def network_observability(topology, meas_idx, V): from pypower.api import makeYbus from pypower.bustypes import bustypes from pypower.dSbus_dV import dSbus_dV from pypower.dSbr_dV import dSbr_dV # build admittances Ybus,Yfrom,Yto = makeYbus(topology["baseMVA"],topology["bus"],topology["branch"]) Ybus, Yfrom, Yto = Ybus.toarray(), Yfrom.toarray(), Yto.toarray() Nk = Ybus.shape[0] # get non-reference buses ref,pv,pq = bustypes(topology["bus"],topology["gen"]) nonref = np.r_[pv,pq] gbus = topology["gen"][:,GEN_BUS].astype(int) # calculate partial derivative dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus,V) dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = dSbr_dV(topology["branch"],Yfrom,Yto,V) # create Jacobian matrix ## submatrices related to line flow dPf_Va = np.real(dSf_dVa)[meas_idx["Pf"],:][:,nonref] dPf_Vm = np.real(dSf_dVm)[meas_idx["Pf"],:][:,nonref] dPt_Va = np.real(dSt_dVa)[meas_idx["Pt"],:][:,nonref] dPt_Vm = np.real(dSt_dVm)[meas_idx["Pt"],:][:,nonref] dQf_Va = np.imag(dSf_dVa)[meas_idx["Qf"],:][:,nonref] dQf_Vm = np.imag(dSf_dVm)[meas_idx["Qf"],:][:,nonref] dQt_Va = np.imag(dSt_dVa)[meas_idx["Qt"],:][:,nonref] dQt_Vm = np.imag(dSt_dVm)[meas_idx["Qt"],:][:,nonref] ## submatrix related to generator output dPg_Va = np.real(dSbus_dVa)[gbus,:][meas_idx["Pg"],:][:,nonref] dPg_Vm = np.real(dSbus_dVm)[gbus,:][meas_idx["Pg"],:][:,nonref] dQg_Va = np.imag(dSbus_dVa)[gbus,:][meas_idx["Qg"],:][:,nonref] dQg_Vm = np.imag(dSbus_dVm)[gbus,:][meas_idx["Qg"],:][:,nonref] ## submatrix related to bus injection dPk_Va = np.real(dSbus_dVa)[meas_idx["Pk"],:][:,nonref] dPk_Vm = np.real(dSbus_dVm)[meas_idx["Pk"],:][:,nonref] dQk_Va = np.imag(dSbus_dVa)[meas_idx["Qk"],:][:,nonref] dQk_Vm = np.imag(dSbus_dVm)[meas_idx["Qk"],:][:,nonref] ## submatrix related to voltage angle dVa_Va = np.eye(Nk)[meas_idx["Va"],:][:,nonref] dVa_Vm = np.zeros((Nk,Nk))[meas_idx["Va"],:][:,nonref] ## submatrix related to voltage magnitude dVm_Va = np.zeros((Nk,Nk))[meas_idx["Vm"],:][:,nonref] dVm_Vm = np.eye(Nk)[meas_idx["Vm"],:][:,nonref] H = np.r_[np.c_[dPf_Va, dPf_Vm],\ np.c_[dPt_Va, dPt_Vm],\ np.c_[dPg_Va, dPg_Vm],\ np.c_[dQf_Va, dQf_Vm],\ np.c_[dQt_Va, dQt_Vm],\ np.c_[dQg_Va, dQg_Vm],\ np.c_[dPk_Va, dPk_Vm],\ np.c_[dQk_Va, dQk_Vm],\ np.c_[dVa_Va, dVa_Vm],\ np.c_[dVm_Va, dVm_Vm]] return isobservable(H,pv,pq)
def voltage2power(V,topology):
"""Calculation of nodal and line power from complex nodal voltages
Returns dictionary of corresponding powers and a dictionary with the partial derivatives
:param V: numpy array of complex nodal voltages
:param topology: dict in pypower casefile format
:returns: dict of calculated power values, dict of jacobians if V is one-dimensional
"""
from pypower.idx_brch import F_BUS,T_BUS
from pypower.idx_gen import GEN_BUS
from pypower.makeYbus import makeYbus
from pypower.idx_bus import PD,QD
from pypower.dSbus_dV import dSbus_dV
from pypower.dSbr_dV import dSbr_dV
assert(str(V.dtype)[:7]=="complex")
list_f = topology["branch"][:,F_BUS].tolist()
list_t = topology["branch"][:,T_BUS].tolist()
Ybus,Yfrom,Yto = makeYbus(topology["baseMVA"],topology["bus"],topology["branch"])
Ybus, Yfrom, Yto = Ybus.toarray(), Yfrom.toarray(), Yto.toarray()
powers = {}
if len(V.shape)==1:
powers["Sf"] = V[list_f]*np.conj(np.dot(Yfrom,V))
powers["St"] = V[list_t]*np.conj(np.dot(Yto,V))
gbus = topology["gen"][:,GEN_BUS].astype(int)
Sgbus= V[gbus]*np.conj(np.dot(Ybus[gbus,:],V))
powers["Sg"] = ( Sgbus*topology["baseMVA"] + topology["bus"][gbus,PD] + 1j*topology["bus"][gbus,QD] ) / topology["baseMVA"]
powers["Sk"] = V*np.conj(np.dot(Ybus,V))
else:
powers = dict([(name,[]) for name in ["Sf","St","Sg","Sk"]])
for k in range(V.shape[0]):
powers["Sf"].append(V[k,list_f]*np.conj(np.dot(Yfrom,V[k,:])))
powers["St"].append(V[k,list_t]*np.conj(np.dot(Yto,V[k,:])))
gbus = topology["gen"][:,GEN_BUS].astype(int)
Sgbus= V[k,gbus]*np.conj(np.dot(Ybus[gbus,:],V[k,:]))
powers["Sg"].append(( Sgbus*topology["baseMVA"] + topology["bus"][gbus,PD] + 1j*topology["bus"][gbus,QD] ) / topology["baseMVA"])
powers["Sk"].append(V[k,:]*np.conj(np.dot(Ybus,V[k,:])))
for key in powers.keys():
powers[key] = np.asarray(powers[key])
# calculate partial derivative
jacobians = dict([])
if len(V.shape)==1:
jacobians["dSbus_dVm"], jacobians["dSbus_dVa"] = dSbus_dV(Ybus,V)
jacobians["dSf_dVa"], jacobians["dSf_dVm"], jacobians["dSt_dVa"], \
jacobians["dSt_dVm"], jacobians["Sf"], jacobians["St"] = dSbr_dV(topology["branch"],Yfrom,Yto,V)
return powers, jacobians
def grad(self, V):
"""
Calculate Jacobi matrix given parameters vector V
Name is such due to compatability with Conjugated Gradients Class
"""
dS_dVm, dS_dVa = dSbus_dV(self.Y, V)
J11 = dS_dVa[self.pvpq[:, None], self.pvpq].real
J12 = dS_dVm[self.pvpq[:, None], self.pq].real
J21 = dS_dVa[self.pq[:, None], self.pvpq].imag
J22 = dS_dVm[self.pq[:, None], self.pq].imag
J = vstack([hstack([J11, J12]), hstack([J21, J22])], format="csr")
return J
def admmopf_consfcn(self, x=None):
if x is None:
x = self.pb['x0']
nx = len(x)
nb = self.region['nb']
ng = self.region['ng']
iv = self.idx['var']
bus = self.region['bus']
gen = self.region['gen']
Ybus = self.region['Ybus']
ridx = self.idx['rbus']['int']
# idx ranges
iVa = iv['iVa']
iVm = iv['iVm']
iPg = iv['iPg']
iQg = iv['iQg']
# grab Pg and Qg
gen[:, PG] = x[iPg]
gen[:, QG] = x[iQg]
# rebuid Sbus
Sbus = makeSbus(1, bus, gen)
# reconstruct V
Va, Vm = x[iVa], x[iVm]
V = Vm * exp(1j * Va)
# evaluate power flow equations
mis = V * conj(Ybus * V) - Sbus
g = r_[mis.real, mis.imag]
row = ridx + [i + nb for i in ridx]
g = g[row]
# ---- evaluate constraint gradients -------
# compute partials of injected bus powers
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V) # w.r.t. V
neg_Cg = sparse((-ones(ng), (gen[:, GEN_BUS], range(ng))), (nb, ng))
# construct Jacobian of equality constraints (power flow) and transpose it
dg = lil_matrix((2 * nb, nx))
blank = sparse((nb, ng))
dg = vstack([ \
#P mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.real, dSbus_dVm.real, neg_Cg, blank]),
# Q mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.imag, dSbus_dVm.imag, blank, neg_Cg])
], "csr")
dg = dg[row, :]
dg = dg.T
h = None
dh = None
return h, g, dh, dg
def cpf_predictor(V, lam, Ybus, Sxfr, pv, pq, step, z, Vprv, lamprv,
parameterization):
# sizes
pvpq = r_[pv, pq]
nb = len(V)
npv = len(pv)
npq = len(pq)
# compute Jacobian for the power flow equations
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V)
j11 = dSbus_dVa[array([pvpq]).T, pvpq].real
j12 = dSbus_dVm[array([pvpq]).T, pq].real
j21 = dSbus_dVa[array([pq]).T, pvpq].imag
j22 = dSbus_dVm[array([pq]).T, pq].imag
J = vstack([hstack([j11, j12]), hstack([j21, j22])], format="csr")
dF_dlam = -r_[Sxfr[pvpq].real, Sxfr[pq].imag].reshape((-1, 1))
dP_dV, dP_dlam = cpf_p_jac(parameterization, z, V, lam, Vprv, lamprv, pv,
pq)
# linear operator for computing the tangent predictor
J = vstack([hstack([J, dF_dlam]), hstack([dP_dV, dP_dlam])], format="csr")
Vaprv = angle(V)
Vmprv = abs(V)
# compute normalized tangent predictor
s = zeros(npv + 2 * npq + 1)
s[-1] = 1
z[r_[pvpq, nb + pq, 2 * nb]] = spsolve(J, s)
z = z / linalg.norm(z)
Va0 = Vaprv
Vm0 = Vmprv
lam0 = lam
# prediction for next step
Va0[pvpq] = Vaprv[pvpq] + step * z[pvpq]
Vm0[pq] = Vmprv[pq] + step * z[nb + pq]
lam0 = lam + step * z[2 * nb]
V0 = Vm0 * exp(1j * Va0)
return V0, lam0, z
def newtonpf(Ybus, Sbus, V0, ref, pv, pq, ppopt=None):
"""Solves the power flow using a full Newton's method.
