def get_scores(self, sol, idx, y=[]): y = np.array(y) if (y.size==0): y=np.array(self.y[idx]) (foo, T) = y.shape N = self.states F = self.dims scores = matrix(0.0, (1, T)) # this is the score of the complete example anom_score = sol.trans()*self.get_joint_feature_map(idx) # transition matrix A = self.get_transition_matrix(sol) # emission matrix without loss em = self.calc_emission_matrix(sol, idx, augment_loss=False); # store scores for each position of the sequence scores[0] = 0.0 + em[int(y[0,0]),0] for t in range(1,T): scores[t] = A[int(y[0,t-1]),int(y[0,t])] + em[int(y[0,t]),t] # transform for better interpretability # transform for better interpretability if max(abs(scores))>10.0**(-15): scores = exp(-abs(4.0*scores/max(abs(scores)))) else: scores = matrix(0.0, (1,T)) return (anom_score, scores)
def testgp(opts):
Aflr = 1000.0
Awall = 100.0
alpha = 0.5
beta = 2.0
gamma = 0.5
delta = 2.0
F = matrix( [[-1., 1., 1., 0., -1., 1., 0., 0.],
[-1., 1., 0., 1., 1., -1., 1., -1.],
[-1., 0., 1., 1., 0., 0., -1., 1.]])
g = log( matrix( [1.0, 2/Awall, 2/Awall, 1/Aflr, alpha, 1/beta, gamma,
1/delta]) )
K = [1, 2, 1, 1, 1, 1, 1]
solvers.options.update(opts)
sol = solvers.gp(K, F, g)
#localcvx.options.update(opts)
#sol = localcvx.gp(K, F, g, kktsolver='chol')
if sol['status'] == 'optimal':
x = sol['x']
print "x=\n", helpers.strSpe(x, "%.17f")
h, w, d = exp(x)
print("\n h = %f, w = %f, d = %f.\n" %(h,w,d))
print "\n *** running GO test ***"
helpers.run_go_test("../testgp", {'x': x})
def F(x=None, z=None): if x is None: return 0, matrix(0.0, (2,1)) w = exp(A*x) f = c.T*x + sum(log(1+w)) grad = c + A.T * div(w, 1+w) if z is None: return f, grad.T H = A.T * spdiag(div(w,(1+w)**2)) * A return f, grad.T, z[0]*H
def F(x=None, z=None): if x is None: return 0, matrix(0.0, (n+1,1)) w = exp(A*x) f = dot(c,x) + sum(log(1+w)) grad = c + A.T * div(w, 1+w) if z is None: return matrix(f), grad.T H = A.T * spdiag(div(w,(1+w)**2)) * A return matrix(f), grad.T, z[0]*H
def test_cvxopt_sparse(self):
m = 100
n = 20
mu = cvxopt.exp(cvxopt.normal(m))
F = cvxopt.normal(m, n)
D = cvxopt.spdiag(cvxopt.uniform(m))
x = Variable(m)
exp = square(norm2(D * x))
def test_cvxopt_sparse(self):
m = 100
n = 20
mu = cvxopt.exp( cvxopt.normal(m) )
F = cvxopt.normal(m, n)
D = cvxopt.spdiag( cvxopt.uniform(m) )
x = Variable(m)
exp = square(norm2(D*x))
def init1(self, dae):
retval = True
# electrical torque in pu
# te = mul(xmu, mul(dae.x[self.irq], dae.y[self.isd]) - mul(dae.x[self.ird], dae.y[self.isq]))
for i in range(self.n):
if self.te0[i] < 0:
self.message(
'Electric power is negative at bus <{}>. Wind speed initialize failed.'
.format(self.bus[i]), ERROR)
retval = False
# wind power in pu
omega = dae.x[self.omega_m]
theta = dae.x[self.theta_p]
pw = mul(self.te0, dae.x[self.omega_m])
dae.y[self.pw] = pw
# wind speed initialization loop
R = 4 * pi * self.system.Settings.freq * mul(self.R, self.ngb,
div(1, self.npole[i]))
AA = pi * self.R**2
vw = 0.9 * self.Vwn
for i in range(self.n):
mis = 1
iter = 0
while abs(mis) > self.system.TDS.tol:
if iter > 50:
self.message(
'Initialization of wind <{}> failed. Try increasing the nominal wind speed.'
.format(self.wind[i]))
retval = False
break
pw_iter, jac = self.windpower(self.ngen[i], self.rho[i], vw[i],
AA[i], R[i], omega[i], theta[i])
mis = pw_iter - pw[i]
inc = -mis / jac[1]
vw[i] += inc
iter += 1
# set wind speed
dae.x[self.vw] = div(vw, self.Vwn)
lamb = div(omega, vw, div(1, R))
ilamb = div(1,
(div(1, lamb + 0.08 * theta) - div(0.035, theta**3 + 1)))
cp = 0.22 * mul(
div(116, ilamb) - 0.4 * theta - 5, exp(div(-12.5, ilamb)))
dae.y[self.lamb] = lamb
dae.y[self.ilamb] = ilamb
dae.y[self.cp] = cp
def init1(self, dae):
# electrical torque in pu
# te = mul(
# xmu,
# mul(dae.x[self.irq], dae.y[self.isd]) - mul(
# dae.x[self.ird], dae.y[self.isq]))
for i in range(self.n):
if self.te0[i] < 0:
logger.error(
'Pe < 0 on bus <{}>. Wind speed initialize failed.'.format(
self.bus[i]))
# wind power in pu
omega = dae.x[self.omega_m]
theta = dae.x[self.theta_p]
pw = mul(self.te0, dae.x[self.omega_m])
dae.y[self.pw] = pw
# wind speed initialization loop
R = 4 * pi * self.system.config.freq * mul(self.R, self.ngb,
div(1, self.npole))
AA = pi * self.R**2
vw = 0.9 * self.Vwn
for i in range(self.n):
mis = 1
niter = 0
while abs(mis) > self.system.tds.config.tol:
if niter > 50:
self.message(
'Wind <{}> init failed. '
'Try increasing the nominal wind speed.'.format(
self.wind[i]))
break
pw_iter, jac = self.windpower(self.ngen[i], self.rho[i], vw[i],
AA[i], R[i], omega[i], theta[i])
mis = pw_iter - pw[i]
inc = -mis / jac[1]
vw[i] += inc
niter += 1
# set wind speed
dae.x[self.vw] = div(vw, self.Vwn)
lamb = div(omega, vw, div(1, R))
ilamb = div(1, lamb + 0.08 * theta) - div(0.035, theta**3 + 1)
cp = 0.22 * mul(
mul(116, ilamb) - 0.4 * theta - 5, exp(mul(-12.5, ilamb)))
dae.y[self.lamb] = lamb
dae.y[self.ilamb] = ilamb
dae.y[self.cp] = cp
def gycall(self, dae):
dae.add_jac(Gy, mul(self.xmu, 1 - dae.x[self.omega_m]), self.vrd,
self.isq)
dae.add_jac(Gy, -mul(self.xmu, 1 - dae.x[self.omega_m]), self.vrq,
self.isd)
dae.add_jac(Gy, -sin(dae.y[self.a]), self.vsd, self.v)
dae.add_jac(Gy, -mul(dae.y[self.v], cos(dae.y[self.a])), self.vsd,
self.a)
dae.add_jac(Gy, cos(dae.y[self.a]), self.vsq, self.v)
dae.add_jac(Gy, -mul(dae.y[self.v], sin(dae.y[self.a])), self.vsq,
self.a)
dae.add_jac(
Gy,
mul(0.5, self.ngen, pi, self.rho, (self.R)**2, (self.Vwn)**3,
div(1, self.mva_mega), (dae.x[self.vw])**3), self.pw, self.cp)
dae.add_jac(
Gy,
mul(-25.52, (dae.y[self.ilamb])**-2,
exp(mul(-12.5, div(1, dae.y[self.ilamb])))) + mul(
12.5, (dae.y[self.ilamb])**-2,
-1.1 + mul(25.52, div(1, dae.y[self.ilamb])) +
mul(-0.088, dae.x[self.theta_p]),
exp(mul(-12.5, div(1, dae.y[self.ilamb])))), self.cp,
self.ilamb)
dae.add_jac(
Gy,
mul((dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p]))**-2,
(div(1, dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p])) +
mul(-0.035, div(1, 1 + (dae.x[self.theta_p])**3)))**-2),
self.ilamb, self.lamb)
dae.add_jac(Gy, -mul(dae.y[self.isd], self.u0), self.a, self.vsd)
dae.add_jac(Gy, -mul(dae.x[self.irq], self.u0), self.a, self.vrq)
dae.add_jac(Gy, -mul(self.u0, dae.y[self.vsq]), self.a, self.isq)
dae.add_jac(Gy, -mul(dae.x[self.ird], self.u0), self.a, self.vrd)
dae.add_jac(Gy, -mul(dae.y[self.isq], self.u0), self.a, self.vsq)
