thus the reduced poles λαr=20 and λαr=λαi=λαi=5, with br10=50, br20=5000, bi10=20, and bi20=20, are chosen in the HIL test. Furthermore, a time delay τ=3 ms has been assumed in the corresponding simulation to consider the effect of the computational delay of the real-time controller.

(1) Case 1: Active and reactive power tracking: The reference of active and reactive power changes at t=0.4 s, t=0.9 s and restores to the original value at t=1.4 s, while DC voltage is regulated at the rated value Vdc1=150 kV. The system responses obtained under the HIL test and simulation are compared by Figure 12, which shows that the HIL test has almost the same results as that of the simulation.

Figure 12.

HIL test results of system responses obtained under the active and reactive power tracking.

(2) Case 2: 5-cycle line-line-line-ground (LLLG) fault at AC bus 1. A 5-cycle LLLG fault occurs at AC bus 1 when t=0.1 s. Figure 13 demonstrates that the system can be rapidly restored and the system responses obtained by the HIL test is similar to that of simulation.

Figure 13.

HIL test results of system responses obtained under the five-cycle LLLG fault at AC bus 1.

(3) Case 3: Weak AC grid connection: The same voltage variation us1=1+0.15sin0.2πt is applied between 0.87 and 2.45 s. It can be readily seen from Figure 14 that the results of the HIL test and simulation match very well.

Figure 14.

HIL test results of system responses obtained with the weak AC grid connection.

The difference of the obtained results between the HIL test and simulation is possibly due to the following two reasons:

  • There exist measurement disturbances in the HIL test, which are, however, not taken into account in the simulation, a filter could be used to remove the measurement disturbances, thus the control performance can be improved.

  • The sampling frequency of VSC-HVDC model and POSMC is the same in simulation (fs=fc=1 kHz) as they are implemented in Matlab of the same computer. In contrast, the sampling frequency of VSC-HVDC model (fs=50 kHz) is significantly increased in the HIL test to make VSC-HVDC model as close to the real plant as possible. Note the sampling frequency of POSMC remains the same (fc=1 kHz) due to the sampling limit of the practical controller.

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6. Conclusion

A POSMC scheme has been developed for the VSC-HVDC system to rapidly compensate the combinatorial effect of nonlinearities, parameter uncertainties, unmodeled dynamics, and time-varying external disturbances. As the upper bound of perturbation is replaced by the smaller bound of its estimation error, an over-conservative control input is avoided such that the tracking accuracy can be improved.

Four case studies have been undertaken to evaluate the control performance of the proposed approach, which verify that POSMC can maintain a consistent control performance with less power overshoot during the power reversal, restore the system rapidly after the AC fault, suppress the oscillation effectively when connected to a weak AC grid, and provide significant robustness in the presence of system parameter uncertainties. At last, a dSPACE-based HIL test has been carried out which validates the implementation feasibility of POSMC.

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Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant Nos. 51477055, 51667010, and 51777078.

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Written By

Bo Yang, Tao Yu, Hongchun Shu and Pulin Cao

Submitted: November 15th, 2017 Reviewed: January 31st, 2018 Published: October 17th, 2018