Line data Source code
1 : /*
2 : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3 : SLEPc - Scalable Library for Eigenvalue Problem Computations
4 : Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain
5 :
6 : This file is part of SLEPc.
7 : SLEPc is distributed under a 2-clause BSD license (see LICENSE).
8 : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
9 : */
10 :
11 : static char help[] = "Power grid small signal stability analysis of WECC 9 bus system.\n\
12 : This example is based on the 9-bus (node) example given in the book Power\n\
13 : Systems Dynamics and Stability (Chapter 8) by P. Sauer and M. A. Pai.\n\
14 : The power grid in this example consists of 9 buses (nodes), 3 generators,\n\
15 : 3 loads, and 9 transmission lines. The network equations are written\n\
16 : in current balance form using rectangular coordinates. It uses the SLEPc\n\
17 : package to calculate the eigenvalues for small signal stability analysis\n\n";
18 :
19 : /*
20 : This example is based on PETSc's ex9bus example (under TS).
21 :
22 : The equations for the stability analysis are described by the DAE
23 :
24 : \dot{x} = f(x,y,t)
25 : 0 = g(x,y,t)
26 :
27 : where the generators are described by differential equations, while the algebraic
28 : constraints define the network equations.
29 :
30 : The generators are modeled with a 4th order differential equation describing the electrical
31 : and mechanical dynamics. Each generator also has an exciter system modeled by 3rd order
32 : diff. eqns. describing the exciter, voltage regulator, and the feedback stabilizer
33 : mechanism.
34 :
35 : The network equations are described by nodal current balance equations.
36 : I(x,y) - Y*V = 0
37 :
38 : where:
39 : I(x,y) is the current injected from generators and loads.
40 : Y is the admittance matrix, and
41 : V is the voltage vector
42 :
43 : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
44 :
45 : The linearized equations for the eigenvalue analysis are
46 :
47 : \dot{\delta{x}} = f_x*\delta{x} + f_y*\delta{y}
48 : 0 = g_x*\delta{x} + g_y*\delta{y}
49 :
50 : This gives the linearized sensitivity matrix
51 : A = | f_x f_y |
52 : | g_x g_y |
53 :
54 : We are interested in the eigenvalues of the Schur complement of A
55 : \hat{A} = f_x - g_x*inv(g_y)*f_y
56 :
57 : Example contributed by: Shrirang Abhyankar
58 : */
59 :
60 : #include <petscdm.h>
61 : #include <petscdmda.h>
62 : #include <petscdmcomposite.h>
63 : #include <slepceps.h>
64 :
65 : #define freq 60
66 : #define w_s (2*PETSC_PI*freq)
67 :
68 : /* Sizes and indices */
69 : const PetscInt nbus = 9; /* Number of network buses */
70 : const PetscInt ngen = 3; /* Number of generators */
71 : const PetscInt nload = 3; /* Number of loads */
72 : const PetscInt gbus[3] = {0,1,2}; /* Buses at which generators are incident */
73 : const PetscInt lbus[3] = {4,5,7}; /* Buses at which loads are incident */
74 :
75 : /* Generator real and reactive powers (found via loadflow) */
76 : const PetscScalar PG[3] = {0.716786142395021,1.630000000000000,0.850000000000000};
77 : const PetscScalar QG[3] = {0.270702180178785,0.066120127797275,-0.108402221791588};
78 : /* Generator constants */
79 : const PetscScalar H[3] = {23.64,6.4,3.01}; /* Inertia constant */
80 : const PetscScalar Rs[3] = {0.0,0.0,0.0}; /* Stator Resistance */
81 : const PetscScalar Xd[3] = {0.146,0.8958,1.3125}; /* d-axis reactance */
82 : const PetscScalar Xdp[3] = {0.0608,0.1198,0.1813}; /* d-axis transient reactance */
83 : const PetscScalar Xq[3] = {0.0969,0.8645,1.2578}; /* q-axis reactance Xq(1) set to 0.4360, value given in text 0.0969 */
84 : const PetscScalar Xqp[3] = {0.0969,0.1969,0.25}; /* q-axis transient reactance */
85 : const PetscScalar Td0p[3] = {8.96,6.0,5.89}; /* d-axis open circuit time constant */
86 : const PetscScalar Tq0p[3] = {0.31,0.535,0.6}; /* q-axis open circuit time constant */
87 : PetscScalar M[3]; /* M = 2*H/w_s */
88 : PetscScalar D[3]; /* D = 0.1*M */
89 :
90 : PetscScalar TM[3]; /* Mechanical Torque */
91 : /* Exciter system constants */
92 : const PetscScalar KA[3] = {20.0,20.0,20.0}; /* Voltage regulartor gain constant */
93 : const PetscScalar TA[3] = {0.2,0.2,0.2}; /* Voltage regulator time constant */
94 : const PetscScalar KE[3] = {1.0,1.0,1.0}; /* Exciter gain constant */
95 : const PetscScalar TE[3] = {0.314,0.314,0.314}; /* Exciter time constant */
96 : const PetscScalar KF[3] = {0.063,0.063,0.063}; /* Feedback stabilizer gain constant */
97 : const PetscScalar TF[3] = {0.35,0.35,0.35}; /* Feedback stabilizer time constant */
98 : const PetscScalar k1[3] = {0.0039,0.0039,0.0039};
99 : const PetscScalar k2[3] = {1.555,1.555,1.555}; /* k1 and k2 for calculating the saturation function SE = k1*exp(k2*Efd) */
100 :
101 : PetscScalar Vref[3];
102 : /* Load constants
103 : We use a composite load model that describes the load and reactive powers at each time instant as follows
104 : P(t) = \sum\limits_{i=0}^ld_nsegsp \ld_alphap_i*P_D0(\frac{V_m(t)}{V_m0})^\ld_betap_i
105 : Q(t) = \sum\limits_{i=0}^ld_nsegsq \ld_alphaq_i*Q_D0(\frac{V_m(t)}{V_m0})^\ld_betaq_i
106 : where
107 : ld_nsegsp,ld_nsegsq - Number of individual load models for real and reactive power loads
108 : ld_alphap,ld_alphap - Percentage contribution (weights) or loads
109 : P_D0 - Real power load
110 : Q_D0 - Reactive power load
111 : V_m(t) - Voltage magnitude at time t
112 : V_m0 - Voltage magnitude at t = 0
113 : ld_betap, ld_betaq - exponents describing the load model for real and reactive part
114 :
115 : Note: All loads have the same characteristic currently.
