Automatic tuning method for the design of ... - IEEE Xplore

www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 29th August 2008 Revised on 9th June 2009 doi: 10.1049/iet-gtd.2008.0424

ISSN 1751-8687

Automatic tuning method for the design of supplementary damping controllers for flexible alternating current transmission system devices R.V. de Oliveira1 R. Kuiava2 R.A. Ramos2 N.G. Bretas2 1

Electromechanical Engineering Department, Federal Technological University of Parana´, CEP 85503-390, Pato Branco, PR- Brazil 2 Engineering School of Sao Carlos, University of Sao Paulo, CEP 13566-590, Sao Carlos, SP- Brazil E-mail: [email protected]

Abstract: The design of supplementary damping controllers to mitigate the effects of electromechanical oscillations in power systems is a highly complex and time-consuming process, which requires a significant amount of knowledge from the part of the designer. In this study, the authors propose an automatic technique that takes the burden of tuning the controller parameters away from the power engineer and places it on the computer. Unlike other approaches that do the same based on robust control theories or evolutionary computing techniques, our proposed procedure uses an optimisation algorithm that works over a formulation of the classical tuning problem in terms of bilinear matrix inequalities. Using this formulation, it is possible to apply linear matrix inequality solvers to find a solution to the tuning problem via an iterative process, with the advantage that these solvers are widely available and have well-known convergence properties. The proposed algorithm is applied to tune the parameters of supplementary controllers for thyristor controlled series capacitors placed in the New England/New York benchmark test system, aiming at the improvement of the damping factor of inter-area modes, under several different operating conditions. The results of the linear analysis are validated by non-linear simulation and demonstrate the effectiveness of the proposed procedure.

1

Introduction

Inter-area oscillations are a common phenomenon observed in power systems world-wide, where groups of synchronous generators are interconnected over long transmission lines. Usually, these lines create a weak electric coupling among the generator groups and must sustain high levels of power flow during normal operation. Consequently, these oscillations can exhibit poor damping in the absence of an adequate stabilising control [1]. In recent years, due to the development of power electronics, flexible alternating current transmission system (FACTS) devices have been successfully used to improve steady-state and dynamic system performance, and became IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

an interesting cost-effective alternative compared to the system expansion [2, 3]. There are several types of FACTS devices that can be used to provide additional damping to inter-area oscillations by inclusion of a supplementary damping controller (SDC) to the device [4]. From the practical viewpoint, there are a number of features that any damping controller has to possess. The following list depicts some of them: 1. the controllers must be robust with respect to the uncertainties in the system operating point; 2. multiple damping controllers operating simultaneously in a system must have a coordinated action; 919

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www.ietdl.org 3. the action of the damping controllers must vanish in steady state, so the controllers do not change the operating point defined by the load flow; 4. a control structure based on dynamic output feedback must be used due to difficulties in obtaining measurements of all state variables of the system; 5. whenever possible, it is preferable to use local control input signals, because the use of remote signals generally increases the cost of the control scheme [wide-area control systems (WACS) have been recently proposed and are still under development, but they are still not widely available and satisfactorily reliable at a reasonable cost]. It is an usual practice to employ classical phase compensation to design SDCs for FACTS devices. These controllers consist basically of a static gain, a washout filter and a phase compensation network [4]. Such a control structure satisfies the requirements (iii)–(v). However, in order to fulfill the requirements (i) and (ii), a careful selection of the phase compensators has to be made, which is not a trivial task. The experience accumulated over years of practice has generated a number of guidelines for the selection of these parameters. The first widely used set of guidelines for this purpose is reported in [5], and the application of them to power system stabiliser (PSS) design is usually called a tuning process. The advent of selective modal analysis [6], and later on the introduction of residue analysis [7] and induced torque coefficients [8] have significantly improved the efficiency of the tuning methods. However, these problems are still very difficult to handle, because they involve several variables and degrees of freedom, and therefore only highly skilled engineers are able to perform an effective tuning of controller parameters for several operating conditions. A number of alternative techniques, based on robust control theories, have been proposed to address these issues. H1 mixedsensitivity formulation [9] and regional pole placement using linear matrix inequalities (LMIs) [10–12] can be cited as examples. These approaches generate very efficient controllers, but most of them provide high-order controller structures. The industry seems to prefer conventional phase compensation structures, as several industry-based researchers adopt them as their structures of choice ([13] is an example). Based on this consideration, this paper proposes an algorithm that aims at relieving the engineer in charge of the controller design from the burden of the trial-and-error process involved in this problem. The designer only has to set up a proper range of controller parameters, based on welldefined equations, and to supply this range as input data to the algorithm, which then applies an iterative method (based on the solution of LMIs) to find an acceptable tuning considering all the operating conditions of interest. 920 & The Institution of Engineering and Technology 2009

The paper is structured as follows: Section 2 depicts the formulation of the tuning problem in terms of bilinear matrix inequalities (BMIs) and Section 3 presents the algorithm that tunes the controller parameters with an iterative LMI solving process, also presenting what is required from the designer and how the burden of manually tuning these parameters is taken off him/her; the results of the application of the proposed algorithm to the New England/New York benchmark system are presented in Section 4, and Section 5 presents the conclusions.