dae.add_jac(Gy, -mul(self.u0, dae.y[self.vsd]), self.a, self.isd)
dae.add_jac(
Gy,
mul(
self.u0,
mul(2, dae.y[self.v], div(1, self.x0)) +
mul(dae.x[self.ird], self.xmu, div(1, self.x0))), self.v,
self.v)
def gcall(self, dae):
toSb = div(self.Sn, self.system.Settings.mva)
dae.g[self.isd] = -dae.y[self.vsd] + mul(
dae.x[self.irq], self.xmu) + mul(dae.y[self.isq], self.x0) - mul(
dae.y[self.isd], self.rs)
dae.g[self.isq] = -dae.y[self.vsq] - mul(
dae.x[self.ird], self.xmu) - mul(dae.y[self.isd], self.x0) - mul(
dae.y[self.isq], self.rs)
dae.g[self.vrd] = -dae.y[self.vrd] + mul(
1 - dae.x[self.omega_m],
mul(dae.x[self.irq], self.x1) +
mul(dae.y[self.isq], self.xmu)) - mul(dae.x[self.ird], self.rr)
dae.g[self.vrq] = -dae.y[self.vrq] - mul(
dae.x[self.irq], self.rr) - mul(
1 - dae.x[self.omega_m],
mul(dae.x[self.ird], self.x1) + mul(dae.y[self.isd], self.xmu))
dae.g[self.vsd] = -dae.y[self.vsd] - mul(dae.y[self.v],
sin(dae.y[self.a]))
dae.g[self.vsq] = -dae.y[self.vsq] + mul(dae.y[self.v],
cos(dae.y[self.a]))
dae.g[self.vref] = self.vref0 - dae.y[self.vref]
# dae.g[self.pwa] = mul(2*dae.x[self.omega_m] - 1, toSb) - dae.y[self.pwa]
dae.g[self.pwa] = mmax(mmin(2 * dae.x[self.omega_m] - 1, 1),
0) - dae.y[self.pwa]
dae.hard_limit(self.pwa, 0, 1)
dae.g[self.pw] = -dae.y[self.pw] + mul(
0.5, dae.y[self.cp], self.ngen, pi, self.rho, (self.R)**2,
(self.Vwn)**3, div(1, self.mva_mega), (dae.x[self.vw])**3)
dae.g[self.cp] = -dae.y[self.cp] + mul(
-1.1 + mul(25.52, div(1, dae.y[self.ilamb])) +
mul(-0.08800000000000001, dae.x[self.theta_p]),
exp(mul(-12.5, div(1, dae.y[self.ilamb]))))
dae.g[self.lamb] = -dae.y[self.lamb] + mul(
4, self.R, self.fn, self.ngb, dae.x[self.omega_m], pi,
div(1, self.Vwn), div(1, self.npole), div(1, dae.x[self.vw]))
dae.g[self.ilamb] = div(
1,
div(1, dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p])) +
mul(-0.035, div(1, 1 +
(dae.x[self.theta_p])**3))) - dae.y[self.ilamb]
dae.g += spmatrix(
mul(
self.u0, -mul(dae.x[self.ird], dae.y[self.vrd]) -
mul(dae.x[self.irq], dae.y[self.vrq]) -
mul(dae.y[self.isd], dae.y[self.vsd]) -
mul(dae.y[self.isq], dae.y[self.vsq])), self.a, [0] * self.n,
(dae.m, 1), 'd')
dae.g += spmatrix(
mul(
self.u0,
mul((dae.y[self.v])**2, div(1, self.x0)) +
mul(dae.x[self.ird], dae.y[self.v], self.xmu, div(
1, self.x0))), self.v, [0] * self.n, (dae.m, 1), 'd')
def gycall(self, dae):
dae.add_jac(
Gy,
mul(0.5, self.ngen, pi, self.rho, (self.R)**2, (self.Vwn)**3,
div(1, self.mva_mega), (dae.x[self.vw])**3), self.pw, self.cp)
dae.add_jac(
Gy,
mul(-25.52, (dae.y[self.ilamb])**-2,
exp(mul(-12.5, div(1, dae.y[self.ilamb])))) + mul(
12.5, (dae.y[self.ilamb])**-2,
-1.1 + mul(25.52, div(1, dae.y[self.ilamb])) +
mul(-0.08800000000000001, dae.x[self.theta_p]),
exp(mul(-12.5, div(1, dae.y[self.ilamb])))), self.cp,
self.ilamb)
dae.add_jac(
Gy,
mul((dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p]))**-2,
(div(1, dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p])) +
mul(-0.035, div(1, 1 + (dae.x[self.theta_p])**3)))**-2),
self.ilamb, self.lamb)
def F(x=None, z=None):
# beta without constant x[:d], constant x[d], t = x[d+1:]
if x is None: return 2 * d, matrix(0.0,(2*d+1,1)) # m = 2 *d is the number of constraints
absE = absA*x[:d+1] # 0 - d contains the constant
absW = exp(absE)
cmpE = cmpA*x[:d+1]
cmpW = exp(cmpE)
f = matrix(0.0,(2*d+1,1))
f[0] = absWeight*(-sum(absE) + sum(log(1+absW))) + cmpWeight*(-sum(cmpE) + sum(log(1+cmpW))) + lamda * sum(x[d+1:])# from d+1 withou the constant
f[1:d+1] = x[:d] - x[d+1:] # beta - t < 0
f[d+1:] = -x[:d] - x[d+1:] # -beta - t <0
Df = matrix(0.0,(2*d+1,2*d+1))
Df[0,:d+1] = absWeight*(matrix(absA.T * (div(absW,1+absW)-1.0))).T + cmpWeight*(matrix(cmpA.T * (div(cmpW,1+cmpW)-1.0))).T
Df[0,d+1:] = lamda
Df[1:d+1,0:d] = spdiag(matrix(1.0,(d,1)))
Df[d+1:, 0:d] = spdiag(matrix(-1.0,(d,1)))
Df[1:d+1,d+1:] = spdiag(matrix(-1.0,(d,1)))
Df[d+1:,d+1:] = spdiag(matrix(-1.0,(d,1)))
if z is None: return f ,Df
H = matrix(0.0,(2*d+1,2*d+1))
H[0:d+1,0:d+1] = absWeight*(absA.T *spdiag(div(absW, (1 + absW) ** 2)) * absA) + cmpWeight*(cmpA.T *spdiag(div(cmpW, (1 +cmpW) ** 2)) * cmpA)
return f, Df, z[0]*H
def gcall(self, dae):
dae.g[self.pw] = -dae.y[self.pw] + mul(
0.5, dae.y[self.cp], self.ngen, pi, self.rho, (self.R)**2,
(self.Vwn)**3, div(1, self.mva_mega), (dae.x[self.vw])**3)
dae.g[self.lamb] = -dae.y[self.lamb] + mul(
4, self.R, self.fn, self.ngb, dae.x[self.omega_m], pi,
div(1, self.Vwn), div(1, self.npole), div(1, dae.x[self.vw]))
dae.g[self.cp] = -dae.y[self.cp] + mul(
-1.1 + mul(25.52, dae.y[self.ilamb]) +
mul(-0.08800000000000001, dae.x[self.theta_p]),
exp(mul(-12.5, dae.y[self.ilamb])))
dae.g[self.ilamb] = div(
1, dae.y[self.lamb] +
mul(0.08, dae.x[self.theta_p])) - dae.y[self.ilamb] + mul(
-0.035, div(1, 1 + (dae.x[self.theta_p])**3))
def windpower(self, ngen, rho, vw, Ar, R, omega, theta, derivative=False):
mva_mega = self.system.mva * 1e6
lamb = omega * R / vw
ilamb = 1 / (1 / (lamb + 0.08 * theta) - 0.035 / (theta**3 + 1))
cp = 0.22 * (116 / ilamb - 0.4 * theta - 5) * exp(-12.5 / ilamb)
pw = 0.5 * ngen * rho * cp * Ar * vw**3 / mva_mega
a1 = exp(-12.5 / ilamb)
a2 = (lamb + 0.08 * theta)**2
a3 = 116. / ilamb - 0.4 * theta - 5
a4 = -9.28 / (lamb + 0.08 * theta) ** 2 + \
12.180 * theta * theta / (theta ** 3 + 1) ** 2 - 0.4
a5 = 1.000 / (lamb + 0.08 * theta) ** 2 - \
1.3125 * theta * theta / (theta ** 3 + 1) ** 2
jac = ones(1, 3)
jac[0] = ngen * R * a1 * rho * vw * vw * Ar * (
-12.760 + 1.3750 * a3) / a2 / mva_mega
jac[1] = ngen * (omega * R * (12.760 - 1.3750 * a3) / a2 +
0.330 * a3 * vw) * vw * Ar * rho * a1 / mva_mega
jac[2] = ngen * 0.110 * rho * (a4 +
a3 * a5) * a1 * Ar * vw**3 / mva_mega
return pw, jac
def get_kernel(X, Y, type='linear', param=1.0):
"""Calculates a kernel given the data X and Y (dims x exms)"""
(Xdims,Xn) = X.size
(Ydims,Yn) = Y.size
kernel = matrix(1.0)
if type=='linear':
print('Calculating linear kernel with size {0}x{1}.'.format(Xn,Yn))
kernel = matrix([ dotu(X[:,i],Y[:,j]) for j in range(Yn) for i in range(Xn)], (Xn,Yn), 'd')
if type=='rbf':
print('Calculating Gaussian kernel with size {0}x{1}.'.format(Xn,Yn))
kernel = matrix([dotu(X[:,i]-Y[:,j],X[:,i]-Y[:,j]) for j in range(Yn) for i in range(Xn)], (Xn,Yn))
kernel = exp(-param*kernel)
return kernel
def computeLoss(Z,Y,D,lam_t,lam_s,T,n_rows,n_cols,lh_trend):
'''
This function evaluates the objective function at a given solution
Z.
