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Indian Journal of Science and Technology, Vol 7(S6), 40–47, October 2014 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 Multi-Machine Small Signal Stability Analysis For Large Scale Power System C. Chennakesavan* and P. Nalandha Department of Electrical and Electronics Engineering, AMET University, Chennai; [email protected]; [email protected] Abstract The main objective of this paper is a fundamental study of the nature of inter-area oscillations in power systems. ­Small-signal stability refers to the ability of a power system to maintain synchronism under small disturbances Instability may arise in two forms increase of rotor angle due to lack of sufficient synchronization torque and rotor oscillations of increasing amplitude due to lack of sufficient damping torque. To study small signal stability analysis synchronous machine model, transmission line model and two area system model and the dynamic state matrix eigenvalues and eigen vectors are constructed and the small signal stability analysis done with the developed algorithm. The 11 bus systems are considered here for study of oscillations. The effects of the system structure, generator modelling, excitation type, and system loads are discussed in detail. In the study, only small signal stability analyses are used to determine the characteristics of the system. In the power system leads to the development of many oscillations at low frequency in the power system. This paper presents the effect of the load model it was easier to identify unstable modes of oscillation. Keywords: Power System Modeling, Eigen Value Analysis, Participation Factors, Small Signal Stability, TYPE-2B MODEL 1. Introduction With the increasing electric power demand, power systems can reach stressed conditions, resulting in undesirable voltage and frequency conditions1. The eigen value analysis was done for small signal stability of 11 bus 2 area system. The drawback of eigenvalue analysis is that the higher order terms neglected from the liniearised set of equation, Here particularly in a system is described by strongly non-linear equations. Modern electrical power systems have grown to a large complexity due to increasing interconnections, installation of large generating units and extra-high voltage tie-lines etc. The system response to such disturbances involves large excursions of generator rotor angles, power flows, bus voltages, and other system variables2. It is ­important that, while steady-state stability is a function only of ­operating conditions. *Author for correspondence Small signal stability is the ability of the system to maintain synchronism under small disturbances which occur continually on the system due to small variations in loads and generation or other small disturbances on the system4. A disturbance is considered to be small if the equations that represent the dynamic performance of the system can be linearized for the purpose of analysis function of both the operating conditions and the disturbance. In this paper a new method for calculation of the two area four generator system is investigated. Stability studies for power system planning, operation and control rely immensely on computer based power system simulation tools. Simulation tools use mathematical models that predict the dynamic performance of the system7. It is crucial that these power system models be modeled accurately to predict the actual performance of the ­system. C. Chennakesavan and P. Nalandha 2. System Description and Modelling are also connected to bus 7 and 9 as shown in the figure 1 below. The system has the fundamental frequency 60 Hz. The system comprises two similar areas connected by a weak tie3. Each area consists of two generators, each having a rating of 900 MVA and 20 kV. We will perform: A standard IEEE 11 bus system1 as shown in Figure 1. 