2 Formulation of the tuning problem in terms of BMIs Since the objective of this paper is to tune SDCs for FACTS devices, the following presentation will assume that the formulation of the problem is suited to this type of design. As will be seen, however, the formulation is general enough to enable the simultaneous tuning of both PSSs and SDCs for FACTS devices. The standard approach to power system modelling for SDC tuning is based on a set of nonlinear differential-algebraic equations in the form x_~ ¼ f (~x, u, ~ z, l) 0 ¼ g(~x, u~ , z, l)

(1) (2)

y~ ¼ h(~x, u~ , z, l)

(3)

where x~ [ Rn is the system state vector, u~ [ Rp is the control input vector, y~ [ Rq is the measured output, z [ Rm is a vector of algebraic variables representing the transmission network coupling among the state variables and l [ Rl is a vector of parameters, representing the load levels and other quantities defining the system operating condition. The algebraic constraints (2) can be eliminated from (1) to (3), and the resulting equations can be linearised around a specific operating condition, which gives x_ j ¼ Aj xj þ Bj uj

(4)

yj ¼ C j xj þ Dj uj

(5)

In (4) and (5), xj [ Rn represents a deviation from an equilibrium point x~ je of (1) – (3), obtained for a particular value of the parameter vector l. In a similar way, uj [ Rp and yj [ Rq represent deviations from u~ je and y~ je , respectively. To ensure the robustness of the designed controllers with respect to variations in the parameter vector l, the industry usually works with a set of models (4) and (5) obtained by linearisation around several different equilibrium points x~ je , j ¼ 1, . . . , L, each of them corresponding to a respective value of interest for l. As mentioned in Section 1, phase compensation is the typical control approach used by the industry to implement both PSSs and SDCs for FACTS devices. The block diagram of such a structure is shown in Fig. 1, where the phase compensation elements were lumped in the middle block, for simplicity. IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

www.ietdl.org into a single one, given by 2

Ac1 ^ ^ ^ 6 . ^ _ x c ¼ A c x c þ Bc y j ¼ 6 4 ..

Figure 1 Typical PSS or SDC block diagram The subscript i in all controller parameters of Fig. 1 denote that this controller is added to the ith FACTS device. The first block in Fig. 1 is known as a washout filter and guarantees that the controller gain is zero under steadystate conditions. This block possesses a derivative action represented by the term sTwi , which can be incorporated into the plant to simplify the design formulation. Note that this does not imply that an ideal derivative signal must be measured in the plant, since this term can be combined with one of the poles of the controller after the design is carried out, and the result can be implemented as a typical washout block, as explained in [11]. To do so, we can define a new system output vector y j [ Rq as  j uj y j ¼ y_ j ¼ C j xj þ D

(6)

 j ¼ C j Bj are matrices with where C j ¼ C j Aj and D dimensions determined by Aj , Bj and C j . We remark that  j can be a non-zero matrix for some type of FACTS D devices due to the sensibility of the line active power flow to variations in the FACTS controllable parameter. Considering a controller with two lead-lag blocks (which corresponds to n ¼ 2), the transfer function corresponding to the block diagram in Fig. 1 can be put in state-space form as follows 2

0 gi

ai 6 x_ ci ¼ 4 gi  ai bi ai b2i  bi gi 

uji ¼ 0 0

gi  bi gi

3 2 3 1 0 7 6 7 0 5xci þ 4 bi 5y ji (7) g i b2i



(8)

Ki xci

where

2

 .. .

0  3

3 0 .. 7 . 7 5xc Acnc

0 .. 7 . 7 5y j    Bcnc 2 3 0 C c1    ^ ^ 6 . ^ ^ .. 7 .. . uj ¼ C c x c ¼ 6 . 7 . 4 . 5xc 0    C cnc Bc1 6 . þ6 4 .. 0

 .. .

