IEEE PES Task Force on Benchmark Systems for Stability Controls

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1 IEEE PES Task Force on Benchmark Systems for Stability Controls Ian Hiskens November 9, 3 Abstract This report summarizes a study of an IEEE -generator, 39-bus system. Three types of analysis were performed: load flow, small disturbance analysis, and dynamic simulation. All analysis was carried out using MATLAB, and this report s objective is to demonstrate how to use MATLAB to obtain results that are comparable to benchmark results from other analysis methods. Data from other methods may be found on the website Contents System Model 3. Generators State Variable Model Model Parameters AVR Model AVR Parameters PSS Model PSS Parameters Governor Model Loads Load Model Load Parameters Lines and Transformers Power and Voltage Setpoints Load Flow Results 3 Small Disturbance Analysis 4 Dynamic Simulation 3 5 Appendix 5 5. Accessing Simulation Data with MATLAB R Variable Indexing Example Application Contents of file 39bus.out List of Figures IEEE 39-bus network AVR block diagram PSS block diagram Graphical depiction of Eigenvalues δ (in radians) versus time for generators

2 Report: 39-bus system (New England Reduced Model) 6 ω (in percent per-unit) for generators Contents of file 39bus.out (Page of 4) List of Tables Generator inertia data Generator data Generator AVR parameters Generator PSS parameters Generator setpoint data Active and reactive power draws for all loads at initial voltage Network data Power and voltage setpoint data Power flow results Eigenvalues calculated through small disturbance analysis Mapping from MATLAB R files to simulation data The IEEE 39-bus system analyzed in this report is commonly known as the -machine New-England Power System. This system s parameters are specified in a paper by T. Athay et al[] and are published in a book titled Energy Function Analysis for Power System Stability []. A diagram of the system is shown in Figure, and system models and parameters are introduced in the following section. G8 <3> G <37> <5> <6> <8> <9> <7> <> <38> <8> <7> <4> G9 <> <3> G6 <6> <35> G <5> <> <> <39> <4> <4> <5> <6> <> <9> <3> <9> <8> <7> G <3> <> <3> <> <3> G3 <> <34> <33> G5 G4 G7 <36> <Bus #> Figure : IEEE 39-bus network

3 Report: 39-bus system (New England Reduced Model) 3 System Model. Generators.. State Variable Model Generator analysis was carried out using a fourth-order model, as defined in the equations below. Equations through 4 model the generator, while the remaining equations relate various parameters. Parameter values for the model are shown in Table on the system base MVA. For more information on this generator model and its parameters, refer to Sauer and Pai pages -3 [3]. Ė q = T do ( E q (x d x d)i d + E fd ) () Ė d = T qo ( E d + (x q x q)i q ) () δ = ω (3) ω = M (T mech (φ d I q φ q I d ) D ω ) (4) = r a I d + φ q + V d (5) = r a I q φ d + V q (6) = φ d x di d + E q (7) = φ q x qi q E d (8) = V d sin δ + V q cos δ V r (9) = V q sin δ V d cos δ V i () = I d sin δ + I q cos δ I r () = I q sin δ I d cos δ I i ().. Model Parameters Generator inertia data is given in Table. Note that the per unit conversion for M associated with ω is in radians per second. Also, we changed the value of T qo for Unit from. to.. Table : Generator inertia data Unit No. M=*H 5./(π) 3.3/(π) /(π) 4 8.6/(π) 5 6./(π) /(π) 7 6.4/(π) 8 4.3/(π) /(π) 4./(π)

4 4 Report: 39-bus system (New England Reduced Model) All other generator parameters are set according to Table. Table : Generator data Unit No. H R a x d x q x d x q T do T qo x l AVR Model All generators in the system are equipped with automatic voltage regulators (AVRs). We chose to use static AVRs with E fd limiters. The model for this controller is shown in Figure below. AVR type smp_lim Figure : AVR block diagram

