Two wind farms are considered: Wind Farm A has 20 WTGs and Wind Farm B has 50 WTGs. The diagrams are shown in Figures I.1 and I.2. The wind farm’s network parameters are presented in Tables I.1 and I.2. Data is partially obtained from [36]. For simplicity, the step-up transformers of WTGs are neglected. Therefore, the WTG’s parameters presented in this appendix are already referred to the high voltage side of the omitted WTGs’ transformers (23 [kV ]). In addition, the parameters of all WTGs are assumed to be identical and are as follows:
Xs = 3.5547 [pu] SW T G= 5 [M V A]
Xr = 3.5859 [pu] Qref = 0 [M var]
Xm = 3.5092 [pu] B = 1.321 × 10−3 [pu s3/m3] Rs= 0.1015 [pu] C = 1.066 × 10−8 [pu s3] Rr = 0.0880 [pu] D = 0.1667 [−] KP 1 = 1 [pu] KI1= 5 [pu] KP 2 = 1 [pu] KI2= 5 [pu] KP 3 = 1 [pu] KI3= 5 [pu] KP 4 = 1 [pu] KI4= 5 [pu] H = 4 [s] (I.1)
Note that if the system’s power base, Sb, is different than SW T G, then Xs,
Xr, Xm, Rs, Rr, KI2, KI4, KP 2 and KP 4 have to be scaled as
P arameternew [pu] = P arameterold [pu] Sb SW T G
Figure I.1: Wind Farm A’s diagram.
In addition, the H-constant of inertia and the turbine’s parameters B and C have to be scaled too as
Hnew [s] = Hold [s]SW T G Sb
(I.3) Bnew [pu s3/m3] = Bold [pu s3/m3]SW T G
Sb
(I.4) Cnew [pu s3] = Cold [pu s3]SW T G
Sb
Table I.1: Parameters of the Wind Farm A’s network From bus To bus R [pu] X [pu] B [pu]
1 PCC 0.0020 0.0400 0.0100 1 2 0.0002 0.0040 0 2 8 0.0185 0.0204 0.0014 2 14 0.0990 0.1050 0.0052 3 4 0.0125 0.0133 0.0007 4 5 0.0137 0.0075 0.0003 4 10 0.0890 0.0391 0.0008 5 6 0.0208 0.0114 0.0004 7 12 0.0211 0.0093 0.0002 8 9 0.0294 0.0129 0.0003 9 10 0.0225 0.0099 0.0002 10 11 0.0488 0.0214 0.0004 11 12 0.0344 0.0151 0.0003 12 13 0.0458 0.0201 0.0004 14 15 0.0120 0.0127 0.0006 15 16 0.0189 0.0200 0.0010 16 17 0.0225 0.0124 0.0004 16 20 0.0694 0.0305 0.0006 17 18 0.0250 0.0137 0.0005 19 20 0.0120 0.0127 0.0006 20 21 0.0381 0.0168 0.0004 21 22 0.0706 0.0310 0.0006 22 23 0.0693 0.0305 0.0006
Table I.2: Parameters of the Wind Farm B’s network From Bus To Bus R [pu] X [pu] B [pu]
1 PCC 0.0020 0.0400 0.0100 1 2 0.0002 0.0040 0 2 13 0.0185 0.0204 0.0014 2 23 0.0552 0.0630 0.0034 2 33 0.0990 0.1050 0.0052 3 4 0.0125 0.0133 0.0007 4 5 0.0137 0.0075 0.0003 4 15 0.0890 0.0391 0.0008 5 6 0.0208 0.0114 0.0004 7 16 0.0555 0.0244 0.0005 8 9 0.0125 0.0133 0.0007 8 18 0.0320 0.0120 0.0004 9 10 0.0135 0.0075 0.0003 10 11 0.0140 0.0133 0.0007 11 12 0.0136 0.0075 0.0003 13 14 0.0294 0.0129 0.0003 14 15 0.0225 0.0099 0.0002 15 16 0.0488 0.0214 0.0004 16 17 0.0802 0.0352 0.0007 17 18 0.0400 0.0200 0.0004 18 19 0.0400 0.0200 0.0004 19 20 0.0400 0.0200 0.0004 20 21 0.0400 0.0200 0.0004 21 22 0.0400 0.0200 0.0004 23 24 0.0400 0.0200 0.0004 24 25 0.0400 0.0200 0.0004 25 26 0.0400 0.0200 0.0004 26 27 0.0400 0.0200 0.0004 27 28 0.0400 0.0200 0.0004 28 29 0.0400 0.0200 0.0004 29 30 0.0400 0.0200 0.0004 30 31 0.0400 0.0200 0.0004