(12)

(13)

The final closed-loop models of the controlled power system can be obtained from the combination of the open-loop system models given by (4) and (6) with the SDCs models given by (12) and (13), which can be written in the form h ^ iT ^ ^ ^ _ ^ x j ¼ Aj x j , where x j ¼ xj x c and ^

2

3

^

Aj

A j ¼4 ^ BC C j

^

Bj C C ^

^

 j CC A C þ BC D

5

(14)

for j ¼ 1, . . . , L. From (14), it becomes clear that the problem of tuning the SDC parameters consists in finding ^

^

^

^

a matrix triplet (Ac , Bc , C c ) ensuring that matrices Aj , j ¼ 1, . . . , L, fulfill some desired performance criterion. The criterion that is most widely accepted by the industry states that the performance of the closed-loop non-linear system can be considered as satisfactory if all eigenvalues of ^

all matrices A j , j ¼ 1, . . . , L, present a damping ratio greater than a certain pre-specified minimum value zmin . ^

This specification defines the loci of eigenvalues of Aj for acceptable performance as a conic sector in the complex plane, as shown in Fig. 2, where u ¼ arccos zmin . Considering the previous background, it is possible to see ^

1 ai ¼ , Twi

T bi ¼ 1i , T2i

1 gi ¼ T2i

(9)

^

^

that the problem of finding a matrix triplet (Ac , Bc , C c ) that fulfills the minimum damping ratio specification can be cast

In (7) and (8), xci [ Rnci is a vector composed by the state variables of the ith SDC. Equations (7) and (8) can be written in a more compact form x_ ci ¼ Aci xci þ Bci y ji

(10)

uji ¼ C ci xci

(11)

Since we have one set of Equations (10) and (11) for each SDC, we can lump all these nc sets of equations (nc being the number FACTS in the system equipped with SDCs) IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

Figure 2 Loci of eigenvalues of A j indicating acceptable performance of the closed-loop system 921

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www.ietdl.org as a search problem in Rcc  Rcp  Rqc , where c  c, c  p and q  c are, respectively, the dimensions of ^

^

^

matrices A c , Bc and C c . Recent works solve this problem using optimisation algorithms based on evolutionary computing techniques (such as, for example, the genetic algorithms used in [14]). These solutions generally achieve good results, despite the intense computational burden involved in these algorithms, which comes from the need to investigate a large number of points in the search space, in order to avoid getting stuck on local optima. This paper opts for a different type of optimisation algorithm that is based on Lyapunov stability theory. The idea is to find, ^

^

^

together with the matrix triplet (A c , Bc , C c ), a set of matrices ^

P j , j ¼ 1, . . . , L, that can prove local stability of each operating condition [15]. Based on Lyapunov theory, each equilibrium x~ je of (1)–(3) is locally stable if and only if there ^

^

^

^

exist matrices Ac , Bc , C c and P j such that ^T

^

Pj ¼ Pj  0

^ ^

^

^T ^

and P j A j þ Aj P j  0

^

^T

(15)

^

 T  ^ ^ ^ ^ 6 sin u Aj P j þ P j Aj 6 6  T T ^ ^ ^ ^ 4 cos u Aj P j  P j Aj

^

^

^

^

linear in P j , j ¼ 1, . . . , L, if matrices A c and C c are fixed. The advantage of transforming (16) into a linear problem is that the search space becomes convex, in such a way that any available LMI solver can handle it. Based on this consideration, it is possible to set up an algorithm to find a solution to (16) by iteratively solving the resulting LMIs ^

^

^

^

when either the P j or the Ac , Bc and C c matrices are fixed, in an alternate manner. Such an algorithm is known as V – K iteration and can be summarised as follows: Algorithm 1 (V – K iteration): ^T

^

Step 1: Choose a set of arbitrary matrices P j ¼ P j  0, j ¼ 1, . . . , L, to initialise the algorithm; ^

^

previous step, minimise s over the entries of Ac and C c subject to 2

 T   T 3 ^ ^ ^ ^ ^ ^ ^ ^ u A P þ P A u A P  P A cos sin j j j j j j j j 7 6 6 7 6  T T  T  7  sI  0 ^ ^ ^ ^ ^ ^ ^ ^ 4 5 cos u Aj P j  P j Aj sin u Aj P j þ P j Aj (17) Step 3: Check whether s , 0; if yes, stop; if not, freeze the

A c , Bc , C c and P j such that P j ¼ P j  0 and 2

^

becomes linear in A c and C c . Conversely, the problem is

^

As mentioned earlier, local stabilisation is not enough to ensure an adequate performance for the closed-loop power system. To fulfill the minimum damping ratio criterion discussed in the previous paragraphs, we must find ^

^

that if matrices P j , j ¼ 1, . . . , L, are fixed, the problem

Step 2: Freeze the values of P j , j ¼ 1, . . . , L, obtained in the

for j ¼ 1, . . . , L, where the notations M  0 and N , 0 indicate positive and negative definiteness of matrices M and N , respectively.

^

this in mind, and looking at (14) and (16), it is easy to see

^

 T 3 ^ ^ ^ ^ cos u A j P j  P j Aj 7 7  T 70 ^ ^ ^ ^ 5 sin u Aj P j þ P j Aj

^

in the previous step, values for A c and C c obtained ^ minimise s over the entries of P j , j ¼ 1, . . . , L, subject to ^T

^

P j ¼ P j  0 and (17); Step 4: Check whether s , 0; If yes, stop; if not, return to Step 2.

