Aldi Mucka Tonin Dodani Marjela Qemali Rajmonda Bualoti Polytechnic University of Tirana, Faculty of Electric Engineering, Republic of Albania

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1 8. СОВЕТУВАЊЕ Охрид, септември Aldi Mucka Tonin Dodani Marjela Qemali Rajmonda Bualoti Polytechnic University of Tirana, Faculty of Electric Engineering, Republic of Albania REHABILITATIONS OF EXCITATION SYSTEM ON SMALL-SIGNAL STABILITY OF POWER SYSTEM ABSTRACT In recent years in Albanian Power System has been made many investments with purpose of his rehabilitation and development. An important one is the construction of interconnection line Tirana Podgorica and the construction of many small witches made the system more powerful. A very important part of investments is the rehabilitation of Fierza HPP and Vau Dejes HPP (main power generation). The rehabilitation consists in replacing the rotation excitation system with static systems and hydro mechanical systems. This paper will analyze the impact of excitation system of these power plants in amelioration of small-signal stability of power system. The study of small-signal stability had been made calculating eigenvalues of a State matrix and participation factor. Therefore through simulations are determined the optimal proportional coefficients of the excitation system so that increase the security factor. Simulations and calculation are performed by software Neplan. Keywords: Power system, Small-signal stability, Eigenvalues. 1 INTRODUCTION Small-signal stability is the ability of the power system to maintain synchronism when subjected to small disturbances. In this context, a disturbance is considered to be small if the equations that describe the resulting response of the system may be linearized for the purpose of analysis. Instability that may result can be of two forms: (i) steady increase in generator rotor angle due to lack of synchronizing torque, or (ii) rotor oscillations of increasing amplitude due to lack of sufficient problem is usually one of insufficient damping of system oscillations. Small-signal analysis using linear techniques provides valuable information about the inherent dynamic characteristics of the power system and assists in its design. Many investments are made in recent years in the Albanian power systems to its rehabilitation and also to modernize its equipment and modern technologies. In these investments we can mention the build of the 400kV line Podgorica-Tirana, the construction of many small hydropowers which all of them inject in the system a total power of 150MW. But one of the most important investments and rehabilitation is the excitation systems, the two main hydropower plant of Albanian power system respectively HPP Fierza and Vau Dejës. Their rehabilitation consists in the replacement of DC excitation systems with static excitation system of model EXPIC1 Siemens, also the installation of AVR and PSS system. This paper with analyze the effect of the new excitation systems on the performance of the small-signal stability of the system. For this case are made a lot of simulations for several different occasions of the system regime. Small-signal stability study was made with the help of software Neplan. The tests of small-signal stability consist in the calculation of the linearized system eigenvalues, whose position on the imaginary axis, gives us the opportunity to consider the question of stability. C2-116R 1/11

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4 MAKO CIGRE 2013 C2-116R 4/11 Figure 4 Reference frame transformation The formulation of the state equations for small-signal analysis involves the development of linearized equations about an operating point and elimination of all variables other than the state variables. The general procedure is similar to that used for a single-machine infinite bus system in the previous sections. However, the need to allow for the representation of extensive transmission networks, loads, a variety of excitation systems and prime mover models, HVDC links, and static var compensators makes the process very complex. Therefore, the formulation of the state equations requires a systematic procedure for treating the wide range device. The following is a description of one such procedure. 2.2 Formulation of the state equations The linearized model of each dynamic device is expressed in the following form: x = A x + BΔv (2) i i i i Δii = Cixi YiΔv (3) Where: x i are the perturbed values of the individual device state variables i i is the current injection into the network from the device v is the vector of the network bus voltages In Equations (2) and (3) B i and Y i have non-zero elements corresponding only to the terminal voltage of the device and any remote bus voltages used to control the device. The current vector i i has two elements corresponding to the real and imaginary components. Similarly, the voltage v has two elements per bus associated with the device. Such state equations for all the dynamic devices in the system may be combined into the form x = A x+ B Δv D D D Δi= C x Y Δv D Where: x is the state vector of the complete system, and A D and C D are block diagonal matrices composed of A i and C i associated with the individual devices. Δi= Y Δv N The interconnecting transmission network is represented by the node equation: Equating Equation (5) associated with the device and Equation (6) associated with the network, we obtain (5) (6) (7)