Table I.2: Continued
From Bus To Bus R [pu] X [pu] B [pu] 31 32 0.0400 0.0200 0.0004 33 34 0.0120 0.0127 0.0006 34 35 0.0189 0.0200 0.0010 35 36 0.0225 0.0124 0.0004 35 44 0.0694 0.0305 0.0006 36 37 0.0250 0.0137 0.0005 37 38 0.0400 0.0200 0.0004 38 39 0.0400 0.0200 0.0004 39 40 0.0400 0.0200 0.0004 40 41 0.0400 0.0200 0.0004 40 51 0.0320 0.0120 0.0004 41 42 0.0400 0.0200 0.0004 43 44 0.0120 0.0127 0.0006 44 45 0.0381 0.0168 0.0004 45 46 0.0706 0.0310 0.0006 46 47 0.0693 0.0305 0.0006 47 48 0.0400 0.0200 0.0004 48 49 0.0400 0.0200 0.0004 49 50 0.0400 0.0200 0.0004 51 52 0.0400 0.0200 0.0004
REFERENCES
[1] International Energy Agency, “World Energy Outlook Report,” 2006. [Online]. Available: http://www.worldenergyoutlook.org
[2] T.E. DyLiacco, “Real-time computer control of power systems,” Pro- ceedings of the IEEE, vol. 62, no. 7, pp. 884–891, 1974.
[3] P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1994.
[4] L. Schleisner, “Life cycle assessment of a wind farm and related exter- nalities,” Renewable Energy, vol. 20, no. 3, pp. 279–288, 2000.
[5] N. Golait, R.M. Moharil, and P.S. Kulkarni, “Wind electric power in the world and perspectives of its development in India,” Renewable and Sustainable Energy Reviews, vol. 13, pp. 233–247, 2009.
[6] World Wind Energy Association, “World Wind Energy Report,” 2008. [Online]. Available: http://www.wwindea.org
[7] Z. Saad-Saoud and N. Jenkins, “Simple wind farm dynamic model,” IEE Proceedings: Generation, Transmission and Distribution, vol. 142, no. 5, pp. 545–548, 1995.
[8] T. Ackermann, Wind Power in Power Systems. Chichester, West Sus- sex, England: John Wiley and Sons, Ltd., 2005.
[9] J.L. Slootweg and W.L. Kling, “Aggregated modelling of wind parks in power system dynamics simulations,” in IEEE Powertech Conference, 2003, June 23-26, Bologna, Italy.
[10] J.L. Slootweg and W.L. Kling, “Modeling of large wind farms in power system simulations,” in IEEE PES Transmission and Distribution Con- ference, vol. 1, 2002, pp. 503–508.
[11] J. Slootweg, S. De Haan, H. Polinder, and W. Kling, “Aggregated mod- elling of wind parks with variable speed wind turbines in power system dynamic simulations,” in 14 Power Systems Computation Conference, June 2002, Seville, Spain.
[12] J.G. Slootweg and W.L. Kling, “The impact of large scale wind power generation on power system oscillations,” Electric Power Systems Re- search, vol. 67, no. 1, pp. 9–20, 2003.