(16) The proof of this statement can be traced from [15, 16] and will not be repeated here due to space limitations. The existence of cross products between the entries of the matrix variables makes this search problem quite complex, because the set of feasible solutions to it is not convex. In the next section, an algorithm to deal with this kind of search will be depicted and tailored to the SDC parameter tuning problem.

3 Algorithm to tune SDC parameters 3.1 BMI solving algorithm ^

As will be seen later, the entries of matrix Bc depend only on the maximum values of phase compensation allowed for the SDCs and, therefore, this matrix is fixed (i.e. its entries are kept constant) during the whole tuning process. Having 922 & The Institution of Engineering and Technology 2009

It is easy to see that both Steps 2 and 3 involve solving a set of LMIs, and that the value of s is non-increasing as the algorithm iterates. The convergence properties of this algorithm are discussed in [17]. Furthermore, it is clear that, once the condition s , 0 is reached, a solution to the original BMI was found, because (17) becomes equivalent to (16) if s , 0. However, global convergence to a solution of (16) might not be expected from this algorithm, as well as from the vast majority of search methods working over non-convex feasible sets. Since the set of feasible solutions is open, additional restrictions must be included to restrict this set to physically and practically meaningful solutions. The engineering part of our proposed approach starts at this point, with the definition of additional constraints on the set of feasible solutions to capture the ranges of SDC parameters that would be suitable for practical implementation. The next section describes the process and tools for determining these constraints. IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

www.ietdl.org 3.2 Defining the ranges of acceptable SDC parameters As previously mentioned, it is necessary to include additional constraints to the search problem presented in Section 3.1 to restrict the set of possible solutions to physically and practically meaningful ones. The phase compensation range that provides additional damping for a particular set of electromechanical modes can be determined by means of residue analysis [7] or induced torque coefficients [8], for example. In this paper, we chose to apply the former, although similar results could be achieved with the latter. In order to enable the application of residue analysis, we initially set the parameters ai and bi , i ¼ 1, . . . , nc, appearing in (7), to constant, pre-determined values. The determination of the values for parameters ai follows well-known industry rules, since these parameters are related to the washout filters – as can be seen in (9) – and defined according to typical values used in most SDC and PSS designs. Common values for the washout time constant are in the range of 3 – 20 s [5], which leads to

ai ¼ ai where ai

const

const ,

0:05 , ai

const

, 0:33

(18)

is the constant value set for ai , i ¼ 1, . . . , nc.

For the determination of bi parameters, we take into account that these parameters are related to the maximum phase compensations wi max that each corresponding SDC can give to its respective input signal. This relation is given by [18] T 1 þ sin(wmax i ) bi ¼ 1i ¼ T2i 1  sin(wmax i )

(19)

for i ¼ 1, . . . , nc. This maximum phase compensation can be specified from a residue analysis of the transfer function relating the output and input signals to the ith SDC, considering the oscillation mode of interest lk (which is usually an inter-area mode to be damped by this SDC). With these fixed values for ai and bi , the actual phase compensation that will be given to the input signal for the ith SDC will be ultimately determined by the gi parameter. Therefore imposing limits on this parameter will result in restricting the set of controllers in the form (7) and (8) that can be provided by our proposed algorithm to contain only practically meaningful ones. These limits can be written as

gi

min

, gi , gi

max

(20)

for i ¼ 1, . . . , nc, where gi min and gi max are, respectively, upper and lower bounds on the allowed values for gi . The upper bounds gi max are calculated by [18]

gi

max

pffiffiffiffiffi ¼ vk bi

mode of interest lk . It is worth mentioning that, if gi is set to gi max , the phase compensation provided by the ith SDC to lk will be wi max . In the other extreme of the range, we must also define a minimum phase compensation wi min to be used in the determination of the lower bound gi min , which can be done using the relation [18] (

1 þ j vi bi g1 i min arg 1 þ j vi g1 i min

) ¼

wi

min

(22)

2

or, equivalently

g2i min tan(wi

min ) þ gi min(vi

 vi bi ) þ v2i bi tan(wi

min ) ¼

0

(23) The static gains Ki , i ¼ 1, . . . , nc, are then the only remaining parameters in (7) and (8) to be constrained. According to [19], using residue analysis it is possible to calculate the gain Ki residue of an SDC as a function of the desired location for the eigenvalue corresponding to lk by Ki

residue

  l  lk  ¼  k des Ri HLD (lk )

(24)

where lk des is the location that provides the desired damping to the eigenvalue corresponding to lk and 

HLD

sGwi ¼ 1 þ sGwi



1 þ sG1i 1 þ sG2i

n (25)

In (25), parameters Gwi , G1i and G2i must be appropriately set to provide the ideal phase compensation for the residue corresponding to lk . Note that (25) is only used to define the bounds on parameters Ki and, therefore, Gwi , G1i and G2i are only instrumental to the process. Considering the previous analysis, it is possible to define a range for lk des that will translate into upper bounds Ki max and lower bounds Ki min for parameters Ki , which can be written as Ki

min

, Ki , Ki

max

(26)

for i ¼ 1, . . . , nc. With the fixed values for ai and bi and the bounded ranges for gi and Ki given by (20) and (26), respectively, the set of feasible solutions to the SDC tuning problem is now closed and practically meaningful, and therefore we are ready to present our proposed algorithm to tune SDCs for FACTS devices, which will be done in the next section.

