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<sup>∘</sup> between themselves. The rectifier dynamics can be written at the angular frequency ω as [14].

$$\begin{cases} \frac{\mathbf{d}i\_{\rm d1}}{\mathbf{d}t} = -\frac{R\_1}{L\_1}i\_{\rm d1} + \omega i\_{\rm q1} + \mu\_{\rm d1} \\ \frac{\mathbf{d}i\_{\rm q1}}{\mathbf{d}t} = -\frac{R\_1}{L\_1}i\_{\rm q1} - \omega i\_{\rm d1} + \mu\_{\rm q1} \\ \frac{\mathbf{d}V\_{\rm dc1}}{\mathbf{d}t} = \frac{3\mu\_{\rm sq1}i\_{\rm q1}}{2C\_1V\_{\rm dc1}} - \frac{i\_{\rm L}}{C\_1} \end{cases} \tag{1}$$

Vdc1i<sup>L</sup> ¼ Vdc2i<sup>L</sup> þ 2R0i

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 Q<sup>2</sup> are the active and reactive powers transmitted from the AC grid to the VSC; Vdc1 and Vdc2 are the DC voltages; and i<sup>L</sup> is the DC

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

<sup>2</sup> <sup>u</sup>sq1iq1 <sup>þ</sup> <sup>u</sup>sd1id1 � � <sup>¼</sup> <sup>3</sup>

<sup>2</sup> <sup>u</sup>sq1id1 � <sup>u</sup>sd1iq1 � � <sup>¼</sup> <sup>3</sup>

<sup>2</sup> <sup>u</sup>sq2iq2 <sup>þ</sup> <sup>u</sup>sd2id2 � � <sup>¼</sup> <sup>3</sup>

<sup>2</sup> <sup>u</sup>sq2id2 � <sup>u</sup>sd2iq2 � � <sup>¼</sup> <sup>3</sup>

2 usq1iq1

2 usq1id1

2 usq2iq2

2 usq2id2

remains as an ideal constant. The power flows from the AC grid can be given as

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

<sup>P</sup><sup>1</sup> <sup>¼</sup> <sup>3</sup>

8

>>>>>>>>>><

>>>>>>>>>>:

3. POSMC design for the VSC-HVDC system

3.1. Perturbation observer-based sliding-mode control

<sup>Q</sup><sup>1</sup> <sup>¼</sup> <sup>3</sup>

<sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>3</sup>

<sup>Q</sup><sup>2</sup> <sup>¼</sup> <sup>3</sup>

where R<sup>0</sup> represents the equivalent DC cable resistance.

Figure 1. A standard two-terminal VSC-HVDC system.

cable current.

2

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

<sup>L</sup> (3)

http://dx.doi.org/10.5772/intechopen.74717

23

(4)

where the rectifier is connected with the AC grid via the equivalent resistance and inductance <sup>R</sup><sup>1</sup> and <sup>L</sup>1, respectively. <sup>C</sup><sup>1</sup> is the DC bus capacitor, <sup>u</sup>d1 <sup>¼</sup> <sup>u</sup>sd1�urd <sup>L</sup><sup>1</sup> and <sup>u</sup>q1 <sup>¼</sup> <sup>u</sup>sq1�urq <sup>L</sup><sup>1</sup> .

The inverter dynamics is written as

$$\begin{cases} \frac{\text{d}\dot{\text{d}}\_{\text{d2}}}{\text{d}t} = -\frac{R\_2}{L\_2}\dot{\text{i}}\_{\text{d2}} + \omega \dot{\text{i}}\_{\text{q2}} + \mu\_{\text{d2}}\\ \frac{\text{d}\dot{\text{i}}\_{\text{q2}}}{\text{d}t} = -\frac{R\_2}{L\_2}\dot{\text{i}}\_{\text{q2}} - \omega \dot{\text{i}}\_{\text{d2}} + \mu\_{\text{q2}}\\ \frac{\text{d}V\_{\text{dc2}}}{\text{d}t} = \frac{3\mu\_{\text{sq2}}\dot{\text{i}}\_{\text{q2}}}{2C\_2V\_{\text{dc2}}} + \frac{\dot{\text{i}}\_{\text{L}}}{C\_2} \end{cases} \tag{2}$$

where the inverter is connected with the AC grid via the equivalent resistance and inductance <sup>R</sup><sup>2</sup> and <sup>L</sup>2, respectively. <sup>C</sup><sup>2</sup> is the DC bus capacitor, <sup>u</sup>d2 <sup>¼</sup> <sup>u</sup>sd2�uid <sup>L</sup><sup>2</sup> and <sup>u</sup>q2 <sup>¼</sup> <sup>u</sup>sq2�uiq <sup>L</sup><sup>2</sup> .

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

Sliding-Mode Perturbation Observer-Based Sliding-Mode Control for VSC-HVDC Systems http://dx.doi.org/10.5772/intechopen.74717 23

Figure 1. A standard two-terminal VSC-HVDC system.

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

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.

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<sup>∘</sup> between

id1 þ ωiq1 þ ud1

iq1 � ωid1 þ uq1

(1)

(2)

� iL C1

id2 þ ωiq2 þ ud2

iq2 � ωid2 þ uq2

þ iL C2 <sup>L</sup><sup>1</sup> and <sup>u</sup>q1 <sup>¼</sup> <sup>u</sup>sq1�urq

<sup>L</sup><sup>2</sup> and <sup>u</sup>q2 <sup>¼</sup> <sup>u</sup>sq2�uiq

<sup>L</sup><sup>1</sup> .

<sup>L</sup><sup>2</sup> .

themselves. The rectifier dynamics can be written at the angular frequency ω as [14].

<sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>3</sup>usq1iq1 2C1Vdc1

where the rectifier is connected with the AC grid via the equivalent resistance and inductance

did1 <sup>d</sup><sup>t</sup> ¼ � <sup>R</sup><sup>1</sup> L1

8 >>>>>>><

>>>>>>>:

<sup>R</sup><sup>1</sup> and <sup>L</sup>1, respectively. <sup>C</sup><sup>1</sup> is the DC bus capacitor, <sup>u</sup>d1 <sup>¼</sup> <sup>u</sup>sd1�urd

diq1 <sup>d</sup><sup>t</sup> ¼ � <sup>R</sup><sup>1</sup> L1

dVdc1

did2 <sup>d</sup><sup>t</sup> ¼ � <sup>R</sup><sup>2</sup> L2

8 >>>>>>><

>>>>>>>:

<sup>R</sup><sup>2</sup> and <sup>L</sup>2, respectively. <sup>C</sup><sup>2</sup> is the DC bus capacitor, <sup>u</sup>d2 <sup>¼</sup> <sup>u</sup>sd2�uid

diq2 <sup>d</sup><sup>t</sup> ¼ � <sup>R</sup><sup>2</sup> L2

dVdc2

<sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>3</sup>usq2iq2 2C2Vdc2

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

where the inverter is connected with the AC grid via the equivalent resistance and inductance

validate its implementation feasibility.

22 Perturbation Methods with Applications in Science and Engineering

Finally, conclusions are drawn in Section 6.

2. VSC-HVDC system modeling

The inverter dynamics is written as

$$V\_{\rm dc1}i\_{\rm L} = V\_{\rm dc2}i\_{\rm L} + 2R\_0i\_{\rm L}^2 \tag{3}$$

where R<sup>0</sup> 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 Q<sup>2</sup> are the active and reactive powers transmitted from the AC grid to the VSC; Vdc1 and Vdc2 are the DC voltages; and i<sup>L</sup> 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

$$\begin{cases} P\_1 = \frac{3}{2} \left( \mu\_{\text{sq1}} i\_{\text{q1}} + \mu\_{\text{sd1}} i\_{\text{d1}} \right) = \frac{3}{2} \mu\_{\text{sq1}} i\_{\text{q1}} \\ Q\_1 = \frac{3}{2} \left( \mu\_{\text{sq1}} i\_{\text{d1}} - \mu\_{\text{sd1}} i\_{\text{q1}} \right) = \frac{3}{2} \mu\_{\text{sq1}} i\_{\text{d1}} \\ P\_2 = \frac{3}{2} \left( \mu\_{\text{sq2}} i\_{\text{q2}} + \mu\_{\text{sd2}} i\_{\text{d2}} \right) = \frac{3}{2} \mu\_{\text{sq2}} i\_{\text{q2}} \\ Q\_2 = \frac{3}{2} \left( \mu\_{\text{sq2}} i\_{\text{d2}} - \mu\_{\text{sd2}} i\_{\text{q2}} \right) = \frac{3}{2} \mu\_{\text{sq2}} i\_{\text{d2}} \end{cases} \tag{4}$$

## 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:

$$\begin{cases} \dot{\mathbf{x}} = A\mathbf{x} + B(a(\mathbf{x}) + b(\mathbf{x})\boldsymbol{\mu} + d(t)) \\ \mathbf{y} = \mathbf{x}\_1 \end{cases} \tag{5}$$

stability with perturbation estimation, while assumption A.3 ensures POSMC can drive the

the worst case, e.g., y ¼ x<sup>1</sup> 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

<sup>b</sup>xn <sup>¼</sup> <sup>Ψ</sup><sup>b</sup> ðÞþ� <sup>α</sup>nx~<sup>1</sup> <sup>þ</sup> knsatð Þþ <sup>x</sup>~<sup>1</sup> <sup>b</sup>0<sup>u</sup>

where <sup>x</sup>~<sup>1</sup> <sup>¼</sup> <sup>x</sup><sup>1</sup> � <sup>b</sup>x1, ki and <sup>α</sup>i, <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, n <sup>þ</sup> 1, are positive coefficients, function satð Þ <sup>x</sup>~<sup>1</sup> is defined as satð Þ¼ x~<sup>1</sup> x~1=∣x~1∣ when ∣x~1∣ > e and satð Þ¼ x~<sup>1</sup> x~1=e when ∣x~1∣ ≤ e. The effect and

• 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, α<sup>i</sup> are chosen such that the root of <sup>s</sup><sup>n</sup>þ<sup>1</sup> <sup>þ</sup> <sup>α</sup>1s<sup>n</sup> <sup>þ</sup> <sup>α</sup>2s<sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> <sup>α</sup><sup>n</sup>þ<sup>1</sup> <sup>¼</sup> ð Þ <sup>s</sup> <sup>þ</sup> λα <sup>n</sup>þ<sup>1</sup> <sup>¼</sup> 0 is in the open lefthalf complex plane. A larger value of α<sup>i</sup> 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

