Contribution Allocation for Voltage Stability In eregulate ower Systems arng M. Huang, Senior Member, I, Kun Men Abstract: With eregulation of power systems, it is of great importance to know who contributes to avoi a voltage collapse, which coul be ue to ifferent parts of the power system generator, control system an transmission part, etc. This paper focuse on how to allocate the responsibility an contribution by bifurcation analysis. We investigate how parameters of the system influence the bifurcation points. Three bifurcations (the singularity inuce bifurcation, sale-noe an Hopf bifurcation [], an their relationship to several commonly use controllers are analyze. Their parameters impact on the bifurcation points is investigate in this paper, from which we foun a way to allocate the contribution by analyzing the relative positions of the bifurcations. We also analyze the influence of other parameters (parameters of transmission an generators) on the bifurcation points. A simple two-bus system is use in this paper to emonstrate our approach. Our result shows that bifurcation analysis is a creible way to allocate the contribution, an the analysis will help us esign the controller an optimize the system to avoi the voltage collapse. Keywors: Contribution, Bifurcation, Voltage Collapse, xciting control system, Voltage Stability, stability margin I. INTROUCTION The eregulate power system is base on transactions; each part of the unbunle systems (generator, control system an transmission part, etc.) has its own contribution to voltage stability. It is of great economic an security importance to allocate these contributions, so appropriate awar can be allocate to encourage infrastructure investments. Analyzing the influence of the parameters of the system on voltage stability will also help us esign an optimize the system. This paper emonstrates that we can allocate the responsibility through bifurcation analysis. A new way is evelope to allocate the responsibility in part III of this paper, an in part IV we also showe that how the parameters of the system will interact with the bifurcation points. The authors gratefully acknowlege the support from Texas Avance Technology rogram an nergy Resource Research rogram at Texas A&M University. arng M. Huang, epartment of lectrical ngineering, Texas A & M University, College Station, TX 77843-328 (huang@ee.tamu.eu) Kun Men, epartment of lectrical ngineering, Texas A & M University, College Station, TX 77843-328 (menkun@ee.tamu.eu) A simple one generator an one loa bus system is use to emonstrate our approach. The system has three basic parts of the power system generator, exciting control system an transmission, which will be use to emonstrate our unbunling of responsibility/contributions.,δ r X, X f xciter Fig.. Simple two bus system In the above system, we assume that the power factor of the loa is constant as the loa changes. We also assume that the voltage ynamic is ecouple from the angle ynamic, which is well behave, so the angle ynamic can be ignore at this scenario. The ynamics of this system can be moele by parameter epenent ifferential-algebraic equations [] as: x = f ( x, y, p), n+ m+ q f : R n R () n+ m+ q m 0 = g( x, y, p), g: R R (2) x X R, y Y R, p R n m q The ifferential equation () represents the control system, the algebraic equation (2) represents the loa flow equation. In this paper we focuse on three commonly use controller controller, I-controller an I controller of a voltage regulator, the loa flow equation of this system an the mathematical moel of these three types of controller are shown in part II. The reuce Jacobian matrix [] of the system can be written as: Fx = fx fg y ygx Through the analysis of the eigenvalue of F x, we emonstrate the influence of the control system in part III. We focus on three types of controllers for regulators an the exciters, an observe that three types of bifurcation usually occurre: Hopf bifurcation, sale-noe an singularity inuce bifurcation [,2], corresponingly we enote these three types of bifurcation as A, B an C in this paper. Then we show how ifferent controllers an their parameters impact on the locations of A, B an C on the V curve. We foun that there are three basic patterns as follows. ) A<B<C; 2) A<C<B; 3) A isappear an B<C when ifferent controllers X (3), δ Loa