'''
n_t=n_rows*n_cols*(T-2);#no. of temporal constraints
n_lh=0
if lh_trend:
n_year=T/52
n_lh=(n_year-2)*n_rows*n_cols
n_s=D.size[0]-n_lh-n_t#no. of spatial constraints
lam_vec=np.array(n_t*[lam_t]+n_s*[lam_s]+n_lh*[lam_t]).reshape((1,-1))
f1=np.sum(np.array(Z+mul(Y**2,exp(-Z))))
f2=float(lam_vec.dot( np.abs(np.array(D*Z)) ))
return f1+f2
def solving_for_cloud_x(m, M):
N, T = m.shape
K = [1]
for i in range(T):
K.append(N)
g = [1]
for i in range(N * T):
g.append(1 / M)
F = np.zeros((N * T + 1, N * T))
F[0, :] = -1 * m.reshape(1, N * T)
for i in range(T):
for j in range(N):
F[N * i + j + 1, j * T + i] = 1
solvers.options['show_progress'] = False
sol_cloud = solvers.gp(K, matrix(F), log(matrix(g, tc='d')))
x_cloud = array(exp(sol_cloud['x'])).reshape((N, T))
return x_cloud
def F(x=None, z=None):
# beta without constant x[:d], constant x[d], t = x[d+1:]
if x is None: return 2 * d, matrix(0.0,(2*d+1,1)) # m = 2 *d is the number of constraints
e = A*x[:d+1] # 0 - d contains the constant
w = exp(e)
f = matrix(0.0,(2*d+1,1))
f[0] = alpha*(-sum(e) + sum(log(1+w))) + lamda * sum(x[d+1::])# from d+1 withou the constant
f[1:d+1] = x[:d] - x[d+1:] # beta - t < 0
f[d+1:] = -x[:d] - x[d+1:] # -beta - t <0
Df = matrix(0.0,(2*d+1,2*d+1))
# Df[0,:d+1] = (matrix(A.T * (div(w,1+w)-1.0))).T
Df[0, :d + 1] = alpha*(matrix(A.T * (div(w, 1 + w) - 1.0))).T
Df[0,d+1:] = lamda
Df[1:d+1,0:d] = spdiag(matrix(1.0,(d,1)))
Df[d+1:, 0:d] = spdiag(matrix(-1.0,(d,1)))
Df[1:d+1,d+1:] = spdiag(matrix(-1.0,(d,1)))
Df[d+1:,d+1:] = spdiag(matrix(-1.0,(d,1)))
if z is None: return f ,Df
H = matrix(0.0,(2*d+1,2*d+1))
H[0:d+1,0:d+1] = alpha*(A.T *spdiag(div(w, (1 + w) ** 2)) * A)
return f, Df, z[0]*H
def get_kernel(X, Y, type='linear', param=1.0):
"""Calculates a kernel given the data X and Y (dims x exms)"""
(Xdims, Xn) = X.size
(Ydims, Yn) = Y.size
kernel = matrix(1.0)
if type == 'linear':
print('Calculating linear kernel with size {0}x{1}.'.format(
Xn, Yn))
kernel = matrix(
[dotu(X[:, i], Y[:, j]) for j in range(Yn) for i in range(Xn)],
(Xn, Yn), 'd')
if type == 'rbf':
print('Calculating Gaussian kernel with size {0}x{1}.'.format(
Xn, Yn))
kernel = matrix([
dotu(X[:, i] - Y[:, j], X[:, i] - Y[:, j]) for j in range(Yn)
for i in range(Xn)
], (Xn, Yn))
kernel = exp(-param * kernel)
return kernel
def test_cvxopt_sparse(self):
m = 100
n = 20
mu = cvxopt.exp( cvxopt.normal(m) )
F = cvxopt.normal(m, n)
D = cvxopt.spdiag( cvxopt.uniform(m) )
x = Variable(m)
exp = square(norm2(D*x))
# # TODO
# # Test scipy sparse matrices.
# def test_scipy_sparse(self):
# m = 100
# n = 20
# mu = cvxopt.exp( cvxopt.normal(m) )
# F = cvxopt.normal(m, n)
# x = Variable(m)
# D = scipy.sparse.spdiags(1.5, 0, m, m )
# exp = square(norm2(D*x))
def fxcall(self, dae):
dae.add_jac(
Gx,
mul(1.5, dae.y[self.cp],
self.ngen, pi, self.rho, (self.R)**2, (self.Vwn)**3,
div(1, self.mva_mega), (dae.x[self.vw])**2), self.pw, self.vw)
dae.add_jac(Gx, mul(-0.088, exp(mul(-12.5, dae.y[self.ilamb]))),
self.cp, self.theta_p)
dae.add_jac(
Gx,
mul(-4, self.R, self.fn, self.ngb, dae.x[self.omega_m], pi,
div(1, self.Vwn), div(1, self.npole), (dae.x[self.vw])**-2),
self.lamb, self.vw)
dae.add_jac(
Gx,
mul(4, self.R, self.fn, self.ngb, pi, div(1, self.Vwn),
div(1, self.npole), div(1, dae.x[self.vw])), self.lamb,
self.omega_m)
dae.add_jac(
Gx,
mul(-0.08,
(dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p]))**-2) +
mul(0.10500000000000001, (dae.x[self.theta_p])**2,
(1 + (dae.x[self.theta_p])**3)**-2), self.ilamb, self.theta_p)
def __init__(self, casefile, **kwargs):
"""
Optional keyword arguments:
branch_rmin (default: -inf )
shunt_gmin (default: -inf )
gen_elim (default: 0.0 )
truncate_gen_bounds (default: None )
line_constraints (default: True )
scale (default: False )
"""
self.to_real = kwargs.get('to_real', True)
self.conversion = kwargs.get('conversion', False)
self.scale = kwargs.get('scale', False)
self.shunt_gmin = kwargs.get('shunt_gmin', -float('inf'))
self.branch_rmin = kwargs.get('branch_rmin', -float('inf'))
self.gen_elim = kwargs.get('gen_elim', 0.0)
self.truncate_gen_bounds = kwargs.get('truncate_gen_bounds',None)
self.line_constraints = kwargs.get('line_constraints', True)
self.pad_constraints = kwargs.get('pad_constraints', True)
self.__verbose = kwargs.get('verbose',0)
self.eigtol = kwargs.get('eigtol',1e5)
### load data
data = _load_case(casefile, verbose = kwargs.get('verbose',0))
assert data['version'] == '2'
if kwargs.get('verbose',0): print("Extracting data from case file.")