5 MAKO CIGRE 2013 C2-116R 5/11 C x Y Δv= Y Δv D D N Hence, Δ v= ( Y + Y ) C x (9) 1 N D D Substituting the above expression for Δv in Equation (6) yields the overall system state equation: x = A x+ B ( Y + Y ) C x = Ax 1 D D N D D Where the state matrix A of the complete system is give by A = A x+ B ( Y + Y ) C (11) 1 D D N D D 2.3 Linearization The behaviour of a dynamic system, such as a power, may be described by a set of n first order nonlinear ordinary differential equations of the following form: i = i 1 2 n 1 2 r x f ( x, x,..., x ; u, u,..., u ; t ) i = 1,2..., n (12) Where n is the order of the system and r is the number of inputs. This can be written in the following form by using vector-matrix notation: x = f(, x u,) t Where: x x1 x xn 2 = u u1 u un 2 = f f1 f fn 2 = The column vector x is referred to as the state vector, and its entries x i as state variables. The column vector u is the vector of inputs to the system. These are the external signals that influence the performance of the system. Time is denoted by t, and the derivative of a state variable x with respect to time is denoted by x. If the derivatives of the state variables are no explicit functions of time, the system is said to be autonomous. In this case, Equation 13 simplifies to x = f(, x u) (14) We are often interested in output variables which can be observed on the system. These may be expressed in terms of the state variables and the input variables in the following form: y= g(, xu) (15) Where: y y1 y yn 2 = g g1 g gn 2 = The column vector y is the vector of outputs, and g is a vector of nonlinear functions relating state and input variables to output variables. We now describe the procedure for linearizing Equation 12. Let x 0 be the initial state vector and u 0 the input vector corresponding to the equilibrium point about which the small-signal performance is to be investigated. Since x 0 and u 0 satisfy Equation 12, we have x = f( x, u ) = 0 (16) (8) (10) (13)

6 MAKO CIGRE 2013 C2-116R 6/11 Let us perturb the system from the above state, by letting x = x + Δx u = u + Δ u (17) 0 0 where the prefix Δ denotes a small deviation. The new state must satisfy Equation 12. Hence, x = x + Δx = f[( x + Δx),( u + Δ u)] (18) As the perturbations are assumed to be small, the nonlinear functions f(x, u) can be expressed in terms of Taylor s series expansion. With terms involving second and higher order powers of Δx and Δu neglected, we may write x = x + Δx = f [( x + Δx),( u + Δu)] i i0 i i 0 0 f f f f = f ( x, u ) + Δx x + Δu Δu i i i i i n 1 r x1 xn u1 ur i Since x 0= f i ( u0, u 0), we obtain f f f f Δ Δ Δ Δ (20) i i i i x i = x xn + u ur x1 xn u1 ur With i = 1, 2,..., n. In a like manner, from Equation 15 we have g g g g Δy Δx x Δu Δ u (21) j j j j j = n r x1 xn u1 ur with j=1,2,...,m. Therefore, the linearized forms of Equations 14 and 25 are Δx = AΔx+ BΔ u (22) Δy = CΔx+ DΔ u (23) (19) Where f1 f1 x1 x n A = fn fn x 1 x n, f1 f1 u1 u n B = fn fn u1 u n, g1 g1 x1 x n C = gn gn x1 x n, g1 g1 u1 u n D = gn gn u1 u n (24) The above partial derivates are evaluated at the equilibrium point about which the small perturbation is being analyzed. By taking the Laplace transform of the above equations, we obtain the state equations in the frequency domain: sδx() s Δx(0) = AΔx() s + BΔ u() s (25) Δy() s = CΔx() s + DΔ u() s (26) Figure 6 shows the block diagram of the state-space representation. Since we are representing the transfer function of the system, the initial condition Δx(0) are assumed to be zero. A formal solution of the state equations can be obtained by solving for Δx(s) and evaluating Δy(s), as follows:

