def Jacobian(Ybus, V, Ibus, pq, pvpq):
"""
Computes the system Jacobian matrix
Args:
Ybus: Admittance matrix
V: Array of nodal voltages
Ibus: Array of nodal current injections
pq: Array with the indices of the PQ buses
pvpq: Array with the indices of the PV and PQ buses
Returns:
The system Jacobian matrix
"""
# dS_dVm, dS_dVa = dSbus_dV(Ybus, V, Ibus) # compute the derivatives
I = Ybus * V - Ibus
diagV = diags(V)
diagI = diags(I)
diagVnorm = diags(V / np.abs(V))
dS_dVm = diagV * conj(Ybus * diagVnorm) + conj(diagI) * diagVnorm
dS_dVa = 1j * diagV * conj(diagI - Ybus * diagV)
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_sp([hstack_sp([J11, J12]), hstack_sp([J21, J22])], format="csr")
return J
def Jrect(G, B, V, pvpq, pqpv, pq, pv):
e = V.real
f = V.imag
ediag = diags(e)
fdiag = diags(f)
J1 = ediag * G + fdiag * B + diags(G * e - B * f)
J2 = fdiag * G - ediag * B + diags(G * f + B * e)
J3 = fdiag * G - ediag * B - diags(G * f + B * e)
J4 = -ediag * G - fdiag * B + diags(G * e - B * f)
J5 = diags(2 * e).tocsc()
J6 = diags(2 * f).tocsc()
J = vstack_sp([
hstack_sp([J1[np.ix_(pvpq, pvpq)], J2[np.ix_(pvpq, pqpv)]]),
hstack_sp([J3[np.ix_(pq, pvpq)], J4[np.ix_(pq, pqpv)]]),
hstack_sp([J5[np.ix_(pv, pvpq)], J6[np.ix_(pv, pqpv)]])
],
format="csc")
return J
def Jacobian_I(Ybus, V, pq, pvpq):
"""
Computes the system Jacobian matrix
Args:
Ybus: Admittance matrix
V: Array of nodal voltages
pq: Array with the indices of the PQ buses
pvpq: Array with the indices of the PV and PQ buses
Returns:
The system Jacobian matrix in current equations
"""
dI_dVm = Ybus * diags(V / np.abs(V))
dI_dVa = 1j * (Ybus * diags(V))
J11 = dI_dVa[array([pvpq]).T, pvpq].real
J12 = dI_dVm[array([pvpq]).T, pq].real
J21 = dI_dVa[array([pq]).T, pvpq].imag
J22 = dI_dVm[array([pq]).T, pq].imag
J = vstack_sp([hstack_sp([J11, J12]), hstack_sp([J21, J22])], format="csr")
return J
from scipy.sparse import hstack as hstack_sp, vstack as vstack_sp
circuit = MultiCircuit()
b1 = Bus(name='B1')
b2 = Bus(name='B2')
l2 = Load(P=1, Q=1)
br1 = Branch(b1, b2, r=0.1, x=1)
circuit.add_bus(b1)
circuit.add_bus(b2)
circuit.add_load(b2, l2)
circuit.add_branch(br1)
islands = circuit.compile_snapshot().compute()
npv = len(islands[0].pv)
npq = len(islands[0].pq)
pvpq = np.r_[islands[0].pv, islands[0].pq]
J = Jacobian(Ybus=islands[0].Ybus,
V=islands[0].Vbus,
Ibus=islands[0].Ibus,
pq=islands[0].pq,
pvpq=pvpq)
ek = np.zeros((1, npv + npq + npq + 1))
ek[0, -1] = 1
K = np.zeros((npv + npq + npq, 1))
J2 = vstack_sp([hstack_sp([J, K]), ek], format="csr")
pass
def ASD1(Ybus, S0, V0, I0, pv, pq, vd, tol, max_it=15, gamma=0.5):
"""
Alternate Search Directions Power Flow
As defined in the paper:
Unified formulation of a family of iterative solvers for power system analysis
by Domenico Borzacchiello et. Al.