5 Report: 39-bus system (New England Reduced Model) 5 AVR parameters are related according to equations 3 to 4. V f = (V T V f ) T R (3) Ė fd = (K A x tgr E fd ) T A (4) ẋ i = x err x tgr { (5) when t > V REF = V T V setpoint when t < (6)..4 AVR Parameters = VT (Vr + Vi ) (7) = T B x tgr T C x err x i (8) = x err (V REF + V P SS V f ) (9) = E fd E fd () { = Upper Limit Switch Upper Limit Detector < () = Upper Limit Detector E fd E fd,max { = Upper Limit Switch Upper Limit Detector > () = Upper Limit Detector (K A x tgr E fd ) { = Lower Limit Switch Lower Limit Detector > (3) = Lower Limit Detector E fd E fd,min { = Lower Limit Switch Lower Limit Detector < (4) = Lower Limit Detector (K A x tgr E fd ) AVR parameters, as defined in the previous section, are specified in Table 3. Table 3: Generator AVR parameters Unit No. T R K A T A T B T C V setpoint E fd,max E fd,min PSS Model Each generator in the system is equipped with a δ and ω type PSS with two phase shift blocks. The model for this PSS is shown in Figure 3 below.

6 6 Report: 39-bus system (New England Reduced Model) PSS type conv Figure 3: PSS block diagram PSS parameters are related according to equations 5 to 33. ẋ W = V W T W ẋ P = V W V P (6) ẋ Q = V P V out (7) (5) = ωk P SS V W x W (8) = V P T V W T x P (9) = V out T 4 V P T 3 x Q (3) = x P SS,Max V out (V P SS,Max V out ) (3) = V out x P SS,Min (V out V P SS,Min ) (3) V P SS x P SS,Max when (V P SS,Max V out ) < = V P SS x P SS,Min when (V out V P SS,Min ) < (33) V P SS V out otherwise..6 PSS Parameters PSS parameters defined in the previous section are specified according to Table 4. Table 4: Generator PSS parameters Unit No. K T W T T T 3 T 4 V P SS,Max V P SS,Min./(π) /(π) /(π) /(π) /(π) /(π) /(π) /(π) /(π) /(π) Governor Model No governor dynamics are included in our analysis, and each generator s mechanical torque is assumed to be constant. Since Unit is the angle reference and resides at the swing node, P set point is determined by the

7 Report: 39-bus system (New England Reduced Model) 7 power flow initialization. The value of P set point is given for each generator in Table 5 on the system base of MVA. Table 5: Generator setpoint data Unit No. P set point Loads.. Load Model We use a constant impedance load model in our analysis. Loads modeled this way are voltage dependent and behave according to the following equations... Load Parameters = P + V r I r + V i I i if t < = Q + V i I r V r I i (34) V = (Vr + Vi ( ) ) α V = P V + Vr I r + V i I i ( ) β if t > V = Q V + Vi I r + V r I i (35) V = (Vr + V Table 6 contains load behavior at initial voltage. Due to the voltage dependence discussed above, these values may not be accurate after voltages change. i )

8 8 Report: 39-bus system (New England Reduced Model) Table 6: Active and reactive power draws for all loads at initial voltage Bus Load P [PU] Q [pu] Lines and Transformers Network data for the system is shown in Table 7. As with generator data, all values are given on the system base MVA at 6 Hz.

9 Report: 39-bus system (New England Reduced Model) 9 Line Data Table 7: Network data Transformer Tap From Bus To Bus R X B Magnitude Angle

10 Report: 39-bus system (New England Reduced Model).4 Power and Voltage Setpoints Table 8 contains power and voltage setpoint data, specified on the system MVA base. Note that Generator is the swing node, and Generator represents the aggregation of a large number of generators. Table 8: Power and voltage setpoint data Bus Type Voltage Load Generator [PU] MW MVar MW MVar Unit No. PQ - PQ - 3 PQ PQ PQ - 6 PQ - 7 PQ PQ PQ - PQ - PQ - PQ PQ - 4 PQ - 5 PQ PQ PQ - 8 PQ PQ - PQ PQ PQ - 3 PQ PQ PQ PQ PQ PQ PQ PV Gen 3 PV Gen 3 PV Gen3 33 PV Gen4 34 PV Gen5 35 PV Gen6 36 PV Gen7 37 PV Gen8 38 PV Gen9 39 PV Gen This completes the description of the system and its model. The remainder of this report focuses on the three analyses performed on the system: load flow, small signal stability assessment via Eigenvalue calculation, and numerical simulation. The results of load flow analysis are presented in the next section.