[13] G. Tsourakis, B.M. Nomikos, and C.D. Vournas, “Effect of wind parks with doubly fed asynchronous generators on small-signal stability,” Elec- tric Power Systems Research, vol. 79, no. 1, pp. 190–200, 2009.
[14] G. Tsourakis, B.M. Nomikos, and C.D. Vournas, “Contribution of dou- bly fed wind generators to oscillation damping,” IEEE Transactions on Energy Conversion, vol. 24, no. 3, pp. 783–791, 2009.
[15] R. Scherer, “Blade design aspects,” Renewable Energy, vol. 16, pp. 1272– 1277, 1999.
[16] R. Zavadil, N. Miller, A. Ellis, E. Muljadi, E. Camm, and B. Kirby, “Queuing up: Interconnecting wind generation into the power system,” IEEE Power and Energy Magazine, vol. 5, no. 6, pp. 47–58, 2007. [17] J. Ekanayake, L. Holdsworth, and N. Jenkins, “Control of DFIG wind
turbines,” IEE Power Engineering, vol. 17, no. 1, pp. 28–32, 2003. [18] N.P. Quang, A. Dittrich, and A. Thieme, “Doubly-fed induction machine
as generator: Control algorithms with decoupling of torque and power factor,” Electrical Engineering, vol. 80, pp. 325–335, 1997.
[19] J.B. Ekanayake, L. Holdsworth, and N. Jenkins, “Comparison of 5th order and 3rd order machine models for doubly-fed inductions generator (DFIG) wind turbines,” Electric Power Systems Research, vol. 67, pp. 207–215, 2003.
[20] Z. Miao and L. Fan, “The art of modeling and simulation of induc- tion generator in wind generation applications using high-order model,” Simulation Modelling Practice and Theory, vol. 16, pp. 1239–1253, 2008. [21] R. Pena, J.C. Clare, and G.M. Asher, “Doubly fed induction genera- tor using back-to-back PWM converters and its application to variable- speed wind-energy generation,” IEE Proceedings - Electric Power Ap- plications, vol. 143, no. 3, pp. 231–241, 1996.
[22] A. Tapia, G. Tapia, J.X. Ostolaza, and J.R. Saenz, “Modeling and con- trol of a wind turbine driven doubly fed induction generator,” IEEE Transactions on Energy Conversion, vol. 18, no. 2, pp. 194–204, 2003. [23] M. Yamamoto and O. Motoyoshi, “Active and reactive power control
for doubly-fed wound rotor induction generator,” IEEE Transactions on Power Electronics, vol. 6, no. 4, pp. 624–629, 1991.
[24] J.G. Slootweg, H. Polinder, and W.L. Kling, “Dynamic modelling of a wind turbine with doubly fed induction generator,” in IEEE PES Transmission and Distribution Conference, 2001, pp. 644–649.
[25] R.G. de Almeida and J.A. Pe¸cas Lopes, “Participation of doubly fed in- duction wind generators in system frequency regulation,” IEEE Trans- actions on Power Systems, vol. 22, no. 3, pp. 944–950, 2007.
[26] M.V.A. Nunes, J.A.P. Lopes, H.H. Zurn, U.H. Bezerra, and R.G. Almeida, “Influence of the variable-speed wind generators in transient stability margin of the conventional generators integrated in electrical grids,” IEEE Transactions on Energy Conversion, vol. 19, no. 4, pp. 692–701, 2004.
[27] M.V.A. Nunes, U.H. Bezerra, and H.H. Zurn, “Transient stability mar- gin of variable versus fixed speed wind systems in electrical grids,” in IEEE Powertech Conference, 2003, Bologna, Italy.
[28] P.C. Krause, O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Ma- chinery and Drive Systems, 2nd ed. Hoboken, NJ, USA: Wiley-IEEE Press, 2002.
[29] P.W. Sauer and M.A. Pai, Power Systems Dynamics and Stability. Up- per Saddle River, NJ, USA: Prentice Hall, 1998.
