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 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_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.
"""
##----- 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