3.3 Proposed SDC parameter tuning algorithm Algorithm 2 (SDC tuning algorithm):

(21)

for i ¼ 1, . . . , nc, where v k is the frequency of the oscillation IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

Step 1: Select an oscillation mode lk to which the SDCs have to provide additional damping and determine the values of 923

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www.ietdl.org the maximum and minimum phase compensations wi and wi min , i ¼ 1, . . ., nc, using residue analysis;

max

Step 2: Calculate the values of parameters ai , using (18), and bi , using (19), the bounds gi max and gi min , using (21) and (23), and the bounds Ki max and Ki min , using (24) and (25), for i ¼ 1, . . ., nc; Step 3: With the values of ai and bi calculated in Step 2, ^

build matrices A j and then choose a set of arbitrary positive ^

definite matrices P j , j ¼ 1, . . . , L; ^

Step 4: Freeze the values of P j , j ¼ 1, . . . , L, obtained ^ ^in the s over the entries of A previous step, and minimise c , B c and ^ C c subject to (20), (26) and 2

 T   T 3 ^ ^ ^ ^ ^ ^ ^ ^ cos u Aj P j  P j Aj 7 6 sin u Aj P j þ P j Aj 6 7 6  T T  T  7  sI  0 ^ ^ ^ ^ ^ ^ ^ ^ 4 5 cos u Aj P j  P j Aj sin u Aj P j þ P j Aj (27)

4 Example of application of the proposed algorithm To demonstrate the effectiveness of our proposed procedure, tests were carried out on the benchmark New England/ New York interconnected system shown in Fig. 3. All the generators were described by a sixth-order model [20] and equipped with a first-order model of a static type AVR, with a gain of 50 p.u./p.u. and a time constant of 0.01 s. The transmission system was modelled as a passive circuit and the system loads as constant impedances. Classical PSSs based on rotor speed signals were placed at generators G2, G3, G5, G7, G9, G10, G11, G12, G14 and G15. These PSSs were tuned based on the residue method given in [7], aiming purely at enhancing damping of the local modes. These conditions are a good representation of what happens in practice, since we have two options to increase damping of the inter-area mode: a major retuning of all stabilisers in the system or the placement of FACTS devices, with their respective SDCs, in strategic points of the system. This paper will assume that the second option was chosen, and thyristor-controlled series capacitors (TCSCs) were selected to perform this task.

^

Step 5: Check if all eigenvalues of Aj , j ¼ 1, . . . , L, have a damping ratio greater than zmin ; If yes, go to Step 6; If not, ^

^

^

freeze the values for A c , Bc and C c obtained in the previous ^

step, and minimise s over the entries of P j , j ¼ 1, . . . , L, ^T

^

subject to P j ¼ P j  0, (17), (20) and (26); then, return to Step 4; Step 6: From the values obtained for ai , bi , gi and Ki , for i ¼ 1, . . ., nc, calculate the values of the SDC parameters Twi , T1i , T2i and KPSSi using (9).

Typical simplifying assumptions [3] were used to model the TCSCs in this example. The input signal to the TCSC is a desired reactance for the device, which compensates the line to generate the desired power flow under steady-state conditions. This input signal is modulated by the output of the SDC, and the net effect of the whole TCSC dynamic behaviour (involving measurement and processing of the input signals to calculate the firing angle and the firing of the thyristors at the calculated angle) over the equivalent reactance of the device was modelled by a first-order linear block, as can be seen in Fig. 4. It is important to remark

Figure 3 New England/New York benchmark test system 924 & The Institution of Engineering and Technology 2009

IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

www.ietdl.org

Figure 4 TCSC dynamic model used in this paper here that a more detailed model of the TCSCs could be used if the study required more precision in the representation of their dynamics. Similarly, other types of FACTS devices (including combined series/shunt devices like the UPFC, for example) could also be handled by our proposed algorithm in any level of detail, the only requirement being that a state-space model is given for them.

Figure 5 Daily load curve of the test system We remark that, although the topology of the network was not varied in these operating points, the proposed algorithm can handle such type of variation as well. The non-linear simulations at the end of this section also show that the tuned SDCs are robust to these network topology variations.

After a careful mode controllability/observability analysis (via residues and participation factors), two TCSCs were placed in the system, one of them being installed between areas #1 and #2, in one of the lines that connect buses #60 and #61. The other was installed between areas #2 and #5, in the line connecting buses #18 and #49. The steady-state compensation levels of both TCSCs correspond to 50% of the impedance of their respective lines.