116 : */
117 : const PetscScalar PD0[3] = {1.25,0.9,1.0};
118 : const PetscScalar QD0[3] = {0.5,0.3,0.35};
119 : const PetscInt ld_nsegsp[3] = {3,3,3};
120 : const PetscScalar ld_alphap[3] = {0.0,0.0,1.0};
121 : const PetscScalar ld_betap[3] = {2.0,1.0,0.0};
122 : const PetscInt ld_nsegsq[3] = {3,3,3};
123 : const PetscScalar ld_alphaq[3] = {0.0,0.0,1.0};
124 : const PetscScalar ld_betaq[3] = {2.0,1.0,0.0};
125 :
126 : typedef struct {
127 : DM dmgen, dmnet; /* DMs to manage generator and network subsystem */
128 : DM dmpgrid; /* Composite DM to manage the entire power grid */
129 : Mat Ybus; /* Network admittance matrix */
130 : Vec V0; /* Initial voltage vector (Power flow solution) */
131 : PetscInt neqs_gen,neqs_net,neqs_pgrid;
132 : IS is_diff; /* indices for differential equations */
133 : IS is_alg; /* indices for algebraic equations */
134 : } Userctx;
135 :
136 : /* Converts from machine frame (dq) to network (phase a real,imag) reference frame */
137 0 : PetscErrorCode dq2ri(PetscScalar Fd,PetscScalar Fq,PetscScalar delta,PetscScalar *Fr,PetscScalar *Fi)
138 : {
139 0 : PetscFunctionBegin;
140 0 : *Fr = Fd*PetscSinScalar(delta) + Fq*PetscCosScalar(delta);
141 0 : *Fi = -Fd*PetscCosScalar(delta) + Fq*PetscSinScalar(delta);
142 0 : PetscFunctionReturn(PETSC_SUCCESS);
143 : }
144 :
145 : /* Converts from network frame ([phase a real,imag) to machine (dq) reference frame */
146 3 : PetscErrorCode ri2dq(PetscScalar Fr,PetscScalar Fi,PetscScalar delta,PetscScalar *Fd,PetscScalar *Fq)
147 : {
148 3 : PetscFunctionBegin;
149 3 : *Fd = Fr*PetscSinScalar(delta) - Fi*PetscCosScalar(delta);
150 3 : *Fq = Fr*PetscCosScalar(delta) + Fi*PetscSinScalar(delta);
151 3 : PetscFunctionReturn(PETSC_SUCCESS);
152 : }
153 :
154 1 : PetscErrorCode SetInitialGuess(Vec X,Userctx *user)
155 : {
156 1 : Vec Xgen,Xnet;
157 1 : PetscScalar *xgen,*xnet;
158 1 : PetscInt i,idx=0;
159 1 : PetscScalar Vr,Vi,IGr,IGi,Vm,Vm2;
160 1 : PetscScalar Eqp,Edp,delta;
161 1 : PetscScalar Efd,RF,VR; /* Exciter variables */
162 1 : PetscScalar Id,Iq; /* Generator dq axis currents */
163 1 : PetscScalar theta,Vd,Vq,SE;
164 :
165 1 : PetscFunctionBegin;
166 1 : M[0] = 2*H[0]/w_s; M[1] = 2*H[1]/w_s; M[2] = 2*H[2]/w_s;
167 : /* D[0] = 0.1*M[0]; D[1] = 0.1*M[1]; D[2] = 0.1*M[2];
168 : */
169 1 : D[0] = D[1] = D[2] = 0.0;
170 1 : PetscCall(DMCompositeGetLocalVectors(user->dmpgrid,&Xgen,&Xnet));
171 :
172 : /* Network subsystem initialization */
173 1 : PetscCall(VecCopy(user->V0,Xnet));
174 :
175 : /* Generator subsystem initialization */
176 1 : PetscCall(VecGetArray(Xgen,&xgen));
177 1 : PetscCall(VecGetArray(Xnet,&xnet));
178 :
179 4 : for (i=0; i < ngen; i++) {
180 3 : Vr = xnet[2*gbus[i]]; /* Real part of generator terminal voltage */
181 3 : Vi = xnet[2*gbus[i]+1]; /* Imaginary part of the generator terminal voltage */
182 3 : Vm = PetscSqrtScalar(Vr*Vr + Vi*Vi); Vm2 = Vm*Vm;
183 3 : IGr = (Vr*PG[i] + Vi*QG[i])/Vm2;
184 3 : IGi = (Vi*PG[i] - Vr*QG[i])/Vm2;
185 :
186 3 : delta = atan2(Vi+Xq[i]*IGr,Vr-Xq[i]*IGi); /* Machine angle */
187 :
188 3 : theta = PETSC_PI/2.0 - delta;
189 :
190 3 : Id = IGr*PetscCosScalar(theta) - IGi*PetscSinScalar(theta); /* d-axis stator current */
191 3 : Iq = IGr*PetscSinScalar(theta) + IGi*PetscCosScalar(theta); /* q-axis stator current */
192 :
193 3 : Vd = Vr*PetscCosScalar(theta) - Vi*PetscSinScalar(theta);
194 3 : Vq = Vr*PetscSinScalar(theta) + Vi*PetscCosScalar(theta);