• The sliding surface constants ki. k<sup>1</sup> ≥ j j x~<sup>2</sup> max must be chosen to guarantee the estimation error of SMSPO (9) will enter into the sliding surface Sspoð Þ¼ ~x x~<sup>1</sup> ¼ 0 at t > t<sup>s</sup> and thereafter remain Sspo ¼ 0, t ≥ t<sup>s</sup> [35, 39]. While the poles of the sliding surface λ<sup>k</sup> are determined by choosing the ratio ki=k1ð Þ i ¼ 2; 3; ⋯; n þ 1 to put the root of pn <sup>þ</sup> ð Þ <sup>k</sup>2=k<sup>1</sup> pn�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> ð Þ kn=k<sup>1</sup> <sup>p</sup> <sup>þ</sup> ð Þ¼ knþ<sup>1</sup>=k<sup>1</sup> ð Þ <sup>p</sup> <sup>þ</sup> <sup>λ</sup><sup>k</sup> <sup>n</sup> <sup>¼</sup> 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

• The layer thickness constant of saturation function e, 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 e 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

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

<sup>b</sup>x<sup>1</sup> <sup>¼</sup> <sup>b</sup>x<sup>2</sup> <sup>þ</sup> <sup>α</sup>1x~<sup>1</sup> <sup>þ</sup> <sup>k</sup>1satð Þ <sup>x</sup>~<sup>1</sup>

Ψb ðÞ¼ � α<sup>n</sup>þ<sup>1</sup>x~<sup>1</sup> þ knþ1satð Þ x~<sup>1</sup>

ð Þ1 <sup>d</sup> ; ⋯; y

ð Þ n�1 d h i<sup>T</sup>

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

[39]. In the consideration of

http://dx.doi.org/10.5772/intechopen.74717

(10)

25

system state x to track a desired state x<sup>d</sup> ¼ yd; y

\_

8 >>>><

>>>>:

setting of the SMSPO parameters are provided as follows:

\_

closed-loop system to ensure a fast online estimation [37].

stability must be made through trial-and-error [39].

recommended.

⋮ \_

as follows:

where <sup>x</sup> <sup>¼</sup> ½ � <sup>x</sup>1; <sup>x</sup>2; <sup>⋯</sup>; xn <sup>T</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> is the state variable vector, <sup>u</sup><sup>∈</sup> <sup>R</sup> and <sup>y</sup> <sup>∈</sup> <sup>R</sup> are the control input and system output, respectively. a xð Þ: <sup>R</sup><sup>n</sup> <sup>↦</sup> <sup>R</sup> and b xð Þ: <sup>R</sup><sup>n</sup> <sup>↦</sup> <sup>R</sup> are unknown smooth functions, and d tð Þ: R<sup>þ</sup> ↦ 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:

$$A = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ 0 & 0 & 0 & \cdots & 0 \end{bmatrix}\_{n \times n}, \qquad B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}\_{n \times 1} \tag{6}$$

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

$$
\Psi(\mathbf{x}, \boldsymbol{\mu}, t) = \boldsymbol{a}(\mathbf{x}) + (b(\mathbf{x}) - b\_0)\boldsymbol{\mu} + d(t) \tag{7}
$$

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

$$
\dot{\mathbf{x}}\_n = a(\mathbf{x}) + (b(\mathbf{x}) - b\_0)\boldsymbol{\mu} + d(t) + b\_0 \boldsymbol{\mu} = \Psi(\mathbf{x}, \boldsymbol{\mu}, t) + b\_0 \boldsymbol{\mu} \tag{8}
$$

Define a fictitious state xnþ<sup>1</sup> ¼ Ψð Þ x; u; t . Then, system (5) can be extended as

$$\begin{cases} \dot{y} = \mathbf{x}\_1 \\ \dot{\mathbf{x}}\_1 = \mathbf{x}\_2 \\ \vdots \\ \dot{\mathbf{x}}\_n = \mathbf{x}\_{n+1} + b\_0 u \\ \dot{\mathbf{x}}\_{n+1} = \dot{\Psi}(\cdot) \end{cases} \tag{9}$$

The new state vector becomes x<sup>e</sup> ¼ ½ � x1; x2; ⋯; xn; xnþ<sup>1</sup> T , and following assumptions are made [35]:


The above three assumptions ensure the effectiveness of such perturbation estimationbased 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 x<sup>d</sup> ¼ yd; y ð Þ1 <sup>d</sup> ; ⋯; y ð Þ n�1 d h i<sup>T</sup> [39]. In the consideration of the worst case, e.g., y ¼ x<sup>1</sup> 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:

x\_ ¼ Ax þ Bax ð Þ ð Þþ b xð Þu þ d tð Þ

where <sup>x</sup> <sup>¼</sup> ½ � <sup>x</sup>1; <sup>x</sup>2; <sup>⋯</sup>; xn <sup>T</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> is the state variable vector, <sup>u</sup><sup>∈</sup> <sup>R</sup> and <sup>y</sup> <sup>∈</sup> <sup>R</sup> are the control input and system output, respectively. a xð Þ: <sup>R</sup><sup>n</sup> <sup>↦</sup> <sup>R</sup> and b xð Þ: <sup>R</sup><sup>n</sup> <sup>↦</sup> <sup>R</sup> are unknown smooth functions, and d tð Þ: R<sup>þ</sup> ↦ R represents the time-varying external disturbance. The n � n matrix A and the

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

, B ¼

x\_<sup>n</sup> ¼ a xð Þþ ð Þ b xð Þ� b<sup>0</sup> u þ d tð Þþ b0u ¼ Ψð Þþ x; u; t b0u (8)

Ψð Þ¼ x; u; t a xð Þþ ð Þ b xð Þ� b<sup>0</sup> u þ d tð Þ (7)

(5)

(6)

(9)

y ¼ x<sup>1</sup>

010 ⋯ 0 001 ⋯ 0 ⋮ ⋮ 000 ⋯ 1 000 ⋯ 0

Define a fictitious state xnþ<sup>1</sup> ¼ Ψð Þ x; u; t . Then, system (5) can be extended as

8 >>>>>><

>>>>>>:

• A.1 b<sup>0</sup> is chosen to satisfy: ∣b xð Þ=b<sup>0</sup> � 1∣ ≤ θ < 1, where θ is a positive constant.

and <sup>Ψ</sup>\_ ð Þ¼ <sup>0</sup>; <sup>0</sup>; <sup>0</sup> 0, where <sup>γ</sup><sup>1</sup> and <sup>γ</sup><sup>2</sup> are positive constants.

y ¼ x<sup>1</sup> x\_ <sup>1</sup> ¼ x<sup>2</sup> ⋮

x\_<sup>n</sup> ¼ xnþ<sup>1</sup> þ b0u <sup>x</sup>\_<sup>n</sup>þ<sup>1</sup> <sup>¼</sup> <sup>Ψ</sup>\_ ð Þ�

• A.2 The functions <sup>Ψ</sup>ð Þ <sup>x</sup>; <sup>u</sup>; <sup>t</sup> : <sup>R</sup><sup>n</sup> � <sup>R</sup> � <sup>R</sup><sup>þ</sup> <sup>↦</sup> <sup>R</sup> and <sup>Ψ</sup>\_ ð Þ <sup>x</sup>; <sup>u</sup>; <sup>t</sup> : <sup>R</sup><sup>n</sup> � <sup>R</sup> � <sup>R</sup><sup>þ</sup> <sup>↦</sup> <sup>R</sup> are bounded over the domain of interest: <sup>∣</sup>Ψð Þ <sup>x</sup>; <sup>u</sup>; <sup>t</sup> <sup>∣</sup> <sup>≤</sup> <sup>γ</sup>1, <sup>∣</sup>Ψ\_ ð Þ <sup>x</sup>; <sup>u</sup>; <sup>t</sup> <sup>∣</sup> <sup>≤</sup> <sup>γ</sup><sup>2</sup> with <sup>Ψ</sup>ð Þ¼ <sup>0</sup>; <sup>0</sup>; <sup>0</sup> <sup>0</sup>

• A.3 The desired trajectory y<sup>d</sup> and its up to nth-order derivative are continuous and

The above three assumptions ensure the effectiveness of such perturbation estimationbased approach. In particular, assumptions A.1 and A.2 guarantee the closed-loop system

T

, and following assumptions are made

�

n � 1 matrix B are of the canonical form as follows:

24 Perturbation Methods with Applications in Science and Engineering

A ¼

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

The new state vector becomes x<sup>e</sup> ¼ ½ � x1; x2; ⋯; xn; xnþ<sup>1</sup>

(6) as follows:

[35]:

bounded.

$$\begin{cases} \dot{\hat{\boldsymbol{x}}}\_{1} = \hat{\mathbf{x}}\_{2} + \alpha\_{1}\tilde{\mathbf{x}}\_{1} + k\_{1}\text{sat}(\tilde{\mathbf{x}}\_{1}) \\ \vdots \\ \dot{\hat{\boldsymbol{x}}}\_{n} = \hat{\Psi}(\cdot) + \alpha\_{n}\tilde{\mathbf{x}}\_{1} + k\_{n}\text{sat}(\tilde{\mathbf{x}}\_{1}) + b\_{0}\boldsymbol{u} \\ \dot{\hat{\Psi}}(\cdot) = a\_{n+1}\tilde{\mathbf{x}}\_{1} + k\_{n+1}\text{sat}(\tilde{\mathbf{x}}\_{1}) \end{cases} \tag{10}$$

where <sup>x</sup>~<sup>1</sup> <sup>¼</sup> <sup>x</sup><sup>1</sup> � <sup>b</sup>x1, ki and <sup>α</sup>i, <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, n <sup>þ</sup> 1, are positive coefficients, function satð Þ <sup>x</sup>~<sup>1</sup> is defined as satð Þ¼ x~<sup>1</sup> x~1=∣x~1∣ when ∣x~1∣ > e and satð Þ¼ x~<sup>1</sup> x~1=e when ∣x~1∣ ≤ e. The effect and setting of the SMSPO parameters are provided as follows:


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

$$\begin{cases} \dot{\tilde{\mathbf{x}}}\_{2} = -\frac{k\_{2}}{k\_{1}}\tilde{\mathbf{x}}\_{2} + \tilde{\mathbf{x}}\_{3} \\ \dot{\tilde{\mathbf{x}}}\_{3} = -\frac{k\_{3}}{k\_{1}}\tilde{\mathbf{x}}\_{2} + \tilde{\mathbf{x}}\_{4} \\ \vdots \\ \dot{\tilde{\mathbf{x}}}\_{n} = -\frac{k\_{n}}{k\_{1}}\tilde{\mathbf{x}}\_{2} + \tilde{\Psi}(\cdot) \\ \dot{\tilde{\Psi}}(\cdot) = -\frac{k\_{n+1}}{k\_{1}}\tilde{\mathbf{x}}\_{2} + \dot{\Psi}(\cdot) \end{cases} \tag{11}$$

Construct a Lyapunov function as follows:

The POSMC for system (5) is designed as

<sup>u</sup> <sup>¼</sup> <sup>1</sup> b0 y ð Þ n <sup>d</sup> �X<sup>n</sup>�<sup>1</sup> i¼1

\_

bS ¼ Ψb ðÞþ� b0u þ

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

which will be fulfilled with the relationship of k<sup>1</sup> if

which, using gains ki, yields

\_ <sup>b</sup><sup>S</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1 ri ki k1

Consequently, the attractiveness of sliding surface can be derived as

ζ∣

ζ∣

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

φ > k<sup>1</sup>

<sup>b</sup>S<sup>∣</sup> <sup>þ</sup> <sup>φ</sup> <sup>&</sup>gt; <sup>X</sup><sup>n</sup>

bS∣ þ φ > k<sup>1</sup>

φ > k<sup>1</sup>

Xn i¼1

riCi�<sup>1</sup> <sup>n</sup> λ<sup>i</sup>�<sup>1</sup>

i¼1 ri ki k1

Xn i¼1 ri ki k1

Xn i¼1 ri ki k1

to be designed to enforce <sup>b</sup><sup>S</sup> \_

sliding surface bS.

dynamics (10), it yields

<sup>V</sup> <sup>¼</sup> <sup>1</sup> 2 bS 2

The attractiveness of sliding surface is achieved if <sup>V</sup>\_ <sup>&</sup>lt; 0 for all <sup>~</sup>x⊈bS, that is, the control <sup>u</sup> needs

ð Þi d � � � <sup>ζ</sup>b<sup>S</sup> � <sup>φ</sup>sat <sup>b</sup><sup>S</sup>

where ζ and φ are control gains which are chosen to fulfill the attractiveness of the estimated

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

<sup>r</sup><sup>i</sup> <sup>b</sup>xiþ<sup>1</sup> � <sup>y</sup>

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

x~<sup>2</sup> � ζbS � φsat bS; ε<sup>c</sup>

ð Þi <sup>d</sup> þ ki k1 x~2 � � (19)

� � (20)

∣x~2∣ (21)

<sup>k</sup> (24)

" #

bS∣ < εc.

http://dx.doi.org/10.5772/intechopen.74717

� � � <sup>Ψ</sup><sup>b</sup> ð Þ�

bS < 0 outside a prescribed manifold ∣

<sup>r</sup><sup>i</sup> <sup>b</sup>xiþ<sup>1</sup> � <sup>y</sup>

kn k1 x~<sup>2</sup> � y ð Þ n <sup>d</sup> <sup>þ</sup>X<sup>n</sup>�<sup>1</sup> i¼1

(17)

27

(18)

(22)

(23)

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~eð Þ0 , the estimation error x~<sup>e</sup> converges exponentially into the neighborhood

$$\|\tilde{\chi}\_{\mathbf{e}}\|\tag{12}$$

In particular,

$$|\tilde{\boldsymbol{x}}\_{i}| \le \frac{\delta}{\lambda\_{\mathbf{k}}^{n+1-i}}, \quad \mathbf{i} = \mathbf{2}, \dots, n+1, \quad \forall t > t\_1. \tag{13}$$

where t<sup>1</sup> 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 γ<sup>2</sup> 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

$$\widehat{S}(\mathbf{x},t) = \sum\_{i=1}^{n} \rho\_i \left(\widehat{\mathbf{x}}\_i - y\_{\mathbf{d}}^{(i-1)}\right) \tag{14}$$

where the estimated sliding surface gains <sup>r</sup><sup>i</sup> <sup>¼</sup> Ci�<sup>1</sup> <sup>n</sup>�<sup>1</sup>λ<sup>n</sup>�<sup>i</sup> <sup>c</sup> , i ¼ 1, ⋯, n, place all poles of the estimated sliding surface at �λc, where λ<sup>c</sup> > 0.

Besides, the actual sliding surface is written by

$$S = \sum\_{i=1}^{n} \rho\_i \left(\mathbf{x}\_i - \mathbf{y}\_d^{(i-1)}\right) \tag{15}$$

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

$$
\hat{S} = S - \widehat{S} = \sum\_{i=1}^{n} \rho\_i \check{\mathbf{x}}\_i \tag{16}
$$

Sliding-Mode Perturbation Observer-Based Sliding-Mode Control for VSC-HVDC Systems http://dx.doi.org/10.5772/intechopen.74717 27

Construct a Lyapunov function as follows:

~\_ <sup>x</sup><sup>2</sup> ¼ � <sup>k</sup><sup>2</sup> k1

8

26 Perturbation Methods with Applications in Science and Engineering

>>>>>>>>>>>>><

~\_ <sup>x</sup><sup>3</sup> ¼ � <sup>k</sup><sup>3</sup> k1

~\_ xn ¼ � kn k1

>>>>>>>>>>>>>:

∣x~i∣ ≤

the observer loop, not in the controller loop.

where the estimated sliding surface gains <sup>r</sup><sup>i</sup> <sup>¼</sup> Ci�<sup>1</sup>

estimated sliding surface at �λc, where λ<sup>c</sup> > 0. Besides, the actual sliding surface is written by

Define an estimated sliding surface as

δ λ<sup>n</sup>þ1�<sup>i</sup> k

<sup>b</sup>S xð Þ¼ ; <sup>t</sup> <sup>X</sup><sup>n</sup>

<sup>S</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1

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

i¼1

<sup>r</sup><sup>i</sup> <sup>b</sup>xi � <sup>y</sup>

r<sup>i</sup> xi � y

i¼1

<sup>S</sup><sup>~</sup> <sup>¼</sup> <sup>S</sup> � <sup>b</sup><sup>S</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>

ð Þ i�1 d � �

<sup>n</sup>�<sup>1</sup>λ<sup>n</sup>�<sup>i</sup>

ð Þ i�1 d � �

where t<sup>1</sup> is the time constant which definition can be found in [39].

borhood

In particular,

~\_

⋮

<sup>Ψ</sup>ðÞ¼� � knþ<sup>1</sup>

x~<sup>2</sup> þ x~<sup>3</sup>

x~<sup>2</sup> þ x~<sup>4</sup>

<sup>x</sup>~<sup>2</sup> <sup>þ</sup> <sup>Ψ</sup><sup>~</sup> ð Þ�

<sup>x</sup>~<sup>2</sup> <sup>þ</sup> <sup>Ψ</sup>\_ ð Þ�

∥x~e∥ ≤ δ (12)

, i ¼ 2, ⋯, n þ 1, ∀t > t1: (13)

<sup>c</sup> , i ¼ 1, ⋯, n, place all poles of the

rix~<sup>i</sup> (16)

(11)

(14)

(15)

k1

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~eð Þ0 , the estimation error x~<sup>e</sup> converges exponentially into the neigh-

Remark 1. When SMSPO is used to estimate the perturbation, the upper bound of the derivative of perturbation γ<sup>2</sup> 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

$$V = \frac{1}{2}\hat{\mathbb{S}}^2\tag{17}$$

The attractiveness of sliding surface is achieved if <sup>V</sup>\_ <sup>&</sup>lt; 0 for all <sup>~</sup>x⊈bS, that is, the control <sup>u</sup> needs to be designed to enforce <sup>b</sup><sup>S</sup> \_ bS < 0 outside a prescribed manifold ∣ bS∣ < εc.

The POSMC for system (5) is designed as

$$\mu = \frac{1}{b\_0} \left[ y\_{\rm d}^{(n)} - \sum\_{i=1}^{n-1} \rho\_i \left( \widehat{\mathbf{x}}\_{i+1} - y\_{\rm d}^{(i)} \right) - \zeta \widehat{\mathbf{S}} - \eta \text{sat} \left( \widehat{\mathbf{S}} \right) - \widehat{\Psi} (\cdot) \right] \tag{18}$$

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

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

$$\dot{\hat{S}} = \hat{\Psi}(\cdot) + b\_0 u + \frac{k\_n}{k\_1} \check{\mathbf{x}}\_2 - y\_\mathbf{d}^{(n)} + \sum\_{i=1}^{n-1} \rho\_i \left( \hat{\mathbf{x}}\_{i+1} - y\_\mathbf{d}^{(i)} + \frac{k\_i}{k\_1} \check{\mathbf{x}}\_2 \right) \tag{19}$$

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

$$\dot{\hat{S}} = \sum\_{i=1}^{n} \rho\_i \frac{k\_i}{k\_1} \ddot{\mathbf{x}}\_2 - \zeta \hat{\mathbf{S}} - \eta \text{sat}\left(\hat{\mathbf{S}}, \varepsilon\_c\right) \tag{20}$$

Consequently, the attractiveness of sliding surface can be derived as

$$\left|\zeta[\hat{S}] + \varphi > \sum\_{i=1}^{n} \rho\_i \frac{k\_i}{k\_1} |\check{x}\_2| \tag{21}$$

which will be fulfilled with the relationship of k<sup>1</sup> if

$$\mathbb{E}|\widehat{S}| + \varphi > k\_1 \sum\_{i=1}^n \rho\_i \frac{k\_i}{k\_1} \tag{22}$$

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

$$
\varphi > k\_1 \sum\_{i=1}^n \rho\_i \frac{k\_i}{k\_1} \tag{23}
$$

which, using gains ki, yields

$$
\varphi > k\_1 \sum\_{i=1}^n \rho\_i \mathbb{C}\_n^{i-1} \lambda\_\mathbf{k}^{i-1} \tag{24}
$$

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

$$\dot{\tilde{S}} = \sum\_{i=1}^{n-1} \rho\_i \tilde{\mathbf{x}}\_{i+1} - \sum\_{i=1}^{n} \rho\_i \frac{k\_i}{k\_1} \tilde{\mathbf{x}}\_2 + \tilde{\Psi}(\cdot) \tag{25}$$

Step 4. Design the ð Þ n þ 1 th-order SMSPO (9) for the extended ð Þ n þ 1 th-order system (8) to obtain the state estimate <sup>b</sup><sup>x</sup> and the perturbation estimate <sup>Ψ</sup><sup>b</sup> ð Þ� by the only measurement of <sup>x</sup>1; Step 5. Design controller (17) for the original nth-order system (5), in which the estimated

of the reactive power and DC voltage, respectively. Define the tracking error

1 <sup>f</sup> r2 � <sup>V</sup>€ <sup>∗</sup> dc1 " # <sup>þ</sup> <sup>B</sup><sup>r</sup>

T , let Q<sup>∗</sup>

<sup>1</sup> and V<sup>∗</sup>

ud1

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

3usq1iq1 2C1Vdc1

� iL C2

sq1= 4C1L<sup>2</sup>

ud1

<sup>0</sup> <sup>b</sup>r20 � � (33)

� iL C1

, differentiate e<sup>r</sup> for rectifier (1) until the control input

dc1 be the given references

(30)

29

(31)

<sup>u</sup>q1 " # (29)

http://dx.doi.org/10.5772/intechopen.74717

<sup>1</sup>Vdc1 � �, which is nonzero within

<sup>u</sup>q1 " # (32)

sliding surface bS is calculated by (13).

