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Liapunov'un ikinci yöntemiyle güç sistemlerinin geçiçi hal kararlılık analizi

Liapunov'un ikinci yöntemiyle güç sistemlerinin geçiçi hal kararlılık analizi

##### Dosyalar

##### Tarih

1990

##### Yazarlar

Bektaş, Cengiz

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu tezde, güç sistemlerinde generatörlerdeki akı değişimlerini de içeren bir matematiksel model kullanıla rak Liapunov'un ikinci yöntemi ile geçici hal kararlılık analizi yapılmıştır. Önce, söz konusu model için Popov ka rarlılık kriterinin genelleştirilmesi, Liapunov fonksiyo nunun kurulması ve sonra da geçici hal kararlılık bölgesi bazı literatüre paralel olarak incelenmiş, ayrıca, bu yöntemle kritik arıza temizleme süresini belirleyen bir bilgisayar programı geliştirilmiştir.

The intuitive idea of stability of a physical system is as follows: Let the system be in some equilibrium state. If on the occurrence of a disturbance, the system eventually returns to the equilibrium position, we say the system is stable. The system is also termed stable if it converges to another equilibrium position generally in the proximity of the initial equilibrium point. This intuitive idea is also applicable to a power system. The stability of an interconnected power system is its ability to return to normal or stable operation after having been subjected to some form of disturbance. Liapunov's second method was introduced to determine whether an equilibrium state of a system is stable or asymptotically stable by Liapunov who was a Russian mathematician. An important advantage of the method is that stability of systems can be determined without having to explicitly study the system solutions. The principle idea of the method is contained in the following reasoning: If the rate of change dV/dt of the energy V(x) of an isolated physical system is negative for every possible state x except for a single equilibrium state x, then the energy will continually decrease until it finally assumes its minumum vaule V(x ). When the description of the system is given in a mathematical form, there is no way of defining energy. Hence, V(x) is a scalar function proportional to the energy. If for a given system, one is able to find a function V(x) such that it is always positive except at x=0 (assuming x =0) where it is zero and its time derivative V(x) is negative except at x=0 where it is zero, then we say the system returns to the origin if it is disturbed, i.e., the system is stable. The function V(x) is called the Liapunov function. Stability considerations have been recognized as an essential part of power systems planning for a long time. With interconnected systems continually growing in size and extending over vast georgaphical regions, it is becoming increasingly more difficult to maintain synchronism between various parts of a power system. The classical stability analysing methods may suitable for off-line studies such as simulation may not be suitable for on-line application since a large number of contingencies have to be simulated in a short time. A technique which offers promise for this purpose is Liapunov's second method. The appeal of this method lies in its ability to compute directly the critical clearing time of circuit breakers for various faults and thus directly assess the degree of stability for a given configuration and operating state. The critical clearing time can also be translated in terms of additional power disturbances that the system can withstand, thus offering a tool for dynamic security assessment. The study of stability in the presence of small