An Advanced LFC Design ConsideringParameter Uncertainties in Power Systems

Satoshi Ohba, Haruka Ohnishi, Student Member, and Shinichi Iwamoto, Member, IEEE

Abstract-- In this paper, we propose a designing method of theload frequency control, which is able to consider uncertainties inpower systems. We design the control system utilizing PIDcontrollers for an H,, control problem. Even though manyadvanced control theories have been established, most industrialcontrollers still use conventional applications such as PI, PID,and simple first-order lag/lead compensators. The HW. controltheory can deal with many robust stability and performanceproblems. We especially pay attention to the inertia constant, thedamping coefficient, and the synchronizing coefficient becausethese parameters change their values or have some errorsdepending on conditions of the power system. As an example, wedesign a case of a two-area system, and carry out simulations tocompare the proposed method with a conventional method (a PItype controller). As the results of the simulations, we canconfirm the effectiveness of the proposed method.

Index Terms-- automatic generator control, load frequencycontrol, robust control theory, H,, control theory, PID control,parameter space

I. INTRODUCTION

THE purpose of the load frequency control (LFC) in thepower system is to maintain the quality of the frequency

by suppressing the frequency deviation and tie-line powerflow deviation in the own area in the range of some tolerances.However, the electric power industry has become morecomplicated than ever in recent years. And, under thederegulated environment, distributed generations, especiallyrenewable energies such as wind power generations have beeninterconnected with the power system. Therefore, reviews ofthe conventional control system are required in order to dealwith the change of such various situations.

As one of the techniques which represent the loadfrequency control, the tie-line bias control (TBC) scheme ismentioned. This is a scheme for adjusting generator outputsto absorb the Area Control Error (ACE) which is calculated toadjust the demand and supply balance in the own area bymeasuring both the frequency deviation and the tie-line powerflow deviation. Many conventional load frequency controlsimulation models use a system model linearized around someoperating point. This system model is a kind of aggregated

S. Ohba is with Tokyo Electric Power Company Inc., Japan.H. Ohnishi and S. Iwamoto are with the Department of Electrical

Engineering and Bioscience, Waseda University, Shinjuku-ku, Tokyo, Japan(e-mail: ohnishiMamwrs.elec.waseda.acjp).

model in which many electric power system components suchas generators, motors, and loads are aggregated as a singlegenerator, motor, and load. However, any parameter value inthe model fluctuates depending on system and power flowconditions which change every minute. A control system isdesired, which can consider these parameter variations due tothe variation of such system conditions. To realize such acontrol system, consideration of robustness at the designingstage is needed.

As a design theory of the robust control system for modelerrors, the HGO control theory was established at the end of the1980's, and applications to various control problems havebeen examined [1]-[4]. However, the degree or dimension ofthe designed controllers has increased more than the necessity,and the problem of the difficulty has been indicated. In recentyears, from such background, the problem for searching suchcontroller has attracted interest, whose degree and structurewere fixed. Researches have been widely carried out, whichrealize the robust control by PID controller [5]-[14]. Since thelargest number of the parameters of the PID controller is three,it is possible to globally catch the effect of the PID gain onstability and performance of the controller, and the optimumgain is obtained, by drawing the region on the parameter spacebased on the parameter space design method [7]-[14].

In this paper, we propose a designing method of a loadfrequency control system using the robust PID controllerbased on a parameter space designing method. Finally, wecarry out simulations for the system in order to compare theproposed method with the conventional method (a PI typecontroller), and confirm the validity of the proposed method.

II. PROPOSED DESIGN METHODOLOGY

A. HIJ Control1) Formulation ofthe HIJ ControlWe consider the control system shown in Fig. 1. Here, w, z,

u, and y represent the disturbances, the controlled variables,the control inputs, and the observed outputs, respectively.The transfer function of the control system is defined by thefollowing equation.