<sup>T</sup> <sup>¼</sup> <sup>Q</sup><sup>1</sup> � <sup>Q</sup><sup>∗</sup>

8 >>>>>>><

>>>>>>>:

Choose the system output <sup>y</sup><sup>r</sup> <sup>¼</sup> <sup>y</sup>r1; <sup>y</sup>r2 � �<sup>T</sup> <sup>¼</sup> <sup>Q</sup><sup>1</sup> ½ � ; <sup>V</sup>dc1

� �<sup>T</sup>

<sup>f</sup> r1 <sup>¼</sup> <sup>3</sup>usq1

<sup>f</sup> r2 <sup>¼</sup> <sup>3</sup>usq1

� <sup>1</sup> 2R0C<sup>1</sup>

The determinant of matrix <sup>B</sup><sup>r</sup> is obtained as <sup>∣</sup>Br<sup>∣</sup> <sup>¼</sup> <sup>9</sup>u<sup>2</sup>

where the constant control gain Br0 is given by

<sup>1</sup>; <sup>V</sup>dc1 � <sup>V</sup><sup>∗</sup>

dc1

€r2 � � <sup>¼</sup> <sup>f</sup> r1 � <sup>Q</sup>\_ <sup>∗</sup>

�ωid1 � <sup>R</sup><sup>1</sup>

3usq1 2L<sup>1</sup>

Assume all the nonlinearities are unknown, define the perturbations Ψr1ð Þ� and Ψr2ð Þ� as

<sup>B</sup>r0 <sup>¼</sup> <sup>b</sup>r10 <sup>0</sup>

<sup>¼</sup> <sup>f</sup> r1 f r2 � �

0

3usq1iq1 2C1Vdc1

B<sup>r</sup> ¼

the operation range of the rectifier, thus system (28) is linearizable.

Ψr1ð Þ� Ψr2ð Þ� � � L1

� �

0

� iL C1 <sup>i</sup>q1 � <sup>i</sup>q1 Vdc1

� <sup>3</sup>usq2iq2 2C2Vdc2

3usq1 2C1L1Vdc1

þ ð Þ B<sup>r</sup> � Br0

� � � �

e\_r1 e

<sup>2</sup> � <sup>R</sup><sup>1</sup> L1 <sup>i</sup>d1 <sup>þ</sup> <sup>ω</sup>iq1 � �

2C1Vdc1

3.2. Rectifier controller design

appears explicitly, yields

e<sup>r</sup> ¼ ½ � er1;er2

where

and

As <sup>b</sup><sup>S</sup> <sup>¼</sup> <sup>S</sup> � <sup>S</sup>~, the actual <sup>S</sup>-dynamics of sliding surface can be obtained with dynamics (19) as

$$\dot{S} + \left(\zeta + \frac{\varrho}{\varepsilon\_{\rm c}}\right) S = \left(\zeta + \frac{\varrho}{\varepsilon\_{\rm c}}\right) \sum\_{i=1}^{n} \rho\_i \ddot{\mathbf{x}}\_i + \sum\_{i=1}^{n-1} \rho\_i \ddot{\mathbf{x}}\_{i+1} + \ddot{\Psi}(\cdot) \tag{26}$$

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

$$\begin{aligned} \|\widehat{S} \mid \text{ é } \varepsilon\_{\varepsilon} \Rightarrow \quad |\mathbb{S} - \widetilde{\mathbb{S}}| \ \leq \varepsilon\_{\varepsilon} \Rightarrow |\mathbb{S}| \ \leq \ |\widehat{\mathbb{S}}| + \varepsilon\_{\varepsilon} \Rightarrow |\mathbb{S}| \ \leq \ |\sum\_{i=1}^{n} \rho\_{i}\widetilde{\mathbf{x}}\_{i}| + \varepsilon\_{\varepsilon} \leq \frac{\delta}{\lambda\_{\mathbf{k}}^{n+1}} \sum\_{i=2}^{n} \rho\_{i}\lambda\_{\mathbf{k}}^{i} + \varepsilon\_{\varepsilon} \quad \forall t > t\_{1}. \end{aligned} \tag{27}$$

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

$$|\mathbf{x}^{(i)}(t) - \mathbf{x}\_{\mathbf{d}}^{(i)}(t)| \le (2\lambda\_{\mathbf{c}})^i \frac{\varepsilon\_{\mathbf{c}}}{\lambda\_{\mathbf{c}}^n} + \frac{\delta}{\lambda\_{\mathbf{k}}^{n+1}} \sum\_{j=2}^n \left(\frac{\lambda\_{\mathbf{k}}}{\lambda\_{\mathbf{c}}}\right)^j \mathsf{C}\_{n-1\prime}^j \qquad \mathbf{i} = \mathbf{0}, 1, \cdots, n-1. \tag{28}$$

Note that POSMC does not require an accurate system model, and only one state measurement y ¼ x<sup>1</sup> is needed. As the upper bound of perturbation Ψð Þ� is replaced by the smaller bound of its estimation error <sup>Ψ</sup><sup>~</sup> ð Þ� , 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 highperformance 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 closedloop 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 nth-order system (5);

Step 2. Define a fictitious state xnþ<sup>1</sup> ¼ Ψð Þ� to represent perturbation (6);

Step 3. Extend the original nth-order system (5) into the extended ð Þ n þ 1 th-order system (8);

Step 4. Design the ð Þ n þ 1 th-order SMSPO (9) for the extended ð Þ n þ 1 th-order system (8) to obtain the state estimate <sup>b</sup><sup>x</sup> and the perturbation estimate <sup>Ψ</sup><sup>b</sup> ð Þ� by the only measurement of <sup>x</sup>1; Step 5. Design controller (17) for the original nth-order system (5), in which the estimated sliding surface bS is calculated by (13).

#### 3.2. Rectifier controller design

Choose the system output <sup>y</sup><sup>r</sup> <sup>¼</sup> <sup>y</sup>r1; <sup>y</sup>r2 � �<sup>T</sup> <sup>¼</sup> <sup>Q</sup><sup>1</sup> ½ � ; <sup>V</sup>dc1 T , let Q<sup>∗</sup> <sup>1</sup> and V<sup>∗</sup> dc1 be the given references of the reactive power and DC voltage, respectively. Define the tracking error e<sup>r</sup> ¼ ½ � er1;er2 <sup>T</sup> <sup>¼</sup> <sup>Q</sup><sup>1</sup> � <sup>Q</sup><sup>∗</sup> <sup>1</sup>; <sup>V</sup>dc1 � <sup>V</sup><sup>∗</sup> dc1 � �<sup>T</sup> , differentiate e<sup>r</sup> for rectifier (1) until the control input appears explicitly, yields

$$
\begin{bmatrix}
\dot{\mathcal{e}}\_{\rm r1} \\
\ddot{\mathcal{e}}\_{\rm r2}
\end{bmatrix} = \begin{bmatrix}
f\_{\rm r1} - \dot{\mathcal{Q}}\_{\rm 1}^{\*} \\
f\_{\rm r2} - \ddot{V}\_{\rm dc1}^{\*}
\end{bmatrix} + B\_{\rm r} \begin{bmatrix}
u\_{\rm d1} \\
u\_{\rm q1}
\end{bmatrix} \tag{29}
$$

where

This condition ensures the existence of a sliding mode on the boundary layer ∣

<sup>S</sup> <sup>¼</sup> <sup>ζ</sup> <sup>þ</sup> <sup>φ</sup>

and the perturbation. The bounds of the sliding surface can be calculated by

<sup>r</sup>ix~<sup>i</sup>þ<sup>1</sup> �X<sup>n</sup>

i¼1 ri ki k1

> i¼1 ri

It is definite that the driving term of S-dynamics is the sum of the estimation errors of states

Based on bounds (26), together with the polynomial gains ri, the states tracking error satisfies

Xn j¼2

Note that POSMC does not require an accurate system model, and only one state measurement y ¼ x<sup>1</sup> is needed. As the upper bound of perturbation Ψð Þ� is replaced by the smaller bound of its estimation error <sup>Ψ</sup><sup>~</sup> ð Þ� , a smaller control gain is needed such that the over-conservativeness

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 highperformance 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 closedloop system stability compared to that of ADRC which uses a nonlinear observer [28], while

λk λc � �<sup>j</sup> Cj

<sup>x</sup>~<sup>i</sup> <sup>þ</sup>X<sup>n</sup>�<sup>1</sup> i¼1

Xn i¼1 ri

x~i∣ þ ε<sup>c</sup> ≤

δ λ<sup>n</sup>þ<sup>1</sup> k

Xn i¼2 riλ<sup>i</sup>

<sup>n</sup>�<sup>1</sup>, i <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>⋯</sup>, n � <sup>1</sup>: (28)

As <sup>b</sup><sup>S</sup> <sup>¼</sup> <sup>S</sup> � <sup>S</sup>~, the actual <sup>S</sup>-dynamics of sliding surface can be obtained with dynamics (19) as

εc � �X<sup>n</sup>

~\_ <sup>S</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>�<sup>1</sup> i¼1

εc � �

<sup>S</sup>\_ <sup>þ</sup> <sup>ζ</sup> <sup>þ</sup> <sup>φ</sup>

28 Perturbation Methods with Applications in Science and Engineering

<sup>∣</sup>b<sup>S</sup> <sup>∣</sup> <sup>≤</sup> <sup>ε</sup><sup>c</sup> ) <sup>∣</sup><sup>S</sup> � <sup>S</sup>~<sup>∣</sup> <sup>≤</sup> <sup>ε</sup><sup>c</sup> ) <sup>∣</sup>S<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>S~<sup>∣</sup> <sup>þ</sup> <sup>ε</sup><sup>c</sup> ) <sup>∣</sup>S<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>

system (15) one can easily calculate

the following relationship [41]

of SMC can be avoided [35].