disturbances constitutes what is known as "dynamic stability analysis" in the literature. The mathematical model for such a study is a set of linear time-invariant differantial equations. When the disturbances are large, the nonlinearities inherent in the power system can no longer be ignored and the study of stability under such circumstances constitutes what is known as "transient stability analysis". The mathemathical model for such a study is a set of nonlinear differential equations coupled with a set of nonlinear algebraic equations. The mathemathical model of a power system including flux decay effects is as follows: mi d26 Î -T + di dt d6\ x dt = P mi - P ex for i=l,2,..., n (1) !, dE. T, ^- r^ = E-,. - E. - (x,. - x'.) i doi dt fdi qi di dx di where, for generator i, mx ei m. x v^i E fdi : mechanical power input : electric power output : angular momentum constant : damping power coefficient : voltage related with the internal voltage as in Fig.l, and 6! indicates a rotor angle relative to a reference frame rotating at synchronous speed. : excitation voltage vx xdi d-axis armature current x,.,Xj.: d-axis synchronous, transient reactances dx di Tl. : d-axis transient open-circuit time constant dox Fig.l. Relations between generator variables In the Fig.l, where E. 76. : internal voltage V\. : terminal voltage ti s I. : armature current x x. : q-axis synchronous reactance qx H J This system can be represented by the Fig. 2, where the matrix W(s) is the linear part and the matrix F(0) is the nonlinear part of the system. For this system, stability criterion (generalized Popov criterian) is as follows: vxx Fig. 2. Nonlinear system model that Theorem: If there exist real matrices N and Q such Z(s) = (N+Qs)W(s) (2) is positive real, then the system shown in Fig 2 is stable, where (N+Qs) does not cause pole-zero cancella tions wxth W(s). as Liapunov function for this system can be chosen V(x) = xTPx + 2V1(o) (3) wherex is the state vector of the system. The positive defınıteness of P and the positive semi-def initeness of V^CJ) ensure that V(x) is positive for every nonzert.o x. mus To construct a Liapunov function, three assumptions t be made: (i) Excitation voltage is constant. (ii) Each internal voltage E lags behind the q-axis by a constant angle 0± all the time. (iii) Transfer conductances G.. are all negligible. Vlll Under these assumptions, (1) reduces to d%. m W T + d. dö. i dt I B..(E?E° sin 5°.-E,E,sin5..) j=l x3 i J 13 x J 13 i= 1,2,..., n (4), dE. t,. -rj- = -(e,-e: doi dt ¦?) - (xh-Xj.) I B.,(E,cos5?.-E.cos6..) x' di dx ^ ijv J xj j ijy and if damping torques are uniform or zero the Liapunov function is obtained as n n V(x) = (1/2 I m,) I I m.m, (w,-w.)' i=l i=l j=l x J x J n n + î I B., [e,E.(cos6°. - cosö.. ) 13 -(6..-S?.)E?E°sin6°.] 13 13 1 3 13 n + I (a./e-XE.-Ep' i = l = Vk(w) + Vp(S,E) + Vf(E) (5) where w. B. 1 : angular velocity, dS./dt : suceptance component of transfer admittance Y.. between the i. and j. generators, Y.. = G.. + jB.. ij 13 13 13 '«1- sj <*! » 8^ : some parameters The superscript "o" denotes the stable equilibrium state of the post-fault system. V, and V are kinetic and potential energies, respectively. xx To estimating critical fault clearing time, it is necessary that the critical value of V for a given disturbance is defined by the value of V at the instant when the system crosses a boundary of the transient stability region. This instant can be detected by checking the sign of dV,/dt which changes from negative to positive at the instant. Lastly, the computer program computing critical fault clearing time is developed and an example is given.

