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)
@author: Richard Lincoln
Modified by University of Kassel (Friederike Meier): Bugfix in line 173
"""
##----- 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)
if len(ipolp):
d2f_dPg2[ipolp] = \
baseMVA**2 * polycost(pcost[ipolp, :], Pg[ipolp] * baseMVA, 2)
if qcost.any(): ## 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: # pragma: no cover
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
if Yf.size:
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:
Hfaa = Hfav = Hfva = Hfvv = Htaa = Htav = Htva = Htvv = sparse(
zeros((nb, nb)))
else: # pragma: no cover
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 _dSbr_dv(self, V):
dSf_dth, dSf_dv, dSt_dth, dSt_dv, _, _ = dSbr_dV(
self.eppci['branch'], self.Yf, self.Yt, V)
return dSf_dth, dSf_dv, dSt_dth, dSt_dv
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], 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