195 :
196 3 : Edp = Vd + Rs[i]*Id - Xqp[i]*Iq; /* d-axis transient EMF */
197 3 : Eqp = Vq + Rs[i]*Iq + Xdp[i]*Id; /* q-axis transient EMF */
198 :
199 3 : TM[i] = PG[i];
200 :
201 : /* The generator variables are ordered as [Eqp,Edp,delta,w,Id,Iq] */
202 3 : xgen[idx] = Eqp;
203 3 : xgen[idx+1] = Edp;
204 3 : xgen[idx+2] = delta;
205 3 : xgen[idx+3] = w_s;
206 :
207 3 : idx = idx + 4;
208 :
209 3 : xgen[idx] = Id;
210 3 : xgen[idx+1] = Iq;
211 :
212 3 : idx = idx + 2;
213 :
214 : /* Exciter */
215 3 : Efd = Eqp + (Xd[i] - Xdp[i])*Id;
216 3 : SE = k1[i]*PetscExpScalar(k2[i]*Efd);
217 3 : VR = KE[i]*Efd + SE;
218 3 : RF = KF[i]*Efd/TF[i];
219 :
220 3 : xgen[idx] = Efd;
221 3 : xgen[idx+1] = RF;
222 3 : xgen[idx+2] = VR;
223 :
224 3 : Vref[i] = Vm + (VR/KA[i]);
225 :
226 3 : idx = idx + 3;
227 : }
228 :
229 1 : PetscCall(VecRestoreArray(Xgen,&xgen));
230 1 : PetscCall(VecRestoreArray(Xnet,&xnet));
231 :
232 : /* PetscCall(VecView(Xgen,0)); */
233 1 : PetscCall(DMCompositeGather(user->dmpgrid,INSERT_VALUES,X,Xgen,Xnet));
234 1 : PetscCall(DMCompositeRestoreLocalVectors(user->dmpgrid,&Xgen,&Xnet));
235 1 : PetscFunctionReturn(PETSC_SUCCESS);
236 : }
237 :
238 1 : PetscErrorCode PreallocateJacobian(Mat J,Userctx *user)
239 : {
240 1 : PetscInt *d_nnz;
241 1 : PetscInt i,idx=0,start=0;
242 1 : PetscInt ncols;
243 :
244 1 : PetscFunctionBegin;
245 1 : PetscCall(PetscMalloc1(user->neqs_pgrid,&d_nnz));
246 46 : for (i=0; i<user->neqs_pgrid; i++) d_nnz[i] = 0;
247 : /* Generator subsystem */
248 4 : for (i=0; i < ngen; i++) {
249 :
250 3 : d_nnz[idx] += 3;
251 3 : d_nnz[idx+1] += 2;
252 3 : d_nnz[idx+2] += 2;
253 3 : d_nnz[idx+3] += 5;
254 3 : d_nnz[idx+4] += 6;
255 3 : d_nnz[idx+5] += 6;
256 :
257 3 : d_nnz[user->neqs_gen+2*gbus[i]] += 3;
258 3 : d_nnz[user->neqs_gen+2*gbus[i]+1] += 3;
259 :
260 3 : d_nnz[idx+6] += 2;
261 3 : d_nnz[idx+7] += 2;
262 3 : d_nnz[idx+8] += 5;
263 :
264 3 : idx = idx + 9;
265 : }
266 :
267 1 : start = user->neqs_gen;
268 :
269 10 : for (i=0; i < nbus; i++) {
270 9 : PetscCall(MatGetRow(user->Ybus,2*i,&ncols,NULL,NULL));
271 9 : d_nnz[start+2*i] += ncols;
272 9 : d_nnz[start+2*i+1] += ncols;
273 9 : PetscCall(MatRestoreRow(user->Ybus,2*i,&ncols,NULL,NULL));
274 : }
275 :
276 1 : PetscCall(MatSeqAIJSetPreallocation(J,0,d_nnz));
277 :
278 1 : PetscCall(PetscFree(d_nnz));
279 1 : PetscFunctionReturn(PETSC_SUCCESS);
280 : }
281 :
282 : /*
283 : J = [-df_dx, -df_dy
284 : dg_dx, dg_dy]
285 : */
286 1 : PetscErrorCode ResidualJacobian(Vec X,Mat J,void *ctx)
287 : {
288 1 : Userctx *user=(Userctx*)ctx;
289 1 : Vec Xgen,Xnet;
290 1 : PetscScalar *xgen,*xnet;
291 1 : PetscInt i,idx=0;
292 1 : PetscScalar Vr,Vi,Vm,Vm2;
293 1 : PetscScalar Eqp,Edp,delta; /* Generator variables */
294 1 : PetscScalar Efd;
295 1 : PetscScalar Id,Iq; /* Generator dq axis currents */
296 1 : PetscScalar Vd,Vq;
297 1 : PetscScalar val[10];
298 1 : PetscInt row[2],col[10];
299 1 : PetscInt net_start=user->neqs_gen;
300 1 : PetscScalar Zdq_inv[4],det;
301 1 : PetscScalar dVd_dVr,dVd_dVi,dVq_dVr,dVq_dVi,dVd_ddelta,dVq_ddelta;
302 1 : PetscScalar dIGr_ddelta,dIGi_ddelta,dIGr_dId,dIGr_dIq,dIGi_dId,dIGi_dIq;
303 1 : PetscScalar dSE_dEfd;
304 1 : PetscScalar dVm_dVd,dVm_dVq,dVm_dVr,dVm_dVi;
305 1 : PetscInt ncols;
306 1 : const PetscInt *cols;
307 1 : const PetscScalar *yvals;
308 1 : PetscInt k;
309 1 : PetscScalar PD,QD,Vm0,*v0,Vm4;
