Determine Power Transfer Limits of An SMIB System through Linear System Analysis with Nonlinear Simulation Validation

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1 Determine Power Transfer Limits of An SMIB System through Linear System Analysis with Nonlinear Simulation Valiation Yin Li, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper extens the materials presente in Chapter 7 of a classic textbook Power System Analysis [] use in EEL 6936 Power Systems II at University of South Floria. This research examines the effect of automatic voltage regulator (AVR) an the power system stabilizer (PSS) on the power transfer level. The system analyze in this paper inclues a synchronous generator connecte to an infinite bus. Their effect on stability is investigate through plotting eigenvalues of closeloop transfer functions along with RouthHurwitz stability criterion examination. Three ifferent moels are consiere. The basic moel inclues a swing equation with one orer rotor flux ynamics. This moel is classifie as an electromagnetic moel (EMT). Then, the voltage control part is ae (EMTAVR). Finally, the power system stabilizer is ae to the system (EMTAVRPSS). These moels are linearize for analysis. Nonlinear simulation results base on the nonlinear moels built in Matlab/Simulink valiate the analysis results. Inex Terms Infinite bus, linear analysis, nonlinear simulation, power level, AVR, PSS. I. INTRODUCTION This paper extens the materials presente in Chapter 7 of a classic textbook Power System Analysis [] use in EEL 6936 Power Systems II at University of South Floria. This research examines the effect of automatic voltage regulator (AVR) an the power system stabilizer (PSS) on the power transfer level. The system analyze in this paper inclues a synchronous generator connecte to an infinite bus. Their effect on stability is investigate through plotting eigenvalues of closeloop transfer functions along with RouthHurwitz stability criterion examination. Three ifferent moels are consiere. The basic moel inclues a swing equation with one orer rotor flux ynamics. This moel is classifie as an electromagnetic moel (EMT). Then, the voltage control part is ae (EMTAVR). Finally, the power system stabilizer is ae to the system (EMTAVRPSS). These moels are linearize for analysis. Nonlinear simulation results base on the nonlinear moels built in Matlab/Simulink valiate the analysis results. Steay state analysis normally uses linear moels [2]. To fin the power transfer level limit of each moel in the steay state, there are three linear moels analyze, EMT, EMTAVR, EMTAVRPSS. Different initial rotor angles represent ifferent power levels. When the require power is larger than the power transfer level, the system will be Y. Li an L. Fan are with Dept. of Electrical Engineering, University of South Floria, Tampa FL linglingfan@usf.eu. unstable. Although the analysis uses linear moels, the initial value or operating point calculation requires the nonlinear algebraic equations. Hence, the nonlinear algebraic equations are erive first using the phasor iagram. The ynamic moels are linearize to buil the linear moels. The stability of linear moels is etermine by two ways, closeloop transfer function pole location map an Routh Hurwitz. Furthermore, the simulation section shows the ynamic responses of the corresponing nonlinear moels. By comparing the linear an nonlinear results, the power level can be etermine an the effects of AVR an PSS can be verifie. II. LINEAR ANALYSIS The system consiere in this paper is a generator connecte to an infinite bus through a transmission line shown in Fig.. The synchronous generator in the system has a salientpole rotor an its amper is negligible (i D i Q 0). It inclues rotor resistance an inuctance, r f an L