Solves for bus voltages given the full system admittance matrix (for
all buses), the complex bus power injection vector (for all buses),
the initial vector of complex bus voltages, and column vectors with
the lists of bus indices for the swing bus, PV buses, and PQ buses,
respectively. The bus voltage vector contains the set point for
generator (including ref bus) buses, and the reference angle of the
swing bus, as well as an initial guess for remaining magnitudes and
angles. C{ppopt} is a PYPOWER options vector which can be used to
set the termination tolerance, maximum number of iterations, and
output options (see L{ppoption} for details). Uses default options if
this parameter is not given. Returns the final complex voltages, a
flag which indicates whether it converged or not, and the number of
iterations performed.
@see: L{runpf}
@author: Ray Zimmerman (PSERC Cornell)
@author: Richard Lincoln
"""
## default arguments
if ppopt is None:
ppopt = ppoption()
## options
tol = ppopt['PF_TOL']
max_it = ppopt['PF_MAX_IT']
verbose = ppopt['VERBOSE']
## initialize
converged = 0
i = 0
V = V0
Va = angle(V)
Vm = abs(V)
## set up indexing for updating V
pvpq = r_[pv, pq]
npv = len(pv)
npq = len(pq)
j1 = 0; j2 = npv ## j1:j2 - V angle of pv buses
j3 = j2; j4 = j2 + npq ## j3:j4 - V angle of pq buses
j5 = j4; j6 = j4 + npq ## j5:j6 - V mag of pq buses
## evaluate F(x0)
mis = V * conj(Ybus * V) - Sbus
F = r_[ mis[pv].real,
mis[pq].real,
mis[pq].imag ]
## check tolerance
normF = linalg.norm(F, Inf)
if verbose > 1:
sys.stdout.write('\n it max P & Q mismatch (p.u.)')
sys.stdout.write('\n---- ---------------------------')
sys.stdout.write('\n%3d %10.3e' % (i, normF))
if normF < tol:
converged = 1
if verbose > 1:
sys.stdout.write('\nConverged!\n')
## do Newton iterations
while (not converged and i < max_it):
## update iteration counter
i = i + 1
## evaluate Jacobian
dS_dVm, dS_dVa = dSbus_dV(Ybus, V)
J11 = dS_dVa[array([pvpq]).T, pvpq].real
J12 = dS_dVm[array([pvpq]).T, pq].real
J21 = dS_dVa[array([pq]).T, pvpq].imag
J22 = dS_dVm[array([pq]).T, pq].imag
J = vstack([
hstack([J11, J12]),
hstack([J21, J22])
], format="csr")
## compute update step
dx = -1 * spsolve(J, F)
## update voltage
if npv:
Va[pv] = Va[pv] + dx[j1:j2]
if npq:
Va[pq] = Va[pq] + dx[j3:j4]
Vm[pq] = Vm[pq] + dx[j5:j6]
V = Vm * exp(1j * Va)
Vm = abs(V) ## update Vm and Va again in case
Va = angle(V) ## we wrapped around with a negative Vm
## evalute F(x)
mis = V * conj(Ybus * V) - Sbus
F = r_[ mis[pv].real,
mis[pq].real,
mis[pq].imag ]
## check for convergence
normF = linalg.norm(F, Inf)
if verbose > 1:
sys.stdout.write('\n%3d %10.3e' % (i, normF))
if normF < tol:
converged = 1
if verbose:
sys.stdout.write("\nNewton's method power flow converged in "
"%d iterations.\n" % i)
if verbose:
if not converged:
sys.stdout.write("\nNewton's method power did not converge in %d "
"iterations.\n" % i)
return V, converged, i
def t_hessian(quiet=False):
"""Numerical tests of 2nd derivative code.