<sup>∣</sup>xð Þ<sup>i</sup> ð Þ� <sup>t</sup> <sup>x</sup>

ð Þi

<sup>d</sup> ð Þ<sup>t</sup> <sup>∣</sup> <sup>≤</sup> ð Þ <sup>2</sup>λ<sup>c</sup> <sup>i</sup> <sup>ε</sup><sup>c</sup>

λn c þ δ λ<sup>n</sup>þ<sup>1</sup> k

they can provide almost the same performance of perturbation estimation.

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

Step 2. Define a fictitious state xnþ<sup>1</sup> ¼ Ψð Þ� to represent perturbation (6);

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

Step 3. Extend the original nth-order system (5) into the extended ð Þ n þ 1 th-order system (8);

bS∣ ≤ εc. From

<sup>k</sup> þ εc, ∀t > t1:

(27)

<sup>x</sup>~<sup>2</sup> <sup>þ</sup> <sup>Ψ</sup><sup>~</sup> ð Þ� (25)

<sup>r</sup>ix~<sup>i</sup>þ<sup>1</sup> <sup>þ</sup> <sup>Ψ</sup><sup>~</sup> ð Þ� (26)

$$\begin{cases} f\_{r1} &= \frac{3u\_{\text{sq1}}}{2} \left( -\frac{R\_{\text{l}}}{L\_{\text{l}}} i\_{\text{dl}} + a i\_{\text{q1}} \right) \\ f\_{r2} &= \frac{3u\_{\text{sq1}}}{2\overline{\text{C}}\_{1}V\_{\text{dc1}}} \left[ -a i\_{\text{dl}1} - \frac{R\_{\text{l}}}{L\_{\text{l}}} i\_{\text{q1}} - \frac{i\_{\text{q1}}}{V\_{\text{dc1}}} \left( \frac{3u\_{\text{sq1}} i\_{\text{q1}}}{2\overline{\text{C}}\_{1} V\_{\text{dc1}}} - \frac{i\_{\text{L}}}{\overline{\text{C}}\_{1}} \right) \right] \\ & \quad - \frac{1}{2R\_{\text{C}}C\_{1}} \left( \frac{3u\_{\text{sq1}} i\_{\text{q1}}}{2\overline{\text{C}}\_{1} V\_{\text{dc1}}} - \frac{i\_{\text{L}}}{C\_{1}} - \frac{3u\_{\text{sq2}} i\_{\text{q2}}}{2C\_{2} V\_{\text{dc2}}} - \frac{i\_{\text{L}}}{C\_{2}} \right) \end{cases} \tag{30}$$

and

$$B\_{\rm r} = \begin{bmatrix} \frac{3u\_{\rm sq1}}{2L\_1} & 0\\ 0 & \frac{3u\_{\rm sq1}}{2C\_1L\_1V\_{\rm dc1}} \end{bmatrix} \tag{31}$$

The determinant of matrix <sup>B</sup><sup>r</sup> is obtained as <sup>∣</sup>Br<sup>∣</sup> <sup>¼</sup> <sup>9</sup>u<sup>2</sup> sq1= 4C1L<sup>2</sup> <sup>1</sup>Vdc1 � �, which is nonzero within the operation range of the rectifier, thus system (28) is linearizable.

Assume all the nonlinearities are unknown, define the perturbations Ψr1ð Þ� and Ψr2ð Þ� as

$$
\begin{bmatrix}
\Psi\_{\rm r1}(\cdot) \\
\Psi\_{\rm r2}(\cdot)
\end{bmatrix} = \begin{bmatrix}
f\_{\rm r1} \\ f\_{\rm r2}
\end{bmatrix} + (B\_{\rm r} - B\_{\rm r0}) \begin{bmatrix}
u\_{\rm d1} \\ \mu\_{\rm q1}
\end{bmatrix} \tag{32}
$$

where the constant control gain Br0 is given by

$$B\_{\rm r0} = \begin{bmatrix} b\_{\rm r10} & 0 \\ 0 & b\_{\rm r20} \end{bmatrix} \tag{33}$$

Then system (28) can be rewritten as

$$
\begin{bmatrix}
\dot{\mathcal{e}}\_{\rm r1} \\
\ddot{\mathcal{e}}\_{\rm r2}
\end{bmatrix} = \begin{bmatrix}
\Psi\_{\rm r1}(\cdot) \\
\Psi\_{\rm r2}(\cdot)
\end{bmatrix} + B\_{\rm r0} \begin{bmatrix}
u\_{\rm d1} \\
u\_{\rm q1}
\end{bmatrix} - \begin{bmatrix}
\dot{Q}\_{1}^{\*} \\
\ddot{V}\_{\rm dc1}^{\*}
\end{bmatrix} \tag{34}
$$

3.3. Inverter controller design

Q∗

where

and

<sup>2</sup>; <sup>P</sup><sup>2</sup> � <sup>P</sup><sup>∗</sup> 2�

Choose the system output <sup>y</sup><sup>i</sup> <sup>¼</sup> <sup>y</sup>i1; <sup>y</sup>i2 � �<sup>T</sup> <sup>¼</sup> <sup>Q</sup><sup>2</sup> ½ � ; <sup>P</sup><sup>2</sup>

T, let Q<sup>∗</sup>

T, differentiate e<sup>i</sup> for inverter (2) until the control input appears explicitly, yields

þ B<sup>i</sup>

ud2

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

2

<sup>f</sup> i2 � <sup>P</sup>\_ <sup>∗</sup> 2

<sup>2</sup> � <sup>R</sup><sup>2</sup> L2 <sup>i</sup>d2 <sup>þ</sup> <sup>ω</sup>iq2 � �

<sup>2</sup> � <sup>R</sup><sup>2</sup> L2 <sup>i</sup>q2 � <sup>ω</sup>id2 � �

3usq2 2L<sup>2</sup>

0

Assume all the nonlinearities are unknown, define the perturbations Ψi1ð Þ� and Ψi2ð Þ� as

<sup>B</sup>i0 <sup>¼</sup> <sup>b</sup>i10 <sup>0</sup>

þ Bi0

ud2 <sup>u</sup>q2 " # � <sup>Q</sup>\_ <sup>∗</sup>

<sup>21</sup> ¼ Q<sup>2</sup> and z<sup>21</sup> ¼ P2, two second-order SMPOs are used to estimate Ψi1ð Þ�

<sup>¼</sup> <sup>f</sup> i1 f i2 � �

0

3usq2 2L<sup>2</sup>

þ ð Þ B<sup>i</sup> � Bi0

s2= 4L<sup>2</sup> 2

ud2

" #

the reactive and active power, respectively. Define the tracking error e<sup>i</sup> ¼ ½ � ei1;ei2

<sup>e</sup>\_i2 � � <sup>¼</sup> <sup>f</sup> i1 � <sup>Q</sup>\_ <sup>∗</sup>

<sup>f</sup> i1 <sup>¼</sup> <sup>3</sup>usq2

<sup>f</sup> i2 <sup>¼</sup> <sup>3</sup>usq2

B<sup>i</sup> ¼

e\_i1

8 >>><

>>>:

The determinant of matrix <sup>B</sup><sup>i</sup> is obtained as <sup>∣</sup>Bi<sup>∣</sup> <sup>¼</sup> <sup>9</sup>u<sup>2</sup>

where the constant control gain Bi0 is given by

Then system (35) can be rewritten as

Similarly, define z<sup>0</sup>

and Ψi2ð Þ� , respectively, as

operation range of the inverter, thus system (35) is linearizable.

Ψi1ð Þ� Ψi2ð Þ� � �

e\_i1

<sup>e</sup>\_i2 � � <sup>¼</sup> <sup>Ψ</sup>i1ð Þ�

Ψi2ð Þ� � � <sup>2</sup> and P<sup>∗</sup>

<sup>2</sup> be the given references of

<sup>u</sup>q2 " # (39)

http://dx.doi.org/10.5772/intechopen.74717

� �, which is nonzero within the

<sup>u</sup>q2 " # (42)

<sup>0</sup> <sup>b</sup>i20 � � (43)

2 P\_ ∗ 2

" #

<sup>T</sup> <sup>¼</sup> <sup>½</sup>Q2�

31

(40)

(41)

(44)

Define z<sup>0</sup> <sup>11</sup> ¼ Q1, a second-order sliding-mode perturbation observer (SMPO) is used to estimate Ψr1ð Þ� as

$$\begin{cases} \dot{\hat{z}}\_{\rm r1}^{\prime} = \hat{\Psi}\_{\rm r1}(\cdot) + a\_{\rm r1}^{\prime} \tilde{Q}\_{1} + k\_{\rm r1}^{\prime} \text{sat}\left(\tilde{Q}\_{1}\right) + b\_{\rm r10} u\_{\rm d1} \\ \dot{\hat{\Psi}}\_{\rm r1}(\cdot) = a\_{\rm r2}^{\prime} \tilde{Q}\_{1} + k\_{\rm r2}^{\prime} \text{sat}\left(\tilde{Q}\_{1}\right) \end{cases} \tag{35}$$

where observer gains k 0 r1, k<sup>0</sup> r2, α<sup>0</sup> r1, and α<sup>0</sup> r2 are all positive constants.