G( ) GI I GI 2 (1)C(s)30 G21 G22

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w

Fig. 1. Control System

For the generalized plant G(s), the feedback control isperformed with controller K(s) expressed by the followingequation.

u = Ky (2)Then, the closed loop transfer function from w to z is

represented by the following equation.

Tz = GI I(s)+ GI2(s)G21(s)K (s) (3)1 -G22(s)K(s)

Because the control objective is to suppress the controlledvariable z against the disturbance w, it is important to designthe controller which suppresses the norm of the transferfunction by (3). It is the HGO control theory that utilizes the HGOnorm as a scale of the size, and the HGO norm of the stabletransfer function T-, will be defined as follows.

||T|| = sup 12 (4)

2) Robust Sensitivity Control ProblemIn order to consider characteristic variation of control

system, the real characteristic of controlled object is definedby using G(s) and model error A(s) as follows:

C(s) = (1 + A(s))G(s), IA(jw)l < I(5)

Here, sensitivity function S and complementationsensitivity function T in Fig. 1 are defined in the followingequations.

1 + K(s)G(s)

T(s) K(s)G(s)1+ K(s)G(s)

(6)

For the plant expressed in (5), equation (7) must besatisfied in order to guarantee the robust stability and therobust performance of the closed-loop system.

|SGO)|+ 1 (7)a(w) /A(0)

Here, oc(oo) and Pf(o) are appropriate weight functions.

B. Parameter Space Designing Method1) Definition ofthe PID Controller [15]Although the controller designed in this paper is PID

controller, we use the imperfect differentiator shown asKDI(1+Ts) for the perfect derivative action, because thedifferentiator is realizable in practice.

It means that the perfect differentiation is approximated bychoosing X as a minute or very small value. Therefore, thePID controller designed in this paper will be expressed as

follows.

K(s) =KPK+ KDs 1+± s

Here, KP, KI, and KD represent the proportional gain, theintegral gain, and the derivative gain, respectively.

2) Application to the Robust Sensitivity Control Problem[12]A plant and a controller in the frequency domain are,IG(jc) = a(0) + jb (a) (10)UK(jt) = r(cosO+ j sinO)

By substituting (10) into (7), we obtain the next inequality.

K/2 12(a2+b2)r2

L2-a os022 ~ acosO±bsinO r± I 0

alfl~ ~ a2

(11)

By obtaining the range of 0 in which this inequality has thesolution of 0 > 0, the solution set is obtained.

Here, (11) is written as (12).Ar2 +2Br+C > 0 (12)

where,

A(a2±b2{1b 212

B acosO- bsinO-a ±ba/8

C=1-I

a2In the following, we consider this inequality.

a) Case ]Discriminant D (=b2 - 4ac) of (12) is positive in all 0.

Therefore,

* IfA>O and C<O, r>-

* IfA<O and C>A, 0 < r

A

-B -

AFrom the existence interval of r, it is found that in the

former, there is the permission region outside of boundarylines, and that in the latter, there is the permission regioninside of boundary lines.

b) Case 2* If B>O and D>A, O<r* If B>O and D<O, O<r

0 If B<O and D>A, 0 < r B -D -B±+IA A

r

* If B<O and D<O, O<rTherefore, the boundary can be decided only when the

third condition is satisfied.c) Case 3

* If B>O and D>A, r does not exist.* If B>O and D<O, r does not exist.

* If B<O andD>A B-D0 r -.,IA A

* If B<O and D<O, r does not exist.

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Therefore, the boundary can be decided only when thethird condition is satisfied.

d) Case 40 If B>O and C>A, O<r

0 If B>O and C<O, r >C2B

0 If B<O and CA>, 0 < r < C2B

* IfB<O and C<O, r does not exist.Therefore, the boundary can be decided when either the

second or the third condition is satisfied.e) Case 5

* IfA>O and B>O, O<r

0 IfA>O and B<O, r> 2BA

* IfA<O and B>O, O< r <2B

A* IfA<O and B<O, r does not exist.Therefore, the boundary can be decided when either the

second or the third condition is satisfied.As B and D are functions of 0, the interval of 0 with the

solution of 0 > 0 for each case will be obtained. By giving 0in the interval, and calculating the existence interval of rwhich satisfies (11), (r, 0) which gives the boundary of the setis obtained.