2.1 Synchronous Machine Model The synchronous generators are modelled as classical machines with δ and ω as state variables. The generator is represented by the Norton equivalent for network solution. The admittance of the generator is included in the main diagonals of Y matrix. J dw = Tm - Te dt (1) dd = w - w0 dt (2) 2.2 Transmission Line Model (Short Line) A transmission line is modeled as a short line. It consists of a series impedance comprised of a resistance R and reactance X between the terminal nodes. 2.3 Load Model The loads are modeled as a constant admittance in ­transient stability studies. 2.4 Introduction of Two-Area System The machines are considered to be TYPE -2B Model. The system contains eleven buses and two areas, connected by a weak tie between bus 7 and 9. Totally two loads are applied to the system at bus 7 and 9. Two shunt capacitors ∗ Power-flow calculation ∗ Linear Analysis and Modal Analysis ∗ Time-domain simulation. The left half of the system is identified as area 1 and the right half is identified as area 2. The saturation of the synchronous machines are not identical. This chapter describes the methodology for incorporation of the model for small-signal analysis of single machine-infinite bus and multi-machine configurations. This is followed by relevant results obtained for these ­configurations. 3. Small Signal Stability Small Signal Stability is the ability of the power system to maintain Synchronism under small perturbations. Such perturbations occur continuously on the system because of small variations in load and generation where the system continuously adjusts itself to changing conditions. Restoring forces acting on the machine help them to maintain stable conditions. The system must be able to operate satisfactory under these conditions and successfully supply the maximum load. The “Small Signal” disturbances are considered suffiently small for linearization of system equations to be permissible for the purpose of analysis. The most common form of synchronism, monotonically, in the first few seconds following the fault due to lack of synchronizing torque and damping torque. The stability of the following types of oscillations is of concern. Local modes are associated with the oscillations of generating units at a particular station with respect to the rest of system. These oscillations are localized in a small part of the power system. Inter area modes are associated with the oscillations of many machines in one part of the system against machines in the other parts7. 3.1 State Equation Figure 1. IEEE two area 11 Bus System Vol 7 (S6) | October 2014 | www.indjst.org For a classical model of the synchronous machine the states are incremental changes in rotor speed and angle. Indian Journal of Science and Technology 41 Multi-Machine Small Signal Stability Analysis For Large Scale Power System In our analysis the super bar notation on the per unit quantities are dropped out. The state equations for the SMIB system shown in the Figure 2 are given by È D.w ˘ È - K D Í ˙ = Í 2H Í . ˙ Í ÍÎ Dd ˙˚ Î w 0 -K S ˘ È 1 ˘ È Dw ˘ Í ˙ 2H Í ˙ + 2H ˙ DTm ˙ Dd 0 ˚ Î ˚ ÍÎ 0 ˙˚ (3) This is the form of X = Ax + Bu. The elements of the state matrix A are seen to be dependent on the system parameters KD, H, XT and the initial operating condition represented by the values of E1 and δ0. Where, ∆ω is in per unit ∆δ is in radians The synchronizing torque coefficient is given by KS = E ¢ V• ( Xd¢ + Xtr + Xline ) cos d (4) 3.2 Multimachine Model The base system is symmetric; it consists of two identical areas connected through a relatively weak tie. Each area including two generating units with equal outputs. The full symmetry of the base system clarifies the effect that various factors have on the inter-area mode. Dynamic data for the generator and excitation systems used in the study are given in Appendix-A. In setting up the various power flows used in the studies, capacitors were added as necessary to ensure that the systems voltage profile was satisfactory. The electro-mechanical modes of oscillation are present in this system; two inter-area modes, one in each area, and one inter area low frequency mode, in which the generating units in one area oscillate against those in other area. Our experience with large inter connected ­systems confirms some of the results of our studies using the small system, and we are confident that the general conclusions drawn from our work, will apply to large systems. The ­following equations pertain to the fourmachine system. 