[30] S. Ahmed-Zaid and M. Taleb, “Structural modeling of small and large induction machines using integral manifolds,” IEEE Transactions on Energy Conversion, vol. 6, no. 3, pp. 529–535, 1991.
[31] E. Drennan, S. Ahmed-Zaid, and P.W. Sauer, “Invariant manifolds and start-up dynamics of induction machines,” in North American Power Symposium, 1989, pp. 129–138.
[32] L. Jiao, B.-T. Ooi, G. Joos, and F. Zhou, “Doubly-fed induction gener- ator (DFIG) as a hybrid of asynchronous and synchronous machines,” Electric Power Systems Research, vol. 76, pp. 33–37, 1997.
[33] H. Pulgar-Painemal and P.W. Sauer, “Doubly-fed induction machine in wind power generation,” in Electrical Manufacturing and Coil Winding Exposition, 2009, Nashville, Tennessee.
[34] D. Santos-Martin, S. Arnaltes, and J.L. Rodriguez-Amenedo, “Reactive power capability of doubly fed asynchronous generators,” Electric Power Systems Research, vol. 78, pp. 1837–1840, 2008.
[35] T. Lund and P. Sorensen, “Reactive power capability of a wind turbine with doubly fed induction generator,” Wind Energy, vol. 10, pp. 379– 394, 2007.
[36] M. Bradt (Program Director), “Course: Fundamentals of Wind Power Plant Design,” 2009, Tab 3: Electric Power Generation and Control Fundamentals by Ian Hiskens, University of Wisconsin-Madison.
[37] W.D. Rosehart and C.A. Ca˜nizares, “Bifurcation analysis of various power system models,” Electrical Power and Energy Systems, vol. 21, pp. 171–182, 1999.
[38] H. Pulgar-Painemal and P.W. Sauer, “Power system modal analysis con- sidering doubly-fed induction generators,” in VIII IREP Symposium - Bulk Power System Dynamics and Control, 2010, Buzios, Rio de Janeiro, Brazil.
[39] P. Gon¸calves, “Behaviour modes, pathways and overall trajectories: Eigenvector and eigenvalue analysis of dynamic systems,” System Dy- namic Review, vol. 25, no. 1, pp. 35–62, 2009.
[40] H. Pulgar-Painemal and P.W. Sauer, “Bifurcations and loadability issues in power systems,” in IEEE Powertech Conference, 2009, Bucharest, Romania.
[41] J. Arrillaga and N.R. Watson, Computer Modelling of Electrical Power Systems, 2nd ed. Chichester, West Sussex, England: John Wiley and Sons, 2001.
[42] T. Van Cutsem, Voltage Stability of Electric Power Systems. Norwell, MA, USA: Kluwer Academic Publishers, 1998.
[43] K.H. Ang, G. Chong, and Y. Li, “PID control system analysis, design and technology,” IEEE Transactions on Control Systems Technology, vol. 13, pp. 559–576, 2005.
[44] J.I. Perez-Arriaga, “Selective modal analysis with applications to electric power systems,” Ph.D. dissertation, Massachusetts Institute of Technol- ogy, 1981.
[45] V. Akhmatov, H. Knudsen, A.H. Nielsen, J.K. Pedersen, and N.K. Poulsen, “Modelling and transient stability of large wind farms,” Elec- trical Power and Energy Systems, vol. 25, pp. 123–144, 2003.
[46] N.W. Miller, W.W. Price, and J.J. Sanchez-Gasca, “Dynamic modeling of GE 1.5 and 3.6 wind turbine-generators,” October 27 2003. [Online]. Available: http://www.easthavenwindfarm.com
[47] Renewable Energy World, “ECR Completes 780-MW Roscoe Wind Farm,” October 2 2009. [Online]. Available: http://www.renewableenergyworld.com
[48] J. Qiao and Z. Lu, “A novel dynamic equivalence method for grid- connected wind farm,” Journal of Zhejiang University: Science A, vol. 9, no. 4, pp. 558–563, 2008.