11 Report: 39-bus system (New England Reduced Model) Load Flow Results Load flow for the system was calculated using MATLAB. The results are in Table 9. Note that all voltages, active power values, and reactive power values are given in per unit on the system MVA base. For a more complete description of power flow, including flow on each line, see Section 5. in the Appendix. Table 9: Power flow results Bus V Angle [deg] Bus Total Load Generator P Q P Q P Q Unit No

12 Report: 39-bus system (New England Reduced Model) 3 Small Disturbance Analysis Small signal stability was assessed by calculating Eigenvalues of the system. To find Eigenvalues, we first calculated reduced A matrices as follows: Linearize: ẋ = f(x, y) = g(x, y) Differential eq. Algebraic eq. ẋ = f x x + f y y = g x x + g y y Eliminate algebraic equations: ẋ = (f x f y gy g x ) x Reduced A matrices Eigenvalues, which correspond to machine oscillatory modes, are calculated from the reduced A matrices. Some Eigenvalues of our system are given in Table below. To access all Eigenvalues calculated in our analysis, see Section 5. in the Appendix. Table : Eigenvalues calculated through small disturbance analysis Index Real part Imaginary part j j j j j j j j j.576 The Eigenvalues in Table are plotted in Figure Imaginary Axis Real Axis Figure 4: Graphical depiction of Eigenvalues Student Version of MATLAB

13 Report: 39-bus system (New England Reduced Model) 3 4 Dynamic Simulation Dynamic simulation was performed for a three-phase fault at Bus 6 occurring at t =.5s. The fault impedance is. PU, and it is cleared at t =.7s. The simulation method used was numerical trapezoidal integration with a time step of ms. Figure 5 shows the generator state δ as a function of time for the ten generators, with Unit as the angle reference. Figure 6 on the following page shows the generator state ω as a function of time for the ten generators. Gen 3 Gen Delta [rad].5.5 Delta [rad] 3 Gen3 3 Gen4 Delta [rad] Delta [rad] 4 Gen5 3 Gen6 Delta [rad] Delta [rad] 3 Gen7 3 Gen8 Delta [rad] Delta [rad] 3 Gen9 Gen Delta [rad] Delta [rad] Time [sec] Time [sec] Figure 5: δ (in radians) versus time for generators Student Version of MATLAB

14 4 Report: 39-bus system (New England Reduced Model).5 Gen.5 Gen Omega [% PU].5 Omega [% PU] Gen3 Gen4 Omega [% PU].5 Omega [% PU].5 4 Gen5 Gen6 Omega [% PU] Omega [% PU] Gen7 Gen8 Omega [% PU] Omega [% PU] Gen9.5 Gen Omega [% PU].5 Omega [% PU].5.5 Time [sec].5 Time [sec] Figure 6: ω (in percent per-unit) for generators To generate these and other figures from the MATLAB files availablestudent on theversion website, of MATLAB see Accessing Simulation Data with MATLAB in the appendix.

15 Report: 39-bus system (New England Reduced Model) 5 5 Appendix 5. Accessing Simulation Data with MATLAB R There are seven MATLAB R.mat files available for download. Each.mat file has a csv counterpart for those who prefer to work with Excel or another CSV-friendly program (the procedure described here may be adapted for such programs). Each file is described below. time.mat is a row vector with 677 elements. This vector represents the simulation time period with increments of ms. To observe the variation of any model variable with respect to time, simply plot the corresponding row of x.mat or y.mat versus time.mat. x.mat contains 9 row vectors, each corresponding to a model state variable. See Table. x.mat contains 9 elements corresponding to initial values of model state variables. It is indexed the same as x.mat. y.mat contains 67 row vectors, each corresponding to a non-state variable in the model. See Table. y.mat contains 67 elements corresponding to initial values of non-state variables. It is indexed the same as y.mat. Aeig.mat contains 9 complex elements corresponding to system Eigenvalues. To obtain any entry of Table, one need only extract the element of Aeig corresponding to that entry s Index. Of course, there are many more Eigenvalues stored in Aeig than those listed in Table, and all Eigenvalues may be easily plotted so long as care is taken in isolating the real and imaginary parts. Ared.mat is a 9-by-9 variable containing reduced A matrix data for the system. matrices are discussed in Section 3.) (Reduced A Model state variables are stored in x.mat while non-state variables are located in y.mat. Table maps from these files to specific model variables by specifying the range of variables corresponding to each component of the model. The order of variables within a component s range is discussed in the next five subsections. Table : Mapping from MATLAB R files to simulation data Model Model ID x range y range Description from to from to Generator G x states and 3 y variables G AVR G x states and y variables G PSS G x states and 7 y variables G Load Bus x states and 4 y variables Bus (x states are real and reactive power) Network Bus y variables Bus Variable Indexing This section defines all variables contained in the MATLAB files. When writing code to extract specific variables, refer to this section to determine which indices to use.