It is possible to determine the PID gain using (14) obtainedfrom (13). Concretely, by fixing KD at some values, thepermission region is described in the (KP, KI) plane.

r(cosO+ jsinO) = Kp + KI + jKD (13)jo) 1±+]o)r

Kp = rcosO

tKD I = rsinO (14)

An example of a permission region and boundary lines isshown in Fig. 2.

It is shown that this figure is an example of Case. 1.

ii V

III. PROPOSED METHOD

A. Control ModelWe propose a method for designing an LFC controller that

has a tie-line bias control (TBC) scheme. Fig. 3 shows alinear model of the power system (area i) used in the variousresearch generally, which has a governor and a reheat steamturbine represented as a first-order model [16]. In this model,it is assumed that neighboring areas are interconnected by onetie line, and that the interconnection of individual generatorsin each system is firm, and the transmission line in the systemcan be disregarded. And, in each system the dynamiccharacteristic is represented by the load characteristic whichrepresents the relationship between the frequency change andthe load transition and by the inertia which represents eachgenerator in the system. In this paper, we design a controllersystem and run simulations using this model.

Fig. 3. Linear Model (area i)

However, it is necessary to guarantee controllability whenwe use this model and design a controller systeminterconnecting multiple areas. Therefore in this paper, weutilize the frequency deviations of neighboring areas as thestate variable, as shown in Fig. 4, in order to design thecontrol system whose controllability is guaranteed [17].Originally, this technique may be applied only to the case inwhich it is interconnected with multiple areas. In this paper,however, by using frequency variation information of theneighboring area, we design the load frequency control systemwhich considers additional information of electric powersystem.

(Note that Kp and Ki do not have any unit, because they are parametersdefined in the Parameter Space Designing Method.)

Fig. 2. Sample of the Permissible Region

2007 39th North American Power Symposium (NAPS 2007)

--- I

a.N,--rm

=;WCi x

I

632

Fig. 4. Linear Model with neighboring areas

characteristics of the power system (inertia constants anddamping coefficients) and also the synchronizing coefficientsbetween the areas. For the above parameters, the assumedranges of the considered variations are as follows:

1 Afi 0 Inertia constant (inverse) 1/M ±20%sM + Di 0 Damping coefficient D ±20%System i * Synchronizing coefficient Tij ±15%

2) Block Diagram with the Designed ControllerBy using constants shown in Table I, the region of the

2 /7Tij 4-0-)solution set which satisfies (11) is described on the parameters ji-plane. The block diagram with the designed PID controller is

Afj shown in Fig. 5.3) Gain Calculations

1 Based on the design procedure shown in section II, thesM + Dj controller was designed for each area. As results of the gainSystem j calculation, the PID gains converged on the values shown in

Table II.

Additionally, in this paper, we pay attention to inertiaconstants, damping coefficients, and synchronizingcoefficients because they are some of the parameters in thismodel that change their values depending on the conditions ofthe power system. Therefore, we consider the changes ofthese parameters beforehand at the stage of designing thecontroller, and carry out stabilizing control of the system.

In addition, we use the area control errors (ACEs) asobserved quantities. The ACE in area i is expressed asfollows:

ACEi = BiAfi + APt%e,iWhere Bi represents the frequency bias value in area i, and

is expressed as follows:

Bi =Di +-IRi

B. Design ModelWe have used a two-area interconnected system [18]. In

this example system, it is assumed that the system capacity ofboth areas is equal. The constant parameters of both areas areshown in Table I.