310 1 : PetscScalar dPD_dVr,dPD_dVi,dQD_dVr,dQD_dVi;
311 1 : PetscScalar dIDr_dVr,dIDr_dVi,dIDi_dVr,dIDi_dVi;
312 :
313 1 : PetscFunctionBegin;
314 1 : PetscCall(MatZeroEntries(J));
315 1 : PetscCall(DMCompositeGetLocalVectors(user->dmpgrid,&Xgen,&Xnet));
316 1 : PetscCall(DMCompositeScatter(user->dmpgrid,X,Xgen,Xnet));
317 :
318 1 : PetscCall(VecGetArray(Xgen,&xgen));
319 1 : PetscCall(VecGetArray(Xnet,&xnet));
320 :
321 : /* Generator subsystem */
322 4 : for (i=0; i < ngen; i++) {
323 3 : Eqp = xgen[idx];
324 3 : Edp = xgen[idx+1];
325 3 : delta = xgen[idx+2];
326 3 : Id = xgen[idx+4];
327 3 : Iq = xgen[idx+5];
328 3 : Efd = xgen[idx+6];
329 :
330 : /* fgen[idx] = (Eqp + (Xd[i] - Xdp[i])*Id - Efd)/Td0p[i]; */
331 3 : row[0] = idx;
332 3 : col[0] = idx; col[1] = idx+4; col[2] = idx+6;
333 3 : val[0] = 1/ Td0p[i]; val[1] = (Xd[i] - Xdp[i])/ Td0p[i]; val[2] = -1/Td0p[i];
334 :
335 3 : PetscCall(MatSetValues(J,1,row,3,col,val,INSERT_VALUES));
336 :
337 : /* fgen[idx+1] = (Edp - (Xq[i] - Xqp[i])*Iq)/Tq0p[i]; */
338 3 : row[0] = idx + 1;
339 3 : col[0] = idx + 1; col[1] = idx+5;
340 3 : val[0] = 1/Tq0p[i]; val[1] = -(Xq[i] - Xqp[i])/Tq0p[i];
341 3 : PetscCall(MatSetValues(J,1,row,2,col,val,INSERT_VALUES));
342 :
343 : /* fgen[idx+2] = - w + w_s; */
344 3 : row[0] = idx + 2;
345 3 : col[0] = idx + 2; col[1] = idx + 3;
346 3 : val[0] = 0; val[1] = -1;
347 3 : PetscCall(MatSetValues(J,1,row,2,col,val,INSERT_VALUES));
348 :
349 : /* fgen[idx+3] = (-TM[i] + Edp*Id + Eqp*Iq + (Xqp[i] - Xdp[i])*Id*Iq + D[i]*(w - w_s))/M[i]; */
350 3 : row[0] = idx + 3;
351 3 : col[0] = idx; col[1] = idx + 1; col[2] = idx + 3; col[3] = idx + 4; col[4] = idx + 5;
352 3 : val[0] = Iq/M[i]; val[1] = Id/M[i]; val[2] = D[i]/M[i]; val[3] = (Edp + (Xqp[i]-Xdp[i])*Iq)/M[i]; val[4] = (Eqp + (Xqp[i] - Xdp[i])*Id)/M[i];
353 3 : PetscCall(MatSetValues(J,1,row,5,col,val,INSERT_VALUES));
354 :
355 3 : Vr = xnet[2*gbus[i]]; /* Real part of generator terminal voltage */
356 3 : Vi = xnet[2*gbus[i]+1]; /* Imaginary part of the generator terminal voltage */
357 3 : PetscCall(ri2dq(Vr,Vi,delta,&Vd,&Vq));
358 :
359 3 : det = Rs[i]*Rs[i] + Xdp[i]*Xqp[i];
360 :
361 3 : Zdq_inv[0] = Rs[i]/det;
362 3 : Zdq_inv[1] = Xqp[i]/det;
363 3 : Zdq_inv[2] = -Xdp[i]/det;
364 3 : Zdq_inv[3] = Rs[i]/det;
365 :
366 3 : dVd_dVr = PetscSinScalar(delta); dVd_dVi = -PetscCosScalar(delta);
367 3 : dVq_dVr = PetscCosScalar(delta); dVq_dVi = PetscSinScalar(delta);
368 3 : dVd_ddelta = Vr*PetscCosScalar(delta) + Vi*PetscSinScalar(delta);
369 3 : dVq_ddelta = -Vr*PetscSinScalar(delta) + Vi*PetscCosScalar(delta);
370 :
371 : /* fgen[idx+4] = Zdq_inv[0]*(-Edp + Vd) + Zdq_inv[1]*(-Eqp + Vq) + Id; */
372 3 : row[0] = idx+4;
373 3 : col[0] = idx; col[1] = idx+1; col[2] = idx + 2;
374 3 : val[0] = -Zdq_inv[1]; val[1] = -Zdq_inv[0]; val[2] = Zdq_inv[0]*dVd_ddelta + Zdq_inv[1]*dVq_ddelta;
375 3 : col[3] = idx + 4; col[4] = net_start+2*gbus[i]; col[5] = net_start + 2*gbus[i]+1;
376 3 : val[3] = 1; val[4] = Zdq_inv[0]*dVd_dVr + Zdq_inv[1]*dVq_dVr; val[5] = Zdq_inv[0]*dVd_dVi + Zdq_inv[1]*dVq_dVi;
377 3 : PetscCall(MatSetValues(J,1,row,6,col,val,INSERT_VALUES));
378 :
379 : /* fgen[idx+5] = Zdq_inv[2]*(-Edp + Vd) + Zdq_inv[3]*(-Eqp + Vq) + Iq; */
380 3 : row[0] = idx+5;
381 3 : col[0] = idx; col[1] = idx+1; col[2] = idx + 2;
382 3 : val[0] = -Zdq_inv[3]; val[1] = -Zdq_inv[2]; val[2] = Zdq_inv[2]*dVd_ddelta + Zdq_inv[3]*dVq_ddelta;