Define z<sup>11</sup> ¼ Vdc1 and z<sup>12</sup> ¼ z\_11, a third-order SMSPO is used to estimate Ψr2ð Þ� as

$$\begin{cases} \dot{\hat{z}}\_{11} = \hat{z}\_{12} + \alpha\_{\text{r1}} \ddot{V}\_{\text{dc1}} + k\_{\text{r1}} \text{sat}\left(\ddot{V}\_{\text{dc1}}\right) \\ \dot{\hat{z}}\_{12} = \hat{\Psi}\_{\text{r2}}(\cdot) + \alpha\_{\text{r2}} \ddot{V}\_{\text{dc1}} + k\_{\text{t2}} \text{sat}\left(\ddot{V}\_{\text{dc1}}\right) + b\_{\text{t2}0} u\_{\text{q1}} \\ \dot{\hat{\Psi}}\_{\text{r2}}(\cdot) = \alpha\_{\text{r3}} \ddot{V}\_{\text{dc1}} + k\_{\text{r3}} \text{sat}\left(\ddot{V}\_{\text{dc1}}\right) \end{cases} \tag{36}$$

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 Q<sup>1</sup> 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

$$
\begin{bmatrix}
\hat{\mathbf{S}}\_{\rm r1} \\
\hat{\mathbf{S}}\_{\rm r2}
\end{bmatrix} = \begin{bmatrix}
\hat{\mathbf{z}}\_{\rm 11}^{\prime} - \mathbf{Q}\_{\rm 1}^{\*} \\
\rho\_1 \langle \hat{\mathbf{z}}\_{\rm 11} - \mathbf{V}\_{\rm dc1}^{\*} \rangle + \rho\_2 \langle \hat{\mathbf{z}}\_{\rm 12} - \dot{\mathbf{V}}\_{\rm dc1}^{\*} \rangle
\end{bmatrix} \tag{37}
$$

where r<sup>1</sup> and r<sup>2</sup> are the positive sliding surface gains. The attractiveness of the estimated sliding surface (33) ensures reactive power Q<sup>1</sup> and DC voltage Vdc1 can track to their reference.

The POSMC of system (28) is designed as

$$\begin{bmatrix} u\_{\rm d1} \\ u\_{\rm q1} \end{bmatrix} = B\_{\rm r0}^{-1} \begin{bmatrix} -\widehat{\Psi}\_{\rm r1}(\cdot) + \dot{Q}\_{\rm r}^{\ast} - \zeta\_{\rm r}^{\prime} \widehat{S}\_{\rm r1} - \varrho\_{\rm r}^{\prime} \text{sat}\left(\widehat{S}\_{\rm r1}\right) \\ -\widehat{\Psi}\_{\rm r2}(\cdot) + \ddot{V}\_{\rm dc1}^{\ast} - \rho\_{\rm 1}(\widehat{z}\_{12} - \dot{V}\_{\rm dc1}^{\ast} - \zeta\_{\rm r} \widehat{S}\_{\rm r2} - \varrho\_{\rm r} \text{sat}\left(\widehat{S}\_{\rm t2}\right)) \end{bmatrix} \tag{38}$$

where positive control gains ζr, ζ<sup>0</sup> <sup>r</sup>, φr, r1, and φ<sup>0</sup> <sup>r</sup> 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 <sup>∣</sup>bz<sup>0</sup> <sup>11</sup>∣ ≤ ∣Q<sup>∗</sup> <sup>1</sup>∣, <sup>∣</sup>Ψ<sup>b</sup> r1ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>Q<sup>∗</sup> <sup>1</sup>∣=Δ, <sup>∣</sup>bz11<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>V<sup>∗</sup> dc1∣, <sup>∣</sup>bz12<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>V<sup>∗</sup> dc1∣=Δ, and <sup>∣</sup>Ψ<sup>b</sup> r2ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>V<sup>∗</sup> dc1∣=Δ<sup>2</sup> , respectively.

#### 3.3. Inverter controller design

Choose the system output <sup>y</sup><sup>i</sup> <sup>¼</sup> <sup>y</sup>i1; <sup>y</sup>i2 � �<sup>T</sup> <sup>¼</sup> <sup>Q</sup><sup>2</sup> ½ � ; <sup>P</sup><sup>2</sup> T, let Q<sup>∗</sup> <sup>2</sup> and P<sup>∗</sup> <sup>2</sup> be the given references of the reactive and active power, respectively. Define the tracking error e<sup>i</sup> ¼ ½ � ei1;ei2 <sup>T</sup> <sup>¼</sup> <sup>½</sup>Q2� Q∗ <sup>2</sup>; <sup>P</sup><sup>2</sup> � <sup>P</sup><sup>∗</sup> 2� T, differentiate e<sup>i</sup> for inverter (2) until the control input appears explicitly, yields

$$
\begin{bmatrix}
\dot{e}\_{i1} \\
\dot{e}\_{i2}
\end{bmatrix} = \begin{bmatrix}
f\_{i1} - \dot{Q}\_{2}^{\*} \\
f\_{i2} - \dot{P}\_{2}^{\*}
\end{bmatrix} + B\_{i} \begin{bmatrix}
u\_{d2} \\
u\_{q2}
\end{bmatrix} \tag{39}
$$

where

Then system (28) can be rewritten as

Define z<sup>0</sup>

mate Ψr1ð Þ� as

where observer gains k

e\_r1 e €r2 � �

\_ bz 0

8 < :

30 Perturbation Methods with Applications in Science and Engineering

0 r1, k<sup>0</sup> r2, α<sup>0</sup>

\_

8 >><

>>:

\_

\_

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

bSr1 bSr2

The POSMC of system (28) is designed as

<sup>¼</sup> <sup>B</sup>�<sup>1</sup> r0

2 6 4

ud1 uq1 " #

where positive control gains ζr, ζ<sup>0</sup>

dc1∣, <sup>∣</sup>bz12<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>V<sup>∗</sup>

estimated sliding surface (33).

<sup>∣</sup>bz11<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>V<sup>∗</sup>

" #

\_ Ψb r1ðÞ¼ � α<sup>0</sup>

<sup>¼</sup> <sup>Ψ</sup>r1ð Þ� Ψr2ð Þ� � �

<sup>11</sup> ¼ Ψb r1ðÞþ� α<sup>0</sup>

r1, and α<sup>0</sup>

þ Br0

r1Q<sup>~</sup> <sup>1</sup> <sup>þ</sup> <sup>k</sup> 0 r1sat <sup>Q</sup><sup>~</sup> <sup>1</sup>

r2Q<sup>~</sup> <sup>1</sup> <sup>þ</sup> <sup>k</sup> 0 r2sat <sup>Q</sup><sup>~</sup> <sup>1</sup> � �

Define z<sup>11</sup> ¼ Vdc1 and z<sup>12</sup> ¼ z\_11, a third-order SMSPO is used to estimate Ψr2ð Þ� as

<sup>b</sup>z<sup>11</sup> <sup>¼</sup> <sup>b</sup>z<sup>12</sup> <sup>þ</sup> <sup>α</sup>r1V<sup>~</sup> dc1 <sup>þ</sup> <sup>k</sup>r1sat <sup>V</sup><sup>~</sup> dc1

<sup>Ψ</sup><sup>b</sup> r2ðÞ¼ � <sup>α</sup>r3V<sup>~</sup> dc1 <sup>þ</sup> <sup>k</sup>r3sat <sup>V</sup><sup>~</sup> dc1

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

voltage Vdc1 at the rectifier side, which can be directly obtained in practice.

<sup>11</sup> � <sup>Q</sup><sup>∗</sup> 1 <sup>r</sup><sup>1</sup> <sup>b</sup>z<sup>11</sup> � <sup>V</sup><sup>∗</sup>

> <sup>1</sup> � ζ<sup>0</sup> r bSr1 � φ<sup>0</sup>

<sup>r</sup>, φr, r1, and φ<sup>0</sup>

<sup>¼</sup> <sup>b</sup>z<sup>0</sup>

�Ψ<sup>b</sup> r1ðÞþ� <sup>Q</sup>\_ <sup>∗</sup>

�Ψ<sup>b</sup> r2ðÞþ� <sup>V</sup>€<sup>∗</sup>

system state and perturbation estimates can be obtained as <sup>∣</sup>bz<sup>0</sup>

dc1∣=Δ, and <sup>∣</sup>Ψ<sup>b</sup> r2ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>V<sup>∗</sup>

<sup>b</sup>z<sup>12</sup> <sup>¼</sup> <sup>Ψ</sup><sup>b</sup> r2ðÞþ� <sup>α</sup>r2V<sup>~</sup> dc1 <sup>þ</sup> <sup>k</sup>r2sat <sup>V</sup><sup>~</sup> dc1

The above observers (31) and (32) only need the measurement of reactive power Q<sup>1</sup> and DC

dc1 � � <sup>þ</sup> <sup>r</sup><sup>2</sup> <sup>b</sup>z<sup>12</sup> � <sup>V</sup>\_ <sup>∗</sup>

where r<sup>1</sup> and r<sup>2</sup> are the positive sliding surface gains. The attractiveness of the estimated sliding surface (33) ensures reactive power Q<sup>1</sup> and DC voltage Vdc1 can track to their reference.

dc1 � <sup>r</sup>1ðbz<sup>12</sup> � <sup>V</sup>\_

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

" #

<sup>r</sup>sat bSr1 � �

> ∗ dc1 � ζ<sup>r</sup>

dc1∣=Δ<sup>2</sup>

ud1 uq1 " #

<sup>11</sup> ¼ Q1, a second-order sliding-mode perturbation observer (SMPO) is used to esti-

r2 are all positive constants.

� �

� �

� <sup>Q</sup>\_ <sup>∗</sup> 1 V€ ∗ dc1

" #

� � <sup>þ</sup> <sup>b</sup>r10ud1

� � <sup>þ</sup> <sup>b</sup>r20uq1

dc1 � �

bSr2 � φrsat bSr2

<sup>r</sup> are chosen to ensure the attractiveness of

<sup>11</sup>∣ ≤ ∣Q<sup>∗</sup>

, respectively.