f. The stator impeance is represente by the q components, L, L q, while the stator resistance is zero. The transmission line is purely inuctive, L L. X is the impeance of the corresponing inuctance. E a jx L Fig.. A generator connecte to an infinite bus. A. EMT moel V For EMT moel, there are three ynamic variables, transient stator voltage, E a, stator frequency, ω, an the phase angle of the stator voltage, δ. The set of ifferential equations of ω an δ is name swing equation. Incluing the ifferential equation of E a, the ynamic equations can be expresse by (). E a t (E f E a ) /T o ω t (P m P e Dω 0 (ω ))/(2H) () δ t ω 0(ω ) where T o is the transient time constant, L f r f, D is the amper coefficient an H is proportional to the inertial of the

2 2 generator. E f, is proportional to the fiel voltage, v f, an is a constant input in EMT moel. P m is the mechanical power or emane power which is another input of the system with a small stepchange. E a is the stator voltage an P e is the electrical power. Both of them can be represente by E a using phase iagram shown in Fig. 2. Thus, EMT moel is consiere as a thirorer system. P e E a V E a E a V cos(δ) (2) ( sin(δ) V 2 2 ) sin(2δ) (3) q where X is the transient stator impeance in axis. means this impeance incluing X L. I aq I a Fig. 2. Phasor iagram. I a δ E a V V a E a j(x X L)I a j(x qx L)I a j(x X X L)I a In the steay state, E a is only relate to E f. E a (E f V cos(δ)) V cos(δ) (4) Base on the steaystate equations, all of initial values of the system can be calculate incluing E a an P e, if the initial value of the phase angle of the stator voltage, δ 0, is provie. A specific δ 0 is corresponing to a specific P e, so δ 0 can be consiere as the power level of the system. Dynamic equations an steaystate equations are employe to buil the nonlinear moel because they are nonlinear. For the linear analysis of EMT moel, all of them have to be linearize using linear ifferential equations of variables. To linearize the ynamic equations, a a small perturbations of to δ 0 to write the linear ifferential equation. Linearizing (2): E a E a E a E a E a δ E a ( ) E a V sin(δ) E a k 3 E a k 4 (5) Substituting (5) into E a ifferential equation of () to fin the linear relationship between E a an, T o E a E f E t k a k 4 3 E a k 3 (T o k 3)s ( E f k 4 ) (6) Using the same way to linearize (3), P e P e E a E a P e δ P e V sin(δ) E a [ ( ) ] E a V cos(δ) V 2 cos(2δ) q P e k 2 E a T (7) Linearizing the swing equations by substituting (7) into them, { ω ( P m (k 2 E a T ) Dω 0 ω)/(2h) δ ω 0 ω (8) Combining them to obtain the seconorer transfer function, 2H ω 0 Pm k 2 E a T D (Ms 2 Ds T ) P m k 2 E a P m k 2 E a Ms 2 Ds T After linearizing the ynamic equations of EMT moel, its linearize moel is esigne using block iagram shown in Fig. 3. P m is consiere as the input an is the output. Because of the constant E f in this system, E f is zero. Base on Fig. 3, the closeloop transfer function of this ΔE f K 3 K 3 T o s Δ E a k 4 k 2 ΔP M Fig. 3. Block iagram of linearize EMT system. system can be erive to fin the pole locations. Ms 2 DsT T EM (s) Ms 2 Ds T T EM (s) T EMT (s) k3 T EM (s) k 4 (T o k3)sk 2 (T o k 3)s (T o k 3s )(Ms 2 Ds T ) k 2 k 3 k 4 ) (0) When δ 0 is given, the gains an coefficients of T EMT (s) can be calculate; then, MATLAB coe pzplot(t EMT (s)) is use to plot the polezero locations. If there is any pole locate in RHP, the system will be unstable. Therefore, the largest δ 0 which makes the system stable can be consiere as the power (9) Δδ