383 3 : col[3] = idx + 5; col[4] = net_start+2*gbus[i]; col[5] = net_start + 2*gbus[i]+1;
384 3 : val[3] = 1; val[4] = Zdq_inv[2]*dVd_dVr + Zdq_inv[3]*dVq_dVr; val[5] = Zdq_inv[2]*dVd_dVi + Zdq_inv[3]*dVq_dVi;
385 3 : PetscCall(MatSetValues(J,1,row,6,col,val,INSERT_VALUES));
386 :
387 3 : dIGr_ddelta = Id*PetscCosScalar(delta) - Iq*PetscSinScalar(delta);
388 3 : dIGi_ddelta = Id*PetscSinScalar(delta) + Iq*PetscCosScalar(delta);
389 3 : dIGr_dId = PetscSinScalar(delta); dIGr_dIq = PetscCosScalar(delta);
390 3 : dIGi_dId = -PetscCosScalar(delta); dIGi_dIq = PetscSinScalar(delta);
391 :
392 : /* fnet[2*gbus[i]] -= IGi; */
393 3 : row[0] = net_start + 2*gbus[i];
394 3 : col[0] = idx+2; col[1] = idx + 4; col[2] = idx + 5;
395 3 : val[0] = -dIGi_ddelta; val[1] = -dIGi_dId; val[2] = -dIGi_dIq;
396 3 : PetscCall(MatSetValues(J,1,row,3,col,val,INSERT_VALUES));
397 :
398 : /* fnet[2*gbus[i]+1] -= IGr; */
399 3 : row[0] = net_start + 2*gbus[i]+1;
400 3 : col[0] = idx+2; col[1] = idx + 4; col[2] = idx + 5;
401 3 : val[0] = -dIGr_ddelta; val[1] = -dIGr_dId; val[2] = -dIGr_dIq;
402 3 : PetscCall(MatSetValues(J,1,row,3,col,val,INSERT_VALUES));
403 :
404 3 : Vm = PetscSqrtScalar(Vd*Vd + Vq*Vq); Vm2 = Vm*Vm;
405 :
406 : /* fgen[idx+6] = (KE[i]*Efd + SE - VR)/TE[i]; */
407 : /* SE = k1[i]*PetscExpScalar(k2[i]*Efd); */
408 :
409 3 : dSE_dEfd = k1[i]*k2[i]*PetscExpScalar(k2[i]*Efd);
410 :
411 3 : row[0] = idx + 6;
412 3 : col[0] = idx + 6; col[1] = idx + 8;
413 3 : val[0] = (KE[i] + dSE_dEfd)/TE[i]; val[1] = -1/TE[i];
414 3 : PetscCall(MatSetValues(J,1,row,2,col,val,INSERT_VALUES));
415 :
416 : /* Exciter differential equations */
417 :
418 : /* fgen[idx+7] = (RF - KF[i]*Efd/TF[i])/TF[i]; */
419 3 : row[0] = idx + 7;
420 3 : col[0] = idx + 6; col[1] = idx + 7;
421 3 : val[0] = (-KF[i]/TF[i])/TF[i]; val[1] = 1/TF[i];
422 3 : PetscCall(MatSetValues(J,1,row,2,col,val,INSERT_VALUES));
423 :
424 : /* fgen[idx+8] = (VR - KA[i]*RF + KA[i]*KF[i]*Efd/TF[i] - KA[i]*(Vref[i] - Vm))/TA[i]; */
425 : /* Vm = (Vd^2 + Vq^2)^0.5; */
426 :
427 3 : dVm_dVd = Vd/Vm; dVm_dVq = Vq/Vm;
428 3 : dVm_dVr = dVm_dVd*dVd_dVr + dVm_dVq*dVq_dVr;
429 3 : dVm_dVi = dVm_dVd*dVd_dVi + dVm_dVq*dVq_dVi;
430 3 : row[0] = idx + 8;
431 3 : col[0] = idx + 6; col[1] = idx + 7; col[2] = idx + 8;
432 3 : val[0] = (KA[i]*KF[i]/TF[i])/TA[i]; val[1] = -KA[i]/TA[i]; val[2] = 1/TA[i];
433 3 : col[3] = net_start + 2*gbus[i]; col[4] = net_start + 2*gbus[i]+1;
434 3 : val[3] = KA[i]*dVm_dVr/TA[i]; val[4] = KA[i]*dVm_dVi/TA[i];
435 3 : PetscCall(MatSetValues(J,1,row,5,col,val,INSERT_VALUES));
436 3 : idx = idx + 9;
437 : }
438 :
439 10 : for (i=0; i<nbus; i++) {
440 9 : PetscCall(MatGetRow(user->Ybus,2*i,&ncols,&cols,&yvals));
441 9 : row[0] = net_start + 2*i;
442 63 : for (k=0; k<ncols; k++) {
443 54 : col[k] = net_start + cols[k];
444 54 : val[k] = yvals[k];
445 : }
446 9 : PetscCall(MatSetValues(J,1,row,ncols,col,val,INSERT_VALUES));
447 9 : PetscCall(MatRestoreRow(user->Ybus,2*i,&ncols,&cols,&yvals));
448 :
449 9 : PetscCall(MatGetRow(user->Ybus,2*i+1,&ncols,&cols,&yvals));
450 9 : row[0] = net_start + 2*i+1;
451 63 : for (k=0; k<ncols; k++) {
452 54 : col[k] = net_start + cols[k];
453 54 : val[k] = yvals[k];
454 : }
455 9 : PetscCall(MatSetValues(J,1,row,ncols,col,val,INSERT_VALUES));
456 9 : PetscCall(MatRestoreRow(user->Ybus,2*i+1,&ncols,&cols,&yvals));
457 : }
458 :
459 1 : PetscCall(MatAssemblyBegin(J,MAT_FLUSH_ASSEMBLY));