3.3 State Space Model Our aim in this section is to derive the equations for multi machine system that represent the dynamics of the machine and the controller in the state space form. 42 Vol 7 (S6) | October 2014 | www.indjst.org The state variables for the machine are Dw Dd DEq′ DEd′ DEd′ DEd′′ . The right hand side of the differential equations for the machine and the ­controllers will contain algebraic variables such as current, voltage or power and incremental changes in them should be eliminated to obtain the equations in state variables canonical form. Once the equations in the state variables canonical form are obtained, we can apply eigen value technique to access stability. − K D ∆T 0 0 0 0 ∆T m e ∆.w 2H 0 ws 0 0 0 0 0 ∆w 1 . ∆EFD ∆d 0 1 − 1 0 0 0 ∆d . Td′0 Td′0 Td′0 ∆Eq′ 1 E ∆ ′ q 1 1 ∆ E = + (5) FD − 0 0 0 0 0 . ∆Ed′ Tq′0 ∆E ′ Tq′0 Tq′0 d E ∆ ′ 1 . 0 1 0 0 − 1 0 d ∆EFD ∆Eq′′ Td′0 Tq′0 ∆Ed′′ Td′0 . 1 ∆EFD ∆Ed′′ 0 1 0 0 0 − 1 Tq′0 Tq′0 Td′0 Assumptions: The machines are considered to be ­classical (no ­controllers) Damping ignored Loads are assumed as constant a­ dmittances Preparation: The initial conditions for δ, ω and voltages are obtained from load flow and the past history terms for δ and ω are obtained from the initial conditions. 4. Multi-Machine Small Signal Stability Algorithm for Type-2b Model 1. Taken IEEE 11 bus 2 area 4 machine system 2. Formulate the admittance matrix YV = I (6) 3. After that reduced the network equations -1 red YGG = YGG - YG , NG * YNG , NG * YNG , G (7) 4. Initialize the power, current, voltage in transient ­condition & sub-transient condition etc. Indian Journal of Science and Technology C. Chennakesavan and P. Nalandha 5. Find out the transformation matrix & inverse ­transformation matrix È e jq1 Í Í 0 T=Í Í 0 Í ÍÎ 0 0 e 0 jq 2 0 e 0 0 jq 3 0 È e - jq1 0 ˘ ˙ Í Í 0 0 ˙ ˙ inv T = Í Í 0 0 ˙ ˙ Í jq 4 e ˙˚ ÍÎ 0 e 0 0 - jq 2 0 0 e 0 - jq 3 0 0 ˘ ˙ 0 ˙ ˙ 0 ˙ ˙ - jq 4 e ˙˚ (8) 6. Then the reduced equations into individual machine rotor coordinates. M = inv(T) ∗ YGGred ∗ T (9) 12. Finally our state variables are ∆ω ∆δ ∆Eq′ ∆Ed′ ∆Ed′ ∆Ed′′ (21) 13. Find out the eigen values from above state variables 5. Eigen Analysis With the linearized power system model in matrix form shown in equation the system can now be analyzed through the use of eigenvalues and eigenvectors. To examine the free response of the system the inputs are put to zero. Δx = [A] x (22) I = I0 + DI (10) E ¢ = E0¢ + DE ¢ (11) where x is a state vector and A is the state matrix of size n x n; The state equation given by is further analyzed by taking the Laplace transform. The new equation derived in the s domain is given by M = M0 + DM (12) T = T0 + DT (13) 7. After applying the linearization of network equations in individual machine rotor coordinates. 8. Small change in current is DI = [M0 DE ¢ - j(DdM0 - M0 Dd)E 0¢ ] (14) 9. Linearization of differential equations: The mechanical side differential equations are given by dDw i DTmi - DTei - K D Dw i = 2H i dt (15) 10. Linearization of swing equation is dDd i = w s Dw i dt (16) 11. We illustrate how to include the damper windings in the formulation of state variables2. 1 [DEq¢ + (X d - X d¢ )DId - DEFD ] Tq¢0 (17) 1 [DEd¢ + (X q - X q¢ )DIq ] Tq¢0 (18) . 1 [DEq¢¢ + (X q¢ - X q¢¢)DId - DEq¢ ] Tq¢¢0 (19) . 