TABLE IPARAMETER VALUES

Parameters Value

Inertia Constant M 0.0833 [p.u.sec/Hz]

Damping Coefficient D 0.0083 [p.u./Hz]

Governor Gain Kg 0.08 [sec]

Governor Time Constant Tg 1.0

Turbine Gain K, 0.30 [sec]

Turbine Time Constant T, 1.0

Speed Regulation R 2.4 [Hz/p.u.]

Frequency Bias Value Bs 0.2115 [p.u./Hz]

Synchronizing Coefficient TY 0.544 [p.u./Hz sec]

C. Controller Design1) Setting ofthe parameter variation rangeWe consider two parameters that show the dynamic

System jFig. 5. Designed Linear Model

TABLE IIPID GAIN IN EACH AREA

Parameters Gain Values

KP 0.595

K1 0.430

KD 0.160

IV. SIMULATIONS

In this paper, we carry out the simulations in a two-areainterconnected system assuming that 0.01 [p.u.] stepdisturbance occurs in area 1. To compare the controlperformance, we also carry out the same simulations with theconventional method which is the integral control of the tie-line bias control (TBC) scheme (The integral gain KI = 0.1).

A. The Nominal Control PerformanceWe carry out simulations for the case in which each

parameter is the nominal value. As a response example, eachfrequency deviation and tie line power flow deviation areshown in Fig. 6 (the frequency deviation) and Fig. 7(the tieline power flow deviation).

2007 39th North American Power Symposium (NAPS 2007) 633

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Tine (sec)

(a) area 1

conventbnal-

Tine (sec)

(a) area 1

-onventbnal-

Tine (sec)

(b) area 2

Fig. 6. Frequency Deviations

£0

£0

£0

£O

Tine (sec)

(b)area 2Fig. 8. Frequency Deviations (with parameter fluctuations)

Tine (sec)

Fig. 7. Tie-line Power Flow Deviations

From the waveforms of Fig. 6 and 7, it is found that theproposed method quickly suppresses the frequency deviationsand the tie-line power flow deviations in each area comparedwith the conventional method.

B. The Robust Control PerformanceIt is difficult to use the nominal values for the parameters

considered in the design under the power system, whichchanges the values every moment. And it is appropriate toconsider that the parameter always changes. Therefore, we

also carry out simulations for the cases in which theparameters have the various errors. The simulation results are

shown in Fig. 8 (the frequency deviations) and 9 (the tie linepower flow deviations), where, the parameter values of eacharea are as follows:

* Inertia constant (inverse) 1/M :+15%* Damping constant D :+15%* Synchronizing coefficient Tij +10%

Fig. 9. Tie-line Power Flow Deviations (with parameter fluctuations)

From the waveforms of Fig. 8 and 9, it is found that theproposed method quickly suppresses the frequency deviationsin each area compared with the conventional method in thecase in which each parameter has the variation. And, it isfound that the proposed method quickly suppresses eachdeviation as well as the case of the nominal value, when it hasthe parameter variations, though the conventional method hastaken more time than the case of the nominal value.

C. The Numerical ComparisonHere, to examine numerically the effectiveness of the

proposed method, we utilize the performance indices that are

used in [19]. The performance indices are defined as follows,

Jf= E E lAf(t) (15)time area

JPtie = E ai APIe(t)i (16)time tieline

where ai: The capacity rate of area iThese functions are expressed as the total absolute value of

the deviations in all areas. Therefore, we can find that the

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0£1

0£2

0£3

0£4

)£1

0£3

I1.......

)nventbnal-

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smaller the value of the performance indices, the better controlis carried out. In this case, we add each deviation everysecond, and compare the proposed method with theconventional method. Table III shows the values of theseperformance indices. Here, we are able to find that the valueof the proposed method is smaller than that of theconventional method. Also, the proposed method can controlthe variation without changing the control performance, evenif the parameters fluctuate. Therefore, the proposed methodhas robustness in its control and the validity can be confirmed.