[49] J. Conroy and R. Watson, “Aggregate modelling of wind farms contain- ing full-converter wind turbine generators with permanent magnet syn- chronous machines: Transient stability studies,” IET Renewable Power Generation, vol. 3, no. 1, pp. 39–52, 2009.
[50] A. Perdana, S. Uski-Joutsenvou, O. Carlson, and B. Lemstrom, “Com- parison of an aggregated model of a wind farm consisting of fixed-speed wind turbines with field measurement,” Wind Energy, vol. 11, no. 1, pp. 13–27, 2008.
[51] V. Akhmatov, “An aggregated model of a large wind farm with variable- speed wind turbines equipped with doubly-fed induction generators,” Wind Engineering, vol. 28, no. 4, pp. 479–486, 2004.
[52] L.M. Fernandez, F. Jurado, and J.R. Saenz, “Aggregated dynamic model for wind farms with doubly fed induction generator wind turbines,” Re- newable Energy, vol. 33, no. 1, pp. 129–140, 2008.
[53] L.M. Fernandez, C.A. Garcia, J.R. Saenz, and F. Jurado, “Equivalent models of wind farms by using aggregated wind turbines and equivalent winds,” Energy Conversion and Management, vol. 50, no. 3, pp. 691– 704, 2009.
[54] A. Shafiu, O. Anaya-Lara, G. Bathurst, and N. Jenkins, “Aggregated wind turbine models for power system dynamic studies,” Wind Engi- neering, vol. 30, no. 3, pp. 171–186, 2006.
[55] F. Koch, M. Gresch, F. Shewarega, I. Erlich, and U. Bachmann, “Con- sideration of wind farm wake effect in power system dynamic simula- tion,” in IEEE Powertech Conference, 2005, St. Petersburg, Russia. [56] J.L. Slootweg and W.L. Kling, “Modeling wind turbines for power sys-
tem dynamics simulations: An overview,” Wind Engineering, vol. 28, pp. 7–26, 2004.
[57] G. Andersson, P. Donalek, R. Farmer, N. Hatziargyriou, I. Kamwa, P. Kundur, N. Martins, J. Paserba, P. Pourbeik, J. Sanchez-Gasca, R. Schulz, A. Stankovic, C. Taylor, and V. Vittal, “Causes of the 2003 major grid blackouts in North America and Europe, and recommended means to improve system dynamic performance,” IEEE Transactions on Power Systems, vol. 20, no. 4, pp. 1922–1928, 2008.
[58] H.O. Wang, E.H. Abed, and A.M.A. Hamdan, “Bifurcations, chaos and crises in voltage collapse of a model power system,” IEEE Transac- tions on Circuits and Systems - I: Fundamental Theory and Applica- tions, vol. 41, no. 3, pp. 294–302, 1994.
[59] K.N. Srivastava and S.C. Srivastava, “Elimination of dynamic bifurca- tion and chaos in power systems using FACTS devices,” IEEE Trans- actions on Circuits and Systems - I: Fundamental Theory and Applica- tions, vol. 45, no. 1, pp. 72–78, 1998.
[60] A.M. Harb and N. Abdel-Jabbar, “Controlling Hopf bifurcation and chaos in a small power system,” Chaos, Solutions and Fractals, vol. 18, pp. 1055–1063, 2003.
[61] W.D. Rosenhart and C.A. Ca˜nizares, “Bifurcation analysis of various power system models,” Electric Power and Energy Systems, vol. 21, pp. 171–182, 1999.
[62] N. Mithulananthan and C.A. Ca˜nizares, “Hopf bifurcations and criti- cal mode damping of power systems for different static load models,” in IEEE Power Engineering Society General Meeting, 2004, vol. 2. pp. 1877–1882.
[63] C. Rajagopalan, B.C. Lesieutre, P.W. Sauer, and M.A. Pai, “Dynamic aspects of voltage/power characteristics,” IEEE Transactions on Power Systems, vol. 7, no. 3, pp. 990–1000, 1992.
