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 opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt, il=None, cost_mult=1.0):
"""Evaluates Hessian of Lagrangian for AC OPF.
Hessian evaluation function for AC optimal power flow, suitable
for use with L{pips}.
Examples::
Lxx = opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt)
Lxx = opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt, il)
Lxx = opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt, il, cost_mult)
@param x: optimization vector
@param lmbda: C{eqnonlin} - Lagrange multipliers on power balance
equations. C{ineqnonlin} - Kuhn-Tucker multipliers on constrained
branch flows.
@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}.
@param cost_mult: (optional) Scale factor to be applied to the cost
(default = 1).
@return: Hessian of the Lagrangian.
@see: L{opf_costfcn}, L{opf_consfcn}
@author: Ray Zimmerman (PSERC Cornell)
@author: Carlos E. Murillo-Sanchez (PSERC Cornell & Universidad
Autonoma de Manizales)
"""
##----- initialize -----
## unpack data
ppc = om.get_ppc()
baseMVA, bus, gen, branch, gencost = \
ppc["baseMVA"], ppc["bus"], ppc["gen"], ppc["branch"], ppc["gencost"]
cp = om.get_cost_params()
N, Cw, H, dd, rh, kk, mm = \
cp["N"], cp["Cw"], cp["H"], cp["dd"], cp["rh"], cp["kk"], cp["mm"]
vv, _, _, _ = om.get_idx()
## unpack needed parameters
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
## reconstruct V
Va = x[vv["i1"]["Va"]:vv["iN"]["Va"]]
Vm = x[vv["i1"]["Vm"]:vv["iN"]["Vm"]]
V = Vm * exp(1j * Va)
nxtra = nxyz - 2 * nb
pcost = gencost[arange(ng), :]
if gencost.shape[0] > ng:
qcost = gencost[arange(ng, 2 * ng), :]
else:
qcost = array([])
## ----- evaluate d2f -----
d2f_dPg2 = zeros(ng) #sparse((ng, 1)) ## w.r.t. p.u. Pg
d2f_dQg2 = zeros(ng) #sparse((ng, 1)) ## w.r.t. p.u. Qg
ipolp = find(pcost[:, MODEL] == POLYNOMIAL)
d2f_dPg2[ipolp] = \
baseMVA**2 * polycost(pcost[ipolp, :], Pg[ipolp] * baseMVA, 2)
if any(qcost): ## Qg is not free
ipolq = find(qcost[:, MODEL] == POLYNOMIAL)
d2f_dQg2[ipolq] = \
baseMVA**2 * polycost(qcost[ipolq, :], Qg[ipolq] * baseMVA, 2)
i = r_[arange(vv["i1"]["Pg"], vv["iN"]["Pg"]),
arange(vv["i1"]["Qg"], vv["iN"]["Qg"])]
# d2f = sparse((vstack([d2f_dPg2, d2f_dQg2]).toarray().flatten(),
# (i, i)), shape=(nxyz, nxyz))
d2f = sparse((r_[d2f_dPg2, d2f_dQg2], (i, i)), (nxyz, nxyz))
## generalized cost
if issparse(N) and N.nnz > 0:
nw = N.shape[0]
r = N * x - rh ## Nx - rhat
iLT = find(r < -kk) ## below dead zone
iEQ = find((r == 0) & (kk == 0)) ## dead zone doesn't exist
iGT = find(r > kk) ## above dead zone
iND = r_[iLT, iEQ, iGT] ## rows that are Not in the Dead region
iL = find(dd == 1) ## rows using linear function
iQ = find(dd == 2) ## rows using quadratic function
LL = sparse((ones(len(iL)), (iL, iL)), (nw, nw))
QQ = sparse((ones(len(iQ)), (iQ, iQ)), (nw, nw))
kbar = sparse((r_[ones(len(iLT)),
zeros(len(iEQ)), -ones(len(iGT))], (iND, iND)),
(nw, nw)) * kk
rr = r + kbar ## apply non-dead zone shift
M = sparse((mm[iND], (iND, iND)), (nw, nw)) ## dead zone or scale
diagrr = sparse((rr, (arange(nw), arange(nw))), (nw, nw))
## linear rows multiplied by rr(i), quadratic rows by rr(i)^2
w = M * (LL + QQ * diagrr) * rr
HwC = H * w + Cw
AA = N.T * M * (LL + 2 * QQ * diagrr)