an ifferent control parameters are use. Accoringly, we fin a way to allocate the contribution an responsibility between the controller an transmission by analyzing the eigenvalues of F x.. We emonstrate that our metho is easy an creible. Base on the analysis in section III, we further emonstrate how the parameters of the system impact on the bifurcation points an the maximum loaability point, max, in section IV, where we emonstrate the influence of controller, transmission an generator on voltage stability. II MATHMATIC MOL for IFFRNT CONTROL SYSTM 2. algebraic equations of loa flow [2] = sin δ x 2 + cosδ Q = x = sinδ x 2 + cos δ Q = x (4) (5) quation (4) can be simplifie as : 2 2 0 = ( x ) ( xq + ) (6) Here equation (6) is the g(x,y,p) in equation (2). 2.2 ifferential equations of controller In this paper, we just focus on three types of commonly use controller -controller, I-controller an Icontroller, their mathmatic moel are shown as below: ) -controller [3] : 2 x + x + x x ( x Q) = + + (7) f T x x 0 0 f = ( f f) K ( x) + ( xq + ) (8) r T Here equation (7)~(8) are the f(x,y,p) in equation (). 2) I-controller [8,0] 2 x+ x x x ( + xq ) = + + f T x x 0 (9) 0 f = ( f f) I / TI K( ( x) + ( xq + ) r) T (0) I = ( x) + ( xq + ) r () Here equation (9)~() are the f(x,y,p) in equation (). 3) I-controller For a I controller [2], K + + K s (2) Ts I We know that K s is not practical [2], so if T is small enough, we can use equation (3) [6,7] to replace equation (2): K s K + + (3) Ts + Ts I then the control system can be expresse as below: 2 x+ x x x ( + xq ) = + + f (4) T x x 0 0 K f = ( f f) I / TI ( K+ )( ( x) + ( xq+ ) r T T (5) I = ( x) + ( xq + ) r (6) K = ( ( x) + ( xq+ ) ) (7) 2 r T T II. Here equation (4)~(7) are the f(x,y,p ) in equation (). CONTRIBUTION ANALYSIS for IFFRNT XCITATION CONTROLLRS emonstrating xample: T0 = 5, T =.5, x =.2, x = 0., x = 0.2, Q = 0.5, r =.0 (Note : all cases shown in this paper will use the same system parameters unless specifie otherwise.) Firstly we will show that if there is no voltage regulator control, the V curve of the system will be change rastically: Fig.2 The V curve with an without controller When there is no voltage controller, is constant (here we it equals to.). The V curve is obtaine by solving equation (6). When we use a controller to keep the g= r, the V curve is obtaine by solving equation (5). Fig 2 shows that the max will increase, which implies that the regulator will increase the stability margin. Note that the regulating range of a voltage regulator is ecie by its exciter size [3,4]. We have shown that when the exciter hits the limit, the max eteriorates.
In the following, we will analyze the impacts of ifferent controllers of the regulator on the voltage stability of the system: 3. -controllers in the regulator In this paper we keep as constant as changes ( is the loa) for the voltage regulation function; thus, the f0 in equation (8) is not constant, an will vary with the loa. Bifurcation point C is inuce by singularity, where et(g y ) equals to zero at point C. When r, g y is ecie by the loa flow equation, so C point is only inuce by transmission system. However, point B will vary with the change of K ; B with K ; an B>C when K >5.25; B A max when K =.895; when K <.895, A will isappear; an B 0.735 with K 0. Fig. 3 shows how the locations of bifurcation A, B an C vary with the change of K p : Fig 5, the eigenvalue which is slightly influence by K p Fig 6, the eigenvalue that is strongly influence by K p Fig.3 The location of A, B an C When K p =2.5,5,0, the locations of the bifucation A, B an C an the eigenvalues of reuce matrix are foun by using equations (3), (6)~(8) as shown in Fig 4~6: Fig 7, the eigenvalue igt (K =.8) Fig 4, the locations of the bifurcations when Kp=2.5,5,0 In Fig 5, note that K p has little influence on one of the eigenvalues (enote by igt), while at Fig 6. K p has a substantial impact on the other eigenvalue (enote by igc). When K =.8, Fig 9 shows the location of B an C in V curve. Note that A isappeare. From Figures 4 to 9 we can see that the eigenvalue igt is strongly relate to the loa flow, while the eigenvalue igc is strongly influence by the controller. Fig 8, the eigenvalue igc (K =.8)