@author: Ray Zimmerman (PSERC Cornell)
"""
t_begin(44, quiet)
## run powerflow to get solved case
ppopt = ppoption(VERBOSE=0, OUT_ALL=0)
results, _ = runpf(case30(), ppopt)
baseMVA, bus, gen, branch = \
results['baseMVA'], results['bus'], results['gen'], results['branch']
## switch to internal bus numbering and build admittance matrices
_, bus, gen, branch = ext2int1(bus, gen, branch)
Ybus, Yf, Yt = makeYbus(baseMVA, bus, branch)
Vm = bus[:, VM]
Va = bus[:, VA] * (pi / 180)
V = Vm * exp(1j * Va)
f = branch[:, F_BUS] ## list of "from" buses
t = branch[:, T_BUS] ## list of "to" buses
nl = len(f)
nb = len(V)
Cf = sparse((ones(nl), (range(nl), f)), (nl, nb)) ## connection matrix for line & from buses
Ct = sparse((ones(nl), (range(nl), t)), (nl, nb)) ## connection matrix for line & to buses
pert = 1e-8
##----- check d2Sbus_dV2 code -----
t = ' - d2Sbus_dV2 (complex power injections)'
lam = 10 * random.rand(nb)
num_Haa = zeros((nb, nb), complex)
num_Hav = zeros((nb, nb), complex)
num_Hva = zeros((nb, nb), complex)
num_Hvv = zeros((nb, nb), complex)
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V)
Haa, Hav, Hva, Hvv = d2Sbus_dV2(Ybus, V, lam)
for i in range(nb):
Vap = V.copy()
Vap[i] = Vm[i] * exp(1j * (Va[i] + pert))
dSbus_dVm_ap, dSbus_dVa_ap = dSbus_dV(Ybus, Vap)
num_Haa[:, i] = (dSbus_dVa_ap - dSbus_dVa).T * lam / pert
num_Hva[:, i] = (dSbus_dVm_ap - dSbus_dVm).T * lam / pert
Vmp = V.copy()
Vmp[i] = (Vm[i] + pert) * exp(1j * Va[i])
dSbus_dVm_mp, dSbus_dVa_mp = dSbus_dV(Ybus, Vmp)
num_Hav[:, i] = (dSbus_dVa_mp - dSbus_dVa).T * lam / pert
num_Hvv[:, i] = (dSbus_dVm_mp - dSbus_dVm).T * lam / pert
t_is(Haa.todense(), num_Haa, 4, ['Haa', t])
t_is(Hav.todense(), num_Hav, 4, ['Hav', t])
t_is(Hva.todense(), num_Hva, 4, ['Hva', t])
t_is(Hvv.todense(), num_Hvv, 4, ['Hvv', t])
##----- check d2Sbr_dV2 code -----
t = ' - d2Sbr_dV2 (complex power flows)'
lam = 10 * random.rand(nl)
# lam = [1 zeros(nl-1, 1)]
num_Gfaa = zeros((nb, nb), complex)
num_Gfav = zeros((nb, nb), complex)
num_Gfva = zeros((nb, nb), complex)
num_Gfvv = zeros((nb, nb), complex)
num_Gtaa = zeros((nb, nb), complex)
num_Gtav = zeros((nb, nb), complex)
num_Gtva = zeros((nb, nb), complex)
num_Gtvv = zeros((nb, nb), complex)
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, _, _ = dSbr_dV(branch, Yf, Yt, V)
Gfaa, Gfav, Gfva, Gfvv = d2Sbr_dV2(Cf, Yf, V, lam)
Gtaa, Gtav, Gtva, Gtvv = d2Sbr_dV2(Ct, Yt, V, lam)
for i in range(nb):
Vap = V.copy()
Vap[i] = Vm[i] * exp(1j * (Va[i] + pert))
dSf_dVa_ap, dSf_dVm_ap, dSt_dVa_ap, dSt_dVm_ap, Sf_ap, St_ap = \
dSbr_dV(branch, Yf, Yt, Vap)
num_Gfaa[:, i] = (dSf_dVa_ap - dSf_dVa).T * lam / pert
num_Gfva[:, i] = (dSf_dVm_ap - dSf_dVm).T * lam / pert
num_Gtaa[:, i] = (dSt_dVa_ap - dSt_dVa).T * lam / pert
num_Gtva[:, i] = (dSt_dVm_ap - dSt_dVm).T * lam / pert
Vmp = V.copy()
Vmp[i] = (Vm[i] + pert) * exp(1j * Va[i])
dSf_dVa_mp, dSf_dVm_mp, dSt_dVa_mp, dSt_dVm_mp, Sf_mp, St_mp = \
dSbr_dV(branch, Yf, Yt, Vmp)
num_Gfav[:, i] = (dSf_dVa_mp - dSf_dVa).T * lam / pert
num_Gfvv[:, i] = (dSf_dVm_mp - dSf_dVm).T * lam / pert
num_Gtav[:, i] = (dSt_dVa_mp - dSt_dVa).T * lam / pert
num_Gtvv[:, i] = (dSt_dVm_mp - dSt_dVm).T * lam / pert
t_is(Gfaa.todense(), num_Gfaa, 4, ['Gfaa', t])
t_is(Gfav.todense(), num_Gfav, 4, ['Gfav', t])
t_is(Gfva.todense(), num_Gfva, 4, ['Gfva', t])
t_is(Gfvv.todense(), num_Gfvv, 4, ['Gfvv', t])
t_is(Gtaa.todense(), num_Gtaa, 4, ['Gtaa', t])
t_is(Gtav.todense(), num_Gtav, 4, ['Gtav', t])
t_is(Gtva.todense(), num_Gtva, 4, ['Gtva', t])
t_is(Gtvv.todense(), num_Gtvv, 4, ['Gtvv', t])
##----- check d2Ibr_dV2 code -----
t = ' - d2Ibr_dV2 (complex currents)'
lam = 10 * random.rand(nl)
# lam = [1, zeros(nl-1)]
num_Gfaa = zeros((nb, nb), complex)
num_Gfav = zeros((nb, nb), complex)
num_Gfva = zeros((nb, nb), complex)
num_Gfvv = zeros((nb, nb), complex)
num_Gtaa = zeros((nb, nb), complex)
num_Gtav = zeros((nb, nb), complex)
num_Gtva = zeros((nb, nb), complex)
num_Gtvv = zeros((nb, nb), complex)
dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, _, _ = dIbr_dV(branch, Yf, Yt, V)
Gfaa, Gfav, Gfva, Gfvv = d2Ibr_dV2(Yf, V, lam)
Gtaa, Gtav, Gtva, Gtvv = d2Ibr_dV2(Yt, V, lam)
for i in range(nb):
Vap = V.copy()
Vap[i] = Vm[i] * exp(1j * (Va[i] + pert))
dIf_dVa_ap, dIf_dVm_ap, dIt_dVa_ap, dIt_dVm_ap, If_ap, It_ap = \
dIbr_dV(branch, Yf, Yt, Vap)
num_Gfaa[:, i] = (dIf_dVa_ap - dIf_dVa).T * lam / pert
num_Gfva[:, i] = (dIf_dVm_ap - dIf_dVm).T * lam / pert
num_Gtaa[:, i] = (dIt_dVa_ap - dIt_dVa).T * lam / pert
num_Gtva[:, i] = (dIt_dVm_ap - dIt_dVm).T * lam / pert
Vmp = V.copy()
Vmp[i] = (Vm[i] + pert) * exp(1j * Va[i])
dIf_dVa_mp, dIf_dVm_mp, dIt_dVa_mp, dIt_dVm_mp, If_mp, It_mp = \
dIbr_dV(branch, Yf, Yt, Vmp)
num_Gfav[:, i] = (dIf_dVa_mp - dIf_dVa).T * lam / pert
num_Gfvv[:, i] = (dIf_dVm_mp - dIf_dVm).T * lam / pert
num_Gtav[:, i] = (dIt_dVa_mp - dIt_dVa).T * lam / pert
num_Gtvv[:, i] = (dIt_dVm_mp - dIt_dVm).T * lam / pert
t_is(Gfaa.todense(), num_Gfaa, 4, ['Gfaa', t])
t_is(Gfav.todense(), num_Gfav, 4, ['Gfav', t])
t_is(Gfva.todense(), num_Gfva, 4, ['Gfva', t])
t_is(Gfvv.todense(), num_Gfvv, 4, ['Gfvv', t])
t_is(Gtaa.todense(), num_Gtaa, 4, ['Gtaa', t])
t_is(Gtav.todense(), num_Gtav, 4, ['Gtav', t])
t_is(Gtva.todense(), num_Gtva, 4, ['Gtva', t])
t_is(Gtvv.todense(), num_Gtvv, 4, ['Gtvv', t])
##----- check d2ASbr_dV2 code -----
t = ' - d2ASbr_dV2 (squared apparent power flows)'
lam = 10 * random.rand(nl)
# lam = [1 zeros(nl-1, 1)]
num_Gfaa = zeros((nb, nb), complex)
num_Gfav = zeros((nb, nb), complex)
num_Gfva = zeros((nb, nb), complex)
num_Gfvv = zeros((nb, nb), complex)
num_Gtaa = zeros((nb, nb), complex)
num_Gtav = zeros((nb, nb), complex)
num_Gtva = zeros((nb, nb), complex)
num_Gtvv = zeros((nb, nb), complex)
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = dSbr_dV(branch, Yf, Yt, V)
dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm = \
dAbr_dV(dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St)
Gfaa, Gfav, Gfva, Gfvv = d2ASbr_dV2(dSf_dVa, dSf_dVm, Sf, Cf, Yf, V, lam)