d2f = d2f + AA * H * AA.T + 2 * N.T * M * QQ * \
sparse((HwC, (arange(nw), arange(nw))), (nw, nw)) * N
d2f = d2f * cost_mult
##----- evaluate Hessian of power balance constraints -----
nlam = len(lmbda["eqnonlin"]) / 2
lamP = lmbda["eqnonlin"][:nlam]
lamQ = lmbda["eqnonlin"][nlam:nlam + nlam]
Gpaa, Gpav, Gpva, Gpvv = d2Sbus_dV2(Ybus, V, lamP)
Gqaa, Gqav, Gqva, Gqvv = d2Sbus_dV2(Ybus, V, lamQ)
d2G = vstack([
hstack([
vstack([hstack([Gpaa, Gpav]),
hstack([Gpva, Gpvv])]).real +
vstack([hstack([Gqaa, Gqav]),
hstack([Gqva, Gqvv])]).imag,
sparse((2 * nb, nxtra))
]),
hstack([sparse(
(nxtra, 2 * nb)), sparse((nxtra, nxtra))])
], "csr")
##----- evaluate Hessian of flow constraints -----
nmu = len(lmbda["ineqnonlin"]) / 2
muF = lmbda["ineqnonlin"][:nmu]
muT = lmbda["ineqnonlin"][nmu:nmu + nmu]
if ppopt['OPF_FLOW_LIM'] == 2: ## current
dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It = dIbr_dV(branch, Yf, Yt, V)
Hfaa, Hfav, Hfva, Hfvv = d2AIbr_dV2(dIf_dVa, dIf_dVm, If, Yf, V, muF)
Htaa, Htav, Htva, Htvv = d2AIbr_dV2(dIt_dVa, dIt_dVm, It, Yt, V, muT)
else:
f = branch[il, F_BUS].astype(int) ## list of "from" buses
t = branch[il, T_BUS].astype(int) ## list of "to" buses
## connection matrix for line & from buses
Cf = sparse((ones(nl2), (arange(nl2), f)), (nl2, nb))
## connection matrix for line & to buses
Ct = sparse((ones(nl2), (arange(nl2), t)), (nl2, nb))
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = \
dSbr_dV(branch[il,:], Yf, Yt, V)
if ppopt['OPF_FLOW_LIM'] == 1: ## real power
Hfaa, Hfav, Hfva, Hfvv = d2ASbr_dV2(dSf_dVa.real, dSf_dVm.real,
Sf.real, Cf, Yf, V, muF)
Htaa, Htav, Htva, Htvv = d2ASbr_dV2(dSt_dVa.real, dSt_dVm.real,
St.real, Ct, Yt, V, muT)
else: ## apparent power
Hfaa, Hfav, Hfva, Hfvv = \
d2ASbr_dV2(dSf_dVa, dSf_dVm, Sf, Cf, Yf, V, muF)
Htaa, Htav, Htva, Htvv = \
d2ASbr_dV2(dSt_dVa, dSt_dVm, St, Ct, Yt, V, muT)
d2H = vstack([
hstack([
vstack([hstack([Hfaa, Hfav]),
hstack([Hfva, Hfvv])]) +
vstack([hstack([Htaa, Htav]),
hstack([Htva, Htvv])]),
sparse((2 * nb, nxtra))
]),
hstack([sparse(
(nxtra, 2 * nb)), sparse((nxtra, nxtra))])
], "csr")
##----- do numerical check using (central) finite differences -----
if 0:
nx = len(x)
step = 1e-5
num_d2f = sparse((nx, nx))
num_d2G = sparse((nx, nx))
num_d2H = sparse((nx, nx))
for i in range(nx):
xp = x
xm = x
xp[i] = x[i] + step / 2
xm[i] = x[i] - step / 2
# evaluate cost & gradients
_, dfp = opf_costfcn(xp, om)
_, dfm = opf_costfcn(xm, om)
# evaluate constraints & gradients
_, _, dHp, dGp = opf_consfcn(xp, om, Ybus, Yf, Yt, ppopt, il)
_, _, dHm, dGm = opf_consfcn(xm, om, Ybus, Yf, Yt, ppopt, il)
num_d2f[:, i] = cost_mult * (dfp - dfm) / step
num_d2G[:, i] = (dGp - dGm) * lmbda["eqnonlin"] / step
num_d2H[:, i] = (dHp - dHm) * lmbda["ineqnonlin"] / step
d2f_err = max(max(abs(d2f - num_d2f)))
d2G_err = max(max(abs(d2G - num_d2G)))
d2H_err = max(max(abs(d2H - num_d2H)))
if d2f_err > 1e-6:
print('Max difference in d2f: %g' % d2f_err)
if d2G_err > 1e-5:
print('Max difference in d2G: %g' % d2G_err)
if d2H_err > 1e-6:
print('Max difference in d2H: %g' % d2H_err)
return d2f + d2G + d2H
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 opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt, il=None, cost_mult=1.0):
"""Evaluates Hessian of Lagrangian for AC OPF.