3 3 level of this system. To fin the power level, δ 0 is increase from 0 to 90. The increment is. Base on the closeloop transfer function, one value of δ 0 will generate three poles. Fig. 4 plots all of pole locations for the whole range of δ 0. It is observe that when δ 0 is over 8, the pole aligne on the real axis is locate on RHP, so 8 can be consiere as the power level of EMT moel. Imaginary Axis Pole Zero Map Real Axis Fig. 4. Polezero locations of EMT moel. From 0 to 90 The power level, 8, also can be verifie by the Routh Hurwitz. s 3 T o k3 M T o k 3T D 0 s 2 T o k 3D M T k 2 k 3 k 4 0 s K 0 s 0 T k 2 k 3 k 4 where K T o k 3T D (T o k3 M)(T k 2k 3k 4) (T o k3dm). Base on the initial value calculation, K is always larger than 0. Therefore, the sign of T k 2 k 3 k 4 etermines if the system stable or not. It is at δ 0 8 while it is at δ B. EMTAVR moel When AVR is ae to EMT moel, E f will not be a constant input, but a variable which is controlle by the terminal voltage. E f is generate by an exciter, so the input of the exciter is the error between the terminal voltage, V a, an its reference, V ref. Certainly, V a is epenent on the emane power, P m or δ 0. Before linearizing the system, the equations for calculating the initial values of EMTAVR moel shoul be erive. If the power factor an δ 0 are provie, the stator current, I a, can be calculate because V is constant. In this paper, reactive power, Q, is zero, so power factor is an I a is aligne on V. a V j q I a I a V tan(δ) q () Base on Fig. 2, E a can be expresse by I a, V, an δ 0. In aition, E f is equal to E a in steay state (). E a V q I a E a V cos(δ 0 ) I a sin(δ) E f E a (2) If E f is calculate, E a is obtaine (4); then, the terminal voltage is calculate by E a []. V aq X L E a X V cos(δ) V a X q V sin(δ) q V a (V a ) 2 (V aq ) 2 V ref E f k A V a (3) where k A is the amplify gain. In this paper, its value is 25. Before linearizing the EMTAVR moel, there are two assumptions which are mae to simplify the moel. It is assume that the voltage control system is very fast, so the time constant of the control loop is zero; the exciter is assume as a pure gain, G e (s) k A. Because E f is not a constant in EMTAVR moel an it is controlle by V a, V a has to be linearize by two variables, E a an δ. V a V a δ V a δ E a ( ) X V a V V aq V a sin(δ) X qv a q V a cos(δ) X LV 0 aq V a 0 E a V a k 5 k 6 E a (4) After aing the linearize AVR part to EMT moel, Fig. 5 shows the block iagram of EMTAVR moel. P m is the input, is the output, an V ref is normally zero. ΔV ref Δ V a Δve k A ΔE f k 5 k 6 k 3 k 3T os Δ E a k 4 k 2 ΔP M Fig. 5. Block iagram of linearize EMTAVR system. Ms 2 DsT To reuce Fig. 5 to a single loop, the feeback transfer function is erive first. E a k 3 T o k 3s [ k 4 k A (k 5 k 6 E a )] E a k 3(k 4 k A k 5 ) T o k 3s k 3 k A k 6 a (5) k 2 bs c Δδ

4 4 where a k 2 k 3 (k 4 k e k 5 ), b k 3 T o, c k 3 k e k 6. Using (5) to erive the closeloop transfer function: T EM (s) T AV R (s) T EM (s) a bsc bs c (Ms 2 Ds T )(bs c) a bs c Mbs 3 (Db Mc)s 2 (T b Dc)s T c a (6) ΔV ref Δ V a Δve k A ΔE f γs τs k 5 k 6 k 3 k 3T os Δ E a k 4 k 2 s ΔP M Ms 2 DsT Δδ Theoretically, EMTAVR system will be unstable if supplying a high active power. It is etermine by the same way, Polezero location. Base on T AV R (s), there are three poles an one zero with a specific δ 0. For observing an analyzing easily, Fig. 6 only shows the poles crossing the jω axis an the range of δ 0 is from 40 to 89. It s observe that the thirteenth pole is on RHP, so the power level of EMTAVR moel is 5. This also can be prove by Routh Hurwitz. Imaginary Axis Pole Zero Map Real Axis Fig. 6. Polezero locations of EMTAVR moel. From 40 to 89 s 3 Mb T b Dc 0 s 2 Db Mc T c a 0 s (Mb)(T c a) K 2 T b Dc DbMc 0 s 0 T c a Base on the calculation, K 2 is at δ 0 5 while it is at δ T c a is always larger than 0. Although AVR part can make the system supply more power, it also reuces δ 0 of the system. III. EMTAVRPSS To solve the samll δ 0 of EMTAVR moel, PSS part is ae to stabilize