The intuitive idea of stability of a physical system is as follows: Let the system be in some equilibrium state. If on the occurrence of a disturbance, the system eventually returns to the equilibrium position, we say the system is stable. The system is also termed stable if it converges to another equilibrium position generally in the proximity of the initial equilibrium point. This intuitive idea is also applicable to a power system. The stability of an interconnected power system is its ability to return to normal or stable operation after having been subjected to some form of disturbance. Liapunov's second method was introduced to determine whether an equilibrium state of a system is stable or asymptotically stable by Liapunov who was a Russian mathematician. An important advantage of the method is that stability of systems can be determined without having to explicitly study the system solutions. The principle idea of the method is contained in the following reasoning: If the rate of change dV/dt of the energy V(x) of an isolated physical system is negative for every possible state x except for a single equilibrium state x, then the energy will continually decrease until it finally assumes its minumum vaule V(x ). When the description of the system is given in a mathematical form, there is no way of defining energy. Hence, V(x) is a scalar function proportional to the energy. If for a given system, one is able to find a function V(x) such that it is always positive except at x=0 (assuming x =0) where it is zero and its time derivative V(x) is negative except at x=0 where it is zero, then we say the system returns to the origin if it is disturbed, i.e., the system is stable. The function V(x) is called the Liapunov function. Stability considerations have been recognized as an essential part of power systems planning for a long time. With interconnected systems continually growing in size and extending over vast georgaphical regions, it is becoming increasingly more difficult to maintain synchronism between various parts of a power system. The classical stability analysing methods may suitable for off-line studies such as simulation may not be suitable for on-line application since a large number of contingencies have to be simulated in a short time. A technique which offers promise for this purpose is Liapunov's second method. The appeal of this method lies in its ability to compute directly the critical clearing time of circuit breakers for various faults and thus directly assess the degree of stability for a given configuration and operating state. The critical clearing time can also be translated in terms of additional power disturbances that the system can withstand, thus offering a tool for dynamic security assessment. The study of stability in the presence of small disturbances constitutes what is known as "dynamic stability analysis" in the literature. The mathematical model for such a study is a set of linear time-invariant differantial equations. When the disturbances are large, the nonlinearities inherent in the power system can no longer be ignored and the study of stability under such circumstances constitutes what is known as "transient stability analysis". The mathemathical model for such a study is a set of nonlinear differential equations coupled with a set of nonlinear algebraic equations. The mathemathical model of a power system including flux decay effects is as follows: mi d26 Î -T + di dt d6\ x dt = P mi - P ex for i=l,2,..., n (1) !, dE. T, ^- r^ = E-,. - E. - (x,. - x'.) i doi dt fdi qi di dx di where, for generator i, mx ei m. x v^i E fdi : mechanical power input : electric power output : angular momentum constant : damping power coefficient : voltage related with the internal voltage as in Fig.l, and 6! indicates a rotor angle relative to a reference frame rotating at synchronous speed. : excitation voltage vx xdi d-axis armature current x,.,Xj.: d-axis synchronous, transient reactances dx di Tl. : d-axis transient open-circuit time constant dox Fig.l. Relations between generator variables In the Fig.l, where E. 76. : internal voltage V\. : terminal voltage ti s I. : armature current x x. : q-axis synchronous reactance qx H J This system can be represented by the Fig. 2, where the matrix W(s) is the linear part and the matrix F(0) is the nonlinear part of the system. For this system, stability criterion (generalized Popov criterian) is as follows: vxx Fig. 2. Nonlinear system model that Theorem: If there exist real matrices N and Q such Z(s) = (N+Qs)W(s) (2) is positive real, then the system shown in Fig 2 is stable, where (N+Qs) does not cause pole-zero cancella tions wxth W(s). as Liapunov function for this system can be chosen V(x) = xTPx + 2V1(o) (3) wherex is the state vector of the system. The positive defınıteness of P and the positive semi-def initeness of V^CJ) ensure that V(x) is positive for every nonzert.o x. mus To construct a Liapunov function, three assumptions t be made: (i) Excitation voltage is constant. (ii) Each internal voltage E lags behind the q-axis by a constant angle 0± all the time. (iii) Transfer conductances G.. are all negligible. Vlll Under these assumptions, (1) reduces to d%. m W T + d. dö. i dt I B..(E?E° sin 5°.-E,E,sin5..) j=l x3 i J 13 x J 13 i= 1,2,..., n (4), dE. t,. -rj- = -(e,-e: doi dt ¦?) - (xh-Xj.) I B.,(E,cos5?.-E.cos6..) x' di dx ^ ijv J xj j ijy and if damping torques are uniform or zero the Liapunov function is obtained as n n V(x) = (1/2 I m,) I I m.m, (w,-w.)' i=l i=l j=l x J x J n n + î I B., [e,E.(cos6°. - cosö.. ) 13 -(6..-S?.)E?E°sin6°.] 13 13 1 3 13 n + I (a./e-XE.-Ep' i = l = Vk(w) + Vp(S,E) + Vf(E) (5) where w. B. 1 : angular velocity, dS./dt : suceptance component of transfer admittance Y.. between the i. and j. generators, Y.. = G.. + jB.. ij 13 13 13 '«1- sj <*! » 8^ : some parameters The superscript "o" denotes the stable equilibrium state of the post-fault system. V, and V are kinetic and potential energies, respectively. xx To estimating critical fault clearing time, it is necessary that the critical value of V for a given disturbance is defined by the value of V at the instant when the system crosses a boundary of the transient stability region. This instant can be detected by checking the sign of dV,/dt which changes from negative to positive at the instant. Lastly, the computer program computing critical fault clearing time is developed and an example is given.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1990

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1990

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1990

##### Anahtar kelimeler

Geçici hal kararlılık analizi,
Güç sistemleri,
Liapunov teorisi,
Transient stability analysis,
Power systems,
Liapunov theory