# Small-Signal Stability Assessment of the Cameroonian Southern Interconnected Grid

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Information about Small-Signal Stability Assessment of the Cameroonian Southern...

Published on September 1, 2016

Author: IJOERS

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3. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 33 Modern control theory with multivariate approach could offer some tools for this task. Until now, from the existing generator control systems none is the implementation of modern control theory [9], they were designed using frequency or root locus method, and they were designed to solve a single problem. It was seen that they helped solve one aspect of the problem, but to the detriment of another aspect. Control system design generally begins with the realization of an appropriate model for the dynamic system, which will be used to evaluate its overall performances. Power system operators and planners need effective tools to assess the overall stability of the system and take appropriate actions to enhance its stability over a wide range of operating conditions: the model simulation. Modeling of power plants and their loads is therefore fundamental to the evaluation of their performance by simulation and design of their control. Numerous research works have been done and published on modeling and simulation of power systems for small-signal stability studies. Demello and Concordia [4], Kundur et al. [11], and Chow et al. [12] developed a linearized model of synchronous generator and its excitation system connected to an infinite bus in the form of block diagram, in frequency domain. Baghani and Koochaki [8] developed a linearized state-space model of a synchronous generator and its excitation system. Their model was derived from the steady-state harmonic operation which is linear. Pedro Camilo de Oliveira e Silva et al. [13] developed a linearized state-space model of a complete power plant. For the Cameroonian SIG, state-space models of water flow system, turbine-generator unit, voltage control system, speed control system, and load connected at the generator terminals will be developed. Thank to the architecture of the SIG that is rather a radial than a meshed one, the SIG can be considered as simple, and will be modeled as a one area single machine system. The model must overcome some of the shortcomings of the aforementioned models. Thus, the turbine-generator unit model will be built by physical principles and put in state-space form. Small disturbances hypothesis permits the linearization of the system of equations and the application of all the wealth of the modern control theory. The other devices will be transformed from frequency representation form that is already linear, into state-space representation form. The performance of the integrated overall system will be assessed using MATLAB simulation. The rest of this paper is structured as follows: in section II the feedback control structure of the system is presented. In section III the detailed development of a linearized state-space model of the Cameroon SIG is realized; this linearized form is suitable for practical study of the classical control systems performance. Some illustrative simulation results are presented and discussed in section IV. In the final section of the paper some concluding remarks are mentioned. II. FEEDBACK CONTROL STRUCTURE OF THE POWER SYSTEM The Cameroonian SIG incorporates in reality many autonomous feedback control systems, each consisting of a generating unit (turbine and alternator), and the load. It can be modeled as a one area single machine system. u Voltages RMS Adapter (Rectifier) Ur Pm Load Exciter and Voltage Regulator Speed Regulator and Water Gate Hydraulic Turbine and Water Flow Rate Regulator Three-Phase Alternator (Windings and Rotor Inertia ) Uréf nréf n Y uF iF i Power Measuring Device Préf P Hréf FIG. 1: FEEDBACK CONTROL STRUCTURE OF THE POWER PLANT The structure of the feedback control system of the SIG is shown in fig. 1 [14]. A generating unit incorporates three feedback control loops for the regulation of the three main variables influencing the power output. These variables include the flow rate of water impacting the turbine, the alternator rotor speed and the alternator armature voltage which determines the magnetic flux permeating the armature circuit. Each of these variables is maintained at a reference value by an appropriate feedback control loop.

4. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 34 III. STATE-SPACE MODELING OF THE POWER SYSTEM 3.1 The Turbine-Alternator Sub-system From a system dynamics perspective, the alternator is the most complex component of the power system, due to the presence of several coils in armature and in rotor, which are mounted on separate geometrical axis and are magnetically mutually coupled. The alternators used in hydro electric power generating plants are salient poles synchronous machines. Their permeance varies across the air gap with respect to the pole axis (d-axis) of the polar wheel and to an axis at right angle to the pole axis, the quadrature axis (q-axis) [14]. As a logical consequence of this, an orthogonal system with d-q axis is introduced on the rotor, with the d-axis superimposed on the polar axis and the q-axis lagging the d-axis. The various inductances in the alternator are represented by six coils [3], [15]. The stator contains three coils mounted on three axis separated from each other by an angle of 120°. The inductor coil F is located on the rotor in the d-axis. The damping effect is represented by the short-circuited fictitious coils KD and KQ in the d- and q-axis respectively. Some inductances vary with the angular position of rotor with respect to the axis of armature reference coil  which in turn varies with time. This introduces considerable complexity in the machine differential equations whose coefficients are time variable. The elimination of the time dependency of the inductance terms is obtained by Park’s transformation [15]. Due to this transformation, the magnetizing effect of the three-phase armature currents is replaced by that of their components id(t) and iq(t) flowing in two imaginary coils located on the d- and q-axis of the rotor, and the magnetizing effect of the damping coils is due to the currents iKD(t) and iKQ(t) flowing in coils KD and KQ. 3.1.1 Electrical State-Space Model of the Alternator The electrical state-space model of the three phase alternator in d-q-0 components is [14]:                                                                 0 )t(u )t(u )t(u B )t(i )t(i )t(i )t(i )t(i A dt )t(di dt )t(di dt )t(di dt )t(di dt )t(di F q d ' G KQ KD F q d ' G KQ KD F q d (1) where the zero-sequence voltage u0 and current i0 are zero, since the alternator operation is assumed to be in equilibrium. The system matrices  ' GA and  ' GB are independent of time and are defined in Appendix 1. 3.1.2 Mechanical model of the Rotor of Alternator-Turbine Unit Assuming rigid coupling of turbine and alternator, the motion of the rotor is described by the following equation: emm m t TT ²dt ²d J   (2) where, Jt = moment of inertia of all rotating elements Tm = mechanical torque produced by the turbine Tem = electrical torque opposing the motion of the rotor m = mechanical angle In the Park transformed electrical model of the generator, the electromagnetic torque results from the interaction between the total flux linkages d and q with the fictitious windings in d- and q-axis, and the currents through these windings id and iq. It is expressed as follows [14], [15]:

5. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 35  dqqdem iipT  (3) where, d and q = fluxes permeating the coils d and q, respectively di and qi = currents in the coils p = number of pairs of poles The expressions of d and q are given by [14], [15]:          )t(iM 2 3 )t(iL)t( )t(iM 2 3 )t(iM 2 3 )t(iL)t( KQS,KQqqq KDS,KDFFSddd (4) The electromagnetic torque then has the detailed form:                     KQdS,KQKDqS,KDFqFSqdqdem iiM 2 3 iiM 2 3 piiM 2 3 iiLLpT (5) Even at synchronous speed, the rotor oscillates and EMFs are then induced in the damping coils and in the rotor iron. The currents driven by the induced EMFs (eddy currents in the rotor iron) interact with the flux linkages of the damping coils and produce a torque which dampens the oscillations (Lenz’s law). The effects of eddy-currents in the rotor iron and of currents in the damper coils can be lumped into a damping torque. The second term in equation (5) is obviously the damping torque TD which is moreover proportional to the rate of change of the angle of polar wheel. It is described by the equation: dt d kiiM 2 3 iiM 2 3 pT p DKQdS,KQKDqS,KDD            (6) Substituting for Tem and TD in equation (2) results in the non-linear differential equation that describes the mechanical behavior of the generator rotor and the turbine:   FqFS t 2 qdqd t 2 m t p t Dp iiM 2 3 J p iiLL J p T J p dt d J kp dt d     (7) where dt d ²dt ²d ²dt ²d pp       . 3.1.3 Linearization of the mechanical model During stable and balanced operation of the alternator, the stator phase voltages are sinusoïdal of constant magnitude Vˆ , they form a balanced three phase system. Applying the inverse Park transform to the system of phase voltages results in the components [14] :       ppq ppd cosUcosV3)t(u sinUsinV3)t(u (8) When the system oscillates about an operating point (ud0, uq0, U0, p0) with the small variations U and p of the variables U and p respectively, the small variations of the components ud and uq can be approximated as:     p0p00pq p0p00pd sinUUcosu cosUUsinu   (9)

6. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 36 Linearising equation (1) and ignoring the damping currents iKD and iKQ whose effect has already been taken into account by introduction of the damping torque TD, results in the following state equations of the three phase alternator:                                                      F pG F q d G F q d u U B i i i A dt id dt id dt id (10) where the system matrices  GA and  GB are given in appendix 1. Equation (7) can also be linearized by considering small variations about an operating point (id0, iq0, iF0, n0, Tm0, Pm0). The linearized version of this equation is of the form :                               n n2 P P n2 1 J p iiM 2 3 J p iiM 2 3 iLL J p iiLL J p dt d J kp dt d 2 0 0m m 0t F0qFS t 2 q0FFS0dqd t 2 d0qqd t 2 p t Dp  (11) Combining the state equations (10) and (11), the state-space representation of the turbine-alternator sub-system is obtained in the form:                                                                          m FTG p F q d TG p F q d P u U B n i i i A dt nd dt d dt id dt id dt id (12) where the system matrices  TGA and  TGB are given in appendix 1. 3.1.4 Model of the Hydraulic Turbine The water gate opening Y and the head Hf have a direct influence on the water velocity in the penstock and thus on the flow rate. In stability studies of power systems, if mechanical power Pm is the output variable and gate opening Y the input variable, the non ideal hydraulic turbine can be represented by a transfer function FTH as follows [3], [15] : sTa1 sTb1 K)s(F w1th w1th tTH    (13) where, s= Laplace operator TW= hydraulic time constant for a given load level Kt= gain of hydraulic turbine ath1 and bth1= coefficients of transfer function The canonical state-space representation for small variations about a stable operating point, derived from the function (13) is of the form:

7. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 37 y a bK x a b 1KP y Ta 1 x Ta 1 dt dx 1th 1tht t 1th 1th tm w1th t w1th t           (14) where the coefficients of the state-space model are represented in Appendix 2. 3.2 State-Space Models of the Control Systems As shown in fig. 1, there are three control loops in the system –the flow, the speed and the voltage control systems. 3.2.1 The Flow Control System The block diagram of the flow control system [16] is shown in fig. 2. FIG. 2 : BLOCK DIAGRAM OF THE FLOW CONTROL SYSTEM The system incorporates a PID controller. The variables and parameters in the block diagram are defined as follows: Q = water flow rate Qréf = reference water flow rate KP = proportional gain of the PID flow controller Td = rate time of the PID flow controller Ti = reset time of the PID flow controller Tf = filter time constant of the flow-head converter Kb = rate of slope The closed-loop transfer function of the system is of the form: 43 rb3 2 rb2rb1rb0 2 rb2rb1rb0 RB ssasasaa sbsbb )s(F    (15) where the coefficients irba and irbb are represented in Appendix 3. The corresponding canonical controllable state-space representation is of the form:                                                                                  4rb 3rb 2rb 1rb rb2rb1rb0 réf 4rb 3rb 2rb 1rb rb3rb2rb1rb0 4rb 3rb 2rb 1rb x x x x 0bbbQ Q 1 0 0 0 x x x x aaaa 1000 0100 0010 dt dx dt dx dt dx dt dx (16) Qréf Q fsT1 1 i dP sT sTK 1  2 b s K

8. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 38 3.2.2 The Speed Control System The block diagram of the speed control system is shown in fig. 3 [3], [17]. The system incorporates a PI controller. The variables and parameters in the block diagram are defined as follows: Y: Water gate opening n: Angular speed of the alternator P: Active power supplied to the load by the alternator nréf : Speed set point Préf : Active power set point Tn : Time constant Kc : Proportional gain of PI Ki : Integral gain of PI Controller Bl : gain of limiter Tx : Time constant Ks : Gain of the servo valve Ts and TV : Time constants of the servo valve and valve-wagon units respectively Bt : Transient droop Rp: Permanent droop Td : Minor loop Time constant In addition to the main speed control feedback loop, a subsidiary power feedback loop is applied to the system to limit the active power supplied during transient operation or in situations of overload. The closed-loop transfer function of the system is of the form:    PP ssasasaa sbb NN ssasasaa sbsbb Y réf43 rv3 2 rv2rv1rv0 rvp1rv0 réf43 rv3 2 rv2rv1rv0 2 rvn2rvn1rv0        (17) where the coefficients airv, birvn and birvp are represented in Appendix 4. The canonical state-space representation, derived from the closed-loop transfer function, is of the form: FIG. 3: SPEED CONTROL SYSTEM nréf Y Préf P s K K i c  lB xT sT K s s 1 s 1 dTtB pR sTn1 sT1 1 V

9. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 39                                                                                                                                                 4rv 3rv 2rv 1rv 4rv 3rv 2rv 1rv rvn2rvn1rv0rvp1rv0 réf réf 4rv 3rv 2rv 1rv 4rv 3rv 2rv 1rv rv3rv2rv1rv0 rv3rv2rv1rv0 4rv 3rv 2rv 1rv 4rv 3rv 2rv 1rv x x x x x x x x 0bbb00bbY nn PP 10 00 00 00 01 00 00 00 x x x x x x x x aaaa0000 10000000 01000000 00100000 0000aaaa 00001000 00000100 00000010 dt xd dt xd dt xd dt xd dt dx dt dx dt dx dt dx (18) 3.2.3 The Voltage Regulator The block diagram of the voltage regulator is shown in fig. 4. FIG. 4 : BLOCK DIAGRAM OF THE VOLTAGE CONTROL LOOP The parameters in the block diagram are defined as follows: Uréf : Armature voltage set point U : RMS value of armature voltages IF : Excitation current UF : Excitation voltage Ur : Voltage representing the RMS value of Armature coils voltages K1 : Gain of the voltage regulator 1 : Time constant of the voltage regulator K2 : Gain of the reference voltage amplifier K3 : Slope of the thyristor characteristic K5 : Gain of the excitation voltage regulator f : Time constant of the excitation voltage K6 : Gain of the current regulator 6 : Time Constant of the current regulator Uréf s1 K 1 1  2K s1 K 6 6  s1 K 8 8  3K s1 K f 5  UF IF U Ur

11. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 41             xCy U n P BxAx dt d S réf réf réf SS                (23) where,  x = 18x1 state vector  y = 13x1 output vector  SA = 18x18 matrix (the A-matrix)  SB = 18x3 matrix (the B-matrix)  SC = 13x18 matrix (the C-matrix) The elements of these different matrices and vectors are presented in Appendix 7. IV. SIMULATION RESULTS AND DISCUSSION In this paper, to show the validity of the linearized state-space model of the Cameroonian SIG, and hence to investigate the performance of its PID controller, PID controlled voltage and speed regulators, simulation works have been carried out for the single machine system connected to a load, using MATLAB programs designed by the authors. Most of the model parameters used in the ensuing simulation are given in the appendix and were obtained from technical manuals of the system. Some parameters were obtained from experiments on a prototypal alternator [18]. In order to prove the validity of the model, the results are compared with those given in source books like [3] and [15]. The perturbations considered in this paper are input unit steps which are applied to the system. We consider the operating point given in appendix 8 which also gives the values of sub-systems parameters. The parameterized state-space model is presented in Appendix 7. The simulation results are step responses of water flow deviation, armature voltage deviation, rotor speed deviation and rotor angle deviation. They are depicted respectively in figures 5 to 8. Fig. 5 illustrates the step response of the dam water flow system given in fig.2, to a unit step disturbance of the reference water flow rate. 0 5 10 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Temps (sec) q(pu) FIG. 5 : RESPONSE TO THE SET POINT UNIT STEP OF THE WATER FLOW The PID controller really stabilizes the system. The dynamics of the water flow rate deviation exhibits during post perturbation state many oscillations with large initial amplitude which then decreases slowly, before it settles to its steady state value 1,0. The system is poorly damped and it is not regulated back to zero deviation after a unit step perturbation of the water flow set point. The system with classical PID controller has a long settling time and a bad accuracy. Fig. 6 illustrates the step responses of the turbine-alternator system, following a disturbance of the reference armature voltage. It shows the deviations of the armature voltage U, the rotor speed n and the polar wheel angle p.

12. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 42 FIG. 6: RESPONSES TO THE SET POINT UNIT STEP OF THE ARMATURE VOLTAGE The dynamics of the armature voltage deviation exhibits a first swing with overshoot during post perturbation state, before it decreases, oscillates slowly and settles to its non zero steady state value; it contains high frequencies due to transients in stator voltages. The PID controlled AVR really stabilizes the system, but not accurately; the armature voltage perturbation is poorly damped and not regulated back to zero by the AVR. The dynamics of the rotor speed deviation displays during post perturbation state, an acceleration of the rotor in the first half cycle and its deceleration in the second half cycle and so on, with decreasing amplitude; it oscillates slowly and settles to its zero steady state value; the perturbation is completely cleared by the speed controller. The dynamics of the rotor angle deviation exhibits during post perturbation state a first positive swing followed by a negative swing, before it oscillates slowly after a negative overshoot and settles to its non zero steady state value. Fig. 7 illustrates the step responses of the turbine-alternator system, following a disturbance of the reference rotor speed. It shows the deviations of the rotor speed n, the rotor angle p and the armature voltage U. FIG. 7 : RESPONSES TO THE SET POINT UNIT STEP OF THE ROTOR SPEED The dynamics of the rotor speed deviation exhibits during post perturbation state, first a decrease followed by an increase and oscillations with decreasing amplitude, before it settles to its zero steady state value; the perturbation is completely cleared by the speed controller. The rotor is decelerated in the first half cycle and accelerated in the second half cycle and so on, with decreasing amplitude; it oscillates slowly and settles to its zero steady state value. The PID controlled speed regulator really stabilizes the system, accurately. The rotor angle deviation first exhibits a decrease and then it increases steadily to reach a positive overshoot; it then decreases and oscillates to stabilize at a non zero value. The armature voltage deviation first exhibits a decrease and then it increases steadily to reach a positive peak value and decreases to stabilize at a non zero value. The PID controlled AVR really stabilizes the system, but not accurately. The initial retardation of the rotor has an aftereffect on the rotor angle and the armature voltage. They are caused by high frequency transients in the transformer voltage terms in the stator voltages. Fig. 8 illustrates the step responses of the turbine-alternator system, following a disturbance of the reference active power demand. It shows the deviations of the rotor speed n, the polar wheel angle p and the RMS value of the armature voltages U. 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Temps (sec) U(pu) 0 2 4 6 8 10 12 14 16 18 20 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Temps (sec) n(tr/min.) 0 5 10 15 20 25 30 -10 -8 -6 -4 -2 0 2 4 Temps (sec) p (degrés) 0 2 4 6 8 10 12 14 16 18 20 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Temps (sec) n(tr/min.) 0 5 10 15 20 25 30 -15 -10 -5 0 5 10 15 20 25 Temps (sec) p (degrés) 0 5 10 15 20 25 30 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Temps (sec) U(pu)

13. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 43 FIG. 8 : RESPONSES TO THE SET POINT UNIT STEP OF THE ACTIVE POWER DEMAND The dynamics of the rotor speed deviation exhibits during post perturbation state a high negative peak that means a sharp deceleration in the first half cycle, followed by an acceleration in the second half cycle and so on; it increases to a high positive overshoot and then decreases and oscillates around zero, before it settles to its zero steady state value; the power perturbation is completely cleared by the speed controller: the PID controlled speed regulator really stabilizes the system, accurately. The dynamics of the rotor angle deviation and the armature voltage deviation exhibit a first decrease to negative values (backswing phenomenon) and then a steady increase to positive values. They exhibit overshoots and stabilize at non zero values. The PID controlled regulator does not clear the perturbation of the power demand. The high frequency oscillations affecting the rotor speed are due to the derivative action of the rotating mechanical system of the turbine-alternator system. V. CONCLUSION A linearized dynamic model of the Cameroonian southern grid has been developed, using the state-space approach. The model is modular in structure. Intensive MATLAB simulation has been carried out to investigate the behavior of the southern grid of the Cameroon power system, after small disturbances. The simulation results have shown that classical controllers actually in the southern grid of the Cameroon power system ensure stability of the SIG, but they are not accurate, they are slow. The power system is poorly damped. This assessment can serve as a basis for the design of most efficient additional automatic controls to operate in combination with classical controllers present in the SIG. REFERENCES [1] Kundur P., Morison G.K., « A Review of Definitions and Classification of Stability Problems in Today’s Power Systems », Proc. of IEEE PES Meeting, New York, 2-6 February 1997. [2] Graham Rogers, « Power System Oscillations », Kluwer Academic Publishers, 2002. [3] Kundur P., « Power System Stability and Control », McGraw-Hill, Inc., 1994. [4] Demello F.P., C. Concordia, « Concepts of synchronous machine stability as affected by excitation control », IEEE Trans. PAS, PAS-88, pp.316-329, 1969. [5] Jan Machowski, Janusz W. Bialek, James R. Bumby, « Power System Dynamics: Stability and Control », John Wiley & Sons, Ltd, second edition 2008, ISBN 978-0-470-72558-0. [6] PA. Government Services Inc., « PEAC » Première Etude du Schéma Directeur pour l’Afrique Centrale, Mai 2005, pp3.18-3.23. [7] Crappe Michel., « Commande et régulation des réseaux électriques : Evolution des réseaux électriques européens face à de nouvelles contraintes et impact de la production décentralisée », Chapitre 2, Traité EGEM, Lavoisier 2003, pp 49-99. [8] D. Baghani, A. Koochaki, « Multi-Machine Power System Optimized Fuzzy Stabilizer Designe by Using Cuckoo Search Algorithm », International Electrical Journal (IEEJ), Vol. 7(2016) No.3, pp.2182-2187, ISSN 2078-2365. [9] H. Glavitsch, « Control Of Power Generation And System Control With The Emphasis On Modern Control Theory », IFAC, Proc. of International Conference on Automatic Control in Power Generation Distribution and Protection, Pretoria, South Africa 1980. [10] M. Ouassaid, A. Nejmi, M. Cherkaoui, and M. Maaroufi, « A New Nonlinear Excitation Controller for Transient Stability Enhancement in Power Systems », Proc. of World Academy of Science, Engineering And Technology Volume 8 October 2005 ISSN 1307-6884. [11] Kundur P., Klein M., Rogers G. J., and Zywno M. S., "Application of Power System Stabilizers for Enhancement of overall System Stability", IEEE Transactions on Power Systems, Vol. 4, 1989, pp. 614-626. [12] Chow J. H., Sanchez-Gasca J. J., « Pole-placement Designs of Power System Stabilizers », IEEE Transactions on Power Systems , Vol. 4, No.1, 1989, pp. 271-277. 0 5 10 15 20 25 30 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Temps (sec) n(tr/min.) 0 5 10 15 20 25 30 -5 0 5 10 15 20 25 Temps (sec)  p (degrés) 0 5 10 15 20 25 30 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Temps (sec) U(pu)

14. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 44 [13] Pedro Camilo de Oliveira e Silva, Basile Kawkabani, Sébastien Alligné, Christophe Nicolet, Jean-jacques Simond, and François Avellan, « Stability Study on Hydroelectric Production Site Using Eigenvalues Analysis Method », Journal of Energy and Power Engineering, Vol. 6, 2012, pp. 940-948. [14] J. H. TSOCHOUNIE, « Commande Multivariable du Réseau Interconnecté Sud du Cameroun dans l’Espace d’Etat », Ph.D/Doctoral thesis in Electrical Engineering, National Polytechnic Institute, Yaounde, Cameroon, 2008.. [15] Padiyar K. R., « Power System Dynamics : Stability and Control », John Wiley & Sons (Asia) Pte Ltd, 1996. [16] GEC ALSTHOM, « Automate d’aménagement : Analyse procédé », Aménagement hydroélectrique de SongLoulou, 1999. [17] V. Raeber et L. Bonny, « Régleur électronique », Bulletin technique VEVEY, 1962. [18] Barret Jean-Paul, Bornard Pierre, Meyer Bruno, « Simulation des réseaux électriques », Editions Eyrolles 1997. APPENDIX Appendix 1: Matrices of Generator and Turbine-Generator Unit Models.      G 1 G ' G RLA   and     1 G ' G LB   where,                                  KQS,KQ KDKD,FS,KD KD,FFFS S,KQq S,KDFSd G L00M 2 3 0 0LM0M 2 3 0ML0M 2 3 M 2 3 00L0 0M 2 3 M 2 3 0L L                                       KQS,KQ KDS,KD FFS S,KDFS1d S,KQq1 G R000M 2 3 0R0M 2 3 0 00RM 2 3 0 0M 2 3 M 2 3 RL M 2 3 00LR R      Ld = inductance of the imaginary coil in the d-axis Lq = inductance of the imaginary coil in the q-axis LF= inductance of the inductor coil F LKD = inductance of the damping coil KD LKQ = inductance of the damping coil KQ MFS = maximum mutual inductance between the inductor coil F and the stator coils MKD,S = maximum mutual inductance between the damping coil KD and the stator coils MKQ,S =maximum mutual inductance between the damping coil KQ and the stator coils MF,KD = maximum mutual inductance between the inductor coil F and the damping coil KD                                                   22 0 2 0 10 22 0 2 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 FSFd dF FSFd qFS FSFd FSs q FS qq d FSFd FSF FSFd Fq FSFd F G MLL LR MLL LM MLL MR L M L R L L MLL MR MLL LL MLL LR A                                             2 FSFd d 2 FSFd 0p0FS 2 FSFd 0pFS q 0p0 q 0p 2 FSFd FS 2 FSFd 0p0F 2 FSFd 0pF G M 2 3 LL L M 2 3 LL sinUM 2 3 M 2 3 LL cosM 2 3 0 L sinU L cos M 2 3 LL M 2 3 M 2 3 LL cosUL M 2 3 LL sinL B

15. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 45                                                             xgiFiqid FSFd pFS FSFd dF FSFd qFS FSFd FS q p q FS qq d FSFd pF FSFd FSF FSFd Fq FSFd F TG KKKK p MLL UM MLL LR MLL LM MLL MR L U L M L R L L MLL UL MLL MR MLL LL MLL LR A 0 20000 0 2 3 sin 2 3 2 3 2 3 2 3 2 3 2 3 0 sin2 3 0 2 3 cos 2 3 2 3 2 3 2 3 2 00 22 0 2 1 00 0 10 2 00 22 0 2 1                                               Pm FSFd d FSFd pFS q p FSFd FS FSFd pF TG K MLL L MLL M L MLL M MLL L B 00 000 0 2 3 2 3 cos 2 3 00 cos 0 2 3 2 3 2 3 sin 22 0 0 22 0 Appendix 2: Parameters of Hydraulic Turbine. Non ideal Hydraulic Turbine: 5,1Kt  ; 1,1ath13  ; 57,0P10064,2a 0L 10 t   ; 18,1P10541,4a 0L 9 th21   ; t 21th13th tt K aa ab   . Coefficients of the state-space model: Wt 0th Ta 1 a   ;            t t Wt t 0th a b 1 Ta K b ; t t tth a b Kd  Appendix 3: Coefficients of Transfer Function of Water Flow Control System. if b rb0 TT K b  ; f Pb rb1 T KK b  ; f db rb2 T TK b  ; if b rb0 TT K a  ; f Pb rb1 T KK a  ; f db rb2 T TK a  ; f rb3 T 1 a  Appendix 4: Coefficients of Transfer Function of Rotor Speed Control System. vs isxl rv0 TT KKTB b  ;   vs nicsxl rvn1 TT TKKKTB b   ; vs ncsxl vn12 TT TKKTB b  ; vs csxl rvp1 TT KKTB b  ; vs gisxl rv0 TT RKKTB a  ;   vs victgsxl rv1 TT TKKRRKTB a   ;   vs vctgsxl rv2 TT TKRRKTB1 a   ; vs vs rv3 TT TT a   Appendix 5: Coefficients of Transfer Function of Voltage Control System. f 53 rt KK b   ; f rt0 1 a   ; 6 6 rt1 K a   ; 1 21 rt2 KK a   ; 6 rt3 1 a   ; 1 rt4 1 a   ; 8 8 rt5 K a   ; 8 rt6 1 a   Appendix 6: Coefficients of Power System Load Model. Qd 0 0Q Dd 0 0D d X U u R U u R  ; Qq 0 0Q Dq 0 0D q R U u X U u R  ;   Q 0 0Q D 0 0D U U u U U u U ; fQ 0 0Q Df 0 0D f U U u U U u U  The voltages uD0 and uQ0, and the currents iD0 and iQ0, are Kron’s components of armature voltages va, vb and vc, and currents ia, ib and ic respectively, in a new coordinate system D, Q and 0. They represent steady-state voltages across and currents in two imaginary coils located on the D and Q axis of the stator, for a given operating point, and are obtained as follows:                         c b a 1 0 0 Q D v v v )(K u u u and                         c b a 1 0 0 Q D i i i )(K i i i The Kron’s transformation matrix  )(K 0 is similar to the Park’s.

16. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 46 QDDQQQDD 0pDQ0pQQ Dd GGGG sinBcosG R    ; QDDQQQDD 0pDQ0pQQ Dq GGGG cosBsinG X    ; QDDQQQDD 0pDD0pQD Qd GGGG sinGcosB X    ; QDDQQQDD 0pDD0pQD Qq GGGG cosGsinB R    ; QDDQQQDD 0DDQ0QQQ D GGGG iBiG U    ; QDDQQQDD 0DDD0QQD Q GGGG iGiB U    ; QDDQQQDD QfDQDfQQ Df GGGG KBKG U    ; QDDQQQDD QfDDDfQD fQ GGGG KGKB U                      2 0 0D0Q 2 0 0L 2 0 2 0D 2 0 0L DD U uu )2n( U Q 1 U u )2m( U P G ;                   2 0 0Q0D 2 0 0L 2 0 2 0Q 2 0 0L DQ U uu )2m( U P 1 U u )2n( U Q B                   2 0 0Q0D 2 0 0L 2 0 2 0D 2 0 0L QD U uu )2m( U P 1 U u )2n( U Q B ;                   2 0 0D0Q 2 0 0L 2 0 2 0Q 2 0 0L QQ U uu )2n( U Q 1 U u )2m( U P G qf2 0 0Q0L pf2 0 0D0L Df K U uQ K U uP K  ; qf2 0 0D0L pf2 0 0Q0L Qf K U uQ K U uP K  Appendix 7: Parameterized State-space Model and Elements of Model Matrices. 2 FSFd 0ds bF1,1 MLL sinRR LA    2 FSFd 0qq0s bF2,1 MLL sinRL LA    2 FSFd FS bF3,1 MLL M2/3 RA    2 FSFd 000 bF4,1 MLL sinUcosU LA     2 FSFd 0F bF5,1 MLL sinpU LA    2 FSFd RT0 bFS15,1 MLL b M2/3A   q 0qd0s b1,2 L cosRL A   q 0qs b2,2 L cosRR A   q 0sFs b3,2 L M2/3 A   q 000 b4,2 L cosUsinU A    q 0F b5,2 L cospU A   2 FSFd 0ds FSb1,3 MLL cosRR M2/3A    2 FSFd 0qq0s FSb2,3 MLL cosRL M2/3A    2 FSFd dF b3,3 MLL LR A   2 FSFd 000 FSb4,3 MLL cosUsinU M2/3A     2 FSFd 0F FSb5,3 MLL cospU M2/3A    2 FSFd rt0d b15,3 MLL bL A   b5,4A  id1,5 KA  iq2,5 KA  iF3,5 KA  n5,5 KA  TH0pm6,5 bKA  rv01THpm7,5 bdKA  rv11THpm8,5 bdKA  rv02THpm11,5 bdKA  rv12THpm12,5 bdKA  rv22THpm13,5 bdKA  TH06,6 aA  rv017,6 bA  rv118,6 bA  rv0211,6 bA  rv1212,6 bA  rv2213,6 bA  1A 8,7  1A 9,8  1A 10,9  0 d0L 1,10 U RmP A  0 q0L 2,10 U RmP A  0 0L 4,10 U UmP A   rv07,10 aA  rv18,10 aA  rv29,10 aA  rv310,10 aA 

17. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 47 1A 12,11  1A 13,12  1A 14,13  1A 5,14  rv011,14 aA  rv112,14 aA  rv213,14 aA  rv314,14 aA  rt015,15 aA  rt116,15 aA  rt217,15 aA  1A 3,16  rt316,16 aA  rt417,17 aA  rt518,17 aA  d1,18 RA  q2,18 RA   UA 4,18 f5,18 pUA  rt618,18 aA  N0s f2 ;            sinILcosIRU sinIRcosIL tana 0q0S0S0 0s0q0S 0 . 1B 1,10  1B 2,14  1B 3,17  1C 1,1  1C 2,2  0Qd0Dd1,3 sinXcosRC  0Qq0Dq2,3 sinRcosXC  0q0Qd0Dd4,3 usinUcosUC   0Qf0Df5,3 sinUcosUpC  0Qd0Dd1,4 cosXsinRC  0Qq0Dq2,4 cosRsinXC  0q0Qd0Dd4,4 ucosUsinUC   0Qf0Df5,4 cosUsinUpC  1C 3,5  rt015,6 bC  d1,7 RC  q2,7 RC   UC 4,7 f5,7 pUC    180 C 4,8   2p 60 C b5,9 b5,10 fpC    0qqd1,11 iLL 3 2 C             0FFS0dqd2,11 iM 2 3 iLL 3 2 C 0qFS3,11 iM 3 2 C  0 d0L 1,12 U RmP C  0 q0L 2,12 U RmP C  0 0L 4,12 U UmP C            pf0L 0 f0L 5,12 KP U UmP pC 0 d0L 1,13 U RnQ C  0 q0L 2,13 U RnQ C  0 0L 4,13 U UnQ C            qf0L 0 f0L 5,13 KQ U UnQ pC rv017,14 bC  rv118,14 bC  rv0211,14 bC  rv1212,14 bC  rv2213,14 bC                                                                18,185,184,182,181,18 18,1717,17 16,16 15,155,154,152,151,15 14,1413,1412,1411,14 10,109,108,107,105,104,102,101,10 13,612,611,68,67,66,6 13,512,511,58,57,56,55,53,52,51,5 5,4 15,35,34,33,32,31,3 5,24,23,22,21,2 15,15,14,13,12,11,1 S a000000000000aa0aa aa0000000000000000 00a000000000000100 000a000000000aa0aa 0000aaaa0000010000 000010000000000000 000001000000000000 000000100000000000 00000000aaaa0aa0aa 000000001000000000 000000000100000000 000000000010000000 00000aaa00aaa00000 00000aaa00aaaa0aaa 0000000000000a0000 000a000000000aaaaa 0000000000000aaaaa 000a000000000aaaaa A

18. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 48                                              00000ccc00cc000000 0000000000000cc0cc 0000000000000cc0cc 000000000000000ccc 0000000000000c0000 000000000000001000 0000000000000cc0cc 000b00000000000000 000000000000000100 0000000000000cc0cc 0000000000000cc0cc 000000000000000010 000000000000000001 C 13,1312,1311,138,137,13 5,124,122,121,12 5,114,112,111,11 3,102,101,10 5,9 5,74,72,71,7 rt 5,44,42,41,4 5,34,32,31,3 S                                                                       4 3 2 1 4 3 2 1 4 3 2 1 rt rt rt rt rv rv rv rv rv rv rv rv t p F q d x x x x x x x x x x x x x n i i i x                                                            000 100 000 000 010 000 000 000 001 000 000 000 000 000 000 000 000 000 SB Appendix 8: Values of Parameters of subsystems of State-Space Model. System Operating Point: .u.p2247.1U0  ; .u.p9.0PL0  ; .u.p5578.0QL0  ; 85.0cos 0  ; .u.p7487.0I0  ; .u.p094.1iF0  ; 97878.0G0  ; .min/tr120n0  ;  210p ; .sec9661.4Tw  Synchronous Generator: .u.p4938.1Ld  ; .u.p9959.0Lq  ; .u.p167.0L  ; .u.p3268.1MFS  ; 25p  .u.p852.1LF  ; .u.p0152.0Rs  ; .u.p0002654.0RF  , ²kgm8800000Jt  . Water Flow Control System parameters: 50Kb  ; 916,0Tf  ; 1Kp  ; 5Td  ; 10Ti  ; Speed Regulator System parameters: 50nref  ; 22,2Tn  ; 08,2Kc  ; 416,0Ki  ; 0,1Rp  ; 44,0Bl  ; 344,1Tx  ; 2,0Ks  ; 02,2Td  ; 55,0Bt  ; 07,0Ts  ; 2,0Tv  . Voltage Regulator System parameters:                                                           y Q P T n U u i u u i i y L L em p F F q d q d                 réf réf réf U n P u

19. International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-8, August- 2016] Page | 49 1,0K1  ; 04,01  ; 39,5K2  ; 47,10K3  ; 486,1K5  ; 93,0f  ; 971,0K6  ; 01,06  ; 000971,0K8  ; 01,08  . Power System Load: 2m  ; 2n  ; 3Kpf  ; 2Kqf  .