the system. Using linear analysis, PSS provies two zeros near the original point an one pole to change the root locus of EMTAVR moel, so the block iagram of EMTAVRPSS moel shoul be like the figure in Fig. 7. Fig. 7. Block iagram of linearize EMTAVRPSS system. Using the same to erive the closeloop transfer function of EMTAVRPSS moel. The feeback transfer function is: ( )] E a k 3 γs 2 [k T o k e 3s τs k 5 k 6 E a k 3 T o k 3s k 4 E a E a k 3 k e γs 2 τk 3 (k 5 k e k 4 )s k 3 (k 5 k e k 4 ) τt o k 3s 2 (τk 3 k 6 k e τ T o k 3)s k 3 k 6 k e k 2 fs 2 τas a τbs 2 es c (7) where γ an τ are the coefficients of PSS. e τk 3 k 6 k e τ T o k 3 an f k 3 k e γk 2. The closeloop transfer function is written: T EM (s) T P SS (s) T EM (s) fs2 τas a τbs 2 esc τbs 2 es c (Ms 2 Ds T )(τbs 2 es c) fs 2 τas a (8) Factoring the enominator of T P SS (s): Mτbs 4 (Me Dτb)s 3 (McDeT τbf)s 2 (DcT e τa)s(t c a). Base on Fig. 8, PSS makes the poles near jω axis locate on LHP when δ 0 is increase from to 89. Because δ 0 cannot be 90 an the increment is, the power level can be consiere to approaching 90. Using Routh Hurwitz to prove it: s 4 Mτbs Mc De T τb f T c a 0 s 3 Me Dτb Dc T e τa 0 0 s 2 K 3 T c a 0 s K 4 0 s 0 T c a (Mτb)(DcT e τa) where K 3 McDeT τbf MeDτb, K 4 (MeDτb)(T c a) Dc T e τa K 3. Base on the calculation, K 3 an K 4 are always larger than 0. IV. SIMULATION RESULTS FROM NONLINEAR MODELS To verify the results from the linear moels, the corresponing nonlinear moels are esigne an simulate in MATLAB/Simulink. The screen copy is shown in Fig. 9. The moel builing exploits the vector feature of MATLAB. For example, the output of the integrator is the vector of the three

5 5 Fig. 9. Matlab/Simulink simulation blocks. Pole Zero Map TABLE I PARAMETER OF THE SYSTEM. Imaginary Axis Table Parameter Values Value(p.u.) Rating Power 835MVA Line to line Voltage 26kV Spee 3600r/min X, X q Ω, Ω, 0.7 r f, X f 0.000Ω,0.080Ω 0.05, 0. X L, X Ω, 0.69Ω 0.5, 0.2 H 5.6s 5.6 s.pu Real Axis Fig. 8. Polezero locations of EMTAVRPSS moel. From to 89 state variables E a, δ, ω. The input of the integrator comes from computation of their erivatives, which is also a vector. In aition, the embee MATLAB function also has a vector as input an a vector as an output. Therefore, mux an emux blocks are frequently use in this moel builing. Such moel builing technique has been etaile ocumente in [3] to buil an inuction machine. The moel builing technique has been applie in variety of system builing, e.g., a oubly fe inuction generator with square wave input from rotor [4], win farm series compensate interconnecte system for subsynchronous resonance analysis [5]. This technique leas to a very concise expression. The nonlinear moels are base on the ifferential equa tions (). Before simulating nonlinear moels, the initial values shoul be calculate; then giving a small step change in P m to check the system s stability. For EMT moel, the require initial values are three ynamic variables, E a, δ, ω, one input with step change, P m, an two constant inputs, E f, V. Because of the synchronous generator, ω shoul be p.u. at steay state. E f an V are set as p.u. δ shoul be δ 0 to fin the power level. E a an P m are calculate base on (2), (3). The generator s parameters are list in Table. Although EMTAVR moel an EMTAVRPSS moel have ifferent block iagrams of linear moels, the initial value calculations are the same. For EMTAVR moel, ω an V are still p.u. but V ref replaces E f as the constant input of the system. E a, P m an V ref are calculate by (3)(4), ()(4). A. EMT moel Base on the linear analysis, the power level of EMT moel is 8, so the initial value of δ 0, is set as 8 an one more egree than the power level, 82. To avoi that a large step change will cause the system unstable, there is p.u.

6 6 power ae to P m at 2s. Fig. 0 shows the step response of one ynamic variable, δ. Although δ responses very slowly in both of situation, it is obviously that δ is approaching a steay state at δ 0 8 while it rises faster an faster at δ Hence, it is prove that the power level of EMT moel is δ 0 8. In aition, the pole which is very close to jω axis oes not have the imaginary part base on Fig. 4, so the step response of δ oes not have the oscillation. Fig. 0 also proves this point. C. EMTAVRPSS moel It is alreay known that EMTAVRPSS moel is always stable when δ 0 is uner 90, so it is only require to check the stability of the system at δ The step change is still 0.0p.u. at 2s. Fig. 2 shows that the system is stable at δ 0 89 an the step response is very fast. It proves that the power level of EMTAVRPSS moel is 89 an PSS provie a supplemental amping to cancel the oscillations from AVR. δ(8 ) δ(82 ) Time (sec) Fig. 0. The step response of δ in EMT moel. Upper plot: δ 0 8 ; lower plot: δ B. EMTAVR moel For the obvious ifference, δ 0 has two initial values, 5 an 55. The step change in P m of EMTAVR moel is not very sensitive, so the step change is 0.0p.u. an happens at 2s. Base on Fig., although the step response of δ has a large oscillation in two plots, it is observe that the oscillation is becoming smaller at δ 0 5 while it is becoming larger at δ The reason causing the amping is that the pole which crosses jω axis has an imaginary part in EMTAVR moel. Therefore, Fig. oes not only prove that the power level of EMTAVR moel is 5, but also proves which pole causes the system unstable. δ(5 ) δ(55 ) δ(89 ) Fig. 2. The step response of δ in EMTAVRPSS moel. δ V. CONCLUSION This paper uses the linear analysis to fin the power level of the typically power system, a synchronous generator connecte to an infinite bus. Moreover, it etermines the power levels of this system which is ae by the voltage control, AVR, an the stabilizer, PSS. The simulation results of the nonlinear moels verify all of three power levels analyze from the linear moels. After the analysis an the verification, there are three points conclue. One is that the power level of this system cannot be equal or larger than 90. The secon is that although AVR can make the generator supply more power, the power transfer angle is its shortage. It means that if there is a large active power transfere, the system will be unstable. Last point, PSS is very useful to stabilize the system when there is higher power require to transfer. Finally, a complete power generation system shoul inclue both of AVR an PSS parts. REFERENCES [] A. R. Bergen an V. Vittal, Power systems analysis (secon eition), in Prentice Hall, 999. [2] P. Kunur, Power system stability an control, in McGrawHill, 994. [3] Z. Miao an L. Fan, The art of moeling an simulation of inuction generator in win generation applications using highorer moel, Simulation Moelling Practice an Theory, vol. 6, no. 9, pp , [4] L. Fan, Z. Miao, S. Yuvarajan, an R. Kavasseri, Hybri moeling of figs for win energy conversion systems, Simulation Moelling Practice an Theory, vol. 8, no. 7, pp , 200. [5] L. Fan, R. Kavasseri, Z. L. Miao, an C. Zhu, Moeling of figbase win farms for ssr analysis, Power Delivery, IEEE Transactions on, vol. 25, no. 4, pp , Time (sec) Fig.. The step response of δ in EMTAVR moel. Upper plot: δ 0 5 ; lower plot: δ 0 55.