Fig 9, the location of B an C when K p=.8 Fig, the eigenvalue igc (I-controller) We also can conclue that there are three basic patterns: ) A<B<C. When (A,B), both the eigenvalue igt an igc are positive; when (B,C), only the eigenvalue igt is positive. 2) A<C<B. When (A,C), both the eigenvalue igt an igc are positive; when (C,B), only the eigenvalue igc is positive. 3) A isappear an B<C. Only the eigenvalue igt is positive when (B,C) 3.2 I-controller in a regulator(k p =2.5, T I =5.0/ T I =20) Using equation (3), (6) an (9)~(), we can obtain three eigenvalues of the system: igc (igc is influence by K p an T I, an is mainly influence by T I ) an igt, the other eigenvalue is always negative, an we also foun bifurcation point A,B an C. Fig 0, the eigenvalue ig T(I-controller) From Fig 0~2, we can conclue that I controller behaves very similar to the -controller case as K. When (0,A), all eigenvalues are negative; when (A,C), both eigenvalue igc an igt are positive; when (C,B), only the eigenvalue igc is positive. Accoringly, it follows the basic pattern 2 as escribe in 3.. The ifference is that A point appear earlier in the V curve, here A=.40, (in - controller, A=.466). Thus, the ynamic stability margin ecrease when the I-component is introuce. However, Icontroller will enhance the ynamic response after a small isturbance of from our simulation. Fig 2, the location of A, B an C in V curve for a I/I controller 3.3 I-controllers in a regulator. (Kp=2.5, TI=5.0, K =,T =0.0/ T =0.005) Using equation (3), (6) an (4)~(7), we obtain four eigenvalues of the system: ig C an igt, an other two negative eigenvalues. We also fin three bifurcation points, where A<C<B. When (A,C), both igc an igt are positive; when (C,B), only igc is positive. This follows a similar pattern 2 as iscusse in 3.. From Fig 2, we can see that I behaves similar to a Icontroller, except that A point appears later (A=.454). This implies that although I-component can ecrease the ynamic stability margin, the -component can compensate the ecrease. On the other han, the ynamic response remains as goo as a I controller. Through sections 3., 3.2, 3.3 we can conclue that the three basic orering patterns of bifurcation points A B C as iscusse in 3. are generally true for all controllers. Our experience inicates that no other orering of A B C is possible. Accoringly, we can raw the conclusion: ) A<B<C. When (A, B), both igc an igt are positive; when (B,C), only the igt is positive. From the parameter analysis, we can conclue that the voltage collapse is ue to both controller an transmission when (A,B). The voltage collapse is only cause by transmission part when (B,C). In this case, [A, C] is the unstable area, an A efines the ynamic stability margin bounary. 2) A<C<B. When (A,C), both igc an igt are positive; when (C,B), only the igc is positive. From the parameter analysis we can conclue that the voltage collapse is ue to both controller an transmission when