16 6 Report: 39-bus system (New England Reduced Model) Generator Variables (See Section..) x : E q x : E d x 3 : δ x 4 : ω y : V r y : V i y 3 : I r y 4 : I i y 5 : V d y 6 : V q y 7 : I d y 8 : I q y 9 : ψ d y : ψ q y : E fd y : ω y 3 : T mech AVR Variables (See Section..3) x : V f x : E fd x 3 : x i x 4 : V REF y : V r y : V i y 3 : V T y 4 : V P SS y 5 : x err y 6 : x tgr y 7 : E fd y 8 : Upper Limit Detector y 9 : Lower Limit Detector y : Upper Limit Switch y : Lower Limit Switch PSS Variables (See Section..5) x : x W x : x P x 3 : x Q y : ω (input) y : V W y 3 : V P y 4 : V out y 5 : V P SS y 6 : V P SS,Max V out y 7 : V out V P SS,Min Load Variables (See Section..) x : P x : Q y : V r y : V i y 3 : I r y 4 : I i Network Variables (See??) Network variables are indexed in the MATLAB files as follows: 5.. Example Application The code below will generate Figures 5 and 6. By modifying or adapting this code, the reader may plot or manipulate any variables in x.mat and y.mat. (Note: The placeholders TIMEPATH, XPATH, and YPATH should be replaced with paths to the respective files on your computer.)

17 Report: 39-bus system (New England Reduced Model) 7 %% Import Data % Import time.mat newdata = load( -mat, TIMEPATH\time.mat ); % Create new variables in the base workspace from those fields. vars = fieldnames(newdata); for i = :length(vars) assignin( base, vars{i}, newdata.(vars{i})); end % Import x.mat newdata = load( -mat, XPATH\x.mat ); % Create new variables in the base workspace from those fields. vars = fieldnames(newdata); for i = :length(vars) assignin( base, vars{i}, newdata.(vars{i})); end % Import y.mat newdata = load( -mat, YPATH\y.mat ); % Create new variables in the base workspace from those fields. vars = fieldnames(newdata); for i = :length(vars) assignin( base, vars{i}, newdata.(vars{i})); end clear newdata, clear vars %% Plot x or y vs. time % Plot omega vs time figure( name, Generator Omega vs Time ) for i = : subplot(5,,i); p = plot(time,x(i*4 -,:)./4, k ); % Note factor of /4 set(p, LineWidth,); figtitle = strcat( Gen,numstr(i)); title(figtitle) if or(i==9,i==) xlabel( Time [sec] ) end ylabel( Omega [% PU] ) grid on end % Plot delta vs time figure( name, Generator Delta vs Time ) for i = : subplot(5,,i); p = plot(time,x(i*4 -,:)-x(3,:), k ); % Subtract Unit (angle reference) set(p, LineWidth,);

18 8 Report: 39-bus system (New England Reduced Model) end figtitle = strcat( Gen,numstr(i)); title(figtitle) if or(i==9,i==) xlabel( Time [sec] ) end ylabel( Delta [rad] ) grid on 5. Contents of file 39bus.out The next four pages show the contents of the file 39bus.out. This file contains power flow data with enough granularity to observe flows on all lines.

19 IEEE 39 bus test system 39bus.out BUS Bus..474PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 4.7KV TO Bus TO Gen BUS Bus..7PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.7KV TO Bus TO Bus TO Gen FX BUS Bus..7PU -6.8 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.3KV TO Bus TO Bus TO Bus UN BUS Bus..PU -6.4 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.KV LOAD TO Bus FX TO Bus FX BUS Bus3..43PU -6. CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.4KV TO Bus TO Bus TO Bus UN BUS Bus4..7PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.KV TO Bus TO Bus TO Bus BUS Bus5..54PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.5KV LOAD TO Bus TO Bus BUS Bus6..38PU -6.9 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.KV LOAD TO Bus TO Bus TO Bus TO Bus TO Bus BUS Bus7..336PU -7.3 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.4KV TO Bus TO Bus TO Bus BUS Bus8..39PU -8. CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.KV LOAD TO Bus TO Bus BUS Bus9..499PU -. CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 5.KV TO Bus TO Gen FX TO Bus FX BUS Bus..487PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 4.9KV TO Bus TO Bus TO Bus TO Gen FX BUS Bus..99PU -. CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 99.KV LOAD Page Report: 39-bus system (New England Reduced Model) 9 Figure 7: Contents of file 39bus.out (Page of 4)