Gtaa, Gtav, Gtva, Gtvv = d2ASbr_dV2(dSt_dVa, dSt_dVm, St, Ct, Yt, V, lam)
for i in range(nb):
Vap = V.copy()
Vap[i] = Vm[i] * exp(1j * (Va[i] + pert))
dSf_dVa_ap, dSf_dVm_ap, dSt_dVa_ap, dSt_dVm_ap, Sf_ap, St_ap = \
dSbr_dV(branch, Yf, Yt, Vap)
dAf_dVa_ap, dAf_dVm_ap, dAt_dVa_ap, dAt_dVm_ap = \
dAbr_dV(dSf_dVa_ap, dSf_dVm_ap, dSt_dVa_ap, dSt_dVm_ap, Sf_ap, St_ap)
num_Gfaa[:, i] = (dAf_dVa_ap - dAf_dVa).T * lam / pert
num_Gfva[:, i] = (dAf_dVm_ap - dAf_dVm).T * lam / pert
num_Gtaa[:, i] = (dAt_dVa_ap - dAt_dVa).T * lam / pert
num_Gtva[:, i] = (dAt_dVm_ap - dAt_dVm).T * lam / pert
Vmp = V.copy()
Vmp[i] = (Vm[i] + pert) * exp(1j * Va[i])
dSf_dVa_mp, dSf_dVm_mp, dSt_dVa_mp, dSt_dVm_mp, Sf_mp, St_mp = \
dSbr_dV(branch, Yf, Yt, Vmp)
dAf_dVa_mp, dAf_dVm_mp, dAt_dVa_mp, dAt_dVm_mp = \
dAbr_dV(dSf_dVa_mp, dSf_dVm_mp, dSt_dVa_mp, dSt_dVm_mp, Sf_mp, St_mp)
num_Gfav[:, i] = (dAf_dVa_mp - dAf_dVa).T * lam / pert
num_Gfvv[:, i] = (dAf_dVm_mp - dAf_dVm).T * lam / pert
num_Gtav[:, i] = (dAt_dVa_mp - dAt_dVa).T * lam / pert
num_Gtvv[:, i] = (dAt_dVm_mp - dAt_dVm).T * lam / pert
t_is(Gfaa.todense(), num_Gfaa, 2, ['Gfaa', t])
t_is(Gfav.todense(), num_Gfav, 2, ['Gfav', t])
t_is(Gfva.todense(), num_Gfva, 2, ['Gfva', t])
t_is(Gfvv.todense(), num_Gfvv, 2, ['Gfvv', t])
t_is(Gtaa.todense(), num_Gtaa, 2, ['Gtaa', t])
t_is(Gtav.todense(), num_Gtav, 2, ['Gtav', t])
t_is(Gtva.todense(), num_Gtva, 2, ['Gtva', t])
t_is(Gtvv.todense(), num_Gtvv, 2, ['Gtvv', t])
##----- check d2ASbr_dV2 code -----
t = ' - d2ASbr_dV2 (squared real power flows)'
lam = 10 * random.rand(nl)
# lam = [1 zeros(nl-1, 1)]
num_Gfaa = zeros((nb, nb), complex)
num_Gfav = zeros((nb, nb), complex)
num_Gfva = zeros((nb, nb), complex)
num_Gfvv = zeros((nb, nb), complex)
num_Gtaa = zeros((nb, nb), complex)
num_Gtav = zeros((nb, nb), complex)
num_Gtva = zeros((nb, nb), complex)
num_Gtvv = zeros((nb, nb), complex)
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = dSbr_dV(branch, Yf, Yt, V)
dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm = \
dAbr_dV(dSf_dVa.real, dSf_dVm.real, dSt_dVa.real, dSt_dVm.real, Sf.real, St.real)
Gfaa, Gfav, Gfva, Gfvv = d2ASbr_dV2(dSf_dVa.real, dSf_dVm.real, Sf.real, Cf, Yf, V, lam)
Gtaa, Gtav, Gtva, Gtvv = d2ASbr_dV2(dSt_dVa.real, dSt_dVm.real, St.real, Ct, Yt, V, lam)
for i in range(nb):
Vap = V.copy()
Vap[i] = Vm[i] * exp(1j * (Va[i] + pert))
dSf_dVa_ap, dSf_dVm_ap, dSt_dVa_ap, dSt_dVm_ap, Sf_ap, St_ap = \
dSbr_dV(branch, Yf, Yt, Vap)
dAf_dVa_ap, dAf_dVm_ap, dAt_dVa_ap, dAt_dVm_ap = \
dAbr_dV(dSf_dVa_ap.real, dSf_dVm_ap.real, dSt_dVa_ap.real, dSt_dVm_ap.real, Sf_ap.real, St_ap.real)
num_Gfaa[:, i] = (dAf_dVa_ap - dAf_dVa).T * lam / pert
num_Gfva[:, i] = (dAf_dVm_ap - dAf_dVm).T * lam / pert
num_Gtaa[:, i] = (dAt_dVa_ap - dAt_dVa).T * lam / pert
num_Gtva[:, i] = (dAt_dVm_ap - dAt_dVm).T * lam / pert
Vmp = V.copy()
Vmp[i] = (Vm[i] + pert) * exp(1j * Va[i])
dSf_dVa_mp, dSf_dVm_mp, dSt_dVa_mp, dSt_dVm_mp, Sf_mp, St_mp = \
dSbr_dV(branch, Yf, Yt, Vmp)
dAf_dVa_mp, dAf_dVm_mp, dAt_dVa_mp, dAt_dVm_mp = \
dAbr_dV(dSf_dVa_mp.real, dSf_dVm_mp.real, dSt_dVa_mp.real, dSt_dVm_mp.real, Sf_mp.real, St_mp.real)
num_Gfav[:, i] = (dAf_dVa_mp - dAf_dVa).T * lam / pert
num_Gfvv[:, i] = (dAf_dVm_mp - dAf_dVm).T * lam / pert
num_Gtav[:, i] = (dAt_dVa_mp - dAt_dVa).T * lam / pert
num_Gtvv[:, i] = (dAt_dVm_mp - dAt_dVm).T * lam / pert
t_is(Gfaa.todense(), num_Gfaa, 2, ['Gfaa', t])
t_is(Gfav.todense(), num_Gfav, 2, ['Gfav', t])
t_is(Gfva.todense(), num_Gfva, 2, ['Gfva', t])
t_is(Gfvv.todense(), num_Gfvv, 2, ['Gfvv', t])
t_is(Gtaa.todense(), num_Gtaa, 2, ['Gtaa', t])
t_is(Gtav.todense(), num_Gtav, 2, ['Gtav', t])
t_is(Gtva.todense(), num_Gtva, 2, ['Gtva', t])
t_is(Gtvv.todense(), num_Gtvv, 2, ['Gtvv', t])
##----- check d2AIbr_dV2 code -----
t = ' - d2AIbr_dV2 (squared current magnitudes)'
lam = 10 * random.rand(nl)
# lam = [1 zeros(nl-1, 1)]
num_Gfaa = zeros((nb, nb), complex)
num_Gfav = zeros((nb, nb), complex)
num_Gfva = zeros((nb, nb), complex)
num_Gfvv = zeros((nb, nb), complex)
num_Gtaa = zeros((nb, nb), complex)
num_Gtav = zeros((nb, nb), complex)
num_Gtva = zeros((nb, nb), complex)
num_Gtvv = zeros((nb, nb), complex)
dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It = dIbr_dV(branch, Yf, Yt, V)
dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm = \
dAbr_dV(dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It)
Gfaa, Gfav, Gfva, Gfvv = d2AIbr_dV2(dIf_dVa, dIf_dVm, If, Yf, V, lam)
Gtaa, Gtav, Gtva, Gtvv = d2AIbr_dV2(dIt_dVa, dIt_dVm, It, Yt, V, lam)
for i in range(nb):
Vap = V.copy()
Vap[i] = Vm[i] * exp(1j * (Va[i] + pert))
dIf_dVa_ap, dIf_dVm_ap, dIt_dVa_ap, dIt_dVm_ap, If_ap, It_ap = \
dIbr_dV(branch, Yf, Yt, Vap)
dAf_dVa_ap, dAf_dVm_ap, dAt_dVa_ap, dAt_dVm_ap = \
dAbr_dV(dIf_dVa_ap, dIf_dVm_ap, dIt_dVa_ap, dIt_dVm_ap, If_ap, It_ap)
num_Gfaa[:, i] = (dAf_dVa_ap - dAf_dVa).T * lam / pert
num_Gfva[:, i] = (dAf_dVm_ap - dAf_dVm).T * lam / pert
num_Gtaa[:, i] = (dAt_dVa_ap - dAt_dVa).T * lam / pert
num_Gtva[:, i] = (dAt_dVm_ap - dAt_dVm).T * lam / pert
Vmp = V.copy()
Vmp[i] = (Vm[i] + pert) * exp(1j * Va[i])
dIf_dVa_mp, dIf_dVm_mp, dIt_dVa_mp, dIt_dVm_mp, If_mp, It_mp = \
dIbr_dV(branch, Yf, Yt, Vmp)
dAf_dVa_mp, dAf_dVm_mp, dAt_dVa_mp, dAt_dVm_mp = \
dAbr_dV(dIf_dVa_mp, dIf_dVm_mp, dIt_dVa_mp, dIt_dVm_mp, If_mp, It_mp)
num_Gfav[:, i] = (dAf_dVa_mp - dAf_dVa).T * lam / pert
num_Gfvv[:, i] = (dAf_dVm_mp - dAf_dVm).T * lam / pert
num_Gtav[:, i] = (dAt_dVa_mp - dAt_dVa).T * lam / pert
num_Gtvv[:, i] = (dAt_dVm_mp - dAt_dVm).T * lam / pert
t_is(Gfaa.todense(), num_Gfaa, 3, ['Gfaa', t])
t_is(Gfav.todense(), num_Gfav, 3, ['Gfav', t])
t_is(Gfva.todense(), num_Gfva, 3, ['Gfva', t])
t_is(Gfvv.todense(), num_Gfvv, 2, ['Gfvv', t])
t_is(Gtaa.todense(), num_Gtaa, 3, ['Gtaa', t])
t_is(Gtav.todense(), num_Gtav, 3, ['Gtav', t])
t_is(Gtva.todense(), num_Gtva, 3, ['Gtva', t])
t_is(Gtvv.todense(), num_Gtvv, 2, ['Gtvv', t])
t_end()
def opf_consfcn(x, om, Ybus, Yf, Yt, ppopt, il=None, *args):
"""Evaluates nonlinear constraints and their Jacobian for OPF.