The results of the eigenvalue analysis show that two interarea modes are not adequately damped in all considered operating conditions without the inclusion of SDCs, as can be seen in Table 1. This table shows that modes 3 and 4 are well damped, while mode 1 is poorly damped in the operating conditions from 7 a.m. to 10 p.m. Mode 2 is poorly damped for all points in the entire load curve.

The tests in this paper considered a daily load curve, which is shown in Fig. 5, composed by 24 operating points, each corresponding to a respective hour of the day. To construct this load curve, we assumed that the data given in [20] corresponds to a ‘base case’, which represents the system operation at 3 p.m. in Fig. 5 (in which the load level is approximately 62% of the maximum loading of the system).

To determine the allowable ranges for SDC parameters, a residue analysis of the operating point at 3 p.m. was conducted. The compensation angles required under this condition are 2107.88 for mode 1 and 293.38 for mode 2. Deviations of +308 with respect to these angles were used to set the ranges for phase compensation of both SDCs, and the results were rounded up to the

Table 1 Damping ratios (%) of inter-area modes for the system without the proposed SDCs Hour (a.m.)

Inter-area modes

Hour (p.m.)

#1

#2

#3

#4

0

6.02

3.66

7.86

8.50

1

7.40

3.75

7.80

2

6.30

3.74

3

6.21

4

Inter-area modes 1

2

3

4

12

4.73

3.10

8.76

9.87

8.34

1

4.66

3.08

8.79

9.87

7.76

8.39

2

4.63

3.06

8.82

9.93

3.68

7.77

8.15

3

4.51

3.00

8.91

10.01

6.27

3.61

7.46

8.33

4

4.24

2.90

9.05

10.13

5

6.05

3.12

7.71

8.56

5

3.84

2.72

9.29

10.31

6

5.43

3.38

8.33

9.23

6

3.58

2.62

9.44

9.53

7

4.97

3.21

8.61

9.65

7

3.73

2.68

9.31

10.30

8

4.44

2.50

8.16

9.50

8

3.99

2.79

9.20

10.25

9

4.40

2.98

8.94

10.04

9

4.33

2.94

8.99

10.10

10

4.54

3.03

8.88

9.98

10

4.77

3.12

8.73

9.83

11

4.82

3.14

8.70

9.77

11

5.27

3.33

8.42

9.45

IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

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www.ietdl.org nearest decade, resulting in 21208 , wSDC_1 , 2608 and 21408 , wSDC_2 , 2808. After that, the maximum allowed phase compensations (w1 max ¼ 120 and w2 max ¼ 140) were employed to calculate the bi parameters using (19), which resulted in b1 ¼ 0:072 and b2 ¼ 0:031 . The bounds on the gi variables were calculated from (21) and (23), using the frequencies of the inter-area modes of interest in the operating point at 3 p.m. (which are, respectively, v1 ¼ 4:63 rad/s and v2 ¼ 3:53 rad/s). The ranges 0:95 , g1 , 5:86 and 0:82 , g2 , 5:22 were obtained from these calculations. Based on (24), and again using the data corresponding to the operating point at 3 p.m., it was possible to conclude and that gains around KSDC 1 ’ 1:28 p.u./p.u. KSDC 2 ’ 0:41 p.u./p.u. would be necessary to provide a damping ratio of 5% for the two inter-area modes of interest, under these operating conditions. The maximum values for the gains were set to be at most twice the values given for the operating point at 3 p.m., and the minimum values were set to at least half these values, resulting in 0:65 , KSDC 1 , 2:6 and 0:2 , KSDC 2 , 0:8, respectively. The time constants of the washout filters were set as 10 s for both SDCs (Twi ¼ 10, i ¼ 1, 2), so they do not interfere significantly with the phase compensation. A minimum damping ratio of 5% for all modes in all 24 operating conditions was specified as the objective of the tuning algorithm, which was performed using the ‘mincx’ solver (available in Matlab LMI Control Toolbox) in a laptop with an Intel Dual Core 1.6 GHz processor and 1024 MB of RAM memory. The whole design process took about 15 h and 20 min, finishing after four iterations.

Figure 6 Inter-area modes of interest in the 5 p.m. operating point, as the algorithm iterates

The transfer functions of the designed controllers are presented in appendix. The location of the closed-loop system poles associated with the two least damped inter-area modes, as the algorithm iterates, are presented in Fig. 6. It can be seen that a solution is reached right after the least damped mode crosses the boundary of the region of acceptable performance. The damping ratios of the inter-area modes with the inclusion of the designed SDCs, in all 24 operating conditions, are shown in Table 2. It is possible to see in this table that all modes present damping ratios greater than 5%, thus fulfilling the design objective.