### add data to object
self.baseMVA = data['baseMVA']
self.cost_scale = self.baseMVA
self.nbus = data['bus'].shape[0]
# branches in service
active_branches = [i for i in range(data['branch'].shape[0]) if data['branch'][i,10] > 0]
self.nbranch = len(active_branches)
# generators in service
active_generators = [i for i in range(data['gen'].shape[0]) if data['gen'][i,7] > 0]
self.ngen = len(active_generators)
# bus data
self.busses = []
for k in range(self.nbus):
bus = {'id': int(data['bus'][k,0]),
'type': int(data['bus'][k,1]),
'Pd': data['bus'][k,2] / self.baseMVA,
'Qd': data['bus'][k,3] / self.baseMVA,
'Gs': data['bus'][k,4],
'Bs': data['bus'][k,5],
'area': data['bus'][k,6],
'Vm': data['bus'][k,7],
'Va': data['bus'][k,8],
'baseKV': data['bus'][k,9],
'maxVm': data['bus'][k,11],
'minVm': data['bus'][k,12]}
if bus['Gs'] < self.shunt_gmin: bus['Gs'] = self.shunt_gmin
self.busses.append(bus)
self.bus_id_to_index = {}
for k, bus in enumerate(self.busses):
self.bus_id_to_index[bus['id']] = k
# generator data
self.generators = []
ii_p = 0
ii_q = 0
for k in active_generators:
gen = {'bus_id': int(data['gen'][k,0]),
'Pg': data['gen'][k,1] / self.baseMVA,
'Pmax': data['gen'][k,8] / self.baseMVA,
'Pmin': data['gen'][k,9] / self.baseMVA,
'Qg': data['gen'][k,2] / self.baseMVA,
'Qmax': data['gen'][k,3] / self.baseMVA,
'Qmin': data['gen'][k,4] / self.baseMVA,
'Vg': data['gen'][k,5],
'mBase': data['gen'][k,6]
}
if gen['Pmax'] > gen['Pmin'] + self.gen_elim:
gen['pslack'] = ii_p
ii_p += 1
else:
# eliminate slack variable and set Pmin and Pmax to their average
gen['pslack'] = None
gen['Pmin'] = (gen['Pmax'] + gen['Pmin'])/2.0
gen['Pmax'] = gen['Pmin']
if gen['Qmax'] > gen['Qmin'] + self.gen_elim:
gen['qslack'] = ii_q
ii_q += 1
else:
# eliminate slack variable and set Qmin and Qmax to their average
gen['qslack'] = None
gen['Qmin'] = (gen['Qmax'] + gen['Qmin'])/2.0
gen['Qmax'] = gen['Qmin']
if self.truncate_gen_bounds:
if gen['Pmin'] < -self.truncate_gen_bounds or gen['Pmax'] > self.truncate_gen_bounds:
if kwargs.get('verbose',0):
print("Warning: generator at bus %i with large active bound(s); decreasing bound(s)"%(gen['bus_id']))
gen['Pmin'] = max(-self.truncate_gen_bounds, gen['Pmin'])
gen['Pmax'] = min( self.truncate_gen_bounds, gen['Pmax'])
if gen['Qmin'] < -self.truncate_gen_bounds or gen['Qmax'] > self.truncate_gen_bounds:
if kwargs.get('verbose',0):
print("Warning: generator at bus %i with large reactive bound(s); decreasing bound(s)"%(gen['bus_id']))
gen['Qmin'] = max(-self.truncate_gen_bounds, gen['Qmin'])
gen['Qmax'] = min( self.truncate_gen_bounds, gen['Qmax'])
self.generators.append(gen)
self.bus_id_to_genlist = {}
for k, gen in enumerate(self.generators):
if gen['bus_id'] in self.bus_id_to_genlist:
self.bus_id_to_genlist[gen['bus_id']].append(k)
else:
self.bus_id_to_genlist[gen['bus_id']] = [k]
# branch data
self.branches = []
for k in active_branches:
branch = {'from': int(data['branch'][k,0]),
'to': int(data['branch'][k,1]),
'r': data['branch'][k,2],
'x': data['branch'][k,3],
'b': data['branch'][k,4],
'rateA': data['branch'][k,5],
'rateB': data['branch'][k,6],
'rateC': data['branch'][k,7],
'ratio': data['branch'][k,8],
'angle': data['branch'][k,9],
'angle_min': data['branch'][k,11],
'angle_max': data['branch'][k,12]
}
if branch['r'] < self.branch_rmin:
if kwargs.get('verbose',0): print("Warning: branch (%i:%i->%i) with small resistance; enforcing min. resistance"%(k,branch['from'],branch['to']))
branch['r'] = self.branch_rmin
if not self.line_constraints:
branch['rateA'] = 0.0
if not self.pad_constraints:
branch['angle_min'] = -360.0
branch['angle_max'] = 360.0
elif (branch['angle_min'] > -360.0 and branch['angle_min'] <= -180.0) or (branch['angle_max'] < 360.0 and branch['angle_max'] >= 180.0) or (branch['angle_min']==0.0 and branch['angle_max']==0.0):
if kwargs.get('verbose',0): print("Warning: branch (%i:%i->%i) with unsupported phase angle diff. constraint; dropping constraint"%(k,branch['from'],branch['to']))
branch['angle_min'] = -360.0
branch['angle_max'] = 360.0
self.branches.append(branch)
# gen cost
for i, k in enumerate(active_generators):
gencost = {'model': int(data['gencost'][k,0]),
'startup': data['gencost'][k,1],
'shutdown': data['gencost'][k,2],
'ncoef': int(data['gencost'][k,3]),
'coef': data['gencost'][k,4:].T
}
if gencost['model'] == 1:
raise TypeError("Piecewise linear cost functions are not supported.")
self.generators[i]['Pcost'] = gencost
if data['gencost'].shape[0] == 2*data['gen'].shape[0]:
offset = data['gen'].shape[0]
for i, k in enumerate(active_generators):
gencost = {'model': int(data['gencost'][offset + k,0]),
'startup': data['gencost'][offset + k,1],
'shutdown': data['gencost'][offset + k,2],
'ncoef': int(data['gencost'][offset + k,3]),
'coef': data['gencost'][offset + k,4:].T
}
self.generators[i]['Qcost'] = gencost
### Compute bus admittance matrix and connection matrices
j = complex(0.0,1.0)
r = matrix([branch['r'] for branch in self.branches])
b = matrix([branch['b'] for branch in self.branches])
x = matrix([branch['x'] for branch in self.branches])
Gs = matrix([bus['Gs'] for bus in self.busses])
Bs = matrix([bus['Bs'] for bus in self.busses])
tap = matrix([1.0 if branch['ratio'] == 0.0 else branch['ratio'] for branch in self.branches])
angle = matrix([branch['angle'] for branch in self.branches])
tap = mul(tap, exp(j*pi/180.*angle))
self.tap = tap
Ys = div(1.0, r + j*x)
Ytt = Ys + j*b/2.0
Yff = div(Ytt, mul(tap, tap.H.T))
Yft = -div(Ys, tap.H.T)
Ytf = -div(Ys, tap)
Ysh = (Gs+j*Bs)/self.baseMVA
self.Ybr = []
for k in range(self.nbranch): self.Ybr.append([[Yff[k],Yft[k]],[Ytf[k],Ytt[k]]])
f = matrix([self.bus_id_to_index[branch['from']] for branch in self.branches])
t = matrix([self.bus_id_to_index[branch['to']] for branch in self.branches])
Cf = spmatrix(1.0, range(self.nbranch), f, (self.nbranch,self.nbus))
Ct = spmatrix(1.0, range(self.nbranch), t, (self.nbranch,self.nbus))
Yf = spmatrix(matrix([Yff,Yft]), 2*list(range(self.nbranch)), matrix([f,t]), (self.nbranch,self.nbus))
Yt = spmatrix(matrix([Ytf,Ytt]), 2*list(range(self.nbranch)), matrix([f,t]), (self.nbranch,self.nbus))
Ybus = Cf.T*Yf + Ct.T*Yt + spmatrix(Ysh, range(self.nbus), range(self.nbus), (self.nbus,self.nbus))
self.Cf = Cf
self.Ct = Ct
self.Yf = Yf
self.Yt = Yt
self.Ybus = Ybus
if kwargs.get('verbose',0): print("Building cone LP.")
self._build_conelp()
if self.conversion:
if kwargs.get('verbose',0):
if kwargs.get('coupling','full') == 'full':
print("Applying chordal conversion to cone LP.")
else:
print("Applying partial chordal conversion to cone LP.")
_conelp_convert(self, **kwargs)
if self.scale:
if kwargs.get('verbose',0): print("Scaling cone LP.")
_conelp_scale(self, **kwargs)
if self.to_real:
if kwargs.get('verbose',0): print("Converting to real-valued cone LP.")