Constraint evaluation function for AC optimal power flow, suitable
for use with L{pips}. Computes constraint vectors and their gradients.
@param x: optimization vector
@param om: OPF model object
@param Ybus: bus admittance matrix
@param Yf: admittance matrix for "from" end of constrained branches
@param Yt: admittance matrix for "to" end of constrained branches
@param ppopt: PYPOWER options vector
@param il: (optional) vector of branch indices corresponding to
branches with flow limits (all others are assumed to be
unconstrained). The default is C{range(nl)} (all branches).
C{Yf} and C{Yt} contain only the rows corresponding to C{il}.
@return: C{h} - vector of inequality constraint values (flow limits)
limit^2 - flow^2, where the flow can be apparent power real power or
current, depending on value of C{OPF_FLOW_LIM} in C{ppopt} (only for
constrained lines). C{g} - vector of equality constraint values (power
balances). C{dh} - (optional) inequality constraint gradients, column
j is gradient of h(j). C{dg} - (optional) equality constraint gradients.
@see: L{opf_costfcn}, L{opf_hessfcn}
@author: Carlos E. Murillo-Sanchez (PSERC Cornell & Universidad
Autonoma de Manizales)
@author: Ray Zimmerman (PSERC Cornell)
@author: Richard Lincoln
"""
##----- initialize -----
## unpack data
ppc = om.get_ppc()
baseMVA, bus, gen, branch = \
ppc["baseMVA"], ppc["bus"], ppc["gen"], ppc["branch"]
vv, _, _, _ = om.get_idx()
## problem dimensions
nb = bus.shape[0] ## number of buses
nl = branch.shape[0] ## number of branches
ng = gen.shape[0] ## number of dispatchable injections
nxyz = len(x) ## total number of control vars of all types
## set default constrained lines
if il is None:
il = arange(nl) ## all lines have limits by default
nl2 = len(il) ## number of constrained lines
## grab Pg & Qg
Pg = x[vv["i1"]["Pg"]:vv["iN"]["Pg"]] ## active generation in p.u.
Qg = x[vv["i1"]["Qg"]:vv["iN"]["Qg"]] ## reactive generation in p.u.
## put Pg & Qg back in gen
gen[:, PG] = Pg * baseMVA ## active generation in MW
gen[:, QG] = Qg * baseMVA ## reactive generation in MVAr
## rebuild Sbus
Sbus = makeSbus(baseMVA, bus, gen) ## net injected power in p.u.
## ----- evaluate constraints -----
## reconstruct V
Va = x[vv["i1"]["Va"]:vv["iN"]["Va"]]
Vm = x[vv["i1"]["Vm"]:vv["iN"]["Vm"]]
V = Vm * exp(1j * Va)
## evaluate power flow equations
mis = V * conj(Ybus * V) - Sbus
##----- evaluate constraint function values -----
## first, the equality constraints (power flow)
g = r_[ mis.real, ## active power mismatch for all buses
mis.imag ] ## reactive power mismatch for all buses
## then, the inequality constraints (branch flow limits)
if nl2 > 0:
flow_max = (branch[il, RATE_A] / baseMVA)**2
flow_max[flow_max == 0] = Inf
if ppopt['OPF_FLOW_LIM'] == 2: ## current magnitude limit, |I|
If = Yf * V
It = Yt * V
h = r_[ If * conj(If) - flow_max, ## branch I limits (from bus)
It * conj(It) - flow_max ].real ## branch I limits (to bus)
else:
## compute branch power flows
## complex power injected at "from" bus (p.u.)
Sf = V[ branch[il, F_BUS].astype(int) ] * conj(Yf * V)
## complex power injected at "to" bus (p.u.)
St = V[ branch[il, T_BUS].astype(int) ] * conj(Yt * V)
if ppopt['OPF_FLOW_LIM'] == 1: ## active power limit, P (Pan Wei)
h = r_[ Sf.real**2 - flow_max, ## branch P limits (from bus)
St.real**2 - flow_max ] ## branch P limits (to bus)
else: ## apparent power limit, |S|
h = r_[ Sf * conj(Sf) - flow_max, ## branch S limits (from bus)
St * conj(St) - flow_max ].real ## branch S limits (to bus)
else:
h = zeros((0,1))
##----- evaluate partials of constraints -----
## index ranges
iVa = arange(vv["i1"]["Va"], vv["iN"]["Va"])
iVm = arange(vv["i1"]["Vm"], vv["iN"]["Vm"])
iPg = arange(vv["i1"]["Pg"], vv["iN"]["Pg"])
iQg = arange(vv["i1"]["Qg"], vv["iN"]["Qg"])
iVaVmPgQg = r_[iVa, iVm, iPg, iQg].T
## compute partials of injected bus powers
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V) ## w.r.t. V
## Pbus w.r.t. Pg, Qbus w.r.t. Qg
neg_Cg = sparse((-ones(ng), (gen[:, GEN_BUS], range(ng))), (nb, ng))
## construct Jacobian of equality constraints (power flow) and transpose it
dg = lil_matrix((2 * nb, nxyz))
blank = sparse((nb, ng))
dg[:, iVaVmPgQg] = vstack([
## P mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.real, dSbus_dVm.real, neg_Cg, blank]),
## Q mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.imag, dSbus_dVm.imag, blank, neg_Cg])
], "csr")
dg = dg.T
if nl2 > 0:
## compute partials of Flows w.r.t. V
if ppopt['OPF_FLOW_LIM'] == 2: ## current
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dIbr_dV(branch[il, :], Yf, Yt, V)
else: ## power
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dSbr_dV(branch[il, :], Yf, Yt, V)
if ppopt['OPF_FLOW_LIM'] == 1: ## real part of flow (active power)
dFf_dVa = dFf_dVa.real
dFf_dVm = dFf_dVm.real
dFt_dVa = dFt_dVa.real
dFt_dVm = dFt_dVm.real
Ff = Ff.real
Ft = Ft.real
## squared magnitude of flow (of complex power or current, or real power)
df_dVa, df_dVm, dt_dVa, dt_dVm = \
dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft)
## construct Jacobian of inequality constraints (branch limits)
## and transpose it.
dh = lil_matrix((2 * nl2, nxyz))
dh[:, r_[iVa, iVm].T] = vstack([
hstack([df_dVa, df_dVm]), ## "from" flow limit
hstack([dt_dVa, dt_dVm]) ## "to" flow limit
], "csr")
dh = dh.T
else:
dh = None
return h, g, dh, dg
def newtonpf(Ybus, Sbus, V0, ref, pv, pq, ppopt=None):
"""Solves the power flow using a full Newton's method.
Solves for bus voltages given the full system admittance matrix (for
all buses), the complex bus power injection vector (for all buses),
the initial vector of complex bus voltages, and column vectors with
the lists of bus indices for the swing bus, PV buses, and PQ buses,
respectively. The bus voltage vector contains the set point for
generator (including ref bus) buses, and the reference angle of the
swing bus, as well as an initial guess for remaining magnitudes and
angles. C{ppopt} is a PYPOWER options vector which can be used to
set the termination tolerance, maximum number of iterations, and
output options (see L{ppoption} for details). Uses default options if
this parameter is not given. Returns the final complex voltages, a
flag which indicates whether it converged or not, and the number of
iterations performed.