1 [DEd¢¢ + (X d¢ - X d¢¢)DId - DEd¢ ] Td¢¢0 (20) . DE q¢ = . DE d¢ = - DE q¢¢ = - DE d¢¢ = - Vol 7 (S6) | October 2014 | www.indjst.org det (s[I] – [A]) = 0 (23) The values of s that satisfy (poles of system) are known as eigenvalues of the matrix A. The eigenvalues may be real or complex. For an n x n matrix they are n eigenvalues. If A is real eigenvalues occur in complex conjugates of the form [6] λ = σ ± jω. (24) 5.1 Eigenvalues and Stability The power system is stable if all of the eigenvalues are on the left-hand side of the imaginary axis of the complex plane; otherwise it is unstable. If any of the eigenvalues comes over into the right hand side of the imaginary axis, the corresponding modes are said to be unstable, and so is the system. The desired aim then is to have all eigenvalues in the left hand plane. This stability is confirmed by looking at the time dependent characteristic of the oscillatory modes corresponding to each eigenvalue λ given by e λ t. A real eigenvalue corresponds to a non oscillatory mode. A negative real eigenvalue indicates a mode that decays in time (the larger the magnitude of the eigenvalue the quicker the decay)9. A positive real eigen value indicates a mode that grows with time and is system will experience aperiodic instability. A conjugate pair complex eigenvalues indicate oscillatory modes of response. λ = σ ± jω (25) Indian Journal of Science and Technology 43 Multi-Machine Small Signal Stability Analysis For Large Scale Power System 1. If a conjugate of pair of complex eigenvalues has negative real parts σ this corresponds to an oscillatory mode that decays with time and the system is said to be globally stable. 2. If a pair has positive real parts, the corresponding oscillatory mode grows exponentially with time and eventually dominates the system behavior. Such a system is said to be unstable. 3. If any one of the eigenvalues has a real part, the system will have an undamped oscillatory response. Eigenvalues associated with an unstable or poorly damped oscillatory mode are called dominant modes since their contribution dominates the time response of the system. The real component of an eigen value gives the damping, and the imaginary part gives the frequency of oscillation. Frequency of oscillation (Hz): f = w 2p (26) The damping ratio: z= -s s 2 +w2 (27) Eigen analysis also includes the computation of eigenvectors. For any eigen value λi, the column vector φi that satisfies is called the right eigenvector for λi [A]φi = λ φi i = 1, 2, … n Vol 7 (S6) | October 2014 | www.indjst.org ÈY i1 ˘ ÍY ˙ Í i2 ˙ Yi = Í ˙ Í ˙ Í ˙ ÍY in ˙ Î ˚ (29) The left eigenvectors, together with the initial conditions of the system state vector x, determine the magnitudes of the modes. 5.3 Mode Shape The right eigenvector gives the mode shape. The mode shape is the relative activity of the state variables when a particular mode is excited. Thus the degrees of activity of the Kth state variable xk in the Ith mode is given by the element φki if the right eigenvector φi. The magnitude of the elements if φi gives the extent of the activities of the n state variables in the Ith mode, and the angles of the elements give phase displacement of the state variables with regard to the mode. Thus we can use the mode shape to analyze the magnitude and phase displacement of the speed and rotor angle state variables in an oscillatory mode. The participation factor P is useful in identifying those states which have the most influence on any mode. The participation factor is non-dimensional. (28) The right eigenvector show the distribution of the modes of response (eigenvalues) through the power system state variables. Correspondingly, there is a row vector ψi that satisfies (24) and is called the left eigenvector of [A] associated with the eigen value λi. 44 ψi [A] =λ ψi 6. Participation Factors 5.2 Eigen Vectors Èj1i ˘ Íj ˙ Í 2i ˙ ji = Í ˙ Í ˙ Í ˙ ÍÎj 2i ˙˚ È P1i ˘ È j1iY i1 ˘ Í P ˙ Íj Y ˙ 2i 2i i 2 ˙ Pi = Í ˙ Í Í ˙Í ˙ ˙ Í ˙Í ÎPni ˚ ÎjniY in ˚ (30) The element pki = ψki φik is called a participation f­ actor. It is a measure