Perturbation estimation based coordinated adaptive passive control for multimachine power systems

Control Engineering Practice 44 (2015) 172–192

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Perturbation estimation based coordinated adaptive passive control for multimachine power systems B. Yang a, L. Jiang a, Wei Yao a,b, Q.H. Wu a,c,n a

Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, United Kingdom State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China c School of Electric Power Engineering, South China University of Technology, Guangzhou 510641, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 15 December 2014 Received in revised form 30 July 2015 Accepted 30 July 2015

This paper proposes a perturbation estimation based coordinated adaptive passive control (PECAPC) of generators excitation system and thyristor-controlled series capacitor (TCSC) devices for complex, uncertain and interconnected multimachine power systems. Discussion begins with the PECAPC design, in which the combinatorial effect of system uncertainties, unmodelled dynamics and external disturbances is aggregated into a perturbation term, and estimated online by a perturbation observer (PO). PECAPC aims to achieve a coordinated adaptive control between the excitation controller (EC) and TCSC controller based on the nonlinearly functional estimate of the perturbation. In this control scheme an explicit control Lyapunov function (CLF) and the strict assumption of linearly parametric uncertainties made on system structures can be avoided. A decentralized stabilizing EC for each generator is firstly designed. Then a coordinated TCSC controller is developed to passivize the whole system, which improves system damping through reshaping the distributed energies in power systems. Case studies are carried out on a single machine infinite bus (SMIB) and a three-machine system, respectively. Simulation results show that the PECAPC-based EC and TCSC controller can coordinate each other to improve the power system stability, finally a hardware-in-the-loop (HIL) test is carried out to verify its implementation feasibility. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Coordinated adaptive passive control Perturbation estimation Energy shaping Multimachine power systems

1. Introduction Power system stability is becoming a crucial issue as the size and complexity of power systems increase (Grigsby, 2012). Conventional linear control methods have been widely used, however, they cannot maintain consistent control performance as power systems are highly nonlinear, which operate under a wide range of operating conditions and various disturbances (Revel, Len, Alonso, & Moiola, 2010). Synchronous generator excitation control is one of the most popular methods to enhance the power system stability. On one hand, many nonlinear control approaches have been used for the excitation controller (EC), such as adaptive H∞ control (Wang, Cheng, Liu, & Li, 2004), L-2 disturbance attenuation control (Shen, Mei, Lu, Hu, & Tamura, 2003), nonlinear adaptive control (Jiang, Wu, & Wen, 2004), adaptive dynamic programming (Jiang & Jiang, 2012), and optimal predictive control (Yao, Jiang, Fang, Wen, & Cheng, 2014). On the other hand, proper controllers of flexible AC transmission systems (FACTS) can also improve the power system n Corresponding author at: School of Electric Power Engineering, South China University of Technology, Guangzhou, 510641, China. Tel.: þ 86 20 87112999. E-mail address: [email protected] (Q.H. Wu).

http://dx.doi.org/10.1016/j.conengprac.2015.07.012 0967-0661/& 2015 Elsevier Ltd. All rights reserved.

stability. Many studies have been undertaken on the development of nonlinear controllers for thyristor controlled series capacitor (TCSC) (Manjarekar, Banavar, & Ortega, 2010), static VAR compensator (SVC) (Sun, Tong, & Liu, 2011), and static synchronous compensator (STATCOM) (Panda & Padhy, 2008). However, uncoordinated EC and FACTS controller may deteriorate each other or even lead to instability under large disturbances (Grigsby, 2012). To resolve the above issues, several nonlinear coordinated control approaches of generator excitation systems and FACTS devices have been applied to achieve an optimal control performance of the whole system, such as optimal-variable-aim strategies (Lei, Li, & Povh, 2001), global control (Leung, Hill, & Ni, 2006), and zero dynamics method (Amin, Mohammad, & Moein, 2013). However, these methods require accurate system models in the controller design process, which are difficult to guarantee reliable control performances when implemented in a real multimachine power system. Hence several adaptive and robust coordinated control methods are investigated in Wang, Hill, Middleton, and Gao (2002), Liu, Sun, Shen, and Song (2003), and Mei, Chen, Lu, Yokoyama, and Goto (2004). However, their design is complicated and cancels possible beneficial nonlinearities. Besides, the optimal control performance may not be obtained as the physical property

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

of a power system is ignored. Passivity provides a physical insight for the analysis and design of nonlinear systems, which decomposes a complex nonlinear system into simpler subsystems that, upon interconnection, adds up their local energies to determine the full system's behavior. The action of a controller connected to the dynamical system may also be regarded, in terms of energy, as another separate dynamical system. Thus the control problem can then be treated as finding an interconnection pattern between the controller and the dynamical system. This ‘energy shaping’ approach is the essence of passive control (PC), which takes into account the energy of the system and gives a clear physical meaning, such that the changes of the overall storage function can take a desired form (Isidori, 1995; Ortega, Van der Schaft, Maschke, & Gerardo, 2001). It is well known that the nature of a power system is to produce, transmit and consume energy, and the power flow into the network must be greater than or equal to the rate of change of the energy stored in the network (Verrelli & Damm, 2010). Coordinated passive control (CPC) (Chen, Ji, Wang, & Xi, 2006; Larsena, Jankovi, & Kokotovi, 2003) is an extended passivity-based method (Jing, Guan, & Hill, 2009; Ortega, Schaf, Castanos, & Astolfi, 2008), which requires an accurate system model. Therefore, coordinated adaptive passive control (CAPC) has been developed for the control of generators and TCSC devices by Sun, Zhao, and Dimirovski (2009) and Fang and Wang (2012). However, it is based on the estimation of unknown constant or slow-varying parameters, which updating law can only deal with an uncertain damping coefficient and a nonlinear TCSC unmodelled dynamics satisfying a linear growth condition. These assumptions restrict its application in real multimachine power system operations as the modelling uncertainty consists of the inertial constant, time constant, general unmodelled TCSC dynamics and inter-area oscillation. Furthermore, an explicit control Lyapunov function (CLF) needs to be constructed which is difficult in multimachine power systems. Perturbation observer (PO) has been proposed to estimate the lumped uncertainty not considered in the nominal plant model, based on an extended state space model (Chen, Jiang, Yao, & Wu, 2014; Han, 2009; Jiang et al., 2004; Kwon & Chung, 2004; Wu, Jiang, & Wen, 2004), such as sliding-mode control with perturbation estimation (SMCPE) which uses a sliding-mode perturbation observer (SMPO) to reduce the conservativeness of the sliding-mode control (SMC) (Kwon & Chung, 2004), active disturbance rejection controller (ADRC) which designs nonlinear disturbance observers (Han, 2009), and a high-gain perturbation observer (HGPO) or SMPO as an extended-order linear observer (Chen et al., 2014; Jiang et al., 2004; Wu et al., 2004). This paper adopts the HGPO as it provides convenience of stability analysis of closedloop system, as the SMPO (Kwon & Chung, 2004) may suffer the discontinuity of high-speed switching and the nonlinear observer (Han, 2009) is too complex for stability analysis. Main contributions of this paper are highlighted as follows:

 A perturbation estimation based coordinated adaptive passive

 

control (PECAPC) scheme has been proposed, through designing a PO to estimate and compensate the real perturbation which is a lumped term including nonlinearities, parameter uncertainties, unknown dynamics and external disturbances. The proposed approach can partially release the dependence of system model in the conventional PC. Furthermore, it can handle fast-varying unknown dynamics, comparing with parameters estimation based adaptive passive control (APC) which can only estimate unknown constant or slow-varying parameters. The design of PECAPC only requires the range of CLF, while CAPC requires an explicit CLF which is difficult to find in complex nonlinear systems. The PECAPC extends the application of PC from single machine



173

infinite bus (SMIB) to multimachine power systems, to which PC cannot be applied due to too complex model. The effectiveness of the proposed controller is verified by simulation, and the implementation feasibility is verified by hardware-in-the-loop (HIL) tests.

Analysis begins with decomposing the original power system into several subsystems, in which the TCSC reactance and its modulated input are chosen as the system output and input, respectively, such that the relative degree is one. A decentralized stabilizing EC for each generator is designed at first, the uncertain parameters and unmodelled dynamics of generator are lumped into a perturbation which is estimated by a high-gain state and perturbation observer (HGSPO). Then a coordinated TCSC controller is developed through passivation to ensure the whole system stability, the uncertain TCSC parameters and general unmodelled TCSC dynamics are aggregated into another perturbation, which is estimated by a HGPO. Two case studies are undertaken on an SMIB and a three-machine system to evaluate the effectiveness of PECAPC, respectively. Simulation and HIL test results are provided to verify the effectiveness and implementation feasibility of the proposed approach.

2. Power plant model 2.1. A single machine infinite bus system with a TCSC device An SMIB system with a TCSC device is shown in Fig. 1, of which the system dynamics is described as (Sun et al., 2009)

⎧ δ ̇ = ω − ω0 ⎪ ⎪ Eq′ Vs sin δ ⎞ ω0 ⎛ D ⎜ Pm − ⎟ ⎪ ω̇ = − (ω − ω 0 ) + H H X ⎝ d′ Σ + Xtcsc ⎠ ⎪ ⎪ ⎨ (X − X d′ )(Vs cos δ − Eq′ ) 1 K ⎪ Eq̇ ′ = d − Eq′ + c ufd Td0 (X d′ Σ + Xtcsc ) Td0 Td0 ⎪ ⎪ ⎪ Ẋtcsc = − 1 (Xtcsc − Xtcsc0 ) + KT uc + ζtcsc ⎪ T T ⎩ c

c

(1)

where δ and ω denote the angle and relative speed of the generator rotor, respectively; H the inertia constant; Pm the constant mechanical power on the generator shaft; D the damping coefficient; Eq′ and Vs the inner generator voltage and infinite bus voltage; Td0 the d-axis transient short-circuit time constant; Tc the time constant 1 of TCSC; Xd′ Σ = Xt + Xd′ + 2 (XL1 + XL2 ); Xt the transformer reactance; ′ the d-axis generator transient reactance; XL1 and XL2 the transXd mission line reactance; Xtcsc the TCSC reactance and Xtcsc0 the initial TCSC reactance; Kc the gain of excitation amplifier; ufd the excitation voltage; KT the gain of TCSC regulator; uc the TCSC modulated input; and ζtcsc the unmodelled TCSC dynamics. 2.2. A multimachine power system with a TCSC device A multimachine power system with n machines and a TCSC device, where the nth machine is the reference machine, is

Fig. 1. The SMIB system equipped with a TCSC device.

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described by (Jiang et al., 2004)

⎧ δi̇ = ωi − ω 0 ⎪ ⎞ ⎪ Di ω0 ⎛ ⎪ ω̇ i = 2H ⎜⎝ Pmi − ω (ωi − ω 0 ) − Pei ⎟⎠ 0 i ⎪ ⎨ 1 ̇ ′ (ufdi − Eqi ), i = 1, 2, …, n, ⎪ Eqi = Td0i ⎪ ⎪ 1 K ⎪ Ẋtcsc = − (Xtcsc − Xtcsc0 ) + T uc + ζtcsc Tc Tc ⎩

zn = Ψ1 (·). Then extend the original (n − 1) th-order system (5) into the following nth-order system:

z ė = A1z e + B1u2 + B2 Ψ1̇ (·)

where ze = [z1, z2, … , zn − 1, zn ]T . B1 = [0, 0, … , b10 , 0]T ∈ n and B2 = [0, 0, … , 1]T ∈ n . Matrix A1 is

(2)

with

Eqi = Eq′ i + (x di − x d′ i ) Idi n

∑

Eq′ j Yij cos (δi − δj )

n

∑

Eq′ j Yij sin (δi − δj )

j = 1, j ≠ i

where subscript i denotes the variables of the ith machine; δi the relative rotor angle; ωi the generator rotor speed; Eqi and Eq′ i the voltage and transient voltage on the q-axis respectively; Pmi the constant mechanical power input; Pei the electrical power output; Hi the rotor inertia; Td0i the d-axis transient short-circuit time constant; Idi and Iqi the d-axis and q-axis generator current respectively; Yij the equivalent admittance between the ith and jth nodes, which is modified as Yij′ = 1/(1/Yij + Xtcsc ) when a TCSC is equipped between the ith and jth nodes; and Gii the equivalent self-conductance of the ith machine.

3. Perturbation estimation based coordinated adaptive passive control Consider a two-input system in formal form of passive system as follows (Isidori, 1995; Larsena et al., 2003):

z ̇ = Az + B (a (z, y) + b (z, y) u2 + ξ )

(3)

ẏ = α (z, y) + β1 (z, y) u1 + β2 (z, y) u2 + ζ

(4)

where y ∈  is the output and the relative degree from u1 to y is one, which is the basic form in CPC design. z ∈ n − 1 is the state vector of the internal dynamics; a (z, y ) : n − 1 × ↦ and n − 1 ∞ b (z , y ) :  × ↦ are C unknown smooth functions, ξ ∈  and ζ ∈  are modelling uncertainties. α (z, y ), β1 (z, y ) and β2 (z, y ) are all unknown functions defined on n − 1 × . Matrices A and B are

⎡0 ⎢ ⎢0 A = ⎢⋮ ⎢0 ⎢ ⎣0

1 0 ⋯ 0⎤ ⎥ 0 1 ⋯ 0⎥ ⋮⎥ , 0 0 ⋯ 1⎥ ⎥ 0 0 ⋯ 0⎦(n − 1)×(n − 1)

1 ⋯ ⋯ 0⎤ ⎥ 0 1 ⋯ 0⎥ ⋮⎥ 0 0 ⋯ 1⎥ ⎥ 0 0 ⋯ 0⎦n × n

̇ z^e = A1z^e + B1u2 + H (z1 − z^1)

j = 1, j ≠ i

Iqi = −

⎡0 ⎢ ⎢0 A1 = ⎢ ⋮ ⎢0 ⎢ ⎣0

Throughout this paper, x˜ = x − x^ refers to the estimation error of x whereas x^ represents the estimate of x. An nth-order HGSPO is used for the extended nth-order system (7) as

Pei = Eq′ i Iqi + Eq′ 2i Gii Idi = −

(7)

⎡ 0⎤ ⎢ ⎥ ⎢ 0⎥ B = ⎢ ⋮⎥ ⎢ 0⎥ ⎢⎣ ⎥⎦ 1 (n − 1)× 1

The zero dynamics of system (3) is assumed to be stabilizable by u2 and written as

z ̇ = Az + B (a (z, 0) + b (z, 0) u2 + ξ )

(5)

where H = [α1/ϵ, α2/ϵ2, … , αn − 1/ϵn − 1, αn/ϵn]T is the observer gain with 0 < ϵ < 1. The estimation error of HGSPO (8) is calculated as

z˜ ė = A1z˜ e + B2 Ψ1̇ (·) − H (z1 − z^1) The design procedure of HGSPO can be summarized as the following steps: Step 1: Define the perturbation for system (5) as Eq. (6). Step 2: Define a fictitious state to represent the perturbation as zn = Ψ1 (·). Step 3: Extend the original (n  1)th-order system (5) into the extended nth-order system (7). Step 4: Design the nth-order HGSPO (8) to estimate state z and perturbation Ψ1 (·) for the extended nth-order system (7). Step 5: Choose αi = Cni λi such that the pole of HGSPO (8) can be placed at  λ, where i = 1, 2, … , n and λ > 0.Similarly, the perturbation of system (4) is defined as

Ψ2 (·) = α (z, y) + β2 (z, y) u2 + (β1 (z, y) − b20 ) u1 + ζ

The perturbation of system (5) is defined as

Ψ1 (·) = a (z, 0) + (b (z, 0) − b10 ) u2 + ξ

(6)

Define a fictitious state to represent the perturbation, that is,

(9)

Define a fictitious state to represent the perturbation, that is, y2 = Ψ2 (·). Then extend the original first-order system (4) into the following second-order system:

⎧ ẏ = y + b20 u1 2 ⎨ ⎪ ⎩ y2̇ = Ψ2̇ (·) ⎪

(10)

A second-order HGPO is used for system (10) as

⎧ ^̇ α1′ ^ ^ ⎪ ⎪ y = Ψ2 + ϵ′ (y − y ) + b20 u1 ⎨ α′ ⎪ ^̇ Ψ (·) = 22 (y − y^ ) ⎪ ⎩ 2 ϵ′

(11)

where 0 < ϵ′⪡1. ˜ ϵ′, Define the scaled estimation errors of HGPO (11) as η′1 = y/ η2′ = Ψ˜ 2 (·), and η′ = [η1′, η2′]T , gives

ϵ′η′̇ = A1′ η′ + ϵ′B1′ Ψ2̇ (·)

(12)

with

⎡ − α1′ 1⎤ ⎥, A1′ = ⎢ ⎣ − α2′ 0⎦ 3.1. Design of HGSPO and HGPO (Jiang et al., 2004)

(8)

⎡ 0⎤ B1′ = ⎢ ⎥ ⎣ 1⎦

where α′1 and α′2 are chosen such that A1′ is a Hurwitz matrix. The design procedure of HGPO is similar to that of HGSPO. Normally the pole of HGSPO (8) and HGPO (11) is chosen to be 10 times larger than the dominant pole of the equivalent linear system of (5) and (4), respectively, which can ensure a fast estimation

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

175

of perturbation Ψ1 (·) and Ψ2 (·). Note that one only needs the measurement of state z1 and control input u2 for the design of HGSPO (8), and the measurement of output y and control input u1 for the design of HGPO (11). The effectiveness of the PO has been discussed in our previous work (Chen et al., 2014; Jiang et al., 2004; Wu et al., 2004).

with

Remark 1. It should be mentioned that during the design procedure, ϵ and ϵ′ used in HGSPO (8) and HGPO (11) are required to be some relatively small positive constants only, and the performance of HGSPO and HGPO is not very sensitive to the observer gains, which are determined based on the upper bound of the derivative of perturbation.

where q˜ (z ) represents the zero dynamics, p˜ (z, y ) denotes the difference between the original system and zero dynamics, which will be cancelled later by u1. K = [k1, k2, …, k n − 1] is the control gain which makes matrix A − BK Hurwitzian, such that the following condition can be satisfied:

3.2. Design of stabilizing controller u2 and coordinated controller u1

where α is a class- 2 function, and η = z˜2 …, z˜n ] ∈ n . The proof of inequality (16) is given in Jiang et al. (2004). The structure of stabilizing controller (13) is illustrated in Fig. 2. The nominal plant is disturbed by the perturbation Ψ1 (·), the sta−1 bilizing controller u2 can be separated as u2 = b10 (us1 + us2 ), where u = − Kz^ is the state feedback and choose k = C i ξ i to place the

Based on the standard CPC design procedure (Larsena et al., 2003), the stabilizing controller u2 is designed first as follows:

u2 = γ (z^e ) =

1 ( − Kz^ − z^n ) b10

(13)

which renders system (3) into

z ̇ = Az + B (a (z, y) + b (z, y) γ (z^e ) + ξ ) = q˜(z ) + p˜ (z, y) y

q˜(z ) = Az + B (a (z, 0) + b (z, 0) γ (z^e ) + ξ )

(14)

and

p˜ (z, y) = B (a (z, y) − a (z, 0) + (b (z, y) − b (z, 0)) γ (z^e )) y−1

∂W ∂W (z ) Ẇ = q˜(z ) + T η ̇ ≤ − α (∥ z ∥) ∂η ∂zT

(16) [z˜1/ϵn − 1,

s1

(15)

i

/ϵn − 2,

n−1

pole of the equivalent linear system of (5) at  ξ, where i = 1, …, n − 1 and ξ > 0; and us2 = − z^n compensates the combinatorial effect of parameter uncertainties, external disturbances

Fig. 2. The structure of stabilizing controller u2.

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B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

Σ

Σ

Σ

Fig. 3. The structure of coordinated controller u1.

1.015 EC alone EC+TCSC EC+TCSC

60

Rotor speed ω (p.u.)

Rotor angle δ (deg)

70

app

50 40 30 20 0

1

2 3 Time (sec)

4

app

1.005 1 0.995

1

2 3 Time (sec)

4

5

0.1

5

(p.u.)

EC alone EC+TCSC EC+TCSC

tcsc

app

TCSC reactance X

fd

Excitation voltage u (p.u.)

10

0

−5

−10 0

1.01

0.99 0

5

EC alone EC+TCSC EC+TCSC

1

2 3 Time (sec)

4

5

EC alone EC+TCSC EC+TCSC

0.05

app

0

−0.05

−0.1 0

1

2 3 Time (sec)

4

5

Fig. 4. System responses obtained with the EC alone, coordinated EC and TCSC controller, and approximated EC and TCSC controller in the SMIB system.

and unmodelled dynamics. The coordinated controller u1 is designed as

⎧ ⎛ ^ ⎞ ∂W −1 ⎜ ⎪ u1 = b20 −Ψ2 (·) − k′y − T p˜ (z, y) + ν⎟ ⎨ ⎝ ⎠ ∂z ⎪ ⎩ ν = − ϕ (y)

(17)

where ν is the additional input, ϕ is any smooth function such that ϕ (0) = 0 and yϕ (y ) > 0 for all y ≠ 0. k′ > 1 is the feedback control gain. The structure of coordinated controller (17) is illustrated in Fig. 3. The nominal plant is disturbed by the perturbation Ψ2 (·), the coordinated controller u1 can be separated as

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

70

1.015 PID+LL CAPC PECAPC

50 40 30

1.005 1 0.995

20

0.99

10 0

1

2 3 Time (sec)

4

0.985 0

5

TCSC reactance Xtcsc (p.u.)

PID+LL CAPC PECAPC

5

0

−5

−10 0

1

2 3 Time (sec)

4

5

0.15

10

fd

PID+LL CAPC PECAPC

1.01 Rotor speed ω (p.u.)

Rotor angle δ (deg)

60

Excitation voltage u (p.u.)

177

1

2 3 Time (sec)

4

5

PID+LL CAPC PECAPC

0.1 0.05 0 −0.05 −0.1 −0.15 0

1

2 3 Time (sec)

4

5

Fig. 5. System responses obtained under the nominal model in the SMIB system.

−1 u1 = b20 (uc1 + uc2 + uc3 + uc4 ), where uc1 = − Ψ^2 (·) is the dynamical perturbation compensation; uc2 = − k′y places the pole of the equivalent linear system of (4) at −k′, where k′ > 0; uc3 = (∂W /∂zT ) p˜ (y, z ) coordinates the two controllers by cancelling the difference between the original system and zero dynamics represented by p˜ (y, z ); and uc4 = ν constructs a passive system by introducing an additional input in the form of a sector-nonlinearity ϕ (y ).

controller u2 and coordinated controller u1 is given in Theorem 1. Theorem 1. Consider systems (3) and (4), with controllers (13) and (17), and let assumptions A1 and A2 hold; then ∃ ϵ1⁎, ϵ1⁎ > 0 such that, ∀ ϵ′, 0 < ϵ′ < ϵ1⁎, then the closed-loop system is passive and its origin is stable. The following assumptions are made on systems (7) and (10). A1. b10 and b20 are chosen to satisfy:

Remark 2. Note that (∂W (z ) /∂zT ) p˜ (z, y ) = (∂W (z ) /∂zn − 1) p˜n − 1 (z, y ) as p˜ i (z, y ) = 0, i = 1, 2, … , n − 2, which can be interpreted as the distributed energies in a complex system and needs to be reshaped. One can choose (∂W (z ) /∂zT ) p˜ (z, y ) = czn − 1p˜n − 1 (z, y ) regardless of the system order, where c is called the coordination coefficient.

0 < |b (z, 0)/b10 − 1| ≤ θ1,

To this end, PECAPC design for systems (3) and (4) can be summarized as:

|Ψk (z, y , u)| ≤ γ1,

Step 1: Obtain zero dynamics (5) of system (3) by setting y¼0. Step 2: Extend system (5) into system (7), and design HGSPO (8) to obtain estimates z^ and z^n . Step 3: Design stabilizing controller (13). Step 4: Extend system (4) into system (10), and design HGPO (11) to obtain the estimate Ψ^2 (·). Step 5: Design coordinated controller (17). 3.3. Closed-loop system stability The proof of stability of the closed-loop system with stabilizing

0 < |β1 (z, y)/b20 − 1| ≤ θ 2

where θ1 < 1 and θ 2 < 1 are positive constants. A2. The perturbation Ψk (·) and its derivative Ψk̇ (·) are Lipschitz in their arguments and bounded over the domain of interest and are globally bounded in (z,y):

|Ψk̇ (z, y , u)| ≤ γ2

where γ1 and γ2 are positive constants. In addition, Ψk (0, 0, 0) = 0, Ψk̇ (0, 0, 0) = 0, k ¼1,2. It guarantees that the origin is an equilibrium point of the open-loop system. Proof. The closed-loop system (4) using controller (17) is

⎧ ∂W (z ) ⎪ ẏ = η2′ − k′y − p˜ (z, y) + ν ⎨ ∂zT ⎪ ̇ ⎩ ϵ′η′ = A1′ η′ + ϵ′B1′ Ψ2̇ (·)

(18)

Define a Lyapunov function V (η′) = η′ T P1η′, where P1 is the positive definite solution of the Lyapunov equation P1A1′ + A1′ T P1 = − I . For closed-loop systems (3) and (4) by controllers (13) and (17),

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B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

70

1.015 PID+LL CAPC PECAPC

1.01 Rotor speed ω (p.u.)

Rotor angle δ (deg)

60 50 40 30

1.005 1 0.995

20

0.99

10 0

1

2 3 Time (sec)

4

0.985 0

5

PID+LL CAPC PECAPC

5

TCSC reactance Xtcsc (p.u.)

fd

1

2

3 Time (sec)

4

5

0.15

10 Excitation voltage u (p.u.)

PID+LL CAPC PECAPC

0

−5

−10 0

1

2 3 Time (sec)

4

0.1 0.05 0 −0.05 −0.1 −0.15 0

5

PID+LL CAPC PECAPC

1

2 3 Time (sec)

4

5

Fig. 6. System responses obtained under an unmodelled TCSC dynamics ζtcsc = 10 sin (Xtcsc − Xtcsc0 ) in the SMIB system.

choose a storage function as

where

1 H (z, η, y , η′) = W (z, η) + 2 y2 + βV (η′)

(19)

where β > 0. Furthermore, assuming

∥ Ψ2̇ (·)∥ ≤ L1 ∥ y ∥ + L2 ∥ η′ ∥ where L1 and L2 are Lipschitz constants. By differentiating H (z, η, y, η′) along the trajectory of system (18) and condition (16), it yields

⎛ ⎞ ∂W ∂W ∂W Ḣ = T (q˜(z ) + p˜ (z, y) y) + T η ̇ + y ⎜ η2′ − k′y − T p˜ (z, y) + ν⎟ ⎝ ⎠ ∂z ∂η ∂z ⎞ ∂V ⎛ A ′ η′ + β T ⎜ 1 + B1′ Ψ2̇ (·) ⎟ ⎠ ∂η′ ⎝ ϵ′ ≤ − α (∥ z ∥) + yν − ∥ y

∥2

⎞ 1 2⎛1 − ⎜ + βL1 ∥ P1 ∥⎟, ⎠ 2 ϵ0 ⎝ 2

b2 =

β − 2β (ϵ 0 L1 + L2 )∥ P1 ∥ − ϵ 0 2ϵ′

with ϵ0 > 0. One can choose β small enough and b1 > 0, then choose ϵ0 ≥ ϵ⁎0 = 2 + 4βL1 ∥ P1 ∥ such that ϵ1⁎ = β /(ϵ⁎02 + 4βL2 ∥ P1 ∥), ∀ ϵ′, ϵ′ ≤ ϵ1⁎, and choose the additional input ν = − ϕ (y ), where ϕ (y ) is a sector-nonlinearity satisfying yϕ (y ) > 0. It can be shown that

Ḣ ≤ − α (∥ z ∥) − min (1/2, β/2ϵ′)(∥ y ∥2 + ∥ η′ ∥2 ) + yν ≤ − α (∥ z ∥) − α1 (∥ y ∥) + yν ≤ yν ≤ − yϕ (y) ≤ 0

β − ∥ η′ ∥2 + 2βL2 ∥ P1 ∥∥ η′ ∥2 ϵ′

+ (1 + 2βL1 ∥ P1 ∥)∥ y ∥∥ η′ ∥ ≤ − α (∥ z ∥) + yν − ∥ y ∥2 −

β ∥ η′ ∥2 + 2βL2 ∥ P1 ∥∥ η′ ∥2 ϵ′

+ (1 + 2βL1 ∥ P1 ∥) ⎛ 1 ⎞ ×⎜ ∥ y ∥2 + ϵ 0 ∥ η′ ∥2 ⎟ ⎝ ϵ0 ⎠ ≤ − α (∥ z ∥) + yν −

b1 =

β 1 ∥ y ∥2 − ∥ η′ ∥2 − b1 ∥ y ∥2 − b2 ∥ η′ ∥2 2 2ϵ′

where α1 is a class- 2 function. One can conclude that the closedloop system is passive and its origin is stable.□ Proposition 1. PECAPC can be easily extended into multi-input systems. If there exist m subsystems for system (3) with m control inputs u2j, the vector variables of states and estimation errors for each subsystem are denoted as z ⁎j and ηj⁎, j = 1, 2, … , m, respectively. Controller (13) using HGSPO (8) can decouple each subsystem by rendering their interactions into a perturbation. Hence u2j can stabilize the jth subsystem, which results in an equivalent CLF W (z⁎, η⁎) = W1 + W2 + ⋯ + Wm , and condition (16) becomes

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

1.02

PID+LL CAPC PECAPC

60

Rotor speed ω (p.u.)

Rotor angle δ (deg)

80

40

20

0 0

1

2 3 Time (sec)

4

1

0.99

0.98 0

5

1

2 3 Time (sec)

4

5

0.15

0

−5

−10 0

1

2 3 Time (sec)

4

5

PID+LL CAPC PECAPC

0.1

tcsc

5

(p.u.)

PID+LL CAPC PECAPC

TCSC reactance X

fd

PID+LL CAPC PECAPC

1.01

10 Excitation voltage u (p.u.)

179

0.05 0 −0.05 −0.1 −0.15 0

1

2 3 Time (sec)

4

5

Fig. 7. System responses obtained under an inter-area type disturbance Vs = 1 + 0.1 sin (5t ) in the SMIB system.

⎞ ∂W ∂W q˜ (z ⁎ ) + ⁎ T η⁎j̇ ⎟⎟ ≤ ⁎T j j ∂z j ∂ηj ⎠ j=1 ⎝ m

Table 1 SMIB system parameters (in p.u.).

Ẇ =

Vt = 0.9669 Xt = 0.127 Xd′ = 0.257

H¼7 XL1 = 0.173 Td0 = 6.9

D¼ 0.5 XL2 = 0.3122 Tc = 0.06

Pm = 0.9 Xd = 1.863 KT = 20

Kc = 1 Xtcsc0 = 0

Vs = 1.0

δ0 = 0.5892

ω0 = 100π

⎛

∑ ⎜⎜

m

∑ αj (∥ z⁎j ∥) j=1

Remark 3. For the closed-loop system (18), control gain k′ should be designed to suppress the perturbation estimation error Ψ˜ 2 (·). Compared to the approach without perturbation compensation, in which k′ should be chosen to suppress the perturbation Ψ2 (·). As Ψ2 (·) is normally larger than Ψ˜ 2 (·), a smaller k′ could be resulted in due to the compensation of perturbation by PECAPC. Remark 4. Denote (z0, y0 ) ∈ Γo and z0′ ∈ Γz as the equilibrium point of systems (3) and (5), respectively, where Γo and Γz are their stability region, which are unequal during transient process. However, with a proper coordination coefficient c, coordinated controller (17) will exponentially drive limt →∞ (z0, y0 ) = z0′ and limt →∞Γo = Γz as limt →∞y = 0.

Fig. 8. The three-machine system equipped with a TCSC device.

Remark 5. Compared to the CAPC (Fang & Wang, 2012; Sun et al., 2009) which usually estimates the linearly parametric uncertainties, the PECAPC can be regarded as a nonlinearly functional estimation method, as it can estimate the combinatorial effect of fast time-varying unknown parameters, unknown nonlinear dynamics and external disturbances. If there does not exist uncertainties and external disturbances, and if the accurate system model is available, such a controller provides the same performance as the exact passive controller. Otherwise, such a controller performs much better than the exact passive controller. The use of PO leads to less concern over the measurement and identification of the fast time-varying unknown parameters, unknown nonlinear dynamics and external disturbances. This tends to require less

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

25

Rotor speed difference ω23 (p.u.)

Rotor angle difference δ23 (deg)

180

PID+LL CPC PECAPC

20

15

10

5 0

1

2 3 Time (sec)

4

0.04

0.02

0

−0.02

−0.04 0

5

PID+LL CPC PECAPC

5

TCSC reactance Xtcsc (p.u.)

(p.u.) fd1

Excitation voltage u

1

2 3 Time (sec)

4

5

0.3

10

0

−5

−10 0

PID+LL CPC PECAPC

1

2 3 Time (sec)

4

PID+LL CPC PECAPC

0.2 0.1 0 −0.1 −0.2 −0.3 0

5

1

2 3 Time (sec)

4

5

Fig. 9. System responses obtained under operation Type I and the nominal model in the three-machine system.

control efforts as the perturbation has already included all of this information.

Ψ1 (·) = −

⎡ Eq′ Vs sin δ ⎞ ⎤ D⎢ D ω ⎛ ω 0 Vs ⎟⎟ ⎥ − − (ω − ω 0 ) + 0 ⎜⎜ Pm − H ⎢⎣ H H ⎝ X d′ Σ + Xtcsc0 ⎠ ⎥⎦ H (X d′ Σ + Xtcsc0 )

⎡ ⎞⎤ sin δ ⎛ (X d − X d′ )(Vs cos δ − Eq′ ) ⎜ × ⎢ Eq′ (ω − ω 0 ) cos δ + − Eq′ ⎟ ⎥ ⎢⎣ Td0 ⎝ (X d′ Σ + Xtcsc0 ) ⎠ ⎥⎦

4. PECAPC design of excitation and TCSC

K ω 0 Vs sin δ × c ufd − b10 ufd H (X d′ Σ + Xtcsc0 ) Td0

Controllers (13) and (17) will be applied for both SMIB system (1) and multimachine power system (2).

−

4.1. Controller design for a single machine infinite bus system

Defining a fictitious state as z 4 = Ψ1 (·), and the extended state variable is denoted as ze = [z1, z2, z3, z 4 ]T , the fourth-order state equation is

For SMIB system (1), choose y = Xtcsc − Xtcsc0 , u1 = uc and u2 = ufd . By setting y¼0, the zero dynamics of system (1) is

z ė = A1z e + B1ufd + B2 Ψ1̇ (·)

⎧ δ ̇ = ω − ω0 ⎪ E ′ Vs sin δ ⎞ ⎪ ω ⎛ D ⎪ ω̇ = − (ω − ω 0 ) + 0 ⎜ Pm − q ⎟ ⎨ H H ⎝ X d′ Σ + Xtcsc0 ⎠ ⎪ ⎪ ̇ ′ (X d − X d′ )(Vs cos δ − Eq′ ) K 1 − Eq′ + c ufd ⎪ Eq = Td0 (X d′ Σ + Xtcsc0 ) Td0 Td0 ⎩

(21)

(22)

where

(20)

Choose z1 = δ − δ0 for system (20), where δ0 is the initial generator rotor angle. Differentiating z1 until the excitation control input ufd appears explicitly, the perturbation Ψ1 (·) is obtained as

⎡0 ⎢ 0 A1 = ⎢ ⎢0 ⎢⎣ 0

1 0 0 0

0 1 0 0

0⎤ ⎥ 0⎥ , 1⎥ 0⎥⎦

⎡ 0 ⎤ ⎥ ⎢ 0 ⎥ , B1 = ⎢ ⎢ b10 ⎥ ⎥ ⎢⎣ 0 ⎦

⎡ 0⎤ ⎢ ⎥ 0 B2 = ⎢ ⎥ ⎢ 0⎥ ⎢⎣ 1⎥⎦

A fourth-order HGSPO (8) is given as

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

0.03

−5

3

x 10

z

State z12 (p.u.)

(p.u.) 11

State z

1 0 −1

0.01 0 −0.01 −0.02

−2 −3 0

0.5

1 Time (sec)

1.5

−0.03 0

2

0.5

1 Time (sec)

1.5

2

1000

20

z13 Perturbation Ψ (p.u.)

z

13est

10

z

(p.u.) 13

z12err

0.02

11err

2

State z

181

1

13err

500

0

−10

0 −500

Ψ1 Ψ

−1000

1est

Ψ1err

−20

0

0.5

1 Time (sec)

1.5

2

−1500

0

0.5

1 Time (sec)

1.5

2

Fig. 10. Estimation errors of HGSPO1 for G1.

0.08

y

Perturbation ψtcsc (p.u.)

Output y (p.u.)

300

err

0.06 0.04 0.02 0 −0.02 −0.04 −0.06

0

0.5

1

1.5

2

Ψtcsc Ψtcscest

200

Ψtcscerr

100 0 −100 −200

0

0.5

1 Time (sec)

Time (sec) Fig. 11. Estimation errors of HGPO for TCSC.

1.5

2

182

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

Table 2 Three-machine system generator parameters (in p.u.). Parameters

G1

G2

G3

xd xd′

1.86 0.335

1.67 0.306

1.75 0.342

Td0 H D

6.9 5.25 2.5

6.9 4.5 2.5

6.9 4 2.5

Table 3 Three-machine system transmission line parameters (in p.u.). Line no.

Impedance

5–7 4–5(1) 4–5(2) 4–6 6–9 7–8 8–9

0.405 0.23 0.23 0.205 0.185 0.325 0.255

P L1 = 3.45 P L3 = 1.5

P L2 = 2.55

nominal parameter 50% parameter increase

20

0.04

Rotor speed ω (deg)

0

21

Rotor angle δ21 (deg)

40

−20 −40 −60

0

1

2 3 Time (sec)

4

0.02 0 −0.02 −0.04 −0.06

5

nominal parameter 50% parameter increase 0

1

2 3 Time (sec)

4

5

nominal parameter 50% parameter increase

(p.u.)

0.3

tcsc

5

TCSC reactance X

Excitation voltage u

fd1

(p.u.)

10

0

−5

−10

0

1

2 3 Time (sec)

4

5

nominal parameter 50% parameter increase

0.2 0.1 0 −0.1 −0.2 −0.3

0

1

2 3 Time (sec)

4

5

Fig. 12. The effect of a 50% parameter increase on the dynamic response of proposed controller without perturbation observer in the three-machine system.

20

0.02

15

0.01 Rotor speed ω21 (deg)

21

Rotor angle δ (deg)

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

10 5 0 −5 −10 nominal parameter 50% parameter increase

−15 −20

0

1

2 3 Time (sec)

0 −0.01 −0.02 −0.03 nominal parameter 50% parameter increase

−0.04

4

−0.05

5

183

0

1

2 3 Time (sec)

4

5

nominal parameter 50% parameter increase

(p.u.)

0.3

tcsc

5

TCSC reactance X

Excitation voltage u

fd1

(p.u.)

10

0

−5

−10

0

1

2 3 Time (sec)

4

0.1 0 −0.1 −0.2 −0.3

5

1200

1

2 3 Time (sec)

4

5

nominal parameter 50% parameter increase

(p.u.)

20 10

tcsc

800

Perturbation Ψ

Perturbation Ψ1 (p.u.)

0

30 nominal parameter 50% parameter increase

1000

600 400 200 0

nominal parameter 50% parameter increase

0.2

0 −10 −20

0

1

2 3 Time (sec)

4

5

−30

0

1

2 3 Time (sec)

4

5

Fig. 13. The effect of a 50% parameter increase on the dynamic response of proposed controller with perturbation observer in the three-machine system.

⎧ ^̇ ^ α1 ^ ⎪ z1 = z2 + ϵ (z1 − z1) ⎪ ⎪ z^2̇ = z^3 + α2 (z1 − z^1) ⎪ ϵ2 ⎨ ̇ ^ ^ ⎪ z3 = Ψ1 (·) + α3 (z1 − z^1) + b10 ufd ⎪ ϵ3 ⎪ ̇ α4 ^ ⎪ Ψ1 (·) = 4 (z1 − z^1) ⎩ ϵ

where

Ψtcsc (·) = −

(23)

⎪

⎪

(25)

A second-order HGPO (11) is used to obtain Ψ^tcsc (·) as

Extending the TCSC dynamics, it yields

⎧ ẏ = Ψtcsc (·) + b20 uc ⎨ ̇ (·) ⎩ y2̇ = Ψtcsc

1 K y + T uc − b20 uc + ζtcsc Tc Tc

(24)

⎧ ^̇ α1′ ^ ^ ⎪ ⎪ y = Ψtcsc (·) + ϵ (y − y ) + b20 uc ⎨ α′ ⎪ ^̇ Ψ (·) = 22 (y − y^ ) ⎪ ⎩ tcsc ϵ

(26)

For system (1), one can obtain p˜ (z, y ) from (21) according to (15) as

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

25

Rotor speed difference ω23 (p.u.)

Rotor angle difference δ23 (deg)

184

20 15 10 PID+LL CPC PECAPC

5 0

0

1

2

3

4

0.04

0.02

0

−0.02

−0.04

5

PID+LL CPC PECAPC

0

1

Time (sec)

PID+LL CPC PECAPC

5

TCSC reactance Xtcsc (p.u.)

(p.u.) fd1

Excitation voltage u

3

4

5

0.3

10

0

−5

−10

2

Time (sec)

0

1

2

3

4

5

PID+LL CPC PECAPC

0.2 0.1 0 −0.1 −0.2 −0.3

0

1

2

3

4

5

Time (sec)

Time (sec)

Fig. 14. System responses obtained under operation Type I and the parameter uncertainties in the three-machine system.

p˜ (z, y)

p˜ ⁎ (z, y) =

ω V ⎡ 1 ⎛ −DEq′ sin δ ⎜ = 0 s⎢ + Eq′ (ω − ω 0 ) cos δ H ⎢⎣ XΔ ⎝ H

(27)

where and X′Δ = XΔ = (Xd′ Σ + Xtcsc0 + y )(Xd′ Σ + Xtcsc0 ) (2Xd′ Σ + 2Xtcsc0 + y ) . The PECAPC-based EC and TCSC controller are

⎧ 1 ( − Ψ^1 (·) − k1z^3 − k2 z^2 − k3 z^1) ⎪ ufd = b10 ⎪ ⎪ ⎨ 1 ( − Ψ^tcsc (·) − cz^3 p˜ (z, y) − k′y + ν ) ⎪ uc = b20 ⎪ ⎪ ⎩ ν = − βy

(29)

To this end, replace p˜ (z, y ) by its approximation p˜ ⁎ (z, y ), controller (28) becomes

⎞ sin δ + × ( − Eq′ + Kc ufd ) ⎟ Td0 ⎠ X′ (X d − X d′ )(Vs cos δ − Eq′ ) sin δ ⎤ ⎥ + Δ2 Td0 XΔ ⎦⎥

⎞ sin δ ω 0 Vs ⎛ ( − Eq′ + Kc ufd ) ⎟ ⎜ Eq′ (ω − ω 0 ) cos δ + ⎠ HXΔ ⎝ Td0

⎧ 1 ( − Ψ^1 (·) − k1z^3 − k2 z^2 − k3 z^1) ⎪ ufd = b 10 ⎪ ⎪ ⎨ ⁎ 1 ( − Ψ^tcsc (·) − cz^3 p˜ ⁎ (z, y) − k′y + ν ) ⎪ uc = b 20 ⎪ ⎪ ⎩ ν = − βy

(30)

Based on assumption A1, constants b10 and b20 must satisfy the following inequalities when the generator operates within its normal region:

b10 < − ω 0 Vs Kc sin δ/[2HTd0 (X d′ Σ + Xtcsc0 )] b20 > KT/(2Tc )

(28)

A known p˜ (z, y ) is a fundamental assumption in CPC design (Larsena et al., 2003), which contains the system states and parameters and needs to be cancelled for passivation. In real power system operations, the damping coefficient D is small compared to the system inertia H thus and |DEq′ sin δ /H| ≈ 0, |(X′Δ/XΔ2 )(Xd − Xd′ )(Vs cos δ − Eq′ ) |⪡| − Eq′ + Kc ufd | during the transient process due to the large excitation control input ufd, thus one can approximate p˜ (z, y ) by ignoring the relatively small components. Denoting p˜ ⁎ (z, y ) as its approximation, it gives

During the most severe disturbance, the electric power will reduce from its initial value to around zero within a short period of time, Δ. Thus the boundary values of the estimated system states can be |Ψ^1 (·)| ≤ ω0 Pm/(HΔ), obtained as and |z^3 | ≤ ω0 Pm/H , ̇ ^ 2 |Ψ1 (·)| ≤ ω0 Pm/(HΔ ), respectively. 4.2. Controller design for a multimachine power system For multimachine power system (2), choose y = Xtcsc − Xtcsc0 , u1 = uc and u2i = ufdi . Setting y¼ 0 in the equivalent admittance matrix Y, the zero dynamics can be obtained. Choose zi1 = δ i − δ i0 ,

(p.u.)

30 20 PID+LL CPC PECAPC

10

0

1

2 3 Time (sec)

4

0.02 0 −0.02 −0.04 0

1

2 3 Time (sec)

4

5

0.3

PID+LL CPC PECAPC

5

TCSC reactance Xtcsc (p.u.)

(p.u.) fd1

Excitation voltage u

0.04

−0.06

5

10

0

−5

−10

PID+LL CPC PECAPC

23

40

0

185

0.06

50

Rotor speed difference ω

Rotor angle difference δ23 (deg)

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

0

1

2

3

4

5

PID+LL CPC PECAPC

0.2 0.1 0 −0.1 −0.2 −0.3

0

Time (sec)

1

2

3

4

Time (sec)

Fig. 15. System responses obtained under operation Type II and the parameter uncertainties in the three-machine system.

Fig. 16. The conventional PID þLL controller structure.

5

186

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

Table 4 Three-machine system operation Type I. Generator

Table 7 Three-machine system PID þLL parameters (in p.u.).

P (p.u.)

Q (p.u.)

0.5821 0.3577 0.1486

G1 G2 G3

0.9124 0.8036 0.3040

V (p.u.) ∠θ (deg)

1.1295∠0.0 1.034∠8.9 1.1372∠ − 7.9

TR = 0.01

Kpss = 30

Tw = 10

T1 = 0.2

T2 = 0.1 KD = − 0.5

T3 = 0.3 Ke = 100

T4 = 0.1 KP = − 10

KI = − 5

Table 8 IAE index of different control schemes. Table 5 Three-machine system operation Type II.

The SMIB system case Method Case

Generator

P (p.u.)

Q (p.u.)

V (p.u.) ∠θ (deg)

G1 G2 G3

0.5356 0.6132 0.1103

0.7052 0.8533 0.2965

1.1488∠0.0 1.1171∠17.8 1.1524∠ − 11.9

Nominal model

IAEδ PIDþ LL 18.500 CAPC 16.370 PECAPC 18.030

Table 6 SMIB system PIDþ LL parameters (in p.u.).

TR = 0.01

Kpss = 40

Tw = 10

T1 = 0.3

T3 = 0.3 Ke = 25

T4 = 0.1 KP = − 1

KI = − 1

i = 1, 2, … , n, where δi0 is the initial rotor angle of the ith generator. Differentiating zi1 until the excitation control input ufdi appears explicitly, the perturbation Ψi (·) is obtained as

−

⎤ Iqi + 2G ii Eq′ i dIqi ω 0 ⎡ Di d ω i ⎢ + Eq′ i + ( − Eq′ i − (x di − x d′ i ) Idi ) ⎥ 2Hi ⎣ ω 0 dt dt Td0i ⎦ 2Hi Td0i

ufdi − b10i ufdi

(31)

Defining a fictitious state as zi4 = Ψi (·), and the extended state variable is denoted as zie = [zi1, zi2, zi3, zi4 ]T , the fourth-order state equation is

zi̇ e = Ai1zie + Bi1ufdi + Bi2 Ψi̇ (·)

(32)

where

⎡0 ⎢ 0 A i1 = ⎢ ⎢0 ⎢⎣ 0

1 0 0 0

0 1 0 0

0⎤ ⎥ 0⎥ , 1⎥ ⎥ 0⎦

⎡ 0 ⎤ ⎥ ⎢ 0 ⎥ , Bi1 = ⎢ ⎢ b10i ⎥ ⎥⎦ ⎢⎣ 0

IAEω

IAEδ

IAEω

IAEδ

IAEω

3.221 2.830 1.193

21.480 17.710 17.500

3.890 2.875 1.207

29.300 33.480 14.900

4.760 5.248 1.500

Nominal model

IAEδ

ω 0 (Iqi + 2G ii Eq′ i )

Inter-area type disturbance

The three-machine system case Method Case

T2 = 0.1 KD = − 0.1

Ψi (·) = −

Unmodelled TCSC dynamics

⎡ 0⎤ ⎢ ⎥ 0 Bi2 = ⎢ ⎥ ⎢ 0⎥ ⎢⎣ 1⎥⎦

A fourth-order HGSPO (8) is used for the ith generator as

⎧ ^̇ αi 1 ^ ^ ⎪ zi1 = zi2 + ϵ (zi1 − zi1) ⎪ ⎪ z^i̇ 2 = z^i3 + αi2 (zi1 − z^i1) ⎪ ϵ2 ⎨ ⎪ z^i̇ 3 = Ψ^i (·) + αi3 (zi1 − z^i1) + b10i ufdi ⎪ ϵ3 ⎪ ̇ αi 4 ^ ⎪ Ψi (·) = 4 (zi1 − z^i1) ⎩ ϵ

PIDþ LL 8.082 CPC 4.817 PECAPC 5.942

IAEδ

IAEω

IAEδ

IAEω

6.293 2.617 2.756

8.606 18.650 6.273

6.871 5.849 3.015

22.540 43.650 11.980

12.200 16.790 8.016

(34)

ω 0 XΔ⁎ ⎛ Eq′ k ⎜ (ufdj − Eq′ j ) sin δjk 2Hj ⎝ Td0j +

tcsc

IAEω

where p˜j (z, y ) and p˜k (z, y ) are calculated from (31) according to (15), in which TCSC device is installed between the jth and kth machine. Thus one chooses i¼ j and i¼k in (31) to calculate p˜j (z, y ) and p˜k (z, y ), respectively, using the values given in Appendix A. Similarly, in real power system operations, |Dj Eq′ j Eq′ k sin δjk/(2Hj )| ≈ 0 as damping coefficient Dj ⪡Hj , |(2Gjj /Td0j )(xdj − xd′ j )|⪡|ωjk | as the selfconductance Gjj is much smaller than time constant Td0j . Furthermore n |(xdk − xd′ k )( ∑i = 1, i ≠ k, j Eq′ i Yki cos δki + (XΔ⁎′/XΔ⁎ ) Eq′ j cos δjk )|⪡|ufdk − Eq′ k | and n |(xdj − xd′ j )( ∑i = 1, i ≠ j, k Eq′ i Yji cos δji + (XΔ⁎′/XΔ⁎ ) Eq′ k cos δkj )|⪡|ufdj − Eq′ j | during the transient process due to the large excitation control inputs ufdk and ufdj , thus one can approximate p˜j (z, y ) and p˜k (z, y ) by ignoring the relatively small components. Denoting their approximation as p˜ j⁎ (z, y ) and p˜k⁎ (z, y ), respectively, gives

(33)

The PECAPC-based EC and TCSC controller for system (2) are

Parameter uncertainties (Type II)

⎧ 1 ( − Ψ^i (·) − ki1z^i3 − ki2 z^i2 − ki3 z^i1) ⎪ ufdi = b 10i ⎪ ⎪ ⎨ 1 ( − Ψ^tcsc (·) − cj z^j3 p˜j (z, y) − ck z^k3 p˜ k (z, y) + ν ) ⎪ uc = b 20 ⎪ ⎪ ⎩ ν = − βy , i = 1, 2, …, n

p˜ ⁎j (z, y) =

Consider the equivalent two-machine subsystem involving the TCSC device and denote them as the jth and kth machine, namely, the TCSC device is installed between the jth and kth machine. Furthermore, each machine denoted by the ith machine is equipped with its own EC. The extended TCSC dynamics is the same as system (24), and the same HGPO (26) is used to obtain Ψ^ (·).

Parameter uncertainties (Type I)

p˜k⁎ (z, y) =

Eq′ j Td0k

⎞ (ufdk − Eq′ k ) sin δjk + Eq′ j Eq′ k ωjk cos δjk ⎟ ⎠

ω 0 XΔ⁎ ⎛ Eq′ j ⎜ (ufdk − Eq′ k ) sin δkj 2Hk ⎝ Td0k +

⎞ (ufdj − Eq′ j ) sin δkj + Eq′ k Eq′ j ωkj cos δkj ⎟ Td0j ⎠ Eq′ k

As a result, one can replace

p˜j (z, y ) and

p˜k (z, y ) by their

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

187

Fig. 17. The configuration of the HIL test.

approximation p˜ j⁎ (z, y ) and p˜k⁎ (z, y ), respectively. Controller (34) becomes

⎧ 1 ( − Ψ^i (·) − ki1z^i3 − ki2 z^i2 − ki3 z^i1) ⎪ ufdi = b10i ⎪ ⎪ ⎨ ⁎ 1 ( − Ψ^tcsc (·) − cj z^j3 p˜ j⁎ (z, y) − ck z^k3 p˜k⁎ (z, y) + ν ) ⎪ uc = b20 ⎪ ⎪ ⎩ ν = − βy , i = 1, 2, …, n

(35)

Similar to the SMIB case, constants b10i and b20 are chosen to be

b10i < − ω 0 (Iqi + 2Gii Eq′ i )/(2Hi Td0i ) b20 > KT/(2Tc ) Furthermore, estimates of state and perturbation are bounded as ̇ |z^i3 | ≤ ω0 Pmi/(2Hi ), |Ψ^i (·)| ≤ ω0 Pmi/(2Hi Δ), and |Ψ^i (·)| ≤ ω0 Pmi/(2Hi Δ2 ). To this end, a decentralized EC can be obtained for the ith machine as only the measurement of its rotor angle δi is required, which can effectively handle the modelling uncertainties. The TCSC controller measures the rotor angles δj and δk, excitation control inputs ufdj and ufdk , transient voltages Eq′ j and Eq′ j . The nominal values of time constant Td0j and Td0k , inertial constants Hj and Hk of the jth and kth machine, and transmission line reactance Xjk are used for coordination. Note that the difference between the nominal values and the real values of the system parameters is aggregated into the perturbation, which is estimated by PO.

5. Simulation results 5.1. Evaluation on a single machine infinite bus system A proportional-integral-derivative (PID) based TCSC controller and automatic voltage regulator equipped with lead-lag (LL) based PSS (PID þLL), CAPC used in Sun et al. (2009), and PECAPC are compared. The system parameters, initial operating conditions and

PID þLL parameters are given in Tables 1 and 6 (Fig. 16). Note that system poles must be put to the left half plane to ensure the system stability and dynamical performance. PECAPC parameters are chosen as k1 = 9, k2 = 27, k3 = 27 so as to place the poles of the linear system at ξ = − 3; b10 = − 15, b20 = 100, k′ = 5, β = 15, c¼ 0.001; α2 = 9.6 × 103, α1 = 160, α3 = 2.56 × 105 and 6 α4 = 2.56 × 10 so as to place all the poles of the HGSPO at λ ¼ 40; α′1 = 30 and α′2 = 225 so as to place all the poles of the HGPO at λ′ = − 15, ϵ ¼0.1 and Δ ¼0.05 s. The system variables are used in the per unit (p.u.) system. In the power systems analysis field of electrical engineering, per unit system is the expression of system quantities as fractions of a defined base unit quantity, which means that the values have been normalized (Grigsby, 2012). In this paper, assume that the independent base values are active power P base = 1.0 p. u. and voltage Vbase = 1.0 p. u. . A three-phase short-circuit fault occurs at t¼1.0 s and cleared at t¼1.1 s, where |ufd | ≤ 7 p. u., and |Xtcsc | ≤ 0.1 p. u. such that a maximum 20% compensation ratio is implemented. Fig. 4 shows system responses obtained with the EC alone, coordinated EC and TCSC controller (28) and its approximated controller (30), respectively. From which the effectiveness of coordination is verified as an extra system damping is injected, and the excitation control cost is reduced. Furthermore, the approximation is valid as it can capture the main feature of the exact coordination, hence the approximated controllers will be used for the rest of the studies. System responses obtained under the nominal model is illustrated by Fig. 5. It is found that PECAPC can effectively stabilize the system. Fig. 6 presents system responses under a nonlinear unmodelled TCSC dynamics ζtcsc = 10 sin (Xtcsc − Xtcsc0 ), which is the same unmodelled TCSC dynamics used in Sun et al. (2009) for the purpose of their control performance evaluation. However, PECAPC does not require the unmodelled TCSC dynamics to satisfy this linear growth condition required by CAPC (Sun et al., 2009). In fact, a general unmodelled TCSC dynamics can be estimated online by PO. PID þLL control performance degrades dramatically as it is

188

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

Fig. 18. The experiment platform of the HIL test.

not robust to the TCSC modelling uncertainty. In contrast, both CAPC and PECAPC can maintain a consistent control performance as this uncertainty is considered during the control design. The low frequency inter-area modes oscillation has been well defined in the power system research, which is caused by the dynamic interactions, in a low frequency, between multiple groups of generators. It results in a degrade of power system stability and must be suppressed. It is concerned that the SMIB-designed controller may not perform well in the presence of inter-area modes oscillation. Thus it will be necessary to evaluate the SMIB-designed PECAPC on a more realistic multimachine power system model with different oscillation frequencies. One approach named single machine quasi-infinite bus, where the infinite bus is modulated in magnitude by inter-area-type frequencies. An inter-area type disturbance Vs = 1 + 0.1 sin (5t ) is chosen to a corresponding oscillation frequency of (2.5/π ) Hz . System responses are given in Fig. 7, the control performance of both PID þLL and CAPC degrades due to the external disturbance. In contrast, PECAPC can effectively attenuate the inter-area disturbance. 5.2. Evaluation on a three-machine system The PID þ LL, CPC, where the real value of the states and perturbation is used in the controllers, and PECAPC (35) are then evaluated on a three-machine system as shown in Fig. 8, where a TCSC device is equipped between bus #7 and bus #8. The system parameters, initial operating conditions and PID þLL parameters

are all given in Tables 2-5 and 7. PECAPC parameters are b10i = − 30, k i1 = 15, k i2 = 75, k i3 = 125 so as to place the poles of the linear system at ξ = − 5; b20 = 20, k′ = 10, β = 1 and c2 = c3 = 0.25; αi1 = 200, αi3 = 5 × 105 and αi2 = 1.5 × 104 , αi4 = 6.25 × 106 so as to place all the poles of the HGSPO at λ = − 40; α′1 = 60 and α′2 = 900 so as to place all the poles of the HGPO at λ′ = − 30, ϵ ¼0.1 and Δ ¼ 0.05 s. A three-phase short-circuit occurs on one transmission line between bus #4 and bus #5 marked as point ‘  ’ at t¼0.5 s, the faulty transmission line is switched off at t ¼0.6 s, and switched on again at t¼1.1 s when the fault is cleared, where |ufdi | ≤ 7 p. u., and |Xtcsc | ≤ 0.2 p. u. such that a maximum 20% compensation ratio is implemented. Each generator is equipped with an EC. System responses under operation type I are given by Fig. 9, which shows PECAPC can achieve as satisfactory control performance as CPC when an accurate system model is completely known, their tiny difference is caused by the estimation error. The observer performance during the fault is also monitored, the estimation errors of HGSPO1 and HGPO are given in Figs. 10 and 11, respectively, which show that the observers can provide accurate estimates of system states with a fast tracking rate. However, there exist relatively larger errors in the estimate of perturbation at the instant of t¼0.5 s, t¼0.6 s, and t¼ 1.1 s, this is due to the discontinuity in the equivalent admittance Y12 caused by the disconnection and reconnection of the transmission line 4–5(2) at the instant when the fault occurs. A 50% increase of the generator inertia Hi, time constant Td0i

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

189

1.02

80

Simulation HIL Rotor speed ω (p.u.)

Rotor angle δ (deg)

Simulation HIL 60

40

20

0

0

2

4

6

1.01

1

0.99

0.98

8

0

2

4

Time (sec)

Time (sec) (p.u.)

Simulation HIL

tcsc

5

TCSC reactance X

Excitation voltage ufd (p.u.)

8

0.15

10

0

−5

−10

6

0

2

4 Time (sec)

6

8

Simulation HIL

0.1 0.05 0 −0.05 −0.1 −0.15

0

2

4 Time (sec)

6

8

Fig. 19. PECAPC performances obtained in the HIL test with large observer poles λ = λ′ = 15 ( fs = 50 kHz , fc = 500 Hz and τ ¼2 ms).

and TCSC time constant Tc used in controller (35) has been tested to evaluate the effect of parameter uncertainties on the dynamic response of the proposed control scheme with and without the PO. Fig. 12 shows that the control performance degrades dramatically in the presence of parameter uncertainties without the PO, in contrast the same control performance can be achieved with the PO shown by Fig. 13. This is because the parameter uncertainties of the generator inertia Hi and time constant Td0i are included in perturbation Ψi (·) (31), which is estimated by HGSPO (33) and compensated by controller (35). While the parameter uncertainties of the TCSC time constant Tc are included in perturbation Ψtcsc (·) (25), which is estimated by HGPO (26) and compensated by controller (35). The robustness of PID þLL, CPC and PECAPC has been evaluated by reducing the inertia constant Hi, time constant Td0i of all generators and TCSC time constant Tc by 30% from their nominal values. System responses are provided in Fig. 14, in which a big difference in CPC has been identified, with and without accurate parameters. By contrast, PID þLL and PECAPC remain satisfactory control performance as their design does not require accurate system parameters. In order to test the general effectiveness of PECAPC, Fig. 15 presents system responses obtained under operation Type II and the above parameter uncertainties. In this case a larger active power is transmitted, and the system suffers more severe oscillation when the fault occurs. One can find PID þ LL control performance degrades dramatically as its control parameters are based on local system linearization. On the other hand, a severe oscillation in CPC can be seen due to the system parameter uncertainty. In contrast, PECAPC can maintain a consistent control performance and provide significant robustness. At last, the integral of absolute error (IAE) index is used to

compare the control performance of each control schemes in all studied cases. Here IAEδ =

T

T

∫0 |δ − δ0 | dt and IAEω = ∫0 |ω − ω0 | dt , the units of rotor angle δ and speed ω are deg and rad/s, respec-

tively, the simulation time T ¼5 s. The results are presented in Table 8. Note that PECAPC has a little bit higher IAE than that of CAPC and CPC under nominal model, which is resulted from the estimation error of PO. Furthermore, PECAPC provides much better robustness as it has the lowest IAE in the presence of various modelling uncertainties. In particular, its IAEδ and IAEω are only 44.5% and 28.6% of CAPC, 50.9% and 31.5% of PID þLL to the interarea type disturbance in the SMIB system, while 27.4% and 47.7% of CPC, 53.1% and 56.7% of PID þLL to the parameter uncertainties (Type II) in the three-machine system.

6. Hardware-in-the-loop test An HIL test has been undertaken based on dSPACE systems to verify the implementation feasibility of PECAPC. The configuration and experiment platform of HIL test are shown in Figs. 17 and 18, respectively. The EC and the TCSC controller are implemented on one dSPACE platform (DS1104 board) with a sampling frequency fc = 500 Hz , and the SMIB 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 one as possible. A time delay τ ¼2 ms is used between the controller and the SMIB system in the simulation to take the computational delay of the real-time controller into account. The measurements of the rotor angle δ, rotor speed ω, inner generator voltage Eq′ , infinite bus voltage Vs and TCSC reactance Xtcsc are obtained from the realtime simulation of SMIB system on the DS1006 board, which are

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B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

80

1.02 Simulation HIL Rotor speed ω (p.u.)

Rotor angle δ (deg)

Simulation HIL 60

40

20

0

1.01

1

0.99

0.98

0

2

4

6

0

8

2

Time (sec)

8

0.15 Simulation HIL

TCSC reactance Xtcsc (p.u.)

Excitation voltage ufd (p.u.)

6

Time (sec)

10

5

0

−5

−10 0

4

2

4

6

8

Simulation HIL

0.1 0.05 0 −0.05 −0.1 −0.15 0

Time (sec)

2

4

6

8

Time (sec)

Fig. 20. PECAPC performances obtained in the HIL test with proper observer poles λ = λ′ = 5 ( fs = 50 kHz , fc = 500 Hz and τ ¼2 ms).

sent to two controllers implemented on the DS1104 board for calculating the control outputs, i.e., excitation voltage ufd and TCSC modulated input uc , respectively. The disturbance is set up as a 0.1 s three-phase short-circuit fault occurs at 1.9 s. The total experiment time is 60 s and only the result of the first 8 s is presented for a clear illustration of the transient responses. The following three tests are carried out: Test 1: Same observer parameters used in the previous simulation, which poles λ ¼ 40, λ′ = 15 with b10 = − 150 and b20 = 100; Test 2: Reduced observer poles λ ¼15, λ′ = 15; and Test 3: Further reduced observer poles λ ¼5, λ′ = 5. It has been found from Test 1 that an unexpected high-frequency oscillation occurs in ufd and Xtcsc , which does not appear in the simulation. This is due to the large observer poles result in high gains, which lead to highly sensitive observer dynamics to the measurement disturbances. Hence reduced observer poles are chosen in Test 2. Fig. 19 shows that the rotor angle and speed can be effectively stabilized, but a consistent high-frequency oscillation still exists in both ufd and Xtcsc . Through the trial-and-error it finds that an observer pole in the range of 3–10 can avoid the high-frequency oscillation but with almost similar transient responses, therefore the observer poles are further reduced in Test 3. The system transient responses obtained in simulation and HIL test are given by Fig. 20, one can find that the high-frequency oscillation disappears and the rotor angle and speed are still effectively stabilized. The difference of the obtained results between the simulation and HIL test is possibly caused by the following three reasons:

(a) there exist measurement disturbances in the HIL test which are not considered in the simulation, a filter could be used to remove the measurement disturbances and improve the control performance; (b) the discretization of the HIL test and sampling holding may introduce an additional amount of error compared to continuous control used in the simulation; and (c) the computational delay of the real-time controller, in which exact value is difficult to obtain. A time delay τ ¼ 2 ms has been assumed in the simulation to consider the effect of this computational delay.

7. Discussion It is necessary to study the computational cost of PECAPC as the high performance systems often have high computational cost. As the PECAPC needs to calculate a fourth-order HGSPO (23) and a second-order HGPO (26) together with a nonlinear function (29), it has the highest computational cost. The computational cost of CAPC is higher than that of CPC, as CAPC requires to solving an additional second-order parameter estimator (A.1). For CPC and PIDþ LL, it is difficult to compare the computational cost as CPC has to calculate three complex functions (21), (25) and (29), while PIDþLL has to calculate a first-order integral used in PID, and a second-order LL plus a first-order washout loop. For the multimachine power systems, the computational cost of CPC, CAPC (nonlinear functions (21), (25) and (29) become more complex) and PECAPC (nonlinear function (29) becomes more complex) will be higher than that of the SMIB system, while the computational cost of PIDþ LL does not

B. Yang et al. / Control Engineering Practice 44 (2015) 172–192

change. As the PECAPC is a decentralized controller, it can be easily extended into multimachine power systems as each generator will be equipped with an individual EC, and the TCSC controller is separately implemented in the TCSC device. Finally, the majority of studies related to power system control and operation is so far based on the simulation or HIL test. The paper is concerned with the investigation of a new control method, for which it is difficult to undertake a physical experiment for the multimachine power system due to its significant scale and complexity.

p˜ k (z, y) =

⎡ ω 0 ⎢ ⎛ −Dk Eq′ k Eq′ j sin δkj ⎜ + Eq′ k Eq′ j ωkj cos δkj 2Hk ⎢⎣ ⎝ 2Hk

Eq′ j

2Eq′ j

+

In this paper, a PECAPC has been developed through designing the PO to estimate and compensate the real perturbation, which is a lumped term including nonlinearities, parameter uncertainties, unknown dynamics and external disturbances. The proposed approach can partially release the dependence of system model in the conventional PC. In addition, only the range of CLF is needed by the PECAPC, while an explicit CLF is required by CAPC which is difficult to find in complex nonlinear systems. The PECAPC has been applied to SMIB systems and multimachine power systems. The simulation results show that it can provide similar control performance under nominal model, but much better robustness in the presence of unmodelled dynamics and parameter uncertainties, comparing to PID þLL, CPC and CAPC. The HIL test has been undertaken to verify the implementation feasibility of the proposed approach.

Acknowledgments This work is supported by the China Scholarship Council (CSC), and National Key Basic Research and Development Program (973 Program, No. 2012CB215100), China.

Appendix A. Controller structures and system parameters The p˜j (z, y ) and p˜k (z, y ) used in controller (34) are

p˜j (z, y) =

+

+

Eq′ k Td0j 2Eq′ k Td0j

⎡ ω 0 ⎢ ⎛ −Dj Eq′ j Eq′ k sin δjk ⎜ + Eq′ j Eq′ k ωjk cos δjk 2Hj ⎢⎣ ⎝ 2Hj

× (ufdj − Eq′ j ) sin δjk

Gjj Eq′ j (x dj − x d′ j ) cos δjk + Eq′ j ×

⎛ ⎜ ufdk − E ′ + (x dk − x ′ ) × qk dk ⎜ ⎝ +

Eq′ k sin δjk Td0j

sin δjk Td0k

⎞ ∑ Eq′ i Yki cos δki ⎟⎟ i = 1, i ≠ k, j ⎠ n

⎞ Eq′ i Y ji cos δji ⎟⎟ XΔ⁎ + XΔ⁎′ ⎠ i = 1, i ≠ k, j n

(x dj − x d′ j )

∑

⎛ E ′ 2 sin (2δjk ) ⎞⎤ Eq′ 2k sin (2δjk ) qj × ⎜⎜ (x dk − x d′ k ) + (x dj − x d′ j ) ⎟⎟ ⎥ 2Td0k 2Td0j ⎝ ⎠ ⎥⎦

× (ufdk − Eq′ k ) sin δkj +

Td0k

Gkk Eq′ k (x dk

Td0k sin δkj − x d′ k ) cos δkj + Eq′ k × Td0j

⎛ ⎜ ufdj − E ′ + (x dj − x ′ ) × qj dj ⎜ ⎝ +

8. Conclusion

191

Eq′ j sin δkj Td0k

⎞ Eq′ i Y ji cos δji ⎟⎟ i = 1, i ≠ j, k ⎠ n

∑

⎞ Eq′ i Yki cos δki ⎟⎟ XΔ⁎ + XΔ⁎′ i = 1, i ≠ j, k ⎠ n

(x dk − x d′ k )

∑

⎛ E ′ 2 sin (2δkj ) ⎞⎤ Eq′ 2j sin (2δkj ) qk × ⎜⎜ (x dj − x d′ j ) + (x dk − x d′ k ) ⎟⎟ ⎥ 2Td0j 2Td0k ⎝ ⎠ ⎥⎦ Xjk = Y −jk1,

where

XΔ⁎ = 1/[(Xjk + y + Xtcsc0 ) Xjk ],

and

XΔ⁎′ = (y + Xtcsc0 + 2Xjk ) /[(Xjk + y + Xtcsc0 ) Xjk ]2. The CAPC based EC and TCSC controller used in (Sun et al., 2009) are

⎧ ⎪ ufd ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ uc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ̇ ⎪ θ^ ⎪ ⎪ ⎪ ψ^ ̇ ⎪ ⎩ ν

=

ω 0 Vs sin δ Td0 ⎧ ⎨ z2 Kc ⎩ H (X d′ Σ + Xtcsc0 ) −

(X d − X d′ )(Vs cos δ − Eq′ ) Td0 (X d′ Σ + Xtcsc0 )

Eq′

′ ) cot δ − c3 z3 − x2 (x3⁎ + Eq0 Td0 H (X d′ Σ + Xtcsc0 ) + ω 0 Vs sin δ ⎡ ̇ × ⎢ x2 (1 + c1c2 + θ^) + (c1 + c2 + θ^) ⎢⎣ +

⎛ ω0 ⎪ ω 0 Eq′ Vs sin δ ⎞ ⎤ ⎫ Pm + θ^x2 − ⎜ ⎟⎥⎬ ⎪ H (X d′ Σ + Xtcsc0 ) ⎠ ⎥⎦ ⎭ ⎝ H =

Tc ⎛ −ω 0 Eq′ Vs sin δ ⎜ KT ⎝ HXΔ z2 +

(X d − X d′ )(Vs cos δ − Eq′ ) Td0 XΔ

⎞ z3 − ψ^ y + ν⎟ ⎠

⎛ ⎞ H (X d′ Σ + Xtcsc0 )(θ^ + c1 + c2 ) ⎟ = γ ⎜⎜ z2 − z3 ⎟ x2 ω 0 Vs sin δ ⎝ ⎠ = ρy 2 = − βy

(A.1)

′ , where x3 = Eq′ − Eq0 x2⁎ = − c1x1, x1 = δ − δ0 , x2 = ω − ω0 , z3 = x3 − x3⁎, x3⁎ = [H (Xd′ Σ + Xtcsc0 ) /(ω0 Vs sin δ )] z2 = x2 − x2⁎, ′ . [x1 + (ω0/H ) Pm + (θ^ + c1) x2 + c2 z2 ] − Eq0

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