Table 2 Damping ratios (%) of inter-area modes for the system with the designed SDCs Hour (a.m.)

Inter-area modes

Hour (p.m.)

#1

#2

#3

#4

0

7.43

5.81

7.96

9.35

1

7.83

5.89

9.31

2

6.50

5.84

3

7.69

4

Inter-area modes #1

#2

#3

#4

12

6.21

5.34

8.83

10.67

9.19

1

6.14

5.33

8.86

10.66

7.88

9.26

2

6.12

5.31

8.89

10.71

5.84

7.86

9.00

3

6.01

5.25

8.97

10.79

7.71

5.77

7.91

9.15

4

5.74

5.16

9.12

10.90

5

7.82

5.86

7.95

9.31

5

5.36

5.05

9.35

11.06

6

6.92

5.59

8.40

10.05

6

5.01

5.01

9.50

11.11

7

6.46

5.43

8.69

10.45

7

5.06

5.26

9.37

11.03

8

5.94

5.24

9.00

10.82

8

5.50

5.07

9.26

11.00

9

5.89

5.22

9.03

10.85

9

5.85

5.20

9.06

10.86

10

6.03

5.27

8.95

10.76

10

6.26

5.36

8.80

10.63

11

6.27

5.36

8.86

10.69

11

6.73

5.54

8.50

10.26

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IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

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Figure 7 Comparative analysis between the SDCs from [21] and the ones tuned by the algorithm proposed in this paper

Figure 8 Operating conditions at 4 p.m.; solid line: system operating with SDCs; dashed line: without SDCs It is important to remark that the best efforts of an experienced designed could produce similar results in about the same time taken by our procedure to tune the SDCs. IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

For this reason, we chose to compare the results of our procedure with the ones given by another robust control technique that does not employ the classical concepts of 927

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www.ietdl.org phase compensation for SDC tuning. The method chosen for comparison was initially proposed for PSS design in [10] and [11] and recently adapted to the design of SDCs for FACTS devices in [21]. Fig. 7 presents this comparison, where it is possible to see that, although the SDCs from [21] provide phase lead over the entire range of possible electromechanical oscillations, they do so at the expense of also providing large amplification of signals in the high-frequency range. This is an important drawback of the method proposed in [21], from the practical viewpoint, since measurement noises and other highfrequency unmodelled dynamics will be substantially amplified in the controller responses, deteriorating their performances. In contrast, the SDCs tuned in this paper provide an amount of phase compensation that is sufficient to enhance the damping ratios of the electromechanical modes, and their respective gains in the high-frequency range are considerably lower than the ones of the controllers from [21]. Given that both types of controllers fulfill the desired performance criterion and the ones tuned in this paper have the mentioned practical advantage, the comparison is clearly favourable to the controllers proposed in this paper. Finally, to validate the conclusions taken from the linear analyses and to ensure performance robustness for the designed SDCs, non-linear simulations of the system response to several perturbations were carried out. The results of one of these simulations are shown in Fig. 8, where a three-phase solid fault was applied to one of the lines connecting buses 47 and 48, when the system operates under the conditions corresponding to 4 p.m. in Fig. 5. The fault was applied at t ¼ 2 s and lasted for 80 ms, being cleared by the disconnection of the faulted line. Since the operating condition without this line was not considered in the design stage, the good results shown in Fig. 8 under these unpredicted conditions are an indication that the controller performance is robust with respect to variations in the operating point. Several other simulations confirmed this statement.

5

Conclusions

time to perform other important tasks for his employer. Furthermore, the equations defining the ranges of allowable parameter variation for the SDCs are well-defined and straightforward to solve, which further simplifies the work of the designer with respect to the manual tuning process. The next step of this research is the investigation of the effects of system model dimension over the computational effort demanded by the algorithm. Owing to the large number of extra variables involved in the search procedure, this effort tends to increase exponentially if the dimension of the model grows. An alternative to overcome this difficulty is the a priori application of model order reduction techniques, which is a topic that is under investigation by the authors.

6

Acknowledgment

The authors wish to acknowledge the support of FAPESP to this research, under grants number 2004/04672-4 and 2006/ 05191-5.

7

References

[1] KUNDUR P., PASERBA J., AJJARAPU V., ET AL .: ‘Definition and classification of power system stability’, IEEE Trans. Power Syst., 2004, 19, (3), pp. 1387– 1401 [2] PASERBA J.J.: ‘How FACTS controllers benefit AC transmission systems’, IEEE Power Eng. Soc. General Meet., 2004, 2 pp. 1257– 1262 [3] HINGORANI N.G. , GYUGYI L. : ‘Understanding FACTS: concepts and technology of flexible AC transmission systems’ (IEEE Press, New York, 2000) [4] DEL ROSSO A.D., CANIZARES C.A., DONA V.M.: ‘A study of TCSC controller design for power system stability improvement’, IEEE Trans. Power Syst., 2003, 18, pp. 1487 – 1496 [5] LARSEN E.V., SWANN D.A. : ‘Applying power systems stabilizers, parts i, ii, iii’, AIEE Trans. Power Appar Syst., 1981, 6, pp. 3017 – 3046

As shown throughout the text, an automatic tuning of SDCs for FACTS devices can be carried out by the algorithm proposed in this paper. Although a relatively skilled engineer is still required to apply the software that will perform residue analysis in order to determine the ranges of allowable parameter variations that will be supplied to the algorithm, the burden of manually iterating over a large number of possible values for the SDC parameters is taken away from the designer and placed on the computer.