of the relative participation of the kth state variable in the ith mode, and vice versa5. In effect participation factors are useful in identifying those state variables which have the most influence on any mode. The higher the value of participation factor of a state for a corresponding mode, the more active that state is in that mode when compared with the other states. Thus the values of participation factors can reveal which generators are involved in a particular mode. It can Indian Journal of Science and Technology C. Chennakesavan and P. Nalandha reveal which machine or machines could go out of step for any known mode or modes that might cause problem in the power system in the advent of load variation10. The participation factors are used to identify the areas in the power system where any mode or (oscillation) has most of its effect. 6.1 Eigen Value Sensitivity to Load Changes The sensitivity of an eigen value λi to an element αkj of the state matrix is equal to the product of the left eigenvector element ψki and right eigenvector element φji. ∂li = Y i k ji j ∂a k j (31) In this analysis the load power at a bus in the power system was varied and the system state matrix computed. The elements αkjC of the state matrix that showed the greatest changes due to the load change were identified. Thus the sensitivity of the eigenvalues to these elements of the state matrix A sys can be computed using equation. In computing the eigen value sensitivity the elements that show a large variation with a change in load give an indication of the sensitivity of the eigen value to load changes. Eigen-analysis of the system can then be performed by obtaining the eigenvalues of the state matrix. The eigenvalues are then check for stability. 6.3 Load Model Procedure 1. The type of load at various buses was selected i.e. changing parameters np and nq 2. The system matrix A sys was computed 3. The eigenvalues are computed and check for stability analysis. 7. Simulation Results A standard 11-bus 4 machine 2 area as shown in system. The initial relative rotor angle obtained from the load flow solution or initial condition. Table 1. EIGEN VALUES NO Real Imaginary 1 −0.76e – 3 0.22e – 2 2 – −0.22e – 2 Frequency (HZ) Damping Ratio 0.0003 0.331 – – 0.545 0.032 3 −0.096 – 4 −0.111 3.43 5 −0.111 −3.43 6 −0.117 – – – 7 −0.265 – – – 8 −0.276 – – – 9 −0.492 6.82 10 −0.492 −6.82 11 −0.506 7.02 12 −0.506 −7.02 ∗ Loads system data ∗ Performs load flow of system, calculates initial ­condition, linearize system equation and construct the matrices and finally the system state matrix A sys. ∗ Calculate system’s eigenvalues a check for unstable mode and gets the participation and mode shape for critical Eigenvalues. 13 −3.428 – – – 14 −4.139 – – – 15 −5.287 – – – 16 −5.303 – – – 17 −31.03 – – – 18 −32.45 – – – 19 −34.07 – – – The effect of loading was analyzed by increasing the load at a particular bus or buses. At each change in loading the initial conditions of the state variables were calculated after running a load flow [9]. Next, linearization of the system equation was done and the state matrix A sys is formed. 20 −35.53 – – – 21 −37.89 0.142 22 −37.89 −0.142 23 −38.01 0.038 24 −38.01 −0.038 6.2 Simulation The test system used in this investigation is the IEEE – 11 bus test system. Matlab 7.5 was used to analyze this system. The system is a consists of 4 generators. Software was written to perform the small signal analysis on the system. This program follows three main algorithms: Vol 7 (S6) | October 2014 | www.indjst.org 1.087 0.072 1.117 0.072 0.023 1.0 0.006 1.0 Indian Journal of Science and Technology 45 Multi-Machine Small Signal Stability Analysis For Large Scale Power System Figure 2. (a,b,c) Mode shapes of rotor angle From this table 1 all eigen values are coming negative. So we see that system is Stable. There are three rotor angle modes of oscillation. Their mode shapes are shown in Figure 2(a, b, c) 8. Conclusion