� �

3 7

<sup>1</sup>∣, <sup>∣</sup>Ψ<sup>b</sup> r1ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>Q<sup>∗</sup>

<sup>5</sup> (38)

<sup>1</sup>∣=Δ,

(34)

(35)

(36)

(37)

$$\begin{cases} f\_{\rm i1} &= \frac{3u\_{\rm sq2}}{2} \left( -\frac{R\_2}{L\_2} i\_{\rm d2} + \omega i\_{\rm q2} \right) \\ f\_{\rm i2} &= \frac{3u\_{\rm sq2}}{2} \left( -\frac{R\_2}{L\_2} i\_{\rm q2} - \omega i\_{\rm d2} \right) \end{cases} \tag{40}$$

and

$$\begin{aligned} \,^1B\_1 &= \begin{bmatrix} \frac{3u\_{\text{sq2}}}{2L\_2} & 0\\ 0 & \frac{3u\_{\text{sq2}}}{2L\_2} \end{bmatrix} \end{aligned} \tag{41}$$

The determinant of matrix <sup>B</sup><sup>i</sup> is obtained as <sup>∣</sup>Bi<sup>∣</sup> <sup>¼</sup> <sup>9</sup>u<sup>2</sup> s2= 4L<sup>2</sup> 2 � �, which is nonzero within the operation range of the inverter, thus system (35) is linearizable.

Assume all the nonlinearities are unknown, define the perturbations Ψi1ð Þ� and Ψi2ð Þ� as

$$
\begin{bmatrix}
\Psi\_{\rm i1}(\cdot) \\
\Psi\_{\rm i2}(\cdot)
\end{bmatrix} = \begin{bmatrix}
f\_{\rm i1} \\ f\_{\rm i2}
\end{bmatrix} + (B\_{\rm i} - B\_{\rm i0}) \begin{bmatrix}
\nu\_{\rm d2} \\
\nu\_{\rm q2}
\end{bmatrix} \tag{42}
$$

where the constant control gain Bi0 is given by

$$B\_{i0} = \begin{bmatrix} b\_{i10} & 0 \\ 0 & b\_{i20} \end{bmatrix} \tag{43}$$

Then system (35) can be rewritten as

$$
\begin{bmatrix}
\dot{e}\_{i1} \\
\dot{e}\_{i2}
\end{bmatrix} = \begin{bmatrix}
\Psi\_{i1}(\cdot) \\
\Psi\_{i2}(\cdot)
\end{bmatrix} + B\_{i0} \begin{bmatrix}
u\_{d2} \\
u\_{q2}
\end{bmatrix} - \begin{bmatrix}
\dot{Q}\_2^\* \\
\dot{P}\_2^\*
\end{bmatrix} \tag{44}
$$

Similarly, define z<sup>0</sup> <sup>21</sup> ¼ Q<sup>2</sup> and z<sup>21</sup> ¼ P2, two second-order SMPOs are used to estimate Ψi1ð Þ� and Ψi2ð Þ� , respectively, as

$$\begin{cases} \dot{\hat{z}}\_{21}^{\prime} = \hat{\Psi}\_{\rm i1}(\cdot) + a\_{\rm i1}^{\prime} \tilde{Q}\_{2} + k\_{\rm i1}^{\prime} \text{sat}(\tilde{Q}\_{2}) + b\_{\rm i10} u\_{\rm d2} \\ \dot{\hat{\Psi}}\_{\rm i1}(\cdot) = a\_{\rm i2}^{\prime} \tilde{Q}\_{2} + k\_{\rm i2}^{\prime} \text{sat}(\tilde{Q}\_{2}) \end{cases} \tag{45}$$

where observer gains k 0 i1, k 0 i2, α<sup>0</sup> i1, and α<sup>0</sup> i2 are all positive constants.

$$\begin{cases} \dot{\hat{z}}\_{21} = \hat{\Psi}\_{i2}(\cdot) + a\_{i1}\tilde{P}\_2 + k\_{i1}\text{sat}(\tilde{P}\_2) + b\_{i20}u\_{\text{q2}}\\ \dot{\hat{\Psi}}\_{i2}(\cdot) = a\_{i2}\tilde{P}\_2 + k\_{i2}\text{sat}(\tilde{P}\_2) \end{cases} \tag{46}$$

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 Q<sup>2</sup> and active power P<sup>2</sup> at the inverter side, which can be directly obtained in practice.

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

$$
\begin{bmatrix}
\widehat{\mathbf{S}}\_{\mathrm{i}1} \\
\widehat{\mathbf{S}}\_{\mathrm{i}2}
\end{bmatrix} = \begin{bmatrix}
\widehat{\mathbf{z}}\_{\mathrm{21}}^{\prime} - Q\_{2}^{\*} \\
\widehat{\mathbf{z}}\_{\mathrm{21}} - P\_{2}^{\*}
\end{bmatrix} \tag{47}
$$

4. Simulation results

Table 1. The VSC-HVDC system parameters.

Figure 2. The overall controller structure of the VSC-HVDC system.

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

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

http://dx.doi.org/10.5772/intechopen.74717

33

AC system-based voltage <sup>V</sup>AC base 132 kV DC cable base voltage <sup>V</sup>DC base 150 kV System base power S base 100 MVA AC system resistance (25 km) R1, R<sup>2</sup> 0.05 Ω/km AC system inductance (25 km) L1, L<sup>2</sup> 0.026 mH/km DC cable resistance (50 km) R<sup>0</sup> 0.21 Ω/km DC bus capacitance C1, C<sup>2</sup> 11.94 μF

<sup>r</sup> ¼ λα<sup>i</sup> ¼ λα<sup>0</sup>

<sup>i</sup> ¼ 20,

in Table 2, in which the observer poles are allocated as λα<sup>r</sup> ¼ 100 and λα<sup>0</sup>

Similarly, the attractiveness of the estimated sliding surface (40) ensures the reactive power Q<sup>2</sup> and active power P<sup>2</sup> can track to their reference.

The POSMC of system (35) is designed as

$$\begin{bmatrix} \mu\_{\rm d2} \\\\ \mu\_{\rm q2} \end{bmatrix} = B\_{\rm i0}^{-1} \begin{bmatrix} -\hat{\Psi}\_{\rm i1}(\cdot) + \dot{\mathcal{Q}}\_{2}^{\*} - \zeta\_{\rm i}^{\prime} \hat{\mathbf{S}}\_{\rm i1} - q\zeta\_{\rm i} \text{sat}\left(\hat{\mathbf{S}}\_{\rm i1}\right) \\\\ -\hat{\Psi}\_{\rm i2}(\cdot) + \dot{\hat{P}}\_{2}^{\*} - \zeta\_{\rm i} \hat{\mathbf{S}}\_{\rm i2} - q\zeta\_{\rm i} \text{sat}\left(\hat{\mathbf{S}}\_{\rm i2}\right) \end{bmatrix} \tag{48}$$

where positive control gains ζi, ζi, φ<sup>i</sup> , and φ<sup>0</sup> <sup>i</sup> 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 <sup>∣</sup>bz<sup>0</sup> <sup>21</sup>∣ ≤ ∣Q<sup>∗</sup> <sup>2</sup>∣, <sup>∣</sup>Ψ<sup>b</sup> i1ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>Q<sup>∗</sup> <sup>2</sup>∣=Δ, <sup>∣</sup>bz21<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>P<sup>∗</sup> <sup>2</sup>∣, and <sup>∣</sup>Ψ<sup>b</sup> i2ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>P<sup>∗</sup> <sup>2</sup>∣=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 Q<sup>1</sup> and DC voltage Vdc1 need to be measured for rectifier controller (34), while active power P<sup>2</sup> and reactive power Q<sup>2</sup> 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.

Sliding-Mode Perturbation Observer-Based Sliding-Mode Control for VSC-HVDC Systems http://dx.doi.org/10.5772/intechopen.74717 33

Figure 2. The overall controller structure of the VSC-HVDC system.

## 4. Simulation results

\_ bz 0

8 < :

32 Perturbation Methods with Applications in Science and Engineering

0 i1, k 0 i2, α<sup>0</sup>

where observer gains k

\_ Ψb i1ðÞ¼ � α<sup>0</sup>

\_

8 < :

\_

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

and active power P<sup>2</sup> can track to their reference.

ud2 uq2 " #

<sup>¼</sup> <sup>B</sup>�<sup>1</sup> i0

<sup>2</sup>∣=Δ, <sup>∣</sup>bz21<sup>∣</sup> <sup>≤</sup> <sup>∣</sup>P<sup>∗</sup>

2 6 4

The POSMC of system (35) is designed as

where positive control gains ζi, ζi, φ<sup>i</sup>

<sup>2</sup>∣, <sup>∣</sup>Ψ<sup>b</sup> i1ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>Q<sup>∗</sup>

estimated sliding surface (40).

as <sup>∣</sup>bz<sup>0</sup>

<sup>21</sup>∣ ≤ ∣Q<sup>∗</sup>

inverter controller (41).

which can be seen in Figure 2.

<sup>21</sup> ¼ Ψb i1ðÞþ� α<sup>0</sup>

i1, and α<sup>0</sup>

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

power P<sup>2</sup> at the inverter side, which can be directly obtained in practice.

i2Q<sup>~</sup> <sup>2</sup> <sup>þ</sup> <sup>k</sup> 0 i2sat <sup>Q</sup><sup>~</sup> <sup>2</sup> � �

<sup>b</sup>z<sup>21</sup> <sup>¼</sup> <sup>Ψ</sup><sup>b</sup> i2ðÞþ� <sup>α</sup>i1P~<sup>2</sup> <sup>þ</sup> <sup>k</sup>i1sat <sup>P</sup>~<sup>2</sup>

<sup>Ψ</sup><sup>b</sup> i2ðÞ¼ � <sup>α</sup>i2P~<sup>2</sup> <sup>þ</sup> <sup>k</sup>i2sat <sup>P</sup>~<sup>2</sup>

bSi1 bSi2

" #

i1Q<sup>~</sup> <sup>2</sup> <sup>þ</sup> <sup>k</sup> 0 i1sat <sup>Q</sup><sup>~</sup> <sup>2</sup>

The above observers (38) and (39) only need the measurement of reactive power Q<sup>2</sup> and active

<sup>¼</sup> <sup>b</sup><sup>z</sup> 0 <sup>21</sup> � <sup>Q</sup><sup>∗</sup> 2

Similarly, the attractiveness of the estimated sliding surface (40) ensures the reactive power Q<sup>2</sup>

�Ψ<sup>b</sup> i1ðÞþ� <sup>Q</sup>\_ <sup>∗</sup>

�Ψ<sup>b</sup> i2ðÞþ� <sup>P</sup>\_

, and φ<sup>0</sup>

Similarly, the boundary values of the system state and perturbation estimates can be obtained

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 Q<sup>1</sup> and DC voltage Vdc1 need to be measured for rectifier controller (34), while active power P<sup>2</sup> and reactive power Q<sup>2</sup> for

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,

<sup>2</sup>∣, and <sup>∣</sup>Ψ<sup>b</sup> i2ð Þ� <sup>∣</sup> <sup>≤</sup> <sup>∣</sup>P<sup>∗</sup>

<sup>b</sup>z<sup>21</sup> � <sup>P</sup><sup>∗</sup> 2

> <sup>2</sup> � ζ<sup>0</sup> i bSi1 � φ<sup>0</sup> i sat bSi1 � �

> > bSi2 � φ<sup>i</sup>

sat bSi2 � �

<sup>2</sup>∣=Δ, respectively.