7 MAKO CIGRE 2013 C2-116R 7/11 D Δu B + Δ x + 1 I s Δx C + + Δy A Rearranging Equation 25, we have ( si A) Δx( s) = Δx(0) + BΔu( s) Hence, Δxs si A Δx BΔus 1 () = ( ) [ (0) + ()] Figure 6 Block diagram of the state-space representation adj( si A) = [ x (0) + B u ( s )] det( si A) Δ Δ (27) and correspondingly, Δ adj( si A) ys () = C [ x(0) + B us ()] + D us () det( si A) Δ Δ Δ (28) The Laplace transforms of Δx and Δy are seen to have two components, one dependent on the initial conditions and the other on the inputs. These are the Laplace transforms of the free and zero-state components of the state and output vectors. The poles of Δx(s) and Δy(s) are the roots of equation of the equation det( si A ) = 0 (29) The values of s which satisfy the above are known as eigenvalues of matrix A, and Equation 29 is referred to as the characteristic equation of matrix A. 2.4 Eigenvalues and Eigenvector The Small Signal Stability analysis of an electrical power system is examined by the eigenvalues of the state matrix A. These eigenvalues may be either real or complex values. A real eigenvalue will represent a non-oscillatory mode, while a complex pair of eigenvalues corresponds to an oscillatory mode. The real part of complex eigenvalues provides the damping coefficient, while the imaginary part gives the oscillation frequency. The eigenvalues, λ of the state matrix are given by the non-trivial solutions of the equation (30). det( si A ) = 0 (30) The i th right eigenvector satisfies Au = λ u (31) i i i The i th left eigenvector satisfies va= λ v (32) i i i The most general representation of state-space for linear systems are given in equations 33 and 34

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10 MAKO CIGRE 2013 C2-116R 10/11 Again in this test we see that the system is stable but in this case we must pay attention that reserve of small-signal stability is decreased. Figure 7 The Electric Power System of Albania

11 MAKO CIGRE 2013 C2-116R 11/11 4 CONCLUSION From the tests that had been made, despite of variations in the analysis, it appears that the establishment of stabilizing signals and rehabilitation of the hydro system excitation Fierëz becomes a necessity. Regarding the specification of the values of factor k ω, k c, the real test will be required on the system. As orientation value can be obtained from the evidence that resulted: k ω = 10 and k c = 0.5 While the K A amplification coefficient as we can say that from the simulation, the optimal value result as follows: HHP Koman: K A = 60 HHP Fierza: K A = 50 HHP Vau Dejes: K A = 50 It is important to mention that the rehabilitation of the excitation systems of three main HPP of our system by replacing them with excitation static systems has increased performance of small-signal stability as well as dynamic stability. Also installation of AVR system bringing a fast response of system when occur short circuit near generating sources. In conclusion it should be pointed that the evidence for the isolated system for all variants examined showed good consistency stand state point. 5 LITERATURE [1]. P. Kondur, Power System Stability and Control, Electric Power Research Institute 3412 Hilliview Avenue Palo Alto, California, pp , [2]. D.C Lee and P. Kundur, Advanced Excitation Control for Power System Stability Enhancement, CIGRE 38-01, [3]. Kundur, P. "Power System Stability and Control" Electric Power Research Institute, McGraw-Hill, Inc [4]. W.E. Arnoldi, The Priciple of Minimized Iterations in the Solution of the Matrix Eigenvalue Problem, Quart. Appl. Math., Vol. 9, pp , 1951 [5]. Y. Saad, Variations on Arnoldi s Method for Computing Eigenelements of Large Unsymmetrical Matrices, Linear Algebra and Its Applications, Vol. 34, pp , June [6]. M. Klein, G.J. Rogers, and P. Cundur, A Fundamental Study of Inter-Area Oscillations, IEEE Trans., Vol. PWRS-6, No. 3, pp , August 1991 [7]. M. Klein, G.J. Rogers, S. Moorty, and P. Cundur, Analytical Investigation of Factors Influencing Power System Stabilizers Performance, Paper 92WM016-6EC,Presented at the IEEE PES Winter Meeting, New York, January 1992