In this investigation a dynamic model for the power system was developed to investigate the development of low frequency oscillations in a power system. The test system was used the IEEE 11 bus test system. A power system model was enveloped in Matlab and testing was done to determine the power system stability. Software developed performed the following function: load flow of power system, calculation of the initial values of the power system model and the construction of the system state matrix. The software then applied the linear analysis tools to find Eigenvalues of A sys, Participation factors, and mode shape. The Eigenvalues of A sys, Participation factors, and Mode shape can be used to determine if system could develop oscillations and where they occur in the power system if excited. The purpose of this research was to investigate the Contribution of load to low frequency oscillations in Power System8. As a result the system was analysis by first examining the effect of loading on the development of system oscillations. Secondly the effect of load model on low frequency oscillation was determined. In the power system leads to the development of many oscillations at low frequency in the power system. Finally the effect of the load model was that it was easier to ­identify unstable modes of oscillation. 46 Vol 7 (S6) | October 2014 | www.indjst.org Bus 7 PL = 967MW QL = 100 MWAR QC = 200 MWAR Bus 9 PL = 1,767MW QL = 100 MWAR QC = 350 MWAR 9. Appendix The system consists of two similar areas connected by a weak tie. Each area consists of two coupled units, each having a 900MVA & 20 KV. The generator parameters in per unit on the rated MVA and KV base are as follows. Xd = 1.8 Xq = 1.7 Xl = 0.2 Xd¢ = 0.25 Xq¢ = 0.25 Td0 ¢ = 8.0 S Tq0 ¢ = 0.4 SS Td0 ¢ = 0.03 S Tq0 ¢ = 0.05 S Asat = 0.015 Bsat = 9.6 jT1 = 0.9 kD = 0 Xq¢ = 0.55 Ra = 0.0025 H = 6.5 (for G! & G2) Xd¢ = 0.3 H = 6.175 (for G3 & G4) Each Step-up transformer has an impedance of 0 + j0.15 per unit on 900 MVA and 20/230 KV base, and has an off-nominal ratio of 1.0 The parameters lines in per unit on 900 MVA & 230KV R = 0.0001 pu/km XL = 0.001 pu/km Bc = 0.00175 pu/km Then the generating units loaded as given below. G1 G2 G3 G4 P = 700 MW P = 700 MW P = 719 MW P = 700 MW Q=185MW Q = 235MW Et = 1.03 – 20.2 G1 G2 Et = 1.01 – 10.5 Q = 176MW Et = 1.03 – - 6.8 Q = 202MW Et = 1.01 – - 17.0 G3 G4 The load and reactive power (Qc) by the shunt ­capacitors at b uses 7 and 9 as follows Indian Journal of Science and Technology C. Chennakesavan and P. Nalandha 10. References 1.Kundur P. Klein M. “Analytical Investigation of Factors Influencing power System Stabilizers Performance”, IEEE Transaction on Energy, 1992. 2.Ramanujam R. “Power System Dynamics”, PHI Learning Private Limited Conversion, 2009; 7(3). 3.Padiyar KR. “Analysis of small signal voltage stability in multimachine Systems using Detailed and Reduced Formulations”, IEEE Transaction on Electrical Power and Energy Systems, 2003; 11(4). 4.Anderson PM. Fouad A. “Power System Control and Stability”, The Iowa State University Press, Ames, Iowa. 1977. 5.Franken B and Anderson. “Analysis of Converters Connected to weak AC Systems”, IEEE Transaction on Power systems, 1990; 5(1). Vol 7 (S6) | October 2014 | www.indjst.org 6.Kundur P. “Power System Stability And Control”, McGrawHill, Inc. 1993. 7.M Ghandhari, G Andersson. “Damping of inter-area and local modes by the use of controllable components”, IEEE Transaction on Power Delivery, 2007; 10(4). 8.MA Pai. Energy, “Function Analysis for Power System Stability”, Kluwer Academic Publishers, 1989. 9.Maurice H. Kent, Waynerfrancis a. “Dynamic Modeling of Loads in Stability Studies” ieee Transactions on Power Apparatus and Systems, 1996; (5) p. 88. 10.Elices A, L Rouco, H Bourles. Physical interpretation of state feedback controllers to damp power system oscillations, IEEE Transactions on Power System, 2004;19(1). Indian Journal of Science and Technology 47