@see: L{runpf}
@author: Ray Zimmerman (PSERC Cornell)
"""
## default arguments
if ppopt is None:
ppopt = ppoption()
## options
tol = ppopt['PF_TOL']
max_it = ppopt['PF_MAX_IT']
verbose = ppopt['VERBOSE']
## initialize
converged = 0
i = 0
V = V0
Va = angle(V)
Vm = abs(V)
## set up indexing for updating V
pvpq = r_[pv, pq]
npv = len(pv)
npq = len(pq)
j1 = 0
j2 = npv ## j1:j2 - V angle of pv buses
j3 = j2
j4 = j2 + npq ## j3:j4 - V angle of pq buses
j5 = j4
j6 = j4 + npq ## j5:j6 - V mag of pq buses
## evaluate F(x0)
mis = V * conj(Ybus * V) - Sbus
F = r_[mis[pv].real, mis[pq].real, mis[pq].imag]
## check tolerance
normF = linalg.norm(F, Inf)
if verbose > 1:
sys.stdout.write('\n it max P & Q mismatch (p.u.)')
sys.stdout.write('\n---- ---------------------------')
sys.stdout.write('\n%3d %10.3e' % (i, normF))
if normF < tol:
converged = 1
if verbose > 1:
sys.stdout.write('\nConverged!\n')
## do Newton iterations
while (not converged and i < max_it):
## update iteration counter
i = i + 1
## evaluate Jacobian
dS_dVm, dS_dVa = dSbus_dV(Ybus, V)
J11 = dS_dVa[array([pvpq]).T, pvpq].real
J12 = dS_dVm[array([pvpq]).T, pq].real
J21 = dS_dVa[array([pq]).T, pvpq].imag
J22 = dS_dVm[array([pq]).T, pq].imag
J = vstack([hstack([J11, J12]), hstack([J21, J22])], format="csr")
## compute update step
dx = -1 * spsolve(J, F)
## update voltage
if npv:
Va[pv] = Va[pv] + dx[j1:j2]
if npq:
Va[pq] = Va[pq] + dx[j3:j4]
Vm[pq] = Vm[pq] + dx[j5:j6]
V = Vm * exp(1j * Va)
Vm = abs(V) ## update Vm and Va again in case
Va = angle(V) ## we wrapped around with a negative Vm
## evalute F(x)
mis = V * conj(Ybus * V) - Sbus
F = r_[mis[pv].real, mis[pq].real, mis[pq].imag]
## check for convergence
normF = linalg.norm(F, Inf)
if verbose > 1:
sys.stdout.write('\n%3d %10.3e' % (i, normF))
if normF < tol:
converged = 1
if verbose:
sys.stdout.write("\nNewton's method power flow converged in "
"%d iterations.\n" % i)
if verbose:
if not converged:
sys.stdout.write("\nNewton's method power did not converge in %d "
"iterations.\n" % i)
return V, converged, i
def opf_consfcn(x, om, Ybus, Yf, Yt, ppopt, il=None, *args):
"""Evaluates nonlinear constraints and their Jacobian for OPF.
Constraint evaluation function for AC optimal power flow, suitable
for use with L{pips}. Computes constraint vectors and their gradients.
@param x: optimization vector
@param om: OPF model object
@param Ybus: bus admittance matrix
@param Yf: admittance matrix for "from" end of constrained branches
@param Yt: admittance matrix for "to" end of constrained branches
@param ppopt: PYPOWER options vector
@param il: (optional) vector of branch indices corresponding to
branches with flow limits (all others are assumed to be
unconstrained). The default is C{range(nl)} (all branches).
C{Yf} and C{Yt} contain only the rows corresponding to C{il}.
@return: C{h} - vector of inequality constraint values (flow limits)
limit^2 - flow^2, where the flow can be apparent power real power or
current, depending on value of C{OPF_FLOW_LIM} in C{ppopt} (only for
constrained lines). C{g} - vector of equality constraint values (power
balances). C{dh} - (optional) inequality constraint gradients, column
j is gradient of h(j). C{dg} - (optional) equality constraint gradients.
@see: L{opf_costfcn}, L{opf_hessfcn}
@author: Carlos E. Murillo-Sanchez (PSERC Cornell & Universidad
Autonoma de Manizales)
@author: Ray Zimmerman (PSERC Cornell)
"""
##----- initialize -----
## unpack data
ppc = om.get_ppc()
baseMVA, bus, gen, branch = \
ppc["baseMVA"], ppc["bus"], ppc["gen"], ppc["branch"]
vv, _, _, _ = om.get_idx()
## problem dimensions
nb = bus.shape[0] ## number of buses
nl = branch.shape[0] ## number of branches
ng = gen.shape[0] ## number of dispatchable injections
nxyz = len(x) ## total number of control vars of all types
## set default constrained lines
if il is None:
il = arange(nl) ## all lines have limits by default
nl2 = len(il) ## number of constrained lines
## grab Pg & Qg
Pg = x[vv["i1"]["Pg"]:vv["iN"]["Pg"]] ## active generation in p.u.
Qg = x[vv["i1"]["Qg"]:vv["iN"]["Qg"]] ## reactive generation in p.u.
## put Pg & Qg back in gen
gen[:, PG] = Pg * baseMVA ## active generation in MW
gen[:, QG] = Qg * baseMVA ## reactive generation in MVAr
## rebuild Sbus
Sbus = makeSbus(baseMVA, bus, gen) ## net injected power in p.u.
## ----- evaluate constraints -----
## reconstruct V
Va = x[vv["i1"]["Va"]:vv["iN"]["Va"]]
Vm = x[vv["i1"]["Vm"]:vv["iN"]["Vm"]]
V = Vm * exp(1j * Va)
## evaluate power flow equations
mis = V * conj(Ybus * V) - Sbus
##----- evaluate constraint function values -----
## first, the equality constraints (power flow)
g = r_[mis.real, ## active power mismatch for all buses
mis.imag] ## reactive power mismatch for all buses
## then, the inequality constraints (branch flow limits)
if nl2 > 0:
flow_max = (branch[il, RATE_A] / baseMVA)**2
flow_max[flow_max == 0] = Inf
if ppopt['OPF_FLOW_LIM'] == 2: ## current magnitude limit, |I|
If = Yf * V
It = Yt * V
h = r_[If * conj(If) - flow_max, ## branch I limits (from bus)
It * conj(It) - flow_max].real ## branch I limits (to bus)
else:
## compute branch power flows
## complex power injected at "from" bus (p.u.)
Sf = V[branch[il, F_BUS].astype(int)] * conj(Yf * V)
## complex power injected at "to" bus (p.u.)
St = V[branch[il, T_BUS].astype(int)] * conj(Yt * V)
if ppopt['OPF_FLOW_LIM'] == 1: ## active power limit, P (Pan Wei)
h = r_[Sf.real**2 - flow_max, ## branch P limits (from bus)
St.real**2 - flow_max] ## branch P limits (to bus)
else: ## apparent power limit, |S|
h = r_[Sf * conj(Sf) - flow_max, ## branch S limits (from bus)
St * conj(St) -
flow_max].real ## branch S limits (to bus)
else:
h = zeros((0, 1))
##----- evaluate partials of constraints -----
## index ranges
iVa = arange(vv["i1"]["Va"], vv["iN"]["Va"])
iVm = arange(vv["i1"]["Vm"], vv["iN"]["Vm"])
iPg = arange(vv["i1"]["Pg"], vv["iN"]["Pg"])
iQg = arange(vv["i1"]["Qg"], vv["iN"]["Qg"])
iVaVmPgQg = r_[iVa, iVm, iPg, iQg].T
## compute partials of injected bus powers
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V) ## w.r.t. V
## Pbus w.r.t. Pg, Qbus w.r.t. Qg
neg_Cg = sparse((-ones(ng), (gen[:, GEN_BUS], list(range(ng)))), (nb, ng))
## construct Jacobian of equality constraints (power flow) and transpose it
dg = lil_matrix((2 * nb, nxyz))
blank = sparse((nb, ng))
dg[:, iVaVmPgQg] = vstack(
[
## P mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.real, dSbus_dVm.real, neg_Cg, blank]),
## Q mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.imag, dSbus_dVm.imag, blank, neg_Cg])
],
"csr")
dg = dg.T
if nl2 > 0:
## compute partials of Flows w.r.t. V
if ppopt['OPF_FLOW_LIM'] == 2: ## current
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dIbr_dV(branch[il, :], Yf, Yt, V)
else: ## power
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dSbr_dV(branch[il, :], Yf, Yt, V)
if ppopt['OPF_FLOW_LIM'] == 1: ## real part of flow (active power)
dFf_dVa = dFf_dVa.real
dFf_dVm = dFf_dVm.real
dFt_dVa = dFt_dVa.real
dFt_dVm = dFt_dVm.real
Ff = Ff.real
Ft = Ft.real
## squared magnitude of flow (of complex power or current, or real power)
df_dVa, df_dVm, dt_dVa, dt_dVm = \
dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft)