TABLE IIIVALUE OF PERFORMANCE INDICES

Proposed conventional

Fl F2 Ptie Fl F2 Ptie

Nominal 0.048 0.058 0.015 0.221 0.228 0.056

Fluctuate 0.048 0.059 0.016 0.229 0.236 0.056

V. CONCLUSION

In this paper, we have designed a control system using arobust PID controller based on the parameter space designingmethod. The proposed method is a new designing method ofa load frequency control which regards ACE as the observedoutputs to guarantee the controllability of the control system.We have carried out simulations for the system in order to

compare the proposed method with the conventional method,and confirmed the validity of the proposed method.

From the result of the simulations, the proposed methodhas improved the quality of the ACE responses and proved tobe effective for the control of the output variation consideringthe parameter fluctuations.

[12] M. Saeki and D. Hirayama, "Parameter Space Design Method of PIDController for Robust Sensitivity Minimization Problem," T.SICE,Vol.32, No. 12, pp. 1612-1619, 1996. (in Japanese)

[13] V. Besson and A. T. Shenton, "Interactive parameter space method inmixed sensitivity problems," IEEE Trans. Automat. Contr., Vol.44, No.6,pp.1272, 1999.

[14] M. T. Ho, "Synthesis of H. controllers: a parametric approach,"Automatica, Vol.39, pp.1069-1075, 2003.

[15] N. Suda, "PID Controller," ASAKURASHOTEN, 1992. (in Japanese)[16] Olle I. Elgerd "Electric energy systems theory an introduction,"

McGraw-Hill Book Company[17] T. Ishii, G. Shirai and G. Fujita, "Decentralized Load Frequency Control

Based on H. Control," T.IEEJ-B, Vol.120-B, No.5, pp.655, 2000. (inJapanese)

[18] S. Ohba and S. Iwamoto, "LFC Using Parameter Space Design MethodConsidering Perturbation in Synchronous Coefficient," IEEJ NationalConference, Vol.6, pp.53, 2006. (in Japanese)

[19] K. Yukita, Y. Goto, Y. Mizutani, "Load Frequency Control Based onPower Demand Estimation and Fuzzy Control Considering Effects ofSelf-Regulation of Generator," T.IEE-B, Vol. 116-B, No.1, pp42-51,1996. (in Japanese)

VII. BIOGRAPHIES

Satoshi Ohba was born in 1983. He received his B.E. and M.E. degree fromWaseda University, Tokyo, Japan in 2005, 2007, respectively. His researchinterest is mainly power system stability and load frequency control.

Haruka Ohnishi was born in 1983. He received his B.E. degree from WasedaUniversity, Tokyo, Japan in 2007. His research interest is mainly powersystem stability and load frequency control.

Shinichi Iwamoto was born in 1948. He received his B.E., M.E., and Ph.D.degrees from Waseda University, Tokyo, Japan in 1971, 1975, and 1978,respectively. From 1972 to 1974, he was at Clarkson University, U.S., andreceived his M.E. degree. He is presently a full professor at WasedaUniversity. From 1992 to 1993, he was at the University of Washington as avisiting professor. His research interests are mainly the deregulated powermarket, voltage stability analysis, transient stability analysis and GPSapplications to power systems.

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[2] K. Zhou, J. C. Doyle, K. Glover, "Robust and Optimal Control," PrenticeHall, 1996.

[3] T. Mita, "H,, control," SHOKODO, Japan. (in Japanese)[4] K. Nonami, H. Nishimura, M. Hirata, "The design of the controller using

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[9] M. Saeki, "A Design Method of the Optimal PID Controller for a TwoDisk Type Mixed Sensitivity Problem," T.ISCIE, Vol.7, No.12, pp.520-527, 1994. (in Japanese)

[10] V. Besson and A. T. Shenton, "Interactive control system design by amixed H,-parameter space method," IEEE Trans. Automat. Contr.,Vol.42, No.7, pp.946, 1997.

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