_conelp_to_real(self, **kwargs)
return
def Se(self):
dae = self.system.DAE
vfout = dae.x[self.vfout]
return mul(self.Ae, exp(mul(self.Be, abs(vfout))))
def fxcall(self, dae):
omega = not0(dae.x[self.omega_m])
toSb = div(self.Sn, self.system.mva)
dae.add_jac(Gx, mul(self.x1, 1 - dae.x[self.omega_m]), self.vrd,
self.irq)
dae.add_jac(
Gx,
-mul(dae.x[self.irq], self.x1) - mul(dae.y[self.isq], self.xmu),
self.vrd, self.omega_m)
dae.add_jac(
Gx,
mul(dae.x[self.ird], self.x1) + mul(dae.y[self.isd], self.xmu),
self.vrq, self.omega_m)
dae.add_jac(Gx, -mul(self.x1, 1 - dae.x[self.omega_m]), self.vrq,
self.ird)
dae.add_jac(
Gx,
mul(1.5, dae.y[self.cp],
self.ngen, pi, self.rho, (self.R)**2, (self.Vwn)**3,
div(1, self.mva_mega), (dae.x[self.vw])**2), self.pw, self.vw)
dae.add_jac(Gx, mul(-0.088, exp(mul(-12.5, div(1,
dae.y[self.ilamb])))),
self.cp, self.theta_p)
dae.add_jac(
Gx,
mul(-4, self.R, self.fn, self.ngb, dae.x[self.omega_m], pi,
div(1, self.Vwn), div(1, self.npole), (dae.x[self.vw])**-2),
self.lamb, self.vw)
dae.add_jac(
Gx,
mul(4, self.R, self.fn, self.ngb, pi, div(1, self.Vwn),
div(1, self.npole), div(1, dae.x[self.vw])), self.lamb,
self.omega_m)
dae.add_jac(
Gx,
mul((div(1, dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p])) +
mul(-0.035, div(1, 1 + (dae.x[self.theta_p])**3)))**-2,
mul(0.08,
(dae.y[self.lamb] + mul(0.08, dae.x[self.theta_p]))**-2) +
mul(-0.105, (dae.x[self.theta_p])**2,
(1 + (dae.x[self.theta_p])**3)**-2)), self.ilamb,
self.theta_p)
dae.add_jac(Gx, -mul(self.u0, dae.y[self.vrq]), self.a, self.irq)
dae.add_jac(Gx, -mul(self.u0, dae.y[self.vrd]), self.a, self.ird)
dae.add_jac(Gx, mul(self.u0, dae.y[self.v], self.xmu, div(1, self.x0)),
self.v, self.ird)
dae.add_jac(
Fx,
mul(dae.y[self.pwa], self.x0, toSb, div(1, self.Te),
(dae.x[self.omega_m])**-2, div(1, dae.y[self.v]),
div(1, self.xmu)), self.irq, self.omega_m)
dae.add_jac(
Fy,
mul(dae.y[self.pwa], self.x0, toSb, div(1, self.Te), div(1, omega),
(dae.y[self.v])**-2, div(1, self.xmu)), self.irq, self.v)
dae.add_jac(
Fy, -mul(self.x0, toSb, div(1, self.Te), div(1, omega),
div(1, dae.y[self.v]), div(1, self.xmu)), self.irq,
self.pwa)
dae.add_jac(Fx, mul(0.5, dae.y[self.isq], self.xmu, div(1, self.H)),
self.omega_m, self.ird)
dae.add_jac(
Fx,
mul(-0.5, dae.y[self.pw], div(1, self.H),
(dae.x[self.omega_m])**-2), self.omega_m, self.omega_m)
dae.add_jac(Fx, mul(-0.5, dae.y[self.isd], self.xmu, div(1, self.H)),
self.omega_m, self.irq)
dae.add_jac(Fy, mul(0.5, div(1, self.H), div(1, dae.x[self.omega_m])),
self.omega_m, self.pw)
dae.add_jac(Fy, mul(-0.5, dae.x[self.irq], self.xmu, div(1, self.H)),
self.omega_m, self.isd)
dae.add_jac(Fy, mul(0.5, dae.x[self.ird], self.xmu, div(1, self.H)),
self.omega_m, self.isq)
# The small GP of section 9.3 (Geometric programming).
from cvxopt import matrix, log, exp, solvers
Aflr = 1000.0
Awall = 100.0
alpha = 0.5
beta = 2.0
gamma = 0.5
delta = 2.0
F = matrix([[-1., 1., 1., 0., -1., 1., 0., 0.],
[-1., 1., 0., 1., 1., -1., 1., -1.],
[-1., 0., 1., 1., 0., 0., -1., 1.]])
g = log(
matrix([
1.0, 2 / Awall, 2 / Awall, 1 / Aflr, alpha, 1 / beta, gamma, 1 / delta
]))
K = [1, 2, 1, 1, 1, 1, 1]
h, w, d = exp(solvers.gp(K, F, g)['x'])
print("\n h = %f, w = %f, d = %f.\n" % (h, w, d))
b[6][2] = 0.5 * A[6][2, :] * (C[5] + C[6])
# For regions k=1, ..., 6, let pk be the optimal value of
#
# minimize x'*x
# subject to A*x <= b.
#
# The Chernoff upper bound is 1.0 - sum exp( - pk / (2 sigma^2)).
P = matrix([1.0, 0.0, 0.0, 1.0], (2, 2))
q = matrix(0.0, (2, 1))
optvals = matrix(
[blas.nrm2(solvers.qp(P, q, A[k], b[k])['x'])**2 for k in range(1, 7)])
nopts = 200
sigmas = 0.2 + (0.5 - 0.2) / nopts * matrix(list(range(nopts)), tc='d')
bnds = [1.0 - sum(exp(-optvals / (2 * sigma**2))) for sigma in sigmas]
try:
import pylab
except ImportError:
pass
else:
pylab.figure(facecolor='w')
pylab.plot(sigmas, bnds, '-')
pylab.axis([0.2, 0.5, 0.9, 1.0])
pylab.title('Chernoff lower bound (fig. 7.8)')
pylab.xlabel('sigma')
pylab.ylabel('Probability of correct detection')
pylab.show()
if 0: # uncomment out for the Monte Carlo estimation.
def F(x=None, z=None):
""" Evaluates the objective and nonlinear constraint functions.
"""
if x is None:
# Include power mismatch constraint twice to force equality.
nln_N = self.om.nln_N + neqnln
return nln_N, x0
# Evaluate objective function -------------------------------------
Pgen = x[Pg.i1:Pg.iN + 1] # Active generation in p.u.
Qgen = x[Qg.i1:Qg.iN + 1] # Reactive generation in p.u.
xx = matrix([Pgen, Qgen]) * base_mva
# Evaluate the objective function value.
if len(ipol) > 0:
# FIXME: Implement reactive power costs.
f0 = sum([g.total_cost(xx[i]) for i, g in enumerate(gn)])
else:
f0 = 0
# Piecewise linear cost of P and Q.
if ny:
y = self.om.get_var("y")
ccost = spmatrix(matrix(1.0, (ny, 1)),
range(y.i1, y.iN + 1),
matrix(0.0, (ny, 1)),
size=(nxyz, 1)).T
f0 = f0 + ccost * x
else:
ccost = matrix(0.0, (1, nxyz))
# TODO: Generalised cost term.
# Evaluate cost gradient ------------------------------------------
iPg = range(Pg.i1, Pg.iN + 1)
iQg = range(Qg.i1, Qg.iN + 1)
# Polynomial cost of P and Q.
df_dPgQg = matrix(0.0, (2 * ng, 1)) # w.r.t p.u. Pg and Qg
for i in ipol:
df_dPgQg[i] = \
base_mva * polyval(polyder(list(gn[i].p_cost)), xx[i])
df0 = matrix(0.0, (nxyz, 1))
df0[iPg] = df_dPgQg[:ng]
df0[iQg] = df_dPgQg[ng:ng + ng]
# Piecewise linear cost of P and Q.
df0 = df0 + ccost.T
# TODO: Generalised cost term.
# Evaluate cost Hessian -------------------------------------------
# d2f = None
# Evaluate nonlinear equality constraints -------------------------
for i, g in enumerate(gn):
g.p = Pgen[i] * base_mva # active generation in MW
g.q = Qgen[i] * base_mva # reactive generation in MVAr
# Rebuild the net complex bus power injection vector in p.u.
Sbus = matrix(case.getSbus(bs))
Vang = x[Va.i1:Va.iN + 1]
Vmag = x[Vm.i1:Vm.iN + 1]
V = mul(Vmag, exp(1j * Vang))
# Evaluate the power flow equations.
mis = mul(V, conj(Ybus * V)) - Sbus
# Equality constraints (power flow).
g = matrix([
mis.real(), # active power mismatch for all buses
mis.imag()
]) # reactive power mismatch for all buses
# Evaluate nonlinear inequality constraints -----------------------
# Inequality constraints (branch flow limits).
# (line constraint is actually on square of limit)
flow_max = matrix([(l.rate_a / base_mva)**2 for l in ln])
# FIXME: There must be a more elegant way to do this.
for i, rate in enumerate(flow_max):
if rate == 0.0:
flow_max[i] = 1e5
if self.flow_lim == IFLOW:
If = Yf * V
It = Yt * V
# Branch current limits.
h = matrix([(If * conj(If)) - flow_max,
(If * conj(It)) - flow_max])
else:
i_fbus = [e.from_bus._i for e in ln]
i_tbus = [e.to_bus._i for e in ln]