20 39bus.out TO Gen FX TO Bus UN BUS Bus..38PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.KV LOAD TO Bus TO Bus BUS Bus..498PU.67 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 5.KV TO Bus TO Bus TO Gen FX BUS Bus3..448PU.47 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 4.5KV LOAD TO Bus TO Bus TO Gen FX BUS Bus4..373PU -6.7 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.7KV LOAD TO Bus TO Bus BUS Bus5..576PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 5.8KV LOAD TO Bus TO Bus TO Gen FX BUS Bus6..5PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 5.KV LOAD TO Bus TO Bus TO Bus TO Bus BUS Bus7..377PU -7.5 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.8KV LOAD TO Bus TO Bus BUS Bus8..5PU -. CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 5.KV LOAD TO Bus TO Bus BUS Bus9..499PU.74 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 5.KV LOAD TO Bus TO Bus TO Gen FX BUS Bus3..3PU -8.6 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.KV LOAD TO Bus TO Bus TO Bus BUS Bus4..39PU -9.6 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.4KV LOAD TO Bus TO Bus TO Bus BUS Bus5..53PU -8.6 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.5KV TO Bus TO Bus TO Bus Page Report: 39-bus system (New England Reduced Model) Contents of file 39bus.out (Page of 4)

21 39bus.out BUS Bus6..77PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.8KV TO Bus TO Bus TO Bus TO Gen FX BUS Bus7..997PU -. CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 99.7KV LOAD TO Bus TO Bus BUS Bus8..996PU -.6 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 99.6KV LOAD TO Bus TO Bus TO Bus BUS Bus9..8PU -.3 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.8KV TO Bus TO Gen BUS Gen3..475PU CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 4.8KV GENERATION ( 5..) TO Bus UN BUS Gen3..98PU. CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 98.KV LOAD GENERATION (..) TO Bus UN BUS Gen3..983PU.57 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 98.3KV GENERATION ( 65..) TO Bus UN BUS Gen PU 4.9 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 99.7KV GENERATION ( 63..) TO Bus UN BUS Gen34..3PU 3.8 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.KV GENERATION ( 58..) TO Bus UN BUS Gen PU 5.63 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 4.9KV GENERATION ( 65..) TO Bus UN BUS Gen PU 8.3 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 6.4KV GENERATION ( 56..) TO Bus UN BUS Gen37..78PU.4 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.8KV GENERATION ( 54..) TO Bus UN BUS Gen38..65PU 7.8 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP.6KV GENERATION ( 83..) TO Bus UN BUS Gen39..3PU -.5 CKT MW MVAR MVA MW(NOM) MVR(NOM) MW(LOSS) MVAR(LOSS) TAP 3.KV LOAD GENERATION (..) TO Bus TO Bus SYSTEM TOTALS Page 3 Report: 39-bus system (New England Reduced Model) Contents of file 39bus.out (Page 3 of 4)

22 MW MVAR GENERATION LOAD LOSSES BUS SHUNTS.. LINE SHUNTS.. SWITCHED SHUNTS. MISMATCH.. BASE MVA :. TOLERANCE:. PU SLACK BUSES Gen3. 39bus.out Page 4 Report: 39-bus system (New England Reduced Model) Contents of file 39bus.out (Page 4 of 4)

23 Report: 39-bus system (New England Reduced Model) 3 References [] T. Athay, R. Podmore, and S. Virmani. A Practical Method for the Direct Analysis of Transient Stability. In: IEEE Transactions on Power Apparatus and Systems PAS-98 ( Mar. 979), pp [] M. A. Pai. Energy function analysis for power system stability. The Kluwer international series in engineering and computer science. Power electronics and power systems. Boston: Kluwer Academic Publishers, 989. [3] Peter W Sauer and M. A Pai. Power system dynamics and stability. English. Champaign, IL.: Stipes Publishing L.L.C., 6.