:param Ybus: Admittance matrix
:param S0: Power injections
:param V0: Initial Voltage Vector
:param I0: Current injections @V=1.0 p.u.
:param pv: Array of PV bus indices
:param pq: Array of PQ bus indices
:param vd: Array of Slack bus indices
:param tol: Tolerance
:param max_it: Maximum number of iterations
:param gamma: Reactive power acceleration factor
:return: V, converged, normF, Scalc, iterations, elapsed
"""
start = time.time()
# initialize
V = V0.copy()
Vset_pv = np.abs(V0[pv])
Scalc = S0.copy()
Scalc[pv] = Scalc[pv].real + 0j
normF = 1e20
iterations = 0
converged = False
n = len(V0)
npq = len(pq)
npv = len(pv)
pqpv = np.r_[pq, pv]
npqpv = len(pqpv)
# kron
if npv:
Y11 = Ybus[pq, :][:, pq]
Y12 = Ybus[pq, :][:, pv]
Y21 = Ybus[pv, :][:, pq]
Y22 = Ybus[pv, :][:, pv]
Ykron = Y22 + Y21 * (spsolve(Y11, Y12))
Yred = vstack_sp((hstack_sp((Y11, Y12)), hstack_sp((Y21, Y22))))
else:
Yred = Ybus[pq, :][:, pq]
# reduced system
Sred = S0[pqpv]
I0_red = I0[pqpv]
Yslack = -Ybus[np.ix_(pqpv, vd)] # yes, it is the negative of this
Vslack = V0[vd]
# compute alpha
Vabs = np.ones(npqpv)
alpha = sp.diags(np.conj(Sred) / (Vabs * Vabs))
# compute Y-alpha
Y_alpha = (Yred - alpha)
# compute beta as a vector
B = Y_alpha.diagonal()
beta = diags(B)
Y_beta = Yred - beta
# get the first voltage approximation V0, or simply V(l
Ivd = Yslack * Vslack # slack currents
V_l = spsolve(Y_alpha, Ivd) # slack voltages influence
V_l = V0[pqpv]
while not converged and iterations < max_it:
iterations += 1
# Global step
rhs_global = np.conj(Sred) / np.conj(V_l) - alpha * V_l + I0_red + Ivd
V_l12 = spsolve(Y_alpha, rhs_global)
# PV correction
if npv:
V_pv = V_l12[npq:]
V_pv_abs = np.abs(V_pv)
dV_l12 = (Vset_pv - V_pv_abs) * V_pv / V_pv_abs
dI_l12 = Ykron * dV_l12
dQ_l12 = (V_pv *
np.conj(dI_l12)).imag # correct the reactive power
Sred[npq:] = Sred[npq:].real + 1j * (
Sred[npq:].imag + gamma * dQ_l12) # assign the reactive power
V_l12[npq:] += dV_l12 # correct the voltage
# local step
A = (Y_beta * V_l12 - I0_red - Ivd) / B
Sigma = -np.conj(Sred) / (A * np.conj(A) * B)
U = (-1 - np.sqrt(1 - 4 * (Sigma.imag * Sigma.imag + Sigma.real))
) / 2.0 + 1j * Sigma.imag
V_l = U * A
# Assign the reduced solution
V[pq] = V_l[:npq]
V[pv] = V_l[npq:]
# compute the calculated power injection and the error of the voltage solution
Scalc = V * np.conj(Ybus * V - I0)
mis = Scalc - S0 # complex power mismatch
mismatch = np.r_[mis[pv].real, mis[pq].real,
mis[pq].imag] # concatenate again
normF = np.linalg.norm(mismatch, np.Inf)
converged = normF < tol
print(normF)
end = time.time()
elapsed = end - start
return V, converged, normF, Scalc, iterations, elapsed