(A,C). The voltage collapse is cause by controller when (C,B). In this case, [A, B] is the unstable area, an A efines the ynamic stability margin bounary. 3) A isappear an B<C, only the igt is positive when (B,C). Thus, the voltage collapse is only ue to transmission when (B,C). In this case, [B, C] is the unstable area, an B is the ynamic stability margin bounary. 4) In conclusion, the tuning of the control parameters will influence A an thus the ynamic stability margin of the system. The contribution with an without the controller/regulator can be easily assesse. More impacts analysis will be substantiate in the following section. 5) The analysis also explains why a ynamic stability margin is smaller than the static analysis as observe by our research team in [3]. IV. CONTRIBUTIONS OF OTHR SYSTM ARAMTRS ON STABILITY Following the same argument, we can also unbunle the voltage collapse responsibility of the system by stuying how the parameters of the three parts of the system (generator, controller an transmission part) affect the bifurcation patterns: 4. The parameters of the control system: a) Three types of regulators: As iscusse in 3., impacts of K in the -controller are summarize as follows. K oes not influence C, but have impacts on B an A. When K, B max ; an when K 0, B 0. 735 ; when K =.895, A B; when K <.895, A isappear; when K =5.25, B C. For I an I controllers, we i simulation with ifferent parameters of the controller an the result summarize in table : (Note : for I an I controller, A efines the ynamic stability margin, an the unstable area is [A, B].) TABL -a X K T I K T A B C 2.5 0 0 =0.98 =3.09 =2.5.2 2 =.67 4 =.273 3 =.236 0.00 =.404 3 0. =.405 0.0 =.45 =.46 2 0 =.625 3 =.2839 0 =0.5 8 =0.648 5 =0.685 00 =0.726 From table -a, we can see that K only has influence on A, which is ifferent from -controller. Here B an C remain the same. (With a constant power factor, we prove that B always appear at max when an I-controller is use in the regulator.) A will increase with bigger T I an bigger K. T has little influence on A, but too large or too small T will ecrease A. All para meters of I an I controllers have no impacts on B an C. The steay-state stability margin C is inepenent of the controller as long as the voltage regulator has enough excitation capacity to regulate the terminal voltage. b) The influence of the exciter The equation (7) represents the exciter. Now we investigate the influence of the T 0 on the stability margin: (In this case the regulator is -controller, K =2.5) TABL -b T 0 max A B C 3 3.09.248.584 2.6 5 3.09.46.584 2.6 7 3.09.534.584 2.6 From table -b we can see that T 0 will influence the A point, but it have no influence on B an C. We verifie the fact for all three controllers. A will increase with bigger T 0, that means the ynamic stability margin will increase with bigger T 0. However, the size of the exciter will limit the range of f an thus the voltage regulation range [3]. 4.2 The parameters of a generator: Now we investigate the influence of the x on the stability margin: a) -controller TABL 2-a x max A B C.2 3.09.46.584 2.6 0.3 3.09 2.06 2.79 2.6 b) I-controller TABL 2-b x max A B C.2 3.09.297 3.09 2.6 0.3 3.09 2.005 3.09 2.6 c) I-controller TABL 2-c x max A B C.2 3.09.45 3.09 2.6 0.3 3.09 2.058 3.09 2.6 Through the table 2-a, b an c, we can see that x has influence on A point; an if we use -controller, it also influence the B point. It implies that the ynamic stability margin will increase with smaller x. C, which efines the steay-state stability margin, it is inepenent with x. 4.3 The parameters of the transmission system:
Now we investigate how x influences stability margin : a) -controller TABL 3-a x max A B C 0. 3.09.46.584 2.6 0.2 2.58.360.509.966 b) I-controller TABL 3-b x max A B C 0. 3.09.297 3.09 2.6 0.2 2.58.2506 2.58.966 c) I-controller TABL 3-c x max A B C 0. 3.09.45 3.09 2.6 0.2 2.58.3589 2.58.966 Through the analysis of table 3-a, b an c, we see that x has influence on all of the three bifurcation points. Both the ynamic an steay stability margin will increase with smaller x. V. CONCLUSION This paper evelops a way to unbunle the contribution of voltage stability to generator owners, transmission owners an excitation control owners. We showe that how the parameters of the three parts of the system (generator, controller an transmission) affect the bifurcation patterns, which enable us to esign an optimize the system better an at the same time to allocate the contributions. In the same token, an investment awar system can be built