460 1 : PetscCall(MatAssemblyEnd(J,MAT_FLUSH_ASSEMBLY));
461 :
462 1 : PetscCall(VecGetArray(user->V0,&v0));
463 4 : for (i=0; i < nload; i++) {
464 3 : Vr = xnet[2*lbus[i]]; /* Real part of load bus voltage */
465 3 : Vi = xnet[2*lbus[i]+1]; /* Imaginary part of the load bus voltage */
466 3 : Vm = PetscSqrtScalar(Vr*Vr + Vi*Vi); Vm2 = Vm*Vm; Vm4 = Vm2*Vm2;
467 3 : Vm0 = PetscSqrtScalar(v0[2*lbus[i]]*v0[2*lbus[i]] + v0[2*lbus[i]+1]*v0[2*lbus[i]+1]);
468 3 : PD = QD = 0.0;
469 3 : dPD_dVr = dPD_dVi = dQD_dVr = dQD_dVi = 0.0;
470 12 : for (k=0; k < ld_nsegsp[i]; k++) {
471 9 : PD += ld_alphap[k]*PD0[i]*PetscPowScalar((Vm/Vm0),ld_betap[k]);
472 9 : dPD_dVr += ld_alphap[k]*ld_betap[k]*PD0[i]*PetscPowScalar((1/Vm0),ld_betap[k])*Vr*PetscPowScalar(Vm,(ld_betap[k]-2));
473 9 : dPD_dVi += ld_alphap[k]*ld_betap[k]*PD0[i]*PetscPowScalar((1/Vm0),ld_betap[k])*Vi*PetscPowScalar(Vm,(ld_betap[k]-2));
474 : }
475 12 : for (k=0; k < ld_nsegsq[i]; k++) {
476 9 : QD += ld_alphaq[k]*QD0[i]*PetscPowScalar((Vm/Vm0),ld_betaq[k]);
477 9 : dQD_dVr += ld_alphaq[k]*ld_betaq[k]*QD0[i]*PetscPowScalar((1/Vm0),ld_betaq[k])*Vr*PetscPowScalar(Vm,(ld_betaq[k]-2));
478 9 : dQD_dVi += ld_alphaq[k]*ld_betaq[k]*QD0[i]*PetscPowScalar((1/Vm0),ld_betaq[k])*Vi*PetscPowScalar(Vm,(ld_betaq[k]-2));
479 : }
480 :
481 : /* IDr = (PD*Vr + QD*Vi)/Vm2; */
482 : /* IDi = (-QD*Vr + PD*Vi)/Vm2; */
483 :
484 3 : dIDr_dVr = (dPD_dVr*Vr + dQD_dVr*Vi + PD)/Vm2 - ((PD*Vr + QD*Vi)*2*Vr)/Vm4;
485 3 : dIDr_dVi = (dPD_dVi*Vr + dQD_dVi*Vi + QD)/Vm2 - ((PD*Vr + QD*Vi)*2*Vi)/Vm4;
486 :
487 3 : dIDi_dVr = (-dQD_dVr*Vr + dPD_dVr*Vi - QD)/Vm2 - ((-QD*Vr + PD*Vi)*2*Vr)/Vm4;
488 3 : dIDi_dVi = (-dQD_dVi*Vr + dPD_dVi*Vi + PD)/Vm2 - ((-QD*Vr + PD*Vi)*2*Vi)/Vm4;
489 :
490 : /* fnet[2*lbus[i]] += IDi; */
491 3 : row[0] = net_start + 2*lbus[i];
492 3 : col[0] = net_start + 2*lbus[i]; col[1] = net_start + 2*lbus[i]+1;
493 3 : val[0] = dIDi_dVr; val[1] = dIDi_dVi;
494 3 : PetscCall(MatSetValues(J,1,row,2,col,val,ADD_VALUES));
495 : /* fnet[2*lbus[i]+1] += IDr; */
496 3 : row[0] = net_start + 2*lbus[i]+1;
497 3 : col[0] = net_start + 2*lbus[i]; col[1] = net_start + 2*lbus[i]+1;
498 3 : val[0] = dIDr_dVr; val[1] = dIDr_dVi;
499 3 : PetscCall(MatSetValues(J,1,row,2,col,val,ADD_VALUES));
500 : }
501 1 : PetscCall(VecRestoreArray(user->V0,&v0));
502 :
503 1 : PetscCall(VecRestoreArray(Xgen,&xgen));
504 1 : PetscCall(VecRestoreArray(Xnet,&xnet));
505 :
506 1 : PetscCall(DMCompositeRestoreLocalVectors(user->dmpgrid,&Xgen,&Xnet));
507 :
508 1 : PetscCall(MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY));
509 1 : PetscCall(MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY));
510 1 : PetscFunctionReturn(PETSC_SUCCESS);
511 : }
512 :
513 1 : int main(int argc,char **argv)
514 : {
515 1 : EPS eps;
516 1 : EPSType type;
517 1 : PetscMPIInt size;
518 1 : Userctx user;
519 1 : PetscViewer Xview,Ybusview;
520 1 : Vec X,Xr,Xi;
521 1 : Mat J,Jred=NULL;
522 1 : IS is0,is1;
523 1 : PetscInt i,*idx2,its,nev,nconv;
524 1 : PetscReal error,re,im;
525 1 : PetscScalar kr,ki;
526 1 : PetscBool terse;
527 :
528 1 : PetscFunctionBeginUser;
529 1 : PetscCall(SlepcInitialize(&argc,&argv,NULL,help));
530 1 : PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD,&size));
531 1 : PetscCheck(size==1,PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");
532 : /* show detailed info unless -terse option is given by user */