# Complex power injected at "from" bus (p.u.).
Sf = mul(V[i_fbus], conj(Yf * V))
# Complex power injected at "to" bus (p.u.).
St = mul(V[i_tbus], conj(Yt * V))
if self.flow_lim == PFLOW: # active power limit, P (Pan Wei)
# Branch real power limits.
h = matrix(
[Sf.real()**2 - flow_max,
St.real()**2 - flow_max])
elif self.flow_lim == SFLOW: # apparent power limit, |S|
# Branch apparent power limits.
h = matrix([
mul(Sf, conj(Sf)) - flow_max,
mul(St, conj(St)) - flow_max
]).real()
else:
raise ValueError
# Evaluate partial derivatives of constraints ---------------------
iVa = matrix(range(Va.i1, Va.iN + 1))
iVm = matrix(range(Vm.i1, Vm.iN + 1))
iPg = matrix(range(Pg.i1, Pg.iN + 1))
iQg = matrix(range(Qg.i1, Qg.iN + 1))
iVaVmPgQg = matrix([iVa, iVm, iPg, iQg]).T
# Compute partials of injected bus powers.
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V)
i_gbus = [gen.bus._i for gen in gn]
neg_Cg = spmatrix(matrix(-1.0, (ng, 1)), i_gbus, range(ng),
(nb, ng))
# Transposed Jacobian of the power balance equality constraints.
dg = spmatrix([], [], [], (nxyz, 2 * nb))
blank = spmatrix([], [], [], (nb, ng))
dg[iVaVmPgQg, :] = sparse([[dSbus_dVa.real(),
dSbus_dVa.imag()],
[dSbus_dVm.real(),
dSbus_dVm.imag()], [neg_Cg, blank],
[blank, neg_Cg]]).T
# Compute partials of flows w.r.t V.
if self.flow_lim == IFLOW:
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dIbr_dV(Yf, Yt, V)
else:
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dSbr_dV(Yf, Yt, V, bs, ln)
if self.flow_lim == PFLOW:
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 (complex power, 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 = spmatrix([], [], [], (nxyz, 2 * nl))
dh[matrix([iVa, iVm]).T, :] = sparse([[df_dVa, dt_dVa],
[df_dVm, dt_dVm]]).T
f = matrix([f0, g, h, -g])
df = matrix([[df0], [dg], [dh], [-dg]]).T
if z is None:
return f, df
# Evaluate cost Hessian -------------------------------------------
nxtra = nxyz - (2 * nb)
# Evaluate d2f ----------------------------------------------------
d2f_dPg2 = matrix(0.0, (ng, 1)) # w.r.t p.u. Pg
d2f_dQg2 = matrix(0.0, (ng, 1)) # w.r.t p.u. Qg
for i in ipol:
d2f_dPg2[i, 0] = polyval(polyder(list(gn[i].p_cost), 2),
Pg.v0[i] * base_mva) * base_mva**2
# TODO: Reactive power costs.
# for i in ipol:
# d2f_dQg2[i] = polyval(polyder(list(gn[i].q_cost), 2),
# Qg.v0[i] * base_mva) * base_mva**2
i = matrix([range(Pg.i1, Pg.iN + 1), range(Qg.i1, Qg.iN + 1)])
d2f = spmatrix(matrix([d2f_dPg2, d2f_dQg2]), i, i, (nxyz, nxyz))
# TODO: Generalised cost model.
d2f = d2f * self.opt["cost_mult"]
# Evaluate Hessian of power balance constraints -------------------
eqnonlin = z[1:neqnln + 1]
nlam = len(eqnonlin) / 2
lamP = eqnonlin[:nlam]
lamQ = eqnonlin[nlam:nlam + nlam]
Gpaa, Gpav, Gpva, Gpvv = d2Sbus_dV2(Ybus, V, lamP)
Gqaa, Gqav, Gqva, Gqvv = d2Sbus_dV2(Ybus, V, lamQ)
d2G_1 = sparse([[
sparse([[Gpaa, Gpva], [Gpav, Gpvv]]).real() +
sparse([[Gqaa, Gqva], [Gqav, Gqvv]]).imag()
], [spmatrix([], [], [], (2 * nb, nxtra))]])
d2G_2 = spmatrix([], [], [], (nxtra, 2 * nb + nxtra))
d2G = sparse([d2G_1, d2G_2])
# Evaluate Hessian of flow constraints ---------------------------
ineqnonlin = z[1 + neqnln:1 + neqnln + niqnln]
nmu = len(ineqnonlin) / 2
muF = ineqnonlin[:nmu]
muT = ineqnonlin[nmu:nmu + nmu]
if self.flow_lim == IFLOW:
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:
fr = [e.from_bus._i for e in ln]
to = [e.to_bus._i for e in ln]
# Line-bus connection matrices.
Cf = spmatrix(1.0, range(nl), fr, (nl, nb))
Ct = spmatrix(1.0, range(nl), to, (nl, nb))
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = \
dSbr_dV(Yf, Yt, V, bs, ln)
if self.flow_lim == PFLOW:
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)
elif self.flow_lim == SFLOW:
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)
else:
raise ValueError
d2H_1 = sparse([[
sparse([[Hfaa, Hfva], [Hfav, Hfvv]]) +
sparse([[Htaa, Htva], [Htav, Htvv]])
], [spmatrix([], [], [], (2 * nb, nxtra))]])
d2H_2 = spmatrix([], [], [], (nxtra, 2 * nb + nxtra))
d2H = sparse([d2H_1, d2H_2])
Lxx = d2f + d2G + d2H
return f, df, Lxx
def polar(m, a):
"""Return complex number from polar form m*exp(1j*a)"""
return mul(m, exp(1j * a))
def sigmoid(X):
return (1.0 + co.exp(-X))**(-1)
def dSe(self):
dae = self.system.dae
vfout = dae.x[self.vfout]
return mul(self.Ae, exp(mul(self.Be, abs(vfout)))) \
+ mul(self.Ae, self.Be, abs(vfout), exp(mul(self.Be, abs(vfout))))
import cvxopt
import numpy
from cvxpy import *
from multiprocessing import Pool
from pylab import figure, show
import math
num_assets = 100
num_factors = 20
mu = cvxopt.exp(cvxopt.normal(num_assets))
F = cvxopt.normal(num_assets, num_factors)
D = cvxopt.spdiag(cvxopt.uniform(num_assets))
x = Variable(num_assets)
gamma = Parameter(sign="positive")
expected_return = mu.T * x
variance = square(norm2(F.T * x)) + square(norm2(D * x))
# construct portfolio optimization problem *once*
p = Problem(Maximize(expected_return - gamma * variance), [sum(x) == 1, x >= 0])
# encapsulate the allocation function
def allocate(gamma_value):
gamma.value = gamma_value
p.solve()
w = x.value
expected_return, risk = mu.T * w, w.T * (F * F.T + D * D) * w
return (expected_return[0], math.sqrt(risk[0]))
import cvxopt
import numpy
from cvxpy import *
from multiprocessing import Pool
from pylab import figure, show
import math
num_assets = 100
num_factors = 20
mu = cvxopt.exp( cvxopt.normal(num_assets) )
F = cvxopt.normal(num_assets, num_factors)
D = cvxopt.spdiag( cvxopt.uniform(num_assets) )
x = Variable(num_assets)
gamma = Parameter(sign="positive")
expected_return = mu.T * x
variance = square(norm2(F.T*x)) + square(norm2(D*x))
# construct portfolio optimization problem *once*
p = Problem(
Maximize(expected_return - gamma * variance),
[sum(x) == 1, x >= 0]
)
# encapsulate the allocation function
def allocate(gamma_value):
gamma.value = gamma_value
p.solve()
w = x.value
expected_return, risk = mu.T*w, w.T*(F*F.T + D*D)*w
def sigmoid(X): return (1.0 + co.exp(-X))**(-1)
def init1(self, dae):
"""New initialization function"""
self.servcall(dae)
retval = True
mva = self.system.mva
self.p0 = mul(self.p0, self.gammap)
self.q0 = mul(self.q0, self.gammaq)
dae.y[self.vsd] = mul(dae.y[self.v], -sin(dae.y[self.a]))
dae.y[self.vsq] = mul(dae.y[self.v], cos(dae.y[self.a]))
rs = matrix(self.rs)
rr = matrix(self.rr)
xmu = matrix(self.xmu)
x1 = matrix(self.xs) + xmu
x2 = matrix(self.xr) + xmu
Pg = matrix(self.p0)
Qg = matrix(self.q0)
Vc = dae.y[self.v]
vsq = dae.y[self.vsq]
vsd = dae.y[self.vsd]