[7] PAGOLA F.L., PEREZ-ARRIAGA I.J., VERGHESE G.C. : ‘On sensitivities, residues and participations: applications to oscillatory stability analysis and control’, IEEE Trans. Power Syst., 1989, 4, (1), pp. 278 – 285

The application of this algorithm significantly enhances the efficiency of the SDC tuning process, since the manual tuning is quite tedious and time consuming, and automating this process gives the engineer much more free

[8] POURBEIK P. , GIBBARD M.J., VOWLES D.J. : ‘Proof of the equivalence of residues and induced torque coefficients for use in the calculation of eigenvalue shifts’, IEEE Power Eng. Rev., 2002, 22, (1), pp. 58 – 60

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[6] PE´REZ-ARRIAGA J.I., VERGHESE G.C., SCHWEPPE F.C.: ‘Selective modal analysis with applications to electric power systems, parts I and II’, IEEE Trans. Power Appar Syst., 1982, 9, pp. 3117 – 3134

IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

www.ietdl.org [9] CHAUDHURI B., PAL B.: ‘Robust damping of multiple swing modes employing global stabilizing signals with a TCSC’, IEEE Trans. Power Syst., 2004, 19, (1), pp. 499– 506 [10] RAMOS R.A., ALBERTO L.F.C., BRETAS N.G.: ‘A new methodology for the coordinated design of robust decentralized power system damping controllers’, IEEE Trans. Power Syst., 2004, 19, (1), pp. 444– 454 [11] RAMOS R.A., MARTINS A.C.P., BRETAS N.G.: ‘An improved methodology for the design of power system damping controllers’, IEEE Trans. Power Syst., 2005, 20, (4), pp. 1938 – 1945 [12] XUE C.F. , ZHANG X.P., GODFREY K.R.: ‘Design of STATCOM damping control with multiple operating points: a multimodel LMI approach’, IEE Proc. Gener. Transm. Distrib., 2006, 153, (4), pp. 375– 382 [13] KAMWA I., GRONDIN R., TRUDEL G.: ‘IEEE PSS2B versus PSS4B: the limits of performance of modern power system stabilizers’, IEEE Trans. Power Syst., 2005, 20, (2), pp. 903– 915 [14] ABDEL-MAGID Y.L., ABIDO M.A., AL-BAIYAT S., MANTAWY A.H. : ‘Simultaneous stabilization of multimachine power systems via genetic algorithms’, IEEE Trans. Power Syst., 1999, 14, (4), pp. 1428– 1439

[16] DAVISON E.J., RAMESH N.: ‘A note on the eigenvalues of a real matrix’, IEEE Trans. Autom. Control, 1970, 16, pp. 252– 253 [17] SKELTON R.E., IWASAKI T., GRIGORIADIS K.: ‘A unified algebraic approach to control design’ (Taylor & Francis, London, 1998) [18] OGATA K.: ‘Modern control engineering’ (Prentice Hall, New Jersey, 2002) [19] SADIKOVIC R., KORBA P., ANDERSSON G.: ‘Application of FACTS devices for damping of power system oscillations’, Proc. IEEE Power Tech., 2005 [20] PAL B., CHAUDHURI B.: ‘Robust control in power systems’ (Springer Science þ Business Media, New York, 2005) [21] KUIAVA R. , RAMOS R.A., BRETAS N.G.: ‘Robust control methodology for the design of supplementary damping controllers for FACTS devices’, Revista SBA – Controle Automa., 2009, 20, (2), pp. 192 – 205

7

Appendix

SDC for TCSC installed between areas #1 and #2 SDC1 ¼ 0:200

s10:0 1 þ s0:0170 1 þ s0:0170 1 þ s10:0 1 þ s0:5484 1 þ s0:5484

SDC for the TCSC installed between the areas #2 and #5 [15] CHIALI M., GAHINET P., APKARIAN P.: ‘Robust pole placement in LMI regions’, IEEE Trans. Autom. Control, 1999, 44, (12), pp. 2257 – 2270

IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 10, pp. 919– 929 doi: 10.1049/iet-gtd.2008.0424

SDC2 ¼ 0:885

s10:0 1 þ s0:0350 1 þ s0:0350 1 þ s10:0 1 þ s0:4871 1 þ s0:4871

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