## construct Jacobian of inequality constraints (branch limits)
## and transpose it.
dh = lil_matrix((2 * nl2, nxyz))
dh[:, r_[iVa, iVm].T] = vstack(
[
hstack([df_dVa, df_dVm]), ## "from" flow limit
hstack([dt_dVa, dt_dVm]) ## "to" flow limit
],
"csr")
dh = dh.T
else:
dh = None
return h, g, dh, dg
def cpf_corrector(Ybus, Sbus, V0, ref, pv, pq, lam0, Sxfr, Vprv, lamprv, z,
step, parameterization, ppopt):
# default arguments
if ppopt is None:
ppopt = ppoption(ppopt)
# options
verbose = ppopt["VERBOSE"]
tol = ppopt["PF_TOL"]
max_it = ppopt["PF_MAX_IT"]
# initialize
converged = 0
i = 0
V = V0
Va = angle(V)
Vm = abs(V)
lam = lam0
# set up indexing for updating V
pvpq = r_[pv, pq]
npv = len(pv)
npq = len(pq)
nb = len(V)
j1 = 0
j2 = npv # j1:j2 - V angle of pv buses
j3 = j2
j4 = j2 + npq # j1:j2 - V angle of pv buses
j5 = j4
j6 = j4 + npq # j5:j6 - V mag of pq buses
j7 = j6
j8 = j6 + 1 # j7:j8 - lambda
# evaluate F(x0, lam0), including Sxfr transfer/loading
mis = V * conj(Ybus.dot(V)) - Sbus - lam * Sxfr
F = r_[mis[pvpq].real, mis[pq].imag]
# evaluate P(x0, lambda0)
P = cpf_p(parameterization, step, z, V, lam, Vprv, lamprv, pv, pq)
# augment F(x, lambda) with P(x, lambda)
F = r_[F, P]
# check tolerance
normF = linalg.norm(F, inf)
if verbose > 1:
print('\n it max P & Q mismatch (p.u.)')
print('\n---- ---------------------------')
print('\n%3d %10.3e' % (i, normF))
if normF < tol:
converged = 1
if verbose > 1:
print('\nConverged!\n')
# do Newton iterations
while not converged and i < max_it:
# update iteration counter
i = i + 1
# evaluate Jacobian
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V)
j11 = dSbus_dVa[array([pvpq]).T, pvpq].real
j12 = dSbus_dVm[array([pvpq]).T, pq].real
j21 = dSbus_dVa[array([pq]).T, pvpq].imag
j22 = dSbus_dVm[array([pq]).T, pq].imag
J = vstack([hstack([j11, j12]), hstack([j21, j22])], format="csr")
dF_dlam = -r_[Sxfr[pvpq].real, Sxfr[pq].imag].reshape((-1, 1))
dP_dV, dP_dlam = cpf_p_jac(parameterization, z, V, lam, Vprv, lamprv,
pv, pq)
# augment J with real/imag -Sxfr and z^T
J = vstack([hstack([J, dF_dlam]),
hstack([dP_dV, dP_dlam])],
format="csr")
# compute update step
dx = -1 * spsolve(J, F)
# update voltage
if npv:
Va[pv] = Va[pv] + dx[j1:j2]
if npq:
Va[pq] = Va[pq] + dx[j3:j4]
Vm[pq] = Vm[pq] + dx[j5:j6]
V = Vm * exp(1j * Va)
Vm = abs(V)
Va = angle(V)
# update lambda
lam = lam + dx[j7:j8]
# evalute F(x, lam)
mis = V * conj(Ybus.dot(V)) - Sbus - lam * Sxfr
F = r_[mis[pv].real, mis[pq].real, mis[pq].imag]
# evaluate P(x, lambda)
P = cpf_p(parameterization, step, z, V, lam, Vprv, lamprv, pv, pq)
# augment F(x, lambda) with P(x, lambda)
F = r_[F, P]
# check for convergence
normF = linalg.norm(F, inf)
if verbose > 1:
print('\n%3d %10.3e' % (i, normF))
if normF < tol:
converged = 1
if verbose:
print('\nNewton'
's method corrector converged in %d iterations.\n' % i)
if verbose:
if not converged:
print('\nNewton'
's method corrector did not converge in %d iterations.\n' %
i)
return V, converged, i, lam
def t_jacobian(quiet=False):
"""Numerical tests of partial derivative code.
@author: Ray Zimmerman (PSERC Cornell)
"""
t_begin(28, quiet)
## run powerflow to get solved case
ppopt = ppoption(VERBOSE=0, OUT_ALL=0)
ppc = loadcase(case30())
results, _ = runpf(ppc, ppopt)
baseMVA, bus, gen, branch = \
results['baseMVA'], results['bus'], results['gen'], results['branch']
## switch to internal bus numbering and build admittance matrices
_, bus, gen, branch = ext2int1(bus, gen, branch)
Ybus, Yf, Yt = makeYbus(baseMVA, bus, branch)
Ybus_full = Ybus.todense()
Yf_full = Yf.todense()
Yt_full = Yt.todense()
Vm = bus[:, VM]
Va = bus[:, VA] * (pi / 180)
V = Vm * exp(1j * Va)
f = branch[:, F_BUS].astype(int) ## list of "from" buses
t = branch[:, T_BUS].astype(int) ## list of "to" buses
#nl = len(f)
nb = len(V)
pert = 1e-8
Vm = array([Vm]).T # column array
Va = array([Va]).T # column array
Vc = array([V]).T # column array
##----- check dSbus_dV code -----
## full matrices
dSbus_dVm_full, dSbus_dVa_full = dSbus_dV(Ybus_full, V)
## sparse matrices
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V)
dSbus_dVm_sp = dSbus_dVm.todense()
dSbus_dVa_sp = dSbus_dVa.todense()
## compute numerically to compare
Vmp = (Vm * ones((1, nb)) + pert*eye(nb)) * (exp(1j * Va) * ones((1, nb)))
Vap = (Vm * ones((1, nb))) * (exp(1j * (Va*ones((1, nb)) + pert*eye(nb))))
num_dSbus_dVm = (Vmp * conj(Ybus * Vmp) - Vc * ones((1, nb)) * conj(Ybus * Vc * ones((1, nb)))) / pert
num_dSbus_dVa = (Vap * conj(Ybus * Vap) - Vc * ones((1, nb)) * conj(Ybus * Vc * ones((1, nb)))) / pert
t_is(dSbus_dVm_sp, num_dSbus_dVm, 5, 'dSbus_dVm (sparse)')
t_is(dSbus_dVa_sp, num_dSbus_dVa, 5, 'dSbus_dVa (sparse)')
t_is(dSbus_dVm_full, num_dSbus_dVm, 5, 'dSbus_dVm (full)')
t_is(dSbus_dVa_full, num_dSbus_dVa, 5, 'dSbus_dVa (full)')
##----- check dSbr_dV code -----
## full matrices
dSf_dVa_full, dSf_dVm_full, dSt_dVa_full, dSt_dVm_full, _, _ = \
dSbr_dV(branch, Yf_full, Yt_full, V)
## sparse matrices
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = dSbr_dV(branch, Yf, Yt, V)
dSf_dVa_sp = dSf_dVa.todense()
dSf_dVm_sp = dSf_dVm.todense()
dSt_dVa_sp = dSt_dVa.todense()
dSt_dVm_sp = dSt_dVm.todense()
## compute numerically to compare
Vmpf = Vmp[f, :]
Vapf = Vap[f, :]
Vmpt = Vmp[t, :]
Vapt = Vap[t, :]
Sf2 = (Vc[f] * ones((1, nb))) * conj(Yf * Vc * ones((1, nb)))
St2 = (Vc[t] * ones((1, nb))) * conj(Yt * Vc * ones((1, nb)))
Smpf = Vmpf * conj(Yf * Vmp)
Sapf = Vapf * conj(Yf * Vap)
Smpt = Vmpt * conj(Yt * Vmp)
Sapt = Vapt * conj(Yt * Vap)
num_dSf_dVm = (Smpf - Sf2) / pert
num_dSf_dVa = (Sapf - Sf2) / pert
num_dSt_dVm = (Smpt - St2) / pert
num_dSt_dVa = (Sapt - St2) / pert
t_is(dSf_dVm_sp, num_dSf_dVm, 5, 'dSf_dVm (sparse)')
t_is(dSf_dVa_sp, num_dSf_dVa, 5, 'dSf_dVa (sparse)')
t_is(dSt_dVm_sp, num_dSt_dVm, 5, 'dSt_dVm (sparse)')
t_is(dSt_dVa_sp, num_dSt_dVa, 5, 'dSt_dVa (sparse)')
t_is(dSf_dVm_full, num_dSf_dVm, 5, 'dSf_dVm (full)')
t_is(dSf_dVa_full, num_dSf_dVa, 5, 'dSf_dVa (full)')
t_is(dSt_dVm_full, num_dSt_dVm, 5, 'dSt_dVm (full)')
t_is(dSt_dVa_full, num_dSt_dVa, 5, 'dSt_dVa (full)')
##----- check dAbr_dV code -----
## full matrices
dAf_dVa_full, dAf_dVm_full, dAt_dVa_full, dAt_dVm_full = \
dAbr_dV(dSf_dVa_full, dSf_dVm_full, dSt_dVa_full, dSt_dVm_full, Sf, St)
## sparse matrices
dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm = \
dAbr_dV(dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St)
dAf_dVa_sp = dAf_dVa.todense()
dAf_dVm_sp = dAf_dVm.todense()
dAt_dVa_sp = dAt_dVa.todense()
dAt_dVm_sp = dAt_dVm.todense()
## compute numerically to compare
num_dAf_dVm = (abs(Smpf)**2 - abs(Sf2)**2) / pert
num_dAf_dVa = (abs(Sapf)**2 - abs(Sf2)**2) / pert
num_dAt_dVm = (abs(Smpt)**2 - abs(St2)**2) / pert
num_dAt_dVa = (abs(Sapt)**2 - abs(St2)**2) / pert
t_is(dAf_dVm_sp, num_dAf_dVm, 4, 'dAf_dVm (sparse)')
t_is(dAf_dVa_sp, num_dAf_dVa, 4, 'dAf_dVa (sparse)')
t_is(dAt_dVm_sp, num_dAt_dVm, 4, 'dAt_dVm (sparse)')
t_is(dAt_dVa_sp, num_dAt_dVa, 4, 'dAt_dVa (sparse)')
t_is(dAf_dVm_full, num_dAf_dVm, 4, 'dAf_dVm (full)')
t_is(dAf_dVa_full, num_dAf_dVa, 4, 'dAf_dVa (full)')
t_is(dAt_dVm_full, num_dAt_dVm, 4, 'dAt_dVm (full)')
t_is(dAt_dVa_full, num_dAt_dVa, 4, 'dAt_dVa (full)')
##----- check dIbr_dV code -----
## full matrices
dIf_dVa_full, dIf_dVm_full, dIt_dVa_full, dIt_dVm_full, _, _ = \
dIbr_dV(branch, Yf_full, Yt_full, V)
## sparse matrices
dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, _, _ = dIbr_dV(branch, Yf, Yt, V)
dIf_dVa_sp = dIf_dVa.todense()
dIf_dVm_sp = dIf_dVm.todense()
dIt_dVa_sp = dIt_dVa.todense()
dIt_dVm_sp = dIt_dVm.todense()
## compute numerically to compare
num_dIf_dVm = (Yf * Vmp - Yf * Vc * ones((1, nb))) / pert
num_dIf_dVa = (Yf * Vap - Yf * Vc * ones((1, nb))) / pert
num_dIt_dVm = (Yt * Vmp - Yt * Vc * ones((1, nb))) / pert
num_dIt_dVa = (Yt * Vap - Yt * Vc * ones((1, nb))) / pert
t_is(dIf_dVm_sp, num_dIf_dVm, 5, 'dIf_dVm (sparse)')
t_is(dIf_dVa_sp, num_dIf_dVa, 5, 'dIf_dVa (sparse)')
t_is(dIt_dVm_sp, num_dIt_dVm, 5, 'dIt_dVm (sparse)')
t_is(dIt_dVa_sp, num_dIt_dVa, 5, 'dIt_dVa (sparse)')
t_is(dIf_dVm_full, num_dIf_dVm, 5, 'dIf_dVm (full)')
t_is(dIf_dVa_full, num_dIf_dVa, 5, 'dIf_dVa (full)')
t_is(dIt_dVm_full, num_dIt_dVm, 5, 'dIt_dVm (full)')
t_is(dIt_dVa_full, num_dIt_dVa, 5, 'dIt_dVa (full)')
t_end()
def admmopf_consfcn(self, x=None):
if x is None:
x = self.pb['x0']
nx = len(x)
nb = self.region['nb']
ng = self.region['ng']
iv = self.idx['var']
bus = self.region['bus']
gen = self.region['gen']
branch = self.region['branch']
Ybus = self.region['Ybus']
Yf = self.region['Yf']
Yt = self.region['Yt']
baseMVA = self.region['baseMVA']
ridx = self.idx['rbus']['int']
# idx ranges
iVa = iv['iVa']
iVm = iv['iVm']
iPg = iv['iPg']
iQg = iv['iQg']
# grab Pg and Qg
gen[:, PG] = x[iPg]
gen[:, QG] = x[iQg]
# rebuid Sbus
Sbus = makeSbus(1, bus, gen)
# reconstruct V
Va, Vm = x[iVa], x[iVm]
V = Vm * exp(1j * Va)
# evaluate power flow equations
mis = V * conj(Ybus * V) - Sbus
g = r_[mis.real, mis.imag]
row = ridx + [i + nb for i in ridx]
g = g[row]
il = find((branch[:, RATE_A] != 0) & (branch[:, RATE_A] < 1e10))
nl2 = len(il)
Ybus2, Yf2, Yt2 = makeYbus(baseMVA, bus, branch)
Yf = Yf2[il, :]
Yt = Yt2[il, :]
if nl2 > 0:
flow_max = (branch[il, RATE_A] / baseMVA)**2
flow_max[flow_max == 0] = Inf
## compute branch power flows
## complex power injected at "from" bus (p.u.)
Sf = V[branch[il, F_BUS].astype(int)] * conj(Yf * V)
## complex power injected at "to" bus (p.u.)
St = V[branch[il, T_BUS].astype(int)] * conj(Yt * V)
h = r_[Sf.real**2 - flow_max, ## branch P limits (from bus)
St.real**2 - flow_max] ## branch P limits (to bus)
else:
h = array([])
# ---- evaluate constraint gradients -------
# compute partials of injected bus powers
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V) # w.r.t. V
neg_Cg = sparse((-ones(ng), (gen[:, GEN_BUS], range(ng))), (nb, ng))
# construct Jacobian of equality constraints (power flow) and transpose it
dg = lil_matrix((2 * nb, nx))
blank = sparse((nb, ng))
dg = vstack([ \
#P mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.real, dSbus_dVm.real, neg_Cg, blank]),
# Q mismatch w.r.t Va, Vm, Pg, Qg
hstack([dSbus_dVa.imag, dSbus_dVm.imag, blank, neg_Cg])
], "csr")
dg = dg[row, :]
dg = dg.T
if nl2 > 0:
## compute partials of Flows w.r.t. V ## power
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dSbr_dV(branch[il, :], Yf, Yt, V)
dFf_dVa = dFf_dVa.real
dFf_dVm = dFf_dVm.real
dFt_dVa = dFt_dVa.real
dFt_dVm = dFt_dVm.real
Ff = Ff.real
Ft = Ft.real
## squared magnitude of flow (of complex power or current, or real power)
df_dVa, df_dVm, dt_dVa, dt_dVm = \
dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft)
## construct Jacobian of inequality constraints (branch limits)
## and transpose it.
dh = lil_matrix((2 * nl2, nx))
dh[:, r_[iVa, iVm].T] = vstack(
[
hstack([df_dVa, df_dVm]), ## "from" flow limit
hstack([dt_dVa, dt_dVm]) ## "to" flow limit
],
"csr")
dh = dh.T
else:
dh = None
h = array([])
dh = None
return h, g, dh, dg