[64] P.W. Sauer, B.C. Lesieutre, and M.A. Pai, “Maximum loadability and voltage stability in power systems,” International Journal of Electrical Power and Energy Systems, vol. 15, no. 3, pp. 145–154, 1993.
[65] J.J. Sanchez-Gasca, N.W. Miller, and W.W. Price, “A modal analysis of a two area system with significant wind power penetration,” in IEEE PES Power Systems Conference and Exposition, 2004, pp. 1148–1152. [66] P. Ledesma and C. Gallardo, “Contribution of variable speed wind farms
to damping of power system oscillations,” in IEEE Powertech Confer- ence, 2007, Lausanne, Switzerland.
[67] R.D. Fernandez, R.J. Mantz, and P.E. Battaiotto, “Contribution of wind farms to the network stability,” in IEEE Power Engineering Society General Meeting, 2006.
[68] Z. Miao, L. Fan, D. Osborn, and S. Yuvarajan, “Control of DFIG-based wind generation to improve interarea oscillation damping,” IEEE Trans- actions on Energy Conversion, vol. 24, no. 2, pp. 415–422, 2009.
[69] K. Elkington, V. Knazkins, and M. Ghandhari, “On the stability of power system containing doubly fed induction generator-based genera- tor,” Electric Power Systems Research, vol. 78, no. 9, pp. 1477–1484, 2008.
[70] D.J. Vowles, C. Samarasinghe, M.J. Gibbard, and G. Ancell, “Effect of wind generation on small signal stability - A New Zealand example,” in IEEE Power Engineering Society General Meeting, 2008.
[71] R. A. H.M.Z. El-Din and P. Chakravarti, “Second-order eigenvalue sen- sitivities applied to multivariable control systems,” Proceedings of the IEEE, vol. 65, no. 2, pp. 277–278, 1977.
[72] K.-S. S. Hae-Kon Nam, Yong-Ku Kim and K. Lee, “A new eigen- sensitivity theory of augmented matrix and its applications to power sys- tem stability analysis,” IEEE Transactions on Power Systems, vol. 15, no. 1, pp. 363–369, 2000.
[73] D. Gautam and V. Vittal, “Impact of DFIG based wind turbine genera- tors on transient and small signal stability of power systems,” in IEEE Power and Energy Society General Meeting, 2009, pp. 1–6.
[74] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, and O. Alsac, “Studies on power system load flow equivalencing,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-99, no. 6, pp. 2301–2310, 1980.
[75] L.F. Shampine and M.W. Reichelt, “The matlab ode suite,” SIAM Jour- nal on Scientific Computing, vol. 18, no. 1, pp. 1–22, 1997.
[76] H.K. Khalil, Nonlinear Analysis, 3rd ed. Upper Saddle River, NJ, USA: Prentice-Hall, 2002.
[77] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilib- rium to Chaos, 2nd ed. New York, NY, USA: Springer-Verlag, 1994.
AUTHOR’S BIOGRAPHY
Hector A. Pulgar-Painemal received a B.S. degree and an M.S. degree in Electrical Engineering from the University of Concepcion, Chile, in 2001 and 2003, respectively. During 1999-2000, he was a teaching assistant at the University of Concepcion. In 2001, he was the recipient of the University of Concepcion Award for best student in his class. During 2001-2002, he was hired for the teaching staff at the Federico Santa Maria Technical University, Chile. In 2003, he was promoted to the position of Academic Instructor. Since then, he has taught over five courses, participated in two research projects and guided several student final projects in the Electrical Engineer- ing Program. He entered the Ph.D. program in the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign in 2006 with a Fulbright Fellowship and the support of the Federico Santa Maria Technical University. After finishing his Ph.D. studies, Mr. Pulgar- Painemal plans to return to the Federico Santa Maria Technical University as a professor. His research activities are in the areas of power system dy- namics, power system operation, wind power generation, and power system simulations.