toSn = div(mva, self.Sn) # to machine base
toSb = self.Sn / mva # to system base
# rotor speed
omega = 1 * (ageb(mva * Pg, self.Sn)) + \
mul(0.5 + 0.5 * mul(Pg, toSn),
aandb(agtb(Pg, 0), altb(mva * Pg, self.Sn))) + \
0.5 * (aleb(mva * Pg, 0))
slip = 1 - omega
theta = mul(self.Kp, mround(1000 * (omega - 1)) / 1000)
theta = mmax(theta, 0)
# prepare for the iterations
irq = mul(-x1, toSb, (2 * omega - 1), div(1, Vc), div(1, xmu),
div(1, omega))
isd = zeros(*irq.size)
isq = zeros(*irq.size)
# obtain ird isd isq
for i in range(self.n):
A = sparse([[-rs[i], vsq[i]], [x1[i], -vsd[i]]])
B = matrix([vsd[i] - xmu[i] * irq[i], Qg[i]])
linsolve(A, B)
isd[i] = B[0]
isq[i] = B[1]
ird = -div(vsq + mul(rs, isq) + mul(x1, isd), xmu)
vrd = -mul(rr, ird) + mul(
slip,
mul(x2, irq) + mul(xmu, isq)) # todo: check x1 or x2
vrq = -mul(rr, irq) - mul(slip, mul(x2, ird) + mul(xmu, isd))
# main iterations
for i in range(self.n):
mis = ones(6, 1)
rows = [0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5]
cols = [0, 1, 3, 0, 1, 2, 2, 4, 3, 5, 0, 1, 2]
x = matrix([isd[i], isq[i], ird[i], irq[i], vrd[i], vrq[i]])
# vals = [-rs, x1, xmu, -x1, -rs, -xmu, -rr,
# -1, -rr, -1, vsd, vsq, -xmu * Vc / x1]
vals = [
-rs[i], x1[i], xmu[i], -x1[i], -rs[i], -xmu[i], -rr[i], -1,
-rr[i], -1, vsd[i], vsq[i], -xmu[i] * Vc[i] / x1[i]
]
jac0 = spmatrix(vals, rows, cols, (6, 6), 'd')
niter = 0
while max(abs(mis)) > self.system.tds.config.tol:
if niter > 20:
logger.error('Initialization of DFIG <{}> failed.'.format(
self.name[i]))
retval = False
break
mis[0] = -rs[i] * x[0] + x1[i] * x[1] + xmu[i] * x[3] - vsd[i]
mis[1] = -rs[i] * x[1] - x1[i] * x[0] - xmu[i] * x[2] - vsq[i]
mis[2] = -rr[i] * x[2] + slip[i] * (x2[i] * x[3] +
xmu[i] * x[1]) - x[4]
mis[3] = -rr[i] * x[3] - slip[i] * (x2[i] * x[2] +
xmu[i] * x[0]) - x[5]
mis[4] = vsd[i] * x[0] + vsq[i] * x[1] + x[4] * x[2] + \
x[5] * x[3] - Pg[i]
mis[5] = -xmu[i] * Vc[i] * x[2] / x1[i] - \
Vc[i] * Vc[i] / x1[i] - Qg[i]
rows = [2, 2, 3, 3, 4, 4, 4, 4]
cols = [1, 3, 0, 2, 2, 3, 4, 5]
vals = [
slip[i] * xmu[i], slip[i] * x2[i], -slip[i] * xmu[i],
-slip[i] * x2[i], x[4], x[5], x[2], x[3]
]
jac = jac0 + spmatrix(vals, rows, cols, (6, 6), 'd')
linsolve(jac, mis)
x -= mis
niter += 1
isd[i] = x[0]
isq[i] = x[1]
ird[i] = x[2]
irq[i] = x[3]
vrd[i] = x[4]
vrq[i] = x[5]
dae.x[self.ird] = mul(self.u0, ird)
dae.x[self.irq] = mul(self.u0, irq)
dae.y[self.isd] = isd
dae.y[self.isq] = isq
dae.y[self.vrd] = vrd
dae.y[self.vrq] = vrq
dae.x[self.omega_m] = mul(self.u0, omega)
dae.x[self.theta_p] = mul(self.u0, theta)
dae.y[self.pwa] = mmax(mmin(2 * dae.x[self.omega_m] - 1, 1), 0)
self.vref0 = mul(aneb(self.KV, 0),
Vc - div(ird + div(Vc, xmu), self.KV))
dae.y[self.vref] = self.vref0
k = mul(div(x1, Vc, xmu, omega), toSb)
self.irq_off = -mul(k, mmax(mmin(2 * omega - 1, 1), 0)) - irq
# electrical torque in pu
te = mul(
xmu,
mul(dae.x[self.irq], dae.y[self.isd]) -
mul(dae.x[self.ird], dae.y[self.isq]))
for i in range(self.n):
if te[i] < 0:
logger.error(
'Pe < 0 on bus <{}>. Wind speed initialize failed.'.format(
self.bus[i]))
retval = False
# wind power in pu
pw = mul(te, omega)
dae.y[self.pw] = pw
# wind speed initialization loop
R = 4 * pi * self.system.freq * mul(self.R, self.ngb, div(
1, self.npole))
AA = pi * self.R**2
vw = 0.9 * self.Vwn
for i in range(self.n):
mis = 1
niter = 0
while abs(mis) > self.system.tds.config.tol:
if niter > 50:
logger.error(
'Wind <{}> init failed. '
'Try increasing the nominal wind speed.'.format(
self.wind[i]))
retval = False
break
pw_iter, jac = self.windpower(self.ngen[i], self.rho[i], vw[i],
AA[i], R[i], omega[i], theta[i])
mis = pw_iter - pw[i]
inc = -mis / jac[1]
vw[i] += inc
niter += 1
# set wind speed
dae.x[self.vw] = div(vw, self.Vwn)
lamb = div(omega, vw, div(1, R))
ilamb = div(1,
(div(1, lamb + 0.08 * theta) - div(0.035, theta**3 + 1)))
cp = 0.22 * mul(
div(116, ilamb) - 0.4 * theta - 5, exp(div(-12.5, ilamb)))
dae.y[self.lamb] = lamb
dae.y[self.ilamb] = ilamb
dae.y[self.cp] = cp
self.system.rmgen(self.gen)
if not retval:
logger.error('DFIG initialization failed')
return retval
c = -matrix([sum( uk for uk, yk in zip(u,y) if yk ), sum(y) ])
# minimize c'*x + sum log (1 + exp(A*x))
#
# variable x (2).
def F(x=None, z=None):
if x is None: return 0, matrix(0.0, (2,1))
w = exp(A*x)
f = c.T*x + sum(log(1+w))
grad = c + A.T * div(w, 1+w)
if z is None: return f, grad.T
H = A.T * spdiag(div(w,(1+w)**2)) * A
return f, grad.T, z[0]*H
sol = solvers.cp(F)
a, b = sol['x'][0], sol['x'][1]
try: import pylab
except ImportError: pass
else:
pylab.figure(facecolor='w')
nopts = 200
pts = -1.0 + 12.0/nopts * matrix(list(range(nopts)))
w = exp(a*pts + b)
pylab.plot(u, y, 'o', pts, div(w, 1+w), '-')
pylab.title('Logistic regression (fig. 7.1)')
pylab.axis([-1, 11, -0.1, 1.1])
pylab.xlabel('u')
pylab.ylabel('Prob(y=1)')
pylab.show()
# Basis functions are Gabor pulses: for k = 0,...,K-1,
#
# exp(-(t - k * tau)^2/sigma^2 ) * cos (l*omega0*t), l = 0,...,L
# exp(-(t - k * tau)^2/sigma^2 ) * sin (l*omega0*t), l = 1,...,L
sigma = 0.05
tau = 0.002
omega0 = 5.0
K = 501
L = 30
N = 501 # number of samples of each signal in [0,1]
# Build dictionary matrix
ts = (1.0/N) * matrix(range(N), tc='d')
B = ts[:, K*[0]] - tau * matrix(range(K), (1,K), 'd')[N*[0],:]
B = exp(-(B/sigma)**2)
A = matrix(0.0, (N, K*(2*L+1)))
# First K columns are DC pulses for k = 0,...,K-1
A[:,:K] = B
for l in range(L):
# Cosine pulses for omega = (l+1)*omega0 and k = 0,...,K-1.
A[:, K+l*(2*K) : K+l*(2*K)+K] = mul(B, cos((l+1)*omega0*ts)[:, K*[0]])
# Sine pulses for omega = (l+1)*omega0 and k = 0,...,K-1.
A[:, K+l*(2*K)+K : K+(l+1)*(2*K)] = \
mul(B, sin((l+1)*omega0*ts)[:,K*[0]])
if pylab_installed:
# For regions k=1, ..., 6, let pk be the optimal value of
#
# minimize x'*x
# subject to A*x <= b.
#
# The Chernoff upper bound is 1.0 - sum exp( - pk / (2 sigma^2)).
P = matrix([1.0, 0.0, 0.0, 1.0], (2,2))
q = matrix(0.0, (2,1))
optvals = matrix([ blas.nrm2( solvers.qp(P, q, A[k], b[k] )['x'] )**2
for k in range(1,7) ])
nopts = 200
sigmas = 0.2 + (0.5 - 0.2)/nopts * matrix(list(range(nopts)), tc='d')
bnds = [ 1.0 - sum( exp( - optvals / (2*sigma**2) )) for sigma
in sigmas]
try: import pylab
except ImportError: pass
else:
pylab.figure(facecolor='w')
pylab.plot(sigmas, bnds, '-')
pylab.axis([0.2, 0.5, 0.9, 1.0])
pylab.title('Chernoff lower bound (fig. 7.8)')
pylab.xlabel('sigma')
pylab.ylabel('Probability of correct detection')
pylab.show()
if 0: # uncomment out for the Monte Carlo estimation.