Hessian evaluation function for AC optimal power flow, suitable
for use with L{pips}.
Examples::
Lxx = opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt)
Lxx = opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt, il)
Lxx = opf_hessfcn(x, lmbda, om, Ybus, Yf, Yt, ppopt, il, cost_mult)
@param x: optimization vector
@param lmbda: C{eqnonlin} - Lagrange multipliers on power balance
equations. C{ineqnonlin} - Kuhn-Tucker multipliers on constrained
branch flows.
@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}.
@param cost_mult: (optional) Scale factor to be applied to the cost
(default = 1).
@return: Hessian of the Lagrangian.
@see: L{opf_costfcn}, L{opf_consfcn}
@author: Ray Zimmerman (PSERC Cornell)
@author: Carlos E. Murillo-Sanchez (PSERC Cornell & Universidad
Autonoma de Manizales)
"""
##----- initialize -----
## unpack data
ppc = om.get_ppc()
baseMVA, bus, gen, branch, gencost = \
ppc["baseMVA"], ppc["bus"], ppc["gen"], ppc["branch"], ppc["gencost"]
cp = om.get_cost_params()
N, Cw, H, dd, rh, kk, mm = \
cp["N"], cp["Cw"], cp["H"], cp["dd"], cp["rh"], cp["kk"], cp["mm"]
vv, _, _, _ = om.get_idx()
## unpack needed parameters
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
## reconstruct V
Va = x[vv["i1"]["Va"]:vv["iN"]["Va"]]
Vm = x[vv["i1"]["Vm"]:vv["iN"]["Vm"]]
V = Vm * exp(1j * Va)
nxtra = nxyz - 2 * nb
pcost = gencost[arange(ng), :]
if gencost.shape[0] > ng:
qcost = gencost[arange(ng, 2 * ng), :]
else:
qcost = array([])
## ----- evaluate d2f -----
d2f_dPg2 = zeros(ng)#sparse((ng, 1)) ## w.r.t. p.u. Pg
d2f_dQg2 = zeros(ng)#sparse((ng, 1)) ## w.r.t. p.u. Qg
ipolp = find(pcost[:, MODEL] == POLYNOMIAL)
d2f_dPg2[ipolp] = \
baseMVA**2 * polycost(pcost[ipolp, :], Pg[ipolp] * baseMVA, 2)
if any(qcost): ## Qg is not free
ipolq = find(qcost[:, MODEL] == POLYNOMIAL)
d2f_dQg2[ipolq] = \
baseMVA**2 * polycost(qcost[ipolq, :], Qg[ipolq] * baseMVA, 2)
i = r_[arange(vv["i1"]["Pg"], vv["iN"]["Pg"]),
arange(vv["i1"]["Qg"], vv["iN"]["Qg"])]
# d2f = sparse((vstack([d2f_dPg2, d2f_dQg2]).toarray().flatten(),
# (i, i)), shape=(nxyz, nxyz))
d2f = sparse((r_[d2f_dPg2, d2f_dQg2], (i, i)), (nxyz, nxyz))
## generalized cost
if issparse(N) and N.nnz > 0:
nw = N.shape[0]
r = N * x - rh ## Nx - rhat
iLT = find(r < -kk) ## below dead zone
iEQ = find((r == 0) & (kk == 0)) ## dead zone doesn't exist
iGT = find(r > kk) ## above dead zone
iND = r_[iLT, iEQ, iGT] ## rows that are Not in the Dead region
iL = find(dd == 1) ## rows using linear function
iQ = find(dd == 2) ## rows using quadratic function
LL = sparse((ones(len(iL)), (iL, iL)), (nw, nw))
QQ = sparse((ones(len(iQ)), (iQ, iQ)), (nw, nw))
kbar = sparse((r_[ones(len(iLT)), zeros(len(iEQ)), -ones(len(iGT))],
(iND, iND)), (nw, nw)) * kk
rr = r + kbar ## apply non-dead zone shift
M = sparse((mm[iND], (iND, iND)), (nw, nw)) ## dead zone or scale
diagrr = sparse((rr, (arange(nw), arange(nw))), (nw, nw))
## linear rows multiplied by rr(i), quadratic rows by rr(i)^2
w = M * (LL + QQ * diagrr) * rr
HwC = H * w + Cw
AA = N.T * M * (LL + 2 * QQ * diagrr)
d2f = d2f + AA * H * AA.T + 2 * N.T * M * QQ * \