to awar investments on the corresponing power infrastructure. A simple two-bus system is use in this paper to emonstrate our approach, which shows our approach is easy an creible. VI. ACKNOWLMNTS The authors gratefully acknowlege the support from Texas Avance Technology rogram, SRC an nergy Resource Research rogram at Texas A&M University. [6] K. Kim, M.J. Basler an A.ohwani. "Supplemental Control in a Moern igital xcitation System". annel Session for the 2000 I ower Meeting, Singapore, 2000 [7] Richar C. Schaefer an Kiyong Kim," xcitation Control of the Synchronous enerator", I inustry Applications Magazine, March/April 200, pp.37-43. [8] Riko Safaric, Karel Jezernik an usan Borojevic, "I-Controller for Avoiing Integrator Win-up in System with Input Saturation", I 99. pp:803-806 [9] Carson W. Taylor, "ower System Voltage Stability", Mcraw-Hill, Inc. 994. [0] Yixin Ni, "ynamic ower System Theorem an Analysis", Tsinghua University publisher. [] V.Venkatasubramanian, H.Schaettler an J.Zaborazky, "Local Bifurcations Feasibility Regions in ifferential-algebraic systems", I Transactions on Automatic Control, VOL 40, NO. 2. ec. 995. pp:992-203 [2] Kailath, T.. 980. Linear System. nglewoo Ciffs: rentice-hall. [3] Huang, H Zhang, ynamic Voltage Stability reserve stuies for eregulate environment, roc. I/S Summer Meeting, July 200, Canaa. [4] arng Huang, Tong Zhu, TCSC as a transient voltage stabilizing controller, in roc. of ower ngineering Society Winter Meeting, 200, I, vol. 2, pp628-633, Columbus, OH, Jan. 200. VIII. BIORAHIS r. arng M. Huang receive his B.S. an M.S. in.. from national Chiao Tung University, Hsinchu, Taiwan, R.O.C. in 975, 977 respectively. He receive his octorate egree in System Science an Mathematics form Washington University, St. Louis in 980. He ha been teaching there since then until 984. He joine Texas A&M University, epartment of lectrical ngineering in 984. He is currently a professor an the irector of grauate stuies there. He have been working on many fune research projects, such as mergency Control of Large Interconnecte ower System, HVC Systems, Restoration of Large Scale ower Systems, On-line etection of System Instabilities an On-line stabilization of Large ower System, Fast arallel/istribute Texture Algorithm for ata Network Routing roblem, etc. His current interest is the large-scale systems theory, large scale parallel/istribute computing an control an their applications. r. Huang is a senior Member of I, an a Registere rofessional ngineer of Texas. He has serve as the Technical Committee Chairman of nergy System Control Committee an an associate eitor in the I Automatic Control Society; he has also been serving in a number of committees an subcommittees of I AS Society. r. Huang has publishe more than one hunre papers an reports in the areas of nonlinear, istribute control systems, parallel/istribute computing an their applications to power systems, ata networks an flexible structures. Kun Men receive his BS egree in lectrical ngineering from Xi an Jiaotong University an MS egrees in lectrical ngineering from Tsinghua University, Beijing, China, respectively in 996 an 200. He is presently a h.. stuent at the epartment of lectrical ngineering, Texas A&M University, College Station, Texas. VII. RFRNCS [] arng Huang, Liang Zhao, Xuefeng Song, "A New Bifurcation Analysis for ower System ynamic Voltage Stability Stuies". Accept by I 2002 S, Winter meeting NY. [2] V.Venkatasubramanian, H.Schaettler an J.Zaborazky, "Voltage ynamics: Stuy of a enerator with Voltage Control, Transmission, an Matche MW Loa", I Transaction on Automatic Control, Vol 37, No., November 992. p77-733 [3] I Committee Report, "xcitation System Moel for ower System Stability Stuies", I Trans. ower Apparatus Syst., Vol.AS- 00,No.2, Feb.98. 494-509. [4]. Kokotovic, H.K.Khalil, an J. O Reilly. "Singular erturbation Methos in Control: Analysis an esign". Acaemic ress, 986 [5] Costas Vournas, "Voltage Stability of lectric ower Systems", Kluwer Acaemic ublishers. 998.