533 1 : PetscCall(PetscOptionsHasName(NULL,NULL,"-terse",&terse));
534 :
535 1 : user.neqs_gen = 9*ngen; /* # eqs. for generator subsystem */
536 1 : user.neqs_net = 2*nbus; /* # eqs. for network subsystem */
537 1 : user.neqs_pgrid = user.neqs_gen + user.neqs_net;
538 1 : PetscCall(PetscPrintf(PETSC_COMM_WORLD,"\nStability analysis in a network with %" PetscInt_FMT " buses and %" PetscInt_FMT " generators\n\n",nbus,ngen));
539 :
540 : /* Create indices for differential and algebraic equations */
541 1 : PetscCall(PetscMalloc1(7*ngen,&idx2));
542 4 : for (i=0; i<ngen; i++) {
543 3 : idx2[7*i] = 9*i; idx2[7*i+1] = 9*i+1; idx2[7*i+2] = 9*i+2; idx2[7*i+3] = 9*i+3;
544 3 : idx2[7*i+4] = 9*i+6; idx2[7*i+5] = 9*i+7; idx2[7*i+6] = 9*i+8;
545 : }
546 1 : PetscCall(ISCreateGeneral(PETSC_COMM_WORLD,7*ngen,idx2,PETSC_COPY_VALUES,&user.is_diff));
547 1 : PetscCall(ISComplement(user.is_diff,0,user.neqs_pgrid,&user.is_alg));
548 1 : PetscCall(PetscFree(idx2));
549 :
550 : /* Read initial voltage vector and Ybus */
551 1 : PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,"X.bin",FILE_MODE_READ,&Xview));
552 1 : PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,"Ybus.bin",FILE_MODE_READ,&Ybusview));
553 :
554 1 : PetscCall(VecCreate(PETSC_COMM_WORLD,&user.V0));
555 1 : PetscCall(VecSetSizes(user.V0,PETSC_DECIDE,user.neqs_net));
556 1 : PetscCall(VecLoad(user.V0,Xview));
557 :
558 1 : PetscCall(MatCreate(PETSC_COMM_WORLD,&user.Ybus));
559 1 : PetscCall(MatSetSizes(user.Ybus,PETSC_DECIDE,PETSC_DECIDE,user.neqs_net,user.neqs_net));
560 1 : PetscCall(MatSetType(user.Ybus,MATBAIJ));
561 : /* PetscCall(MatSetBlockSize(user.Ybus,2)); */
562 1 : PetscCall(MatLoad(user.Ybus,Ybusview));
563 :
564 1 : PetscCall(PetscViewerDestroy(&Xview));
565 1 : PetscCall(PetscViewerDestroy(&Ybusview));
566 :
567 : /* Create DMs for generator and network subsystems */
568 1 : PetscCall(DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.neqs_gen,1,1,NULL,&user.dmgen));
569 1 : PetscCall(DMSetOptionsPrefix(user.dmgen,"dmgen_"));
570 1 : PetscCall(DMSetFromOptions(user.dmgen));
571 1 : PetscCall(DMSetUp(user.dmgen));
572 1 : PetscCall(DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.neqs_net,1,1,NULL,&user.dmnet));
573 1 : PetscCall(DMSetOptionsPrefix(user.dmnet,"dmnet_"));
574 1 : PetscCall(DMSetFromOptions(user.dmnet));
575 1 : PetscCall(DMSetUp(user.dmnet));
576 :
577 : /* Create a composite DM packer and add the two DMs */
578 1 : PetscCall(DMCompositeCreate(PETSC_COMM_WORLD,&user.dmpgrid));
579 1 : PetscCall(DMSetOptionsPrefix(user.dmpgrid,"pgrid_"));
580 1 : PetscCall(DMCompositeAddDM(user.dmpgrid,user.dmgen));
581 1 : PetscCall(DMCompositeAddDM(user.dmpgrid,user.dmnet));
582 :
583 1 : PetscCall(DMCreateGlobalVector(user.dmpgrid,&X));
584 :
585 1 : PetscCall(MatCreate(PETSC_COMM_WORLD,&J));
586 1 : PetscCall(MatSetSizes(J,PETSC_DECIDE,PETSC_DECIDE,user.neqs_pgrid,user.neqs_pgrid));
587 1 : PetscCall(MatSetFromOptions(J));
588 1 : PetscCall(PreallocateJacobian(J,&user));
589 :
590 : /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
591 : Set initial conditions
592 : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
593 1 : PetscCall(SetInitialGuess(X,&user));
594 :
595 : /* Form Jacobian */
596 1 : PetscCall(ResidualJacobian(X,J,(void*)&user));
597 1 : PetscCall(MatScale(J,-1));
598 1 : is0 = user.is_diff;
599 1 : is1 = user.is_alg;
600 :