sparse((HwC, (arange(nw), arange(nw))), (nw, nw)) * N
d2f = d2f * cost_mult
##----- evaluate Hessian of power balance constraints -----
nlam = len(lmbda["eqnonlin"]) / 2
lamP = lmbda["eqnonlin"][:nlam]
lamQ = lmbda["eqnonlin"][nlam:nlam + nlam]
Gpaa, Gpav, Gpva, Gpvv = d2Sbus_dV2(Ybus, V, lamP)
Gqaa, Gqav, Gqva, Gqvv = d2Sbus_dV2(Ybus, V, lamQ)
d2G = vstack([
hstack([
vstack([hstack([Gpaa, Gpav]),
hstack([Gpva, Gpvv])]).real +
vstack([hstack([Gqaa, Gqav]),
hstack([Gqva, Gqvv])]).imag,
sparse((2 * nb, nxtra))]),
hstack([
sparse((nxtra, 2 * nb)),
sparse((nxtra, nxtra))
])
], "csr")
##----- evaluate Hessian of flow constraints -----
nmu = len(lmbda["ineqnonlin"]) / 2
muF = lmbda["ineqnonlin"][:nmu]
muT = lmbda["ineqnonlin"][nmu:nmu + nmu]
if ppopt['OPF_FLOW_LIM'] == 2: ## current
dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It = dIbr_dV(Yf, Yt, V)
Hfaa, Hfav, Hfva, Hfvv = d2AIbr_dV2(dIf_dVa, dIf_dVm, If, Yf, V, muF)
Htaa, Htav, Htva, Htvv = d2AIbr_dV2(dIt_dVa, dIt_dVm, It, Yt, V, muT)
else:
f = branch[il, F_BUS].astype(int) ## list of "from" buses
t = branch[il, T_BUS].astype(int) ## list of "to" buses
## connection matrix for line & from buses
Cf = sparse((ones(nl2), (arange(nl2), f)), (nl2, nb))
## connection matrix for line & to buses
Ct = sparse((ones(nl2), (arange(nl2), t)), (nl2, nb))
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = \
dSbr_dV(branch[il,:], Yf, Yt, V)
if ppopt['OPF_FLOW_LIM'] == 1: ## real power
Hfaa, Hfav, Hfva, Hfvv = d2ASbr_dV2(dSf_dVa.real, dSf_dVm.real,
Sf.real, Cf, Yf, V, muF)
Htaa, Htav, Htva, Htvv = d2ASbr_dV2(dSt_dVa.real, dSt_dVm.real,
St.real, Ct, Yt, V, muT)
else: ## apparent power
Hfaa, Hfav, Hfva, Hfvv = \
d2ASbr_dV2(dSf_dVa, dSf_dVm, Sf, Cf, Yf, V, muF)
Htaa, Htav, Htva, Htvv = \
d2ASbr_dV2(dSt_dVa, dSt_dVm, St, Ct, Yt, V, muT)
d2H = vstack([
hstack([
vstack([hstack([Hfaa, Hfav]),
hstack([Hfva, Hfvv])]) +
vstack([hstack([Htaa, Htav]),
hstack([Htva, Htvv])]),
sparse((2 * nb, nxtra))
]),
hstack([
sparse((nxtra, 2 * nb)),
sparse((nxtra, nxtra))
])
], "csr")
##----- do numerical check using (central) finite differences -----
if 0:
nx = len(x)
step = 1e-5
num_d2f = sparse((nx, nx))
num_d2G = sparse((nx, nx))
num_d2H = sparse((nx, nx))
for i in range(nx):
xp = x
xm = x
xp[i] = x[i] + step / 2
xm[i] = x[i] - step / 2
# evaluate cost & gradients
_, dfp = opf_costfcn(xp, om)
_, dfm = opf_costfcn(xm, om)
# evaluate constraints & gradients
_, _, dHp, dGp = opf_consfcn(xp, om, Ybus, Yf, Yt, ppopt, il)
_, _, dHm, dGm = opf_consfcn(xm, om, Ybus, Yf, Yt, ppopt, il)
num_d2f[:, i] = cost_mult * (dfp - dfm) / step
num_d2G[:, i] = (dGp - dGm) * lmbda["eqnonlin"] / step
num_d2H[:, i] = (dHp - dHm) * lmbda["ineqnonlin"] / step
d2f_err = max(max(abs(d2f - num_d2f)))
d2G_err = max(max(abs(d2G - num_d2G)))
d2H_err = max(max(abs(d2H - num_d2H)))
if d2f_err > 1e-6:
print('Max difference in d2f: %g' % d2f_err)
if d2G_err > 1e-5:
print('Max difference in d2G: %g' % d2G_err)
if d2H_err > 1e-6:
print('Max difference in d2H: %g' % d2H_err)
return d2f + d2G + d2H
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 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