3 7

<sup>i</sup> are chosen to ensure the attractiveness of

<sup>5</sup> (48)

∗ <sup>2</sup> � ζ<sup>i</sup>

" #

i2 are all positive constants.

� �

� � <sup>þ</sup> <sup>b</sup>i10ud2

� � <sup>þ</sup> <sup>b</sup>i20uq2

(45)

(46)

(47)

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 λα<sup>r</sup> ¼ 100 and λα<sup>0</sup> <sup>r</sup> ¼ λα<sup>i</sup> ¼ λα<sup>0</sup> <sup>i</sup> ¼ 20,


Table 1. The VSC-HVDC system parameters.


while control inputs are bounded as ∣uqi∣ ≤ 80 kV and ∣udi∣ ≤ 60 kV, where i ¼ 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

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

(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 ¼ 0:2 s, t ¼ 0:4 s and restores to the original value at

are illustrated by Figure 3. One can find that POSMC has the fastest tracking rate and main-

(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 ¼ 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.

dc1 ¼ 150 kV. The system responses

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35

using a personal computer with an IntelR CoreTMi7 CPU at 2.2 GHz and 8 GB of RAM.

tains a consistent control performance under different operation conditions.

<sup>t</sup> <sup>¼</sup> <sup>0</sup>:6 s, while DC voltage is regulated at the rated value <sup>V</sup><sup>∗</sup>

Figure 4. System responses obtained under the five-cycle LLLG fault at AC bus 1.

Table 2. POSMC parameters for the VSC-HVDC system.

Figure 3. System responses obtained under the active and reactive power tracking.

while control inputs are bounded as ∣uqi∣ ≤ 80 kV and ∣udi∣ ≤ 60 kV, where i ¼ 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.

Rectifier controller gains

Rectifier observer gains

Inverter controller gains

Inverter observer gains

k 0

ζ<sup>r</sup> ¼ 20 ζ<sup>0</sup>

αr1 ¼ 300 α<sup>0</sup>

φ<sup>i</sup> ¼ 10 φ<sup>0</sup>

αi1 ¼ 40 α<sup>0</sup>

ki1 ¼ 75 k<sup>0</sup>

r1 <sup>¼</sup> <sup>75</sup> <sup>k</sup>r2 <sup>¼</sup> 105 <sup>k</sup>

34 Perturbation Methods with Applications in Science and Engineering

Table 2. POSMC parameters for the VSC-HVDC system.

<sup>i</sup> ¼ 10

Figure 3. System responses obtained under the active and reactive power tracking.

br10 ¼ 100 br20 ¼ 7000 r<sup>1</sup> ¼ 800 r<sup>2</sup> ¼ 1

<sup>α</sup>r3 <sup>¼</sup> 106 <sup>Δ</sup> <sup>¼</sup> <sup>0</sup>:<sup>01</sup> <sup>e</sup> <sup>¼</sup> <sup>0</sup>:<sup>1</sup> <sup>k</sup>r1 <sup>¼</sup> <sup>100</sup>

bi10 ¼ 50 bi20 ¼ 50 ζ<sup>i</sup> ¼ 10 ζ<sup>0</sup>

<sup>r</sup> ¼ 10 φ<sup>r</sup> ¼ 20 φ<sup>0</sup>

r1 <sup>¼</sup> <sup>40</sup> <sup>α</sup>r2 <sup>¼</sup> <sup>3</sup> � 104 <sup>α</sup><sup>0</sup>

i1 ¼ 40 αi2 ¼ 400 α<sup>0</sup>

i1 <sup>¼</sup> <sup>75</sup> <sup>k</sup>i2 <sup>¼</sup> <sup>3</sup>:<sup>75</sup> � <sup>10</sup><sup>4</sup> <sup>k</sup><sup>0</sup>

0

<sup>r</sup> ¼ 20

r2 ¼ 400

<sup>i</sup> ¼ 10

i2 ¼ 400

i2 <sup>¼</sup> <sup>3</sup>:<sup>75</sup> � <sup>10</sup><sup>4</sup>

r2 <sup>¼</sup> <sup>3</sup>:<sup>75</sup> � <sup>10</sup><sup>4</sup> <sup>k</sup>r3 <sup>¼</sup> <sup>2</sup>:<sup>5</sup> � 107

(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 ¼ 0:2 s, t ¼ 0:4 s and restores to the original value at <sup>t</sup> <sup>¼</sup> <sup>0</sup>:6 s, while DC voltage is regulated at the rated value <sup>V</sup><sup>∗</sup> dc1 ¼ 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.

(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 ¼ 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 timevarying 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 ¼ 1 þ 0:15 sin 0ð Þ :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

Figure 7. The peak active power ∣P2∣ (in p.u.) to a �120 A in the DC cable current i<sup>L</sup> obtained at nominal grid voltage for

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

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37

plant-model mismatches in the range of 20% (one parameter changes and others keep constant).

(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 R<sup>2</sup> and L<sup>2</sup> with �20% uncertainties. All tests are undertaken under the nominal grid voltage and a corresponding �120 A in the DC cable current i<sup>L</sup> 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 R<sup>2</sup> and 8% to L2, respectively. Responses to mismatch of R<sup>2</sup> and L<sup>2</sup> 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.

<sup>0</sup> <sup>∣</sup>P<sup>2</sup> � <sup>P</sup><sup>∗</sup>

The integral of absolute error (IAE) indices of each approach calculated in different cases are

<sup>1</sup>∣dt, IAE<sup>V</sup>dc1 <sup>¼</sup> <sup>Ð</sup> <sup>T</sup>

<sup>2</sup>∣dt. The simulation time T = 3 s. Note that POSMC has

<sup>0</sup> <sup>∣</sup>Vdc1 � <sup>V</sup><sup>∗</sup>

dc1∣dt, IAE<sup>Q</sup><sup>2</sup> ¼

<sup>0</sup> <sup>∣</sup>Q<sup>1</sup> � <sup>Q</sup><sup>∗</sup>

voltage and reactive power.

tabulated in Table 3. Here, IAE<sup>Q</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup>

<sup>2</sup>∣dt, and IAE<sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup>

Ð T

<sup>0</sup> <sup>∣</sup>Q<sup>2</sup> � <sup>Q</sup><sup>∗</sup>

Figure 6. System responses obtained with the weak AC grid connection.

Sliding-Mode Perturbation Observer-Based Sliding-Mode Control for VSC-HVDC Systems http://dx.doi.org/10.5772/intechopen.74717 37

(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 timevarying 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 ¼ 1 þ 0:15 sin 0ð Þ :2πt . System responses are presented in Figure 6, it illustrates that both DC voltage and reactive power are oscillatory,

Figure 5. Estimation errors of the perturbations obtained under the five-cycle LLLG fault at AC bus 1.

36 Perturbation Methods with Applications in Science and Engineering

Figure 6. System responses obtained with the weak AC grid connection.

Figure 7. The peak active power ∣P2∣ (in p.u.) to a �120 A in the DC cable current i<sup>L</sup> obtained at nominal grid voltage for plant-model mismatches in the range of 20% (one parameter changes and others keep constant).

while POSMC can effectively suppress such oscillation with the smallest fluctuation of DC voltage and reactive power.

(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 R<sup>2</sup> and L<sup>2</sup> with �20% uncertainties. All tests are undertaken under the nominal grid voltage and a corresponding �120 A in the DC cable current i<sup>L</sup> 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 R<sup>2</sup> and 8% to L2, respectively. Responses to mismatch of R<sup>2</sup> and L<sup>2</sup> 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.

The integral of absolute error (IAE) indices of each approach calculated in different cases are tabulated in Table 3. Here, IAE<sup>Q</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup> <sup>0</sup> <sup>∣</sup>Q<sup>1</sup> � <sup>Q</sup><sup>∗</sup> <sup>1</sup>∣dt, IAE<sup>V</sup>dc1 <sup>¼</sup> <sup>Ð</sup> <sup>T</sup> <sup>0</sup> <sup>∣</sup>Vdc1 � <sup>V</sup><sup>∗</sup> dc1∣dt, IAE<sup>Q</sup><sup>2</sup> ¼ Ð T <sup>0</sup> <sup>∣</sup>Q<sup>2</sup> � <sup>Q</sup><sup>∗</sup> <sup>2</sup>∣dt, and IAE<sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup> <sup>0</sup> <sup>∣</sup>P<sup>2</sup> � <sup>P</sup><sup>∗</sup> <sup>2</sup>∣dt. 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 IAE<sup>Q</sup><sup>1</sup> and IAE<sup>V</sup>dc1 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

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

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

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

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 f <sup>c</sup> ¼ 1 kHz, and the VSC-HVDC system is simulated on another dSPACE platform (DS1006 board) with the limit sampling frequency

<sup>0</sup> <sup>j</sup>ud1jþjuq1jþjud2jþjuq2<sup>j</sup> � �dt. It is obvious

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costs are illustrated in Figure 9, with IAE<sup>u</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup>

over-conservative control input can be avoided.

5. Hardware-in-the-loop test results

dynamic systems.

Figure 10. The configuration of the HIL test.

Figure 8. The peak active power ∣P2∣ (in p.u.) to a 120 A in the DC cable current i<sup>L</sup> obtained at nominal grid voltage for plant-model mismatches in the range of 20% (different parameters may change at the same time).


Table 3. IAE indices (in p.u.) of different control schemes calculated in different cases.

Figure 9. Overall control costs IAE<sup>u</sup> (in p.u.) obtained in different cases.

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 IAE<sup>Q</sup><sup>1</sup> and IAE<sup>V</sup>dc1 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 IAE<sup>u</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup> <sup>0</sup> <sup>j</sup>ud1jþjuq1jþjud2jþjuq2<sup>j</sup> � �dt. 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.