N = 100000
# Basis functions are Gabor pulses: for k = 0,...,K-1,
#
# exp(-(t - k * tau)^2/sigma^2 ) * cos (l*omega0*t), l = 0,...,L
# exp(-(t - k * tau)^2/sigma^2 ) * sin (l*omega0*t), l = 1,...,L
sigma = 0.05
tau = 0.002
omega0 = 5.0
K = 501
L = 30
N = 501 # number of samples of each signal in [0,1]
# Build dictionary matrix
ts = (1.0 / N) * matrix(range(N), tc='d')
B = ts[:, K * [0]] - tau * matrix(range(K), (1, K), 'd')[N * [0], :]
B = exp(-(B / sigma)**2)
A = matrix(0.0, (N, K * (2 * L + 1)))
# First K columns are DC pulses for k = 0,...,K-1
A[:, :K] = B
for l in range(L):
# Cosine pulses for omega = (l+1)*omega0 and k = 0,...,K-1.
A[:, K + l * (2 * K):K + l * (2 * K) + K] = mul(
B,
cos((l + 1) * omega0 * ts)[:, K * [0]])
# Sine pulses for omega = (l+1)*omega0 and k = 0,...,K-1.
A[:, K+l*(2*K)+K : K+(l+1)*(2*K)] = \
mul(B, sin((l+1)*omega0*ts)[:,K*[0]])
def PA_GP_MCS_minRate(channel_alloc, channel_gain, B, noise_vec, priority,
SU_power, minRate, SNR_gap, QAM_cap, objective_list):
n_su = channel_alloc.shape[0]
n_channel = channel_alloc.shape[1]
channel_gain = copy.deepcopy(channel_gain)
Noise = copy.deepcopy(noise_vec)
channel_gain = channel_gain * (10**8)
Noise = Noise * (10**8)
# non-zero power list
channel_alloc_total = np.sum(channel_alloc, dtype=np.int32)
def A(m, n, j, t):
if (t == n):
return -1
else:
if (t == j):
return 1
else:
return 0
def C(m, n, j):
if (j == n):
return (Noise[n]) / (channel_gain[n, n] / SNR_gap[n])
else:
return channel_gain[n, j] / (channel_gain[n, n] / SNR_gap[n])
# Objective function
F_obj = np.zeros(channel_alloc_total * 2)
i = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
F_obj[channel_alloc_total + i] = priority[n] * B / (10**6)
i = i + 1
g_obj = np.zeros(1)
# Constraints of slack variables (m => n => j)
n_F_slack = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
for j in range(n_su):
if (channel_alloc[j, m] == 1):
n_F_slack = n_F_slack + 1
F_slack = np.zeros((n_F_slack, channel_alloc_total * 2))
index0 = 0
index1 = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
for j in range(n_su):
if (channel_alloc[j, m] == 1):
F_slack[index0, channel_alloc_total + index1] = -1
index3 = 0
for t in range(n_su):
if (channel_alloc[t, m] == 1):
F_slack[index0,
int(np.sum(channel_alloc[:, 0:m])) +
index3] = A(m, n, j, t)
index3 = index3 + 1
index0 = index0 + 1
index1 = index1 + 1
g_slack = np.zeros(n_F_slack)
index0 = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
for j in range(n_su):
if (channel_alloc[j, m] == 1):
g_slack[index0] = np.log(C(m, n, j))
index0 = index0 + 1
# Constraints of maximum power
F_power = np.zeros((channel_alloc_total, channel_alloc_total * 2))
index0 = 0
for n in range(n_su):
for m in range(n_channel):
if (channel_alloc[n, m] == 1):
F_power[index0,
int(np.sum(channel_alloc[:, 0:m])) +
int(np.sum(channel_alloc[:(n + 1), m])) - 1] = 1
index0 = index0 + 1
g_power = np.zeros(channel_alloc_total)
index0 = 0
for n in range(n_su):
for m in range(n_channel):
if (channel_alloc[n, m] == 1):
g_power[index0] = np.log(1 / SU_power[n])
index0 = index0 + 1
# Constraints of minimum data rate constraint
F_minRate = np.zeros((n_su, channel_alloc_total * 2))
index0 = 0
for n in range(n_su):
for m in range(n_channel):
if (channel_alloc[n, m] == 1):
F_minRate[n, channel_alloc_total + index0] = B / (10**6)
index0 = index0 + 1
g_minRate = np.zeros(n_su)
for n in range(n_su):
g_minRate[n] = np.log(2**(minRate[n] / (10**6)))
# Constraints of QAM capability
F_QAM = np.zeros((channel_alloc_total, channel_alloc_total * 2))
index0 = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
F_QAM[index0, channel_alloc_total + index0] = 1
index0 = index0 + 1
g_QAM = np.zeros(channel_alloc_total)
index0 = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
g_QAM[index0] = np.log(2**-(QAM_cap[n] + 1))
index0 = index0 + 1
K_obj = np.ones(1)
K_slack = n_su * np.ones(channel_alloc_total)
index0 = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
K_slack[index0] = np.sum(channel_alloc[:, m])
index0 = index0 + 1
K_power = n_channel * np.ones(n_su)
for n in range(n_su):
K_power[n] = np.sum(channel_alloc[n, :])
K_minRate = np.ones(n_su)
K_QAM = np.ones(channel_alloc_total)
K_arr = np.concatenate((K_obj, K_slack, K_power, K_minRate, K_QAM))
K_arr = K_arr.astype(int)
F_arr = np.vstack((F_obj, F_slack, F_power, F_minRate, F_QAM))
g_arr = np.concatenate((g_obj, g_slack, g_power, g_minRate, g_QAM))
K = K_arr.tolist()
F_arr = F_arr.T
F = cvx.matrix(F_arr.tolist())
g_arr = g_arr.T
g = cvx.matrix(g_arr.tolist())
cvx.solvers.options['show_progress'] = False
sol = cvx.exp(cvx.solvers.gp(K, F, g, quiet=True)['x'])
power_alloc = np.zeros((n_su, n_channel))
index0 = 0
for m in range(n_channel):
for n in range(n_su):
if (channel_alloc[n, m] == 1):
power_alloc[n, m] = sol[index0]
index0 = index0 + 1
objective_list['total'].append(
objective_value(channel_alloc, power_alloc, priority,
channel_gain / (10**8), B, noise_vec, SNR_gap))
return power_alloc
# The small GP of section 9.3 (Geometric programming).
from cvxopt import matrix, log, exp, solvers
Aflr = 1000.0
Awall = 100.0
alpha = 0.5
beta = 2.0
gamma = 0.5
delta = 2.0
F = matrix( [[-1., 1., 1., 0., -1., 1., 0., 0.],
[-1., 1., 0., 1., 1., -1., 1., -1.],
[-1., 0., 1., 1., 0., 0., -1., 1.]])
g = log( matrix( [1.0, 2/Awall, 2/Awall, 1/Aflr, alpha, 1/beta, gamma,
1/delta]) )
K = [1, 2, 1, 1, 1, 1, 1]
h, w, d = exp( solvers.gp(K, F, g)['x'] )
print("\n h = %f, w = %f, d = %f.\n" %(h,w,d))
def F(x=None, z=None):
if x is None: return 0, matrix(0.0, (2, 1))
w = exp(A * x)
f = c.T * x + sum(log(1 + w))
grad = c + A.T * div(w, 1 + w)
if z is None: return f, grad.T
H = A.T * spdiag(div(w, (1 + w)**2)) * A
return f, grad.T, z[0] * H
sol = solvers.cp(F)
a, b = sol['x'][0], sol['x'][1]
try:
import pylab
except ImportError:
pass
else:
pylab.figure(facecolor='w')
nopts = 200
pts = