Robust SVC controller design and analysis for uncertain power systems

ARTICLE IN PRESS Control Engineering Practice 17 (2009) 1280–1290

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Robust SVC controller design and analysis for uncertain power systems Sylwester Robak  Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland

a r t i c l e in f o

a b s t r a c t

Article history: Received 7 January 2008 Accepted 3 June 2009

A Static Var Compensator (SVC) installed in a power transmission network can be effectively used to enhance the damping of electromechanical oscillations [Schweickardt, H. E., Romegialli, G., & Reichert, K. (1978). Closed loop control of static VAR sources (SVS) on EHV transmission lines. IEEE Pes winter power meeting, (paper no A78, pp. 135–136), New York, Jan. 29–Feb. 3]. An adequately designed robust controller, which takes into account variations in the operating conditions, can help to achieve the desired damping control. The proposed approach described in this paper is aimed to achieve damping of electromechanical oscillations by considering a systematic approach, based on interval systems theory and Kharitonov’s Theorem. The method presented allows for the design of a fixed-parameter, low-order controller, given a supposed stability degree of the system. The synthesis of a robust SVC controller is divided into two tasks. The first is the determination of the region of stability in the controller parameter plane by plotting the stability boundary locus. The second task is the optimization of the selected controller parameters from the obtained solutions to the first task. Examples of eigenvalue analysis and time simulation demonstrate the effectiveness and robustness of the designed controller. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Power system stabilization FACTS device Interval system Robust control

1. Introduction The problem of designing controllers to enhance the damping of electromechanical power swings is currently important. Rotor oscillations of synchronous generators result in slow power and frequency oscillations, called electromechanical oscillations. These natural phenomena have been observed in power systems (Kurth & Welfonder, 2005). These oscillations can persist for long periods without a controller. In some cases, such oscillations can endanger the stability of power systems. The stability of a power system can be enhanced by appropriate control of the generator excitation voltage (Chaturvedi & Malik, 2005; Kundur, 1994; Machowski, Bialek, & Bumby, 2008). This control can be realized by applying an additional feedback control loop in the AVR system of selected generating units. This additional controller is called a power system stabilizer. Another method to enhance the stability of a power system is based on flexible AC transmission systems (FACTS). These devices are installed in transmission lines in order to control the network power flow and the voltage at a defined node. These devices can also perform the additional function of controlling the transient state, related to electromechanical oscillations. Considering the large number of FACTS installed in power systems all over the world (Sullivan, 2006), this paper analyzes a Static VAR Compensator (SVC) (connected in shunt), applied for nodal reactive power

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compensation and localized voltage control (Schweickardt, Romegialli, & Reichert, 1978). An analysis of the damping phenomena should be carried out, in order to appropriately select a damping controller based on a FACTS device. Uncertainty in the power system is a key issue, and complicates the process of selecting a controller. System robustness is a controller property that describes the controller’s ability to perform its desired function in the presence of uncertainties in the system; systems characterized by this feature are called robust systems (Bhattacharyya, Chapellat, & Keel, 1995). Modern control theory includes adaptive, optimal, and artificial intelligence-based methods of control design. Each of these methods has been used (with varying levels of success) to design robust controllers for uncertain systems, characterized by a lack of complete knowledge of dynamic features of the system (Ellithy & Al-Naamany, 2000; Pal & Chaudhuri, 2005; Sadikovic, Korba, & Andersson, 2006). The controllers obtained using these methods tend to either be high-order, or contain quite complicated control rules. However, such solutions are rarely satisfying from the practical point of view. The power system utility technical staff is not inclined to use them for the sake of the system’s operation security. They expect simpler solutions, similar to those to which they are used to, and in which they have experience. It is, therefore, essential to work on efficient controller tuning methods that result in typical transfer functions, similar to those that are commonly used in damping controllers applied for power systems. This paper proposes the design of a robust SVC-based controller, using interval system theory and Kharitonov’s Theorem.

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Such an approach has previously been used in the design of power system stabilizers that generate a signal to modulate the reference of the AVR (Khutoryansky & Pai, 1997; Soliman, Elshafei, Shaltout, & Moris, 2000). This paper presents a robust control design based on SVC-type FACTS devices. The presented approach facilitates the design of a low-order, fixed-parameter, robust controller. The solution to the synthesis problem determines the region of stability within the controller parameter space. The final selection of the parameters can be carried out by using an appropriate performance index.

P 4

 Variation of operating conditions of the generation units, in particular those equipped with synchronous generators.

 Variation of the structure of the power system, primarily due 



to changes in the network configuration and the number of operating generation units. Uncertain parameters of the elements of the power system, which are mainly caused by variation of the parameters due to climate changes, variation of the power system operation mode, or simply erroneous parameter assessment. Bad approximations in power system modeling, resulting in unstructured uncertainties. There are two main techniques used in power system modeling: the first method consists of physical or topological reduction of the model (Machowski et al., 2008), while the second is based on linearization of a nonlinear model of the power system.

In this paper, attention is paid to the primary source of power system uncertainty, changes in the operating conditions of power systems. Considering the changes in load consumption, the active power produced by different turbine generators should have as wide a range of variability as large. However, several factors limit this range. Different authors (Adibi & Milanicz, 1994; Nilsson & Mercurio, 1994) have underlined the importance of determining the possible operating range that the generator is capable of in actual operation; in fact, the operating conditions of the generating plant and the electrical power system may effectively limit the region of stability of the generator. These limitations can cause significant reductions from the synchronous machine capability chart furnished by the manufacturer. The capability chart of a typical synchronous generator is depicted in Fig. 1. The prime mover is supposed to limit the active power generated to within a fixed range, according to the curve (AG, CD). The second boundary represents the thermal limitation of the stator current (BC, DE). The third boundary represents the underexcitation limiter (AB) action, the fourth one is related to the rotor current thermal limit (it represents the rotor field thermal

3

2

F''' D

F''

1

L L''' L''

C

F'

E

Pi

B K

L'

F

A

G

2. Power system uncertainty There are many causes of variations in a power system’s operating conditions. The most common causes are the following: continual changes in power consumption, resulting from the behavior of power consumers; changes in operating states resulting from changes in the generation and transmission device structure, due to an inability to store large amounts of energy. Most of the time, energy must be consumed as it is generated. This is why the amount of energy being consumed exerts a great influence on the amount of power generated, and hence on the control of the power system operation. The following causes of uncertainty are a matter of great concern in efforts to design robust controllers for power systems (Lachs & Sutanto, 1995; Makarov, Hill, & Milanovic, 1997; Othman, Chow, & Taranto, 1992; Praprost & Loparo, 1992; Robak, 2008):

1281

Q(ind)

Q(cap) 0 Fig. 1. Capability chart of a synchronous generator.

limit, specified by a DC current rating) (EF), and the last one is the limitation due to steady-state stability (FG). Fig. 1 shows an example of limitations of the generator active power Pi. The area delimited by the line KL represents the reactive capability limitation of the synchronous generator. KL is composed of a few sections; among other sections, section L0 L corresponds to the area gained due to the excitation controller of the generator. The generated active power depends not only on the design parameters of the synchronous machine, but also on the operating conditions of the plant (turbine and thermal elements) and power system. The upper limiter Pg max (segment marked CD) depends on the prime-mover’s MW output. The lower limiter Pg min (AG) depends on the technical minimum of the turbine generator, which depends on the type and quality of fuel, and the additional combustion of mazut.

3. Interval system stability theorem A robust controller based on an SVC is proposed in this paper. The controller is designed with the use of the interval system stability theorem. A feedback control system (Fig. 2) is used for the analysis. The plant is represented by a transfer function gðs; n; dÞ, in which the numerator and the denominator are interval polynomials: nðs; nÞ ¼ ½n0 ; n¯ 0  þ ½n1 ; n¯ 1 s þ ½n2 ; n¯ 2 s2 þ ½n3 ; n¯ 3 s3 þ ½n4 ; n¯ 4 s4 þ    dðs; dÞ ¼ ½d0 ; d¯ 0  þ ½d1 ; d¯ 1 s þ ½d2 ; d¯ 2 s2 þ ½d3 ; d¯ 3 s3 þ ½d4 ; d¯ 4 s4 þ   

(1)

(2)

where n¯ i ; d¯ i and ni ; di are the upper and lower boundary values of coefficients ni ; di . The fixed parameter controller can be assumed to be as follows: GC ðsÞ ¼

n^ 0 þ n^ 1 s d^ þ d^ s 0

(3)

1

The four Kharitonov polynomials associated with the numerator n(s, n) and denominator dðs; dÞ of the plant are in the form of

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plant +

Σ

G

L21

L11

T

g(s,n,d)

L12

s

L22

-

a SVC GC(s) controller

Fig. 3. The SVC equipped test system.

Fig. 2. Uncertain plant with a fixed compensator in the feedback loop.

interval polynomials, expressed as follows: n1 ðsÞ ¼ n¯ 0 þ n¯ 1 s þ n2 s2 þ n3 s3 þ n¯ 4 s4 þ   

Va

n2 ðsÞ ¼ n0 þ n1 s þ n¯ 2 s2 þ n¯ 3 s3 þ n4 s4 þ    n3 ðsÞ ¼ n0 þ n¯ 1 s þ n¯ 2 s2 þ n3 s3 þ n4 s4 þ    n4 ðsÞ ¼ n¯ 0 þ n1 s þ n2 s2 þ n¯ 3 s3 þ n¯ 4 s4 þ   

Varef + Σ +

KSVC

BSVC

1+sTSVC

Vsc

(4)

Fig. 4. The SVC controller block diagram. 1 d ðsÞ ¼ d¯ 0 þ d¯ 1 s þ d2 s2 þ d3 s3 þ d¯ 4 s4 þ    2

d ðsÞ ¼ d0 þ d1 s þ d¯ 2 s2 þ d¯ 3 s3 þ d4 s4 þ    3

d ðsÞ ¼ d0 þ d¯ 1 s þ d¯ 2 s þ d3 s þ d4 s þ    2

3

4

4

d ðsÞ ¼ d¯ 0 þ d1 s þ d2 s2 þ d¯ 3 s3 þ d¯ 4 s4 þ   

(5)

Based on the four Kharitonov polynomials, a set of 16 Kharitonov transfer functions (Ackerman, Kaesbauer, Sienel, & Steinhauser, 1993) can be defined by the following expression: ( ) ni ðsÞ : i; j 2 f1; 2; 3; 4g (6) GK ðsÞ ¼ j d ðsÞ The following theorems are the basis of the robust controller design. Theorem 3.1. (Kharitonov’s Theorem). An real interval polynomial (2) with invariant degree is robustly stable if and only if its four i Kharitonov polynomials d ðsÞ, where i ¼ 1; 2; 3; 4, defined by (5), are stable. Theorem 3.2. (Barmish, Hollot, Kraus, & Tempo, 1992). The closed loop system of Fig. 2, where gðs; n; dÞ is an interval (plant) transfer function, is robustly stable with the first-order fixed compensator (3) if and only if the 16 characteristic polynomials of the set j i ^ ^ ðsÞ þ dðsÞd ðsÞ : i; j 2 f1; 2; 3; 4gg P K ¼ fnðsÞn

of the model are the generator speed deviation Do, the rotor angle d, and the generator transient emfs E0q , E0d . The excitation voltage regulator of the generator is modeled using a second-order transfer function of the following form: GðsÞ ¼

K A 1 þ sT B 1 þ sT A 1 þ sT C

(8)

The mechanical power of the turbine is assumed to be constant, Pm ¼ const (Kundur, 1994). The turbine regulator can be omitted in the modeling process. The linearized seventh-order system model can be described using a classical state variable matrix equation as follows:

Dx_ ¼ ADx þ BDu Dy ¼ CDx

(9) (10)

or by using the operator transfer function given by CadjðsI  AÞ1 B detðsI  AÞ n0 þ n1 s þ n2 s2 þ    þ nw1 sw1 þ sw ¼ d0 þ d1 s þ d2 s2 þ    þ dw1 sw1 þ sw

GðsÞ ¼ CðsI  AÞ1 B ¼

(11)

where w is the system order.

(7)

are stable.

5. Problem formulation

4. Single machine infinite bus (SMIB) system description

5.1. Design tasks

The proposed SVC controller will be illustrated assuming that the analyzed generator-infinite bus system is composed of the four transmission lines L11, L12, L21, and L22, as shown in Fig. 3. These transmission lines connect the generator to the system. An SVC installed at node a increases the voltage control efficiency and the system loadability. The installed SVC is also used to damp power oscillations that occur between the generator and the infinite bus system. In general, the SVC can be considered to be a variable susceptance parallel element BSVC . The SVC for voltage control can be represented using a first-order transfer function (CIGRE, 1999; Taylor et al., 1994), whose input signal is the voltage error, as shown in Fig. 4. The component of the SVC input signal denoted V sc is the output signal of the designed stabilizing controller GC ðsÞ. The generator is modeled using a fourth-order model derived in Kundur (1994) and Machowski et al. (2008). The state variables

The design task is formulated as follows: In the presence of uncertainties in the feedback model of a power system (Fig. 2), a stabilizing controller with fixed parameters should be designed by using interval system theory. The designed controller must be represented by the simplest transfer function GC ðsÞ possible. The controller parameters should be selected by taking into account the requirements of electromechanical oscillations damping and the criteria regarding the quality of the control of the electrical quantities. 5.2. Description of uncertainty of power systems In this paper, the following assumption is made: the transfer function of the analyzed system is in the form of a ratio of polynomials (i.e., polynomials in both the numerator and denominator), which can be defined using the appropriate state

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equations and output matrices for a given operating state. Moreover, the following assumption is also made: the coefficients of the numerator and denominator polynomials are not accurately known a priori, i.e., the system’s transfer function is uncertain. However, the values of these coefficients are constrained to lie within a determined interval. During the analysis and synthesis processes examine cases in which changes in the power system’s operating conditions, caused by changes in the generator operating states, are the reason for uncertainty in the coefficient values of the numerator and denominator of gðs; n; dÞ. An additional simplification is also taken into account: the uncertainty of a defined coefficient in the numerator or denominator polynomial of the system transfer function is considered to be independent of the other coefficients. This assumption introduces an error in computing the parameters of the robust controller; this error can be neglected in an accurate analysis over the range of variability of the power system operating state. In this analysis, the uncertainties of the plant are described as a group of finite sets Ai of the power system’s operating states, where the index i indicates the scenario of the change under consideration. It is assumed that the operating condition pji , which belongs to the set Ai , depends upon a defined factor that influences the power system uncertainty. For a given variant of the analysis, each set Ai is composed of a few or a few tens of representative operating conditions of the power system: Ai ¼ fp1 ; p2 ; . . . ; pji ; . . . ; pJi g;

ji ¼ 1; 2; . . . ; Ji

(12)

where J i indicates the maximal number of operating conditions in the i-th operating condition set. The analysis scenarios are determined by considering the power system’s physical features, after having taken into account the restrictions resulting from technical criteria. Also, subsets of the sets Ai , denoted as Ap , which fulfill the condition Ap  Ai , are introduced. Due to the definition of Ap , an evaluation of the uncertainties over a limited range of variations can be carried out.

5.3. Parameters of the uncertain power system model For a given scenario of operating conditions changes, the uncertain model of the object gðs; n; dÞ and its parameters can be defined by a finite family of transfer functions as follows: Gji ðsÞ ¼ Cji ðsI  Aji Þ1 Bji ;

ji ¼ 1; 2; . . . ; J i

(13)

where

following equations: j

J

j

J

nh ¼ minfn1h ; n2h ; . . . ; nhi ; . . . ; nhi g 1

2

dh ¼ minfdh ; dh ; . . . ; dhi ; . . . ; dhi g j

ji ¼ 1; 2; . . . J i

(17)

ji ¼ 1; 2; . . . ; Ji

(18)

J

n¯ h ¼ maxfn1h ; n2h ; . . . ; nhi ; . . . ; nhi g j J 1 2 d¯ h ¼ maxfdh ; dh ; . . . ; dhi ; . . . ; dhi g

The process modeling the influence of variability in the power system leads to the largest possible range of uncertainties. However, the relative simplicity of the form of this uncertain model facilitates its application in the later stages of the robust controller design. 5.4. Controller type and structure Considering the conservativeness of the power system industry, the designed controller should have a relatively simple block diagram, preferably based on pre-existing controller design schemes. Hence, a controller with a transfer function similar to that of PSS’s (Machowski et al., 2008) is taken into account in this paper. A typical PSS is composed of a measuring element, a differential element, a phase correction element, a gain, and a limiter. The value of the measuring element time constant T p is natural and cannot be set, and its natural value is quite small, in the range of Tp ¼ [0.01, 0.03] s for a modern PSS. Thus, the measuring element is mostly omitted during the process of synthesizing the controller parameters. The differential element is a high-pass filter, used to eliminate the constant component of the input signal. Because of this, the regulator will only generate an output signal when the system is operating in a transient state. The phase correction element, composed of lead–lag elements connected in series, is used to provide the appropriate phase lag. The amplifier is used to achieve the desired damping level. As the order of the controller transfer function GC ðsÞ increases, an increasing number of closed-loop eigenvalues have to be robustly stabilized. It is therefore recommended to use the lowest order controller possible (Ackerman et al., 1993). An additional justification of this choice is that the reliability of the controller decreases as the number of its elements increases. Taking into account the above considerations, a first-order controller, containing a differential element and a signal amplifier, is used. The block diagram of this controller is shown in Fig. 5. 5.5. Controller synthesis

Gji ðsÞ ¼

j

j

j

j

j

j

j

j

j

j

i n0i þ n1i s þ n2i s2 þ    þ nw1 sw1 þ nwi sw i d0i þ d1i s þ d2i s2 þ    þ dw1 sw1 þ dwi sw

;

ji ¼ 1; 2; . . . ; Ji (14)

The assumption is made that the coefficients of the polynomials making up the numerator and denominator of the transfer function gðs; n; dÞ are either constant, or slowly varying compared to the dynamics of the power system, and can therefore be described by their upper and lower boundary values. The following coefficient vectors are introduced: j

j

j

j

ji ¼ 1; 2; . . . ; J i

(15)

j

j

j

j

ji ¼ 1; 2; . . . ; J i

(16)

nji ¼ ½n0i ; n1i ; . . . ; nhi ; . . . ; nwi ; j

1283

d i ¼ ½d0i ; d1i ; . . . ; dhi ; . . . ; dwi ;

Afterwards, the boundary values of the numerator and denominator polynomial coefficients are determined according to the

According to Theorem 3.2, when the first-order controller is used, the synthesis process will consist of determining a set of controller parameters for which all 16 subsystems corresponding to the polynomials (7) are stable. The following references describe several proposed methods to determine the controller parameters. Some proposals are based on a stability analysis via the Routh criterion (Barmish et al., 1992) or D-decomposition (Nejmark, 1978), which can be transformed into linear programming problems, either through a generalized Hermit–Biehler theorem (Ho, Datta, & Bhattacharyya, 1997) or a graphical method (So¨ylemez, Munro, & Baki, 2003).

Yi

sTr 1+sTr

K

Ui

Fig. 5. The block diagram of the SVC stabilizing controller.

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This paper presents a method based on determining a region of stability in the space of regulator parameter values, by using the D-decomposition method along with the controller parameter region analysis (Ackerman et al., 1993; Huang & Wang, 2000; Tan, Kaya, Yeroglu, & Atherton, 2006), and with application of the Hermit–Biehler theorem (Tan, 2003). The proposed method consists of searching a non-conservative Kharitonov region in the controller parameter plane. The curve representing the boundary of the stability region splits the controller coefficient parameter plane into sub-regions. If any point within a given subregion also belongs to the stability region, that sub-region is assigned to the stability region. The stability boundary curves obtained for each of the 16 analyzed subsystems allow for rapid determination of the stability region within the space of controller parameters. The application of the method has previously considered either PI or PID controllers. This paper presents a method for determining the solution space for a FACTS-based stabilizing controller. The following concerns the first-order stabilizing controller presented in Fig. 5; its transfer function is expressed as follows: Gc ðsÞ ¼

Trs n^ 1 s n^ ðsÞ ¼ K¼ ^d þ d^ s d^ ðsÞ 1 þ Trs 0 1

(19)

where Q i ðoÞ ¼ o2 nio ðo2 Þ j

Rj ðoÞ ¼ o2 do ðo2 Þ Si ðoÞ ¼ onie ðo2 Þ j

U j ðoÞ ¼ ode ðo2 Þ j

X j ðoÞ ¼ de ðo2 Þ j

Y j ðoÞ ¼ odo ðo2 Þ

(28)

Finally, the boundaries of the stability region in the space of controller parameters can be defined using the following expressions: n^ 1ij ðoÞ d^ ðoÞ

(29)

T rij ðoÞ ¼ d^ 1ij ðoÞ

(30)

K ij ðoÞ ¼

1ij

Let us assume that s ¼ jo and m ¼ o2 . Decomposing the numerator and denominator of the system’s transfer function into odd and even parts yields:

The curves of expressions (29) and (30) split the controller parameter plane into a finite number of subsets. Assuming that the time constant T rij 40, and ascertaining that any point that belongs to each subset is stable, the set Psij can be defined as the sum of all subsets that guarantee stability, which corresponds to the set of admissible solutions to the system described by the transfer function Gij ðsÞ. The set of admissible solutions to the whole family of polynomials GK ðsÞ, which a common part of all of the different solutions, can be expressed as follows:

ni ðjoÞ ¼ nie ðmÞ þ jonio ðmÞ

Ps ¼ Ps11 \ Ps12 \    \ Psij \    \ Ps44 ;

and the interval system is described by a family of transfer functions GK ðsÞ. Any transfer function belonging to the family GK ðsÞ can be expressed as follows: Gij ðsÞ ¼

j

ni0 þ ni1 s þ ni2 s þ ni3 s þ ni4 s þ    j d0

þ

j d1 s

j

þ

j d2 s

þ

j d3 s

þ

j d4 s

¼

þ 

ni ðsÞ j

d ðsÞ

j

d ðjoÞ ¼ de ðmÞ þ jodo ðmÞ

(20)

(21)

where nie ðmÞ ¼ ni0  ni2 m þ ni4 m2  ni6 m3 þ    nio ðmÞ ¼ ni1  ni3 m þ ni5 m2  ni7 m3 þ   

(22)

The subscript e indicates the even part, and the subscript o indicates the odd part of a given polynomial. The closed-loop system consists of a subsystem with the transfer function Gij ðsÞ, which belongs to the transfer function family GK ðsÞ, and a controller with the transfer function GC ðsÞ. The characteristic polynomail of this system is given by ^ oÞ ^ oÞ þ dj ðjoÞdðj dij ðo; n^ 1 ; d^ 0 ; d^ 1 Þ ¼ ni ðjoÞnðj ¼ ½nie ðo2 Þ þ jonio ðo2 Þjon^ 1 j j þ ½de ðo2 Þ þ jodo ðo2 Þðd^ 0 þ jod^ 1 Þ

(23)

For the considered controller, d^ 0 ¼ 1. Comparing the real and imaginary parts of (23) leads to the following expression: n^ 1 ðo

o ÞÞ þ d^ 1 ðo2 djo ðo2 ÞÞ ¼ dje ðo2 Þ

2 i no ð

2

(31)

An additional requirement may be introduced, in order to force the closed-loop system eigenvalues, obtained from Kharitonov 16 transfer functions (Tan et al., 2006). Then, instead of the variable s, a new variable s þ Z is introduced, in which Z indicates the established stability degree of the system, and is defined as follows:   1 1 (32) Z ¼ ln De ts where ts is the setting time and De the control error. In Pal and Chaudhuri (2005), the authors proposed that the control time be set between 10 and 20 s, and the control error should be in the interval of 0.02–0.05 p.u. Assuming that the control time should be o20 s and the control error should not exceed 0.05, a stability degree of 0.15 is obtained. As a consequence of the preceding paragraph, the general expressions of (26), (27) and (29), (30) are identical to the case of Z ¼ 0. Only the auxiliary variables, described by the expressions (28), are subject to changes, taking the following form: Q i ðoÞ ¼ ð˜nie ðo2 ÞÞZ  o2 ð˜nio ðo2 ÞÞ

(24) j

j

Rj ðoÞ ¼ ðd˜ e ðo2 ÞÞZ  o2 ðd˜ o ðo2 ÞÞ

and j j n^ 1 ðonie ðo2 ÞÞ þ d^ 1 ðode ðo2 ÞÞ ¼ odo ðo2 Þ

i; j 2 ð1; 2; 3; 4Þ

(25)

Si ðoÞ ¼ oð˜nio ðo2 ÞÞZ þ oð˜nie ðo2 ÞÞ

Solving Eqs. (24) and (25) yields:

j

j

U j ðoÞ ¼ oðd˜ o ðo2 ÞÞZ þ oðd˜ e ðo2 ÞÞ

X j ðoÞU j ðoÞ  Y j ðoÞRj ðoÞ n^ 1ij ðoÞ ¼ Q i ðoÞU j ðoÞ  Rj ðoÞSi ðoÞ

(26)

Y j ðoÞQ i ðoÞ  X j ðoÞSj ðoÞ d^ 1ij ðoÞ ¼ Q i ðoÞU j ðoÞ  Rj ðoÞSi ðoÞ

(27)

j

X j ðoÞ ¼ d˜ e ðo2 Þ j

Y j ðoÞ ¼ oðd˜ o ðo2 ÞÞ

(33)

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where i

i

˜ ij ðsÞ ¼ Gij ðs þ ZÞ ¼ n ðs þ ZÞ ¼ n˜ ðsÞ G j j d ðs þ ZÞ d˜ ðsÞ

(34)

1285

method. A detailed description of the procedure of creating the main Matlab program and the necessary additional Matlab functions to perform the optimization is given at www.mathworks.com.

and i i 2 2 ˜ ij ðjoÞ ¼ n˜ e ðo Þ þ jon˜ o ðo Þ G ˜d j ðo2 Þ þ jod˜ j ðo2 Þ e o

(35)

6.1. Case study

5.6. Selection of the controller’s parameters The next step in the process of synthesizing the robust controller is to select its parameters from the admissible solution set Ps . Generally, this design stage is considered to be an optimization problem. In the case of the uncertain power system analyzed here, during the process of designing the SVC-based stabilizing controller, a compromise should be reached between having the best possible electromechanical power swing damping and maintaining appropriate quality of the electrical quantities, which are influenced by the actions of the controller. An analytical approach to selecting the optimal parameters for the robust controller is presented in this paper. In this case, the controller parameters can be chosen with integral performance indexes I, which are based on the time-varying signals obtained after a perturbation. A selected step change in some quantity or non-zero initial conditions are considered as test perturbations of the linear system. Selected control errors, calculated from their steady-state values, are used as the integrated quantities in the integral performance indexes. The voltage at the SVC node and the variation in the generator speed are used as these quantities. Three performance indexes have been selected. The first one, denoted by Io, evaluates the quality of the damping of electromechanical power swings, and is given by the following expression: ! Z tk X 16 (36) ðDom Þ2 dt Io ðK; T r Þ ¼ ko t¼0

m¼1

where ko is the weight coefficient corresponding to the speed variation Do, t k is the integration boundary, and m is the number of subsystems of the Kharitonov family. The SVC stabilizing controller influences the damping of electromechanical oscillation through varying electrical quantities at its switching node. It is essential that the actions of this controller cause no negative consequences, such as controlling quantities that should be controlled by the main SVC controller. Any negative actions by the stabilizing controller, also called interactions, may cause protection devices to operate unnecessarily. Therefore, a second index, which evaluates the SVC stabilizing controller’s influence on the control quantity at the device location, has been introduced. ! Z tk X 16 (37) ðDY m Þ2 dt Ip ðK; T r Þ ¼ kp t¼0

m¼1

where DY is the deviation of the control, equal to the node voltage deviation DY m ¼ U a  U aref for the SVC device. The third index, called Itot, is the sum of all above-defined indexes: Itot ðK; T r Þ ¼ Io ðK; T r Þ þ Ip ðK; T r Þ

6. Example

(38)

The parameters of the designed controller are optimized using the index (38). The minmax algorithm, implemented as Matlabs fminimax in the Matlab package, was used as the optimization

The test system was composed of the following elements: a generator, with an active power rating of Pn ¼ 360 MW and a voltage rating of Un ¼ 22 kV, connected to a transmission line (Un ¼ 220 kV) through a transformer unit. The set of operating conditions used must allow for analysis of the largest possible range of loads, within the network’s and generator’s technical limits. To this end, some representative values of selected system parameters have been arbitrarily chosen. The following quantities were selected: (1) Generator terminal voltage: U g1 ¼ 1:091U n ¼ 24:0 kV, U g2 ¼ 1:023U n ¼ 22:5 kV, and U g3 ¼ 0:955U n ¼ 21:0 kV. (2) Active power generation: Pg1 ¼ P n ¼ 360 MW, P g2 ¼ 0:86P n ¼ 310 MW, P g3 ¼ 0:72P n ¼ 260 MW, Pg4 ¼ 0:58P n ¼ 210 MW, and Pg5 ¼ 0:44P n ¼ 160 MW. (3) Equivalent system node voltage: U s1 ¼ 0:955U n ¼ 210 kV and U s2 ¼ 1:023U n ¼ 225 kV. A set of 30 operating conditions, based on the above quantities, was also created. The following auxiliary subsets were also introduced: (a) Ap1: Comprising operating states fulfilling the following condition: U g2 ¼ 0:955; U n ¼ 21:0 kV and (b) Ap2: Comprising operating states fulfilling the following conditions 0:44Pn  Pg  P n and U g1 ¼ 1:091U n ¼ 24:0 kV or U g2 ¼ 1:023U n ¼ 22:5 kV. The subset Ap1 corresponds to the generator’s capacitive operating conditions, whereas the subset Ap2 corresponds to the generator’s typical operating conditions, when the generated reactive power is inductive. 6.2. The stability area This section presents the results obtained if the controller’s input signal is proportional to deviations in the rotor shaft speed. The stability regions in the controller parameter plane for different degrees of system stability are presented in Fig. 6. The Figure has been prepared with variations over the full range (Fig. 6a) and a partial range of operating conditions (Fig. 6b and c). Fig. 6b shows the areas obtained for the subset Ap2, and Fig. 6c is for the subset Ap1. The analysis results presented in Fig. 6 lead to the conclusion that the stability region narrows as the stability degree increases. Nevertheless, even if the stability degree is very high, some solutions still exist for a very limited range of operating conditions. 6.3. Optimization of controller parameters The results of the controller parameter optimization, based on the index Itot, are presented in this section. The analysis was carried out assuming variations over the full range of operating states. The weight coefficients, necessary to determine the index,

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Table 1 shows the results of the electromechanical eigenvalue analysis; both the eigenvalues l and the corresponding damping coefficients x are included in the table. Fig. 7 presents a few cases of time variable analysis, using rotor angle, generator voltage, and SVC control signal for the following load conditions: P ¼ Pn and Q ¼ 0.67Qn, as a complement to the eigenvalue analysis. A temporary fault occurring at the node of the equivalent system has been simulated as a resulting disturbance in the transient state. The presented results show very strong damping of power swings and rotor angle. The following are conclusions from the results obtained from tests of the features of the designed robust controller:

 The designed controller enhances the damping of electromechanical oscillations.

 The designed controller minimally influences the quality of the generator voltage control.

 The controller executes a bang-bang control during initial transient operation, which means that the designed controller allows for efficient use of the SVC control features.

To further quantify the benefits of the proposed controller, simulations have been performed over a range of inductive and capacitive loading conditions. Table 2 shows the results of integrating, over a 5 s period, the absolute value of the errors in the power angle, the real power, and the terminal voltage. In all cases, the performance is better for lower values of the integrated error.

7. The case of a multimachine power system 7.1. Problem description The analysis presented above was concerned with the simple case of a single machine infinite busbar (SMIB) test system. A real power system is composed of several interconnected generators, operating through a common transmission system; the system is called a multimachine system. No matter what approach is chosen, which methods of uncertainty analysis are used, or how the robust controller is designed, the analysis of such a system is always a daunting challenge. The main difficulties linked to such an analysis are due to the following:

 A multimachine system is characterized by several, local and

Fig. 6. Areas of robust stability for a system equipped with an SVC damping controller.

are the following: ko ¼ 10 000, kp ¼ 2. The optimized values of the SVC controller parameters are the following: K ¼ 46.0 and Tr ¼ 1.0. The evaluation of the designed controller features is based on both eigenvalue analysis and time variable analysis of a non-linear system. The eigenvalue analysis was carried out in four extreme cases of generator operating conditions.



 

inter-area, electromechanical modes (Machowski et al., 2008) existing between different parts of the power system. Different strategies have been used to test the performance of FACTS devices, and determine their best location on different transmission system nodes. FACTS devices can enhance the damping of inter-area oscillations. Due to the multitude of power system components (generators, controllers), a multimachine system is a high-order dynamic system. Thus, in several cases, designing the damping controller requires the use of different model reduction techniques (Machowski et al., 2008; Pal & Chaudhuri, 2005). A multimachine power system is characterized by a large variety of possible operating points. This makes determining the uncertain power system model parameters difficult. The optimization of robust controller parameters should take into account the complex actions of the FACTS device in a multimachine power system.

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1287

Table 1 Electromechanical eigenvalues for different operating conditions. Case

Generator operating conditions

Electromechanical eigenvalues Without SVC controller

1 2 3 4

P ¼ Pn P ¼ Pn P ¼ 0.44Pn P ¼ 0.44Pn

Q ¼ 0.67Qn Q ¼ 0.18Qn Q ¼ 0.59Qn Q ¼ 0.28Qn

l

x

l

x

0.0377j6.991 0.4707j7.145 0.1477j6.152 0.1927j6.520

0.005 0.066 0.024 0.029

3.0867j6.136 1.8177j6.713 3.5237j5.036 1.6437j5.606

0.499 0.261 0.753 0.281

1.0

 [rad]

0.8 0.6 0.4 0.2 0 0

1

2

3

4

5

3

4

5

t [s] 1.5

V [pu]

1.25 1.0 0.75 0.5 0

1

2 t [s]

0.5

Bsvc [pu]

With SVC controller

0

in this paper. A comprehensive solution will be presented in the next paper. The four-machine test system, the parameters of which are in Kundur (1994), was used for this analysis. This test system is appropriate for the analysis of both local (within system 1 or 2) and inter-area multimode electromechanical oscillations (between system 1 and 2). The phenomena exhibited by AVR and generation excitation controllers are neglected, to simplify the analyzed model (Fig. 8). Among several possible variations on the system, the following case has been taken into account: the tie-line active power is assumed to be constant (Ptie ¼ P78 ¼ const), whereas the load and generation powers in system 1 and 2 are assumed to be variable. Data related to different operating conditions are presented in Table 3. The results published in Kundur (1994) show that the analyzed test system is characterized by good damping of local oscillations. However, inter-area oscillations are either slightly damped or not damped at all. The parameters of the uncertain power system are determined by using the method presented in Section 5. The features of the uncertain test system model are determined by analyzing the distribution of Kharitonov transfer function poles pi . A description of this transfer function is given by expression (6). Partial results of the analysis are included in Table 4. The presented results can be summarized as follows: the analyzed system is not robustly stable without an SVC device damping controller. This is shown by the presence of positive complex poles, corresponding to frequency oscillations of f i ¼ Imðpi Þ=2p, appearing for both local and inter-area electromechanical oscillations. A robust damping controller should damp not only the poorly damped inter-area electromechanical oscillations, but also local oscillations or other types of oscillations that may be present.

7.2. Controller design -0.5 0

1

2

3

4

5

t [s] Fig. 7. Power system performance, under 100 ms three-phase to ground fault, occurring at the equivalent system node.

The issues presented above also affect the use of interval polynomial theory in robust controller design. The problems persist, despite intense progress in research in this domain, the results of which are presented in Chapellat and Bhattacharyya (1989). They are concerned with the design of controllers of any order, for either a MISO (multi-input single-output) or a SIMO (single-input multi-output) system. Due to the complexity of the problem, only some preliminary results of the multimachine power system analyses are presented

Theorem 3.2 is not sufficient for the design of an SVC-based robust controller, considering the complexity of the electromechanical oscillation phenomenon. Hence, in order to design a multi-input controller, the generalized Kharitonov theorem (GKT) has been taken into account (Chapellat & Bhattacharyya, 1989). This theorem can be considered to be a basic tool for the design of controllers of any order in the case of a SIMO plant. For the analyzed plant model, the transfer function may be expressed as follows: 0 1 B B gðs; qÞ ¼ dðs; dÞ @

1 n1 ðs; n1 Þ C .. C . A nm ðs; nm Þ

(39)

where m is the number of outputs, and q ¼ ½n1 . . . ni . . . nm d ¼ ½n d is the vector of uncertain parameters in the plant.

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Table 2 Simulation results for a short-circuit in network. Case

Generator operating conditions

Integral of deviation of Without SVC controller

1 2 3 4

P ¼ Pn P ¼ Pn P ¼ 0.44Pn P ¼ 0.44Pn

G1

5

1

Q ¼ 0.67Qn Q ¼ 0.18Qn Q ¼ 0.59Qn Q ¼ 0.28Qn

7

6

8

9

Real power P

Rotor angle d

Voltage V

Real power P

Rotor angle d

Voltage V

0.438 0.146 0.434 0.217

0.391 0.162 0.524 0.539

0.094 0.086 0.128 0.115

0.119 0.068 0.112 0.077

0.175 0.137 0.278 0.458

0.081 0.083 0.112 0.116

10 11

G3

Y1

3

SVC

SL9

Yi

2

+ Ym System 2

Fig. 8. The multimachine test system.

Σ

Generating power

Tie-line exchange power

The analysis results (Chapellat & Bhattacharyya, 1989) show that the closed-loop system, including the plan (39) and the controller (42), is stable if the family of characteristic polynomials

DðsÞ ¼ n^ 1 ðsÞN 1 ðs; n1 Þ þ    þ n^ i ðsÞNi ðs; ni Þ þ    þ n^ m ðsÞNm ðs; nm Þ

System 1 (MVA)

System 2 (MVA)

P78 (MW)

992+j360 1045+j238 1093+j354 1141+j355

1010+j443 1060+j410 1110+j438 1160+j432

200 200 200 200

^ þ dðsÞDðs; dÞ

d1(s) d2(s) d3(s) d4(s)

Local

Inter-area

0.7367j6.472 0.4217j7.289 0.6117j4.591 0.7447j5.574

0.3357j3.272 0.5577j3.787

dðs; dÞ ¼ d0 þ d1 s þ    þ dw1 sw1 þ dw sw w1

(40) w

þ ni;w s

(41)

If we assume that a c-order fixed-parameter controller can be expressed as follows: 1 ^ dðsÞ

Ni ðs; ni Þ is the family of polynomials corresponding to the interval polynomial ni ðs; ni Þ Dðs; dÞ is the family of polynomials corresponding to the interval polynomial dðs; dÞ

Dominant poles

Assuming that the order of the plant is equal to w, the different interval polynomials in the numerators and denominators of the transfer functions can be expressed in the following way:

ni ðs; ni Þ ¼ ni;0 þ ni;1 s þ    þ ni;w1 s

(45)

is stable. The definitions of the different polynomials of (45) are presented in the following:

Table 4 Dominant poles of Kharitonov transfer function, corresponding to different numerator polynomials. Denominator polynomials of GK ðsÞ

GCm(s)

Fig. 9. The structure of the proposed multi-input robust controller.

Table 3 Generation and tie-line active power of the multimachine system.

^ GC ðsÞ ¼ gðsÞ ¼

Vsc

G4

System 1

1 2 3 4

+ +

GCi(s)

4

G2

Case

GC1(s)

C9

C7 SL7

With SVC controller

ðn^ 1 ðsÞ; . . . ; n^ i ðsÞ; . . . ; n^ m ðsÞÞ

(42)

where ^ dðsÞ ¼ d^ 0 þ d^ i;1 s þ    þ d^ c1 sc1 þ d^ c sc

(43)

n^ i ðsÞ ¼ n^ i;0 þ n^ i;1 s þ    þ n^ i;c1 sc1 þ n^ i;c sc

(44)

Fig. 9 depicts the structure of the analyzed multi-input controller.

Moreover, it follows from Chapellat and Bhattacharyya (1989) that the family of polynomials (45) is stable if and only if a prescribed set of k4k line segments is stable, where k ¼ m+1 (and m is the controller input number). This number is independent of the degree, w, of the polynomials. The family of k4k segments is defined as follows. For any fixed integer l between 1 and k, set Pi ðsÞ ¼ K qi ðsÞ

for ial and for some q ¼ 1; 2; 3; 4

(46)

Kw i ðsÞ

where are the four Kharitonov polynomials associated with the family of interval polynomials that correspond to N1 ðs; n1 Þ; . . . ; Ni ðs; ni Þ; . . . ; N m ðs; nm Þ; Dðs; dÞ. For l, suppose that Pl ðsÞ varies within one of the four Kharitonov segments bK 1l ðsÞ; K 2l ðsÞc

or

or bK 2l ðsÞ; K 4l ðsÞc

bK 1l ðsÞ; K 3l ðsÞc or

bK 3l ðsÞ; K 4l ðsÞc

bK 1l ðsÞ; K 2l ðsÞc

where the segment of the following form:

(47)

means all convex combinations

½K 1l ðsÞ; K 2l ðsÞ ¼ lK 1l ðsÞ þ ð1  lÞK 2l ðsÞ;

l 2 h0; 1i

(48)

A complete description of the GKT, including its demonstration and some simple application examples, can be found in Bhattacharyya et al. (1995).

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It follows from the GKT that an increase in the control path number leads to a substantial increase of conditions that should be fulfilled by the robust controller. If the controller has only one control path, the maximal number of conditions is equal to 32, but for two paths it is equal to 192, for three paths it equals 1024, and so on. Hence, a controller with the smallest possible number of control paths is preferred, for the sake of simplicity. At this stage in the analysis, it was assumed that the maximal number of paths was equal to 2. Remarks:

 Practical use of the GKT is difficult, due to a lack of analytical





methods for solving the problem with a large number of robust controller inputs. This leads to the use of heuristic methods, based on grid-search procedures, to solve the controller parameter synthesis problem. The proposed approach for determining the admissible set of controller parameters consists of searching for a solution space by means of the network method, during which it is assumed that the upper and lower boundary values, as well as the parameter variation steps, belong to the controller parameter vector K. The computing procedure starts at the particular case of l ¼ 0, corresponding to the family of Kharitonov transfer functions. The stability of the Kharitonov segments for la0 is checked afterwards. The complexity of the synthesis can be reduced by assuming that the different control paths are composed of first-order real differential controllers GCi ðsÞ ¼ T ri s=ð1 þ T ri sÞ, where the differential constants for the different control paths are equal, T r1 ¼ T r2 ¼ T r .

Δ31

Δ24

1289

-12500

17500

1.2s 1+1.2s

+ UR = Usc Σ +

1.2s 1+1.2s

Fig. 11. Examples of multi-input robust controller.

Table 5 Simulation results for a multimachine test system. System operating conditions Integral of deviation of

Case Case Case Case

1 2 3 4

Without SVC controller

With SVC controller

Id12

Id34

Id31

Id12

Id34

Id31

0.0198 0.0214 0.0227 0.0246

0.0342 0.0358 0.0390 0.0412

0.4365 0.4655 0.5033 0.5356

0.0191 0.0209 0.0215 0.0228

0.0329 0.0346 0.0378 0.0402

0.4080 0.4362 0.4740 0.5068

Results of the time analysis of the non-linear system model are presented in Table 5, in order to appropriately evaluate the controller features. Table 5 includes the results of integrating the absolute values of the errors in rotor angle deviations, where

 Id12, difference between the rotor angles of the system 1 generators;

 Id34, difference between the rotor angles of the system 2 generators; and 7.3. Test results

 Id31, difference between the rotor angles of generators G3 and G1.

Fig. 10 depicts some examples of the results of robust controller synthesis. The results are shown as a set of admissible solutions in the parameter space for the following input signals: input signal 1Do31 ¼ Do3Do1, input signal 2Do24 ¼ Do2Do4. The depicted results were obtained for a case in which both control paths were first-order. Running an optimization procedure, where the performance index Io takes into account not only the local oscillations Do21, Do43, but also the inter-area oscillations Do31, leads to a final form for the controller, depicted in Fig. 11.

A transient fault occurring at node 7 has been modeled as a test fault. The fault lasted 100 ms, and the simulation time was equal to tk ¼ 10. The SVC output signal was limited to 70.25 p.u. for the purposes of analyzing the non-linear system. The results included in Table 5 show that the designed controller enhances the electromechanical oscillation damping for the analyzed operating conditions. The results presented above are only a starting point for further universal analysis. The author is convinced that the development of interval polynomial theory and the analysis of its application in multimachine system models will allow for useful procedures for synthesizing robust FACTS controllers.

8. Conclusions

Fig. 10. The set of admissible solutions, where K1 corresponds to an amplification of the slip signal Do31, whereas K2 corresponds to Do24.

A novel method of designing robust SVC controllers for both generator-infinite bus systems and multimachine systems has been presented in this paper. Uncertainty in the power system was described using a transfer function with interval polynomials in the numerator and denominator. The analyzed uncertainty is the variation in operating conditions due to load variation. The model of the proposed robust controller is based on interval systems theory, which relies on Kharitonov’s Theorem. The advantage of this theory is that it facilitates the design of a fixed-parameter low-order controller. On the other hand, this theory has a certain drawback—the computational burden is much heavier than that required by other methods, especially for determining the boundary values of the uncertain power system’s model parameters.

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The above-presented design method allows the region of stability to be determined, which is calculated in the controller parameter space, where the stability of the model is analyzed. The optimal parameters of the controller are then determined, based on a defined performance index. A detailed solution of the design of a first-order controller for the single machine infinite busbar system is presented in this paper. Due to the complexity of the problem, only some preliminary results of the application of a multi-input controller in a multimachine power system are included. A comprehensive solution will be presented shortly, in the next paper. The results of the analysis of the designed controller features are outlined. They have confirmed that the designed controller is robust, since it efficiently damps electromechanical oscillations at different analyzed power system operating points.

Appendix The parameters of the test system under consideration are: network: rated voltage 220 kV; short-circuit power: 15 000 MVA; line: R ¼ 0.0564 O/km, X ¼ 0.418 O/km, L11, L12 and L21, L22 lines length: 100 and 160 km. Step-up transformer: Sn ¼ 426 MVA, Vn ¼ 22/220 kV, VSHC ¼ 0.12Vn. Generator: Sn ¼ 426 MVA, Vn ¼ 22 kV, cos fn ¼ 0.85, M ¼ 6.45, Xd ¼ 2.6, X 0d ¼ 0:33, X 00d ¼ 0:235, T 00d0 ¼ 0:042, X q ¼ 2:48, X 0q ¼ 0:53, X 00q ¼ 0:29, T 0d0 ¼ 9:2, 0 00 T q0 ¼ 1:095, T q0 ¼ 0:065. Reactances are in p.u., time constants are in seconds. The voltage controller data is: K A ¼ 200, T A ¼ 0:02 s, T B ¼ 1 s, T C ¼ 10 s. Static Var Compensator data is: K SVC ¼ 10 and T SVC ¼ 0:15 s. References Ackerman, J., Kaesbauer, D., Sienel, W., & Steinhauser, R. (1993). Robust control systems with uncertain physical parameters. London: Springer. Adibi, M. M., & Milanicz, D. P. (1994). Reactive capability limitation of synchronous machines. IEEE Transactions on Power Systems, PWRS-9, 29–40. Barmish, B. R., Hollot, C. V., Kraus, F. J., & Tempo, R. (1992). Extreme point results for robust stabilization of interval plants with first order compensators. IEEE Transaction on Automatic Control, 37(11), 707–714. Bhattacharyya, S. P., Chapellat, H., & Keel, L. H. (1995). Robust control. The parametric approach. London: Prentice-Hall. Chapellat, H., & Bhattacharyya, S. P. (1989). A generalization of Kharitonov’s theorem; Robust stability of interval plants. IEEE Transaction on Automatic Control, 34(3), 306–311. Chaturvedi, D. K., & Malik, O. P. (2005). Generalized neuron-based PSS and adaptive PSS. Control Engineering Practice, 13(12), 1507–1514. CIGRE (1999). Modeling of power electronics equipment (FACTS) in load flow and stability programs, TF 38.01.08.

Ellithy, K. A., & Al-Naamany, A. (2000). A hybrid neuro-fuzzy SVC stabilizers for power system damping improvement in the presence of load parameters uncertainty. Electric Power Systems Research, 56(3), 211–223. Ho, M. T., Datta, A., & Bhattacharyya, S. P. (1997). A linear programming characterization of all stabilizing PID controllers. In Proceedings of the American control conference (Vol. 6, pp. 3922–3928), Albuquerque, New Mexico. Huang, Y. J., & Wang, Y.-J. (2000). Robust PID tuning strategy for uncertain plants based on the Kharitonov theorem. ISA Transactions, 39(4), 419–431. Khutoryansky, E., & Pai, M. A. (1997). Parametric robust stability of power systems using generalized Kharitonov’s theorem. In 36th IEEE conference on decision and control (Vol. 4, pp. 3097–3099), San Diego, December 10–12. Kundur, P. (1994). Power system stability and control. New York: McGraw-Hill. Kurth, M., & Welfonder, E. (2005). Oscillation behaviour of the enlarged European power system under deregulated energy market conditions. Control Engineering Practice, 13(12), 1525–1536. Lachs, W. R., & Sutanto, D. (1995). Uncertainty in electricity supply controlled by energy storage. In International conference on energy management and power delivery. Proceedings of EMPD 1995 (Vol. 1, pp. 302–307). Machowski, J., Bialek, J. W., & Bumby, J. R. (2008). Power system dynamics: Stability and control. Chichester: Wiley. Makarov, Y. V., Hill, D. J., & Milanovic, J. V. (1997). Effect of load uncertainty on small disturbance stability margins in open-access power systems. In Proceedings Hawaii international conference on system sciences HICSS-30 (Vol. 5, pp. 648–657), Kihei, Maui, Hawaii, January 7–10. Nejmark, J. J. (1978). Dynamical systems and controlled processes. Moskva: Nauka (Dinamicheskije sistemy i upravljajemyje processy—in Russian). Nilsson, N. E., & Mercurio, J. (1994). Synchronous generator capability curve testing and evaluation. IEEE Transactions on Power Delivery, 9(1), 414–424. Othman, H. A., Chow, J. H., & Taranto, G. N. (1992). Modeling of impedance uncertainty in power system networks. In 31st IEEE conference on decision and control (pp. 591–592), December 12–18. Pal, B. C., & Chaudhuri, B. (2005). Robust control in power systems. New York: Springer. Praprost, K. L., & Loparo, K. A. (1992). Power system transient stability analysis for random initial conditions and parameters. In IEEE international conference on systems, man and cybernetics (Vol. 1, pp. 761–767), October 18–21. Robak, S. (2008). Sources of uncertainty in power system analysis (Z´ro´d"a niepewnos´ci w analizie systemo´w elektroenergetycznych). Przegla˛d Elektrotechniczny (Electrical Review)(1/2008), 54–57 (in Polish). Sadikovic, R., Korba, P., & Andersson, G. (2006). Self-tuning controller for damping of power system oscillations with FACTS devices. IEEE power engineering society general meeting. Schweickardt, H. E., Romegialli, G., & Reichert, K. (1978). Closed loop control of static VAR sources (SVS) on EHV transmission lines. IEEE PES winter power meeting, (paper no A78, pp. 135–136), New York, Jan. 29–Feb. 3. Soliman, H. M., Elshafei, A. L., Shaltout, A. A., & Moris, M. F. (2000). Robust power system stabilizer. IEE Proceedings of the Generation, Transmission, and Distribution, 147(5), 285–291. So¨ylemez, M. T., Munro, N., & Baki, H. (2003). Fast calculation of stabilizing PID controllers. Automatica, 39(1), 121–126. Sullivan, D.J. (2006). Improvements in voltage control and dynamic performance of power transmission systems using static VAR compensators (SVC). Master’s thesis, University of Pittsburgh, School of Engineering. Tan, N. (2003). Computation of stabilizing lag/lead controller parameters. Computers and Electrical Engineering, 29(8), 835–846. Tan, N., Kaya, I., Yeroglu, C., & Atherton, D. P. (2006). Computation of stabilizing PI and PID controllers using stability boundary locus. Energy Conversion and Management, 47(18–19), 3045–3058. Taylor, C. W., et al. (1994). Static var compensator models for power flow and dynamic performance simulation. IEEE Special Stability Controls Working Group. IEEE Transactions on Power Systems, 9(1), 229–240. /www.mathworks.comS.