601 1 : PetscCall(MatGetSchurComplement(J,is1,is1,is0,is0,MAT_IGNORE_MATRIX,NULL,MAT_SCHUR_COMPLEMENT_AINV_DIAG,MAT_INITIAL_MATRIX,&Jred));
602 :
603 1 : if (!terse) PetscCall(MatView(Jred,NULL));
604 :
605 1 : PetscCall(MatCreateVecs(Jred,NULL,&Xr));
606 1 : PetscCall(MatCreateVecs(Jred,NULL,&Xi));
607 :
608 : /* Create the eigensolver and set the various options */
609 1 : PetscCall(EPSCreate(PETSC_COMM_WORLD,&eps));
610 1 : PetscCall(EPSSetOperators(eps,Jred,NULL));
611 1 : PetscCall(EPSSetProblemType(eps,EPS_NHEP));
612 1 : PetscCall(EPSSetFromOptions(eps));
613 :
614 : /* Solve the eigenvalue problem */
615 1 : PetscCall(EPSSolve(eps));
616 :
617 1 : PetscCall(EPSGetIterationNumber(eps,&its));
618 1 : PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the eigensolver: %" PetscInt_FMT "\n",its));
619 1 : PetscCall(EPSGetType(eps,&type));
620 1 : PetscCall(PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n", type));
621 1 : PetscCall(EPSGetDimensions(eps,&nev,NULL,NULL));
622 1 : PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %" PetscInt_FMT "\n",nev));
623 :
624 : /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
625 : Display solution and clean up
626 : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
627 1 : if (terse) PetscCall(EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL));
628 : else {
629 : /* Get number of converged approximate eigenpairs */
630 0 : PetscCall(EPSGetConverged(eps,&nconv));
631 0 : PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %" PetscInt_FMT "\n\n",nconv));
632 :
633 0 : if (nconv>0) {
634 : /* Display eigenvalues and relative errors */
635 0 : PetscCall(PetscPrintf(PETSC_COMM_WORLD,
636 : " k ||Ax-kx||/||kx||\n"
637 : " ----------------- ------------------\n"));
638 :
639 0 : for (i=0;i<nconv;i++) {
640 : /* Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
641 : ki (imaginary part) */
642 0 : PetscCall(EPSGetEigenpair(eps,i,&kr,&ki,Xr,Xi));
643 : /* Compute the relative error associated to each eigenpair */
644 0 : PetscCall(EPSComputeError(eps,i,EPS_ERROR_RELATIVE,&error));
645 :
646 : #if defined(PETSC_USE_COMPLEX)
647 : re = PetscRealPart(kr);
648 : im = PetscImaginaryPart(kr);
649 : #else
650 0 : re = kr;
651 0 : im = ki;
652 : #endif
653 0 : if (im!=0.0) PetscCall(PetscPrintf(PETSC_COMM_WORLD," %9f%+9fi %12g\n",(double)re,(double)im,(double)error));
654 0 : else PetscCall(PetscPrintf(PETSC_COMM_WORLD," %12f %12g\n",(double)re,(double)error));
655 : }
656 0 : PetscCall(PetscPrintf(PETSC_COMM_WORLD,"\n"));
657 : }
658 : }
659 :
660 : /* Free work space */
661 1 : PetscCall(EPSDestroy(&eps));
662 1 : PetscCall(MatDestroy(&J));
663 1 : PetscCall(MatDestroy(&Jred));
664 1 : PetscCall(MatDestroy(&user.Ybus));
665 1 : PetscCall(VecDestroy(&X));
666 1 : PetscCall(VecDestroy(&Xr));
667 1 : PetscCall(VecDestroy(&Xi));
668 1 : PetscCall(VecDestroy(&user.V0));
669 1 : PetscCall(DMDestroy(&user.dmgen));
670 1 : PetscCall(DMDestroy(&user.dmnet));
671 1 : PetscCall(DMDestroy(&user.dmpgrid));
672 1 : PetscCall(ISDestroy(&user.is_diff));
673 1 : PetscCall(ISDestroy(&user.is_alg));
674 1 : PetscCall(SlepcFinalize());
675 : return 0;
676 : }
677 :
678 : /*TEST
679 :
680 : build:
681 : requires: !complex
682 :
683 : test:
684 : suffix: 1
685 : args: -terse
686 : requires: double !complex !defined(PETSC_USE_64BIT_INDICES)
687 : localrunfiles: X.bin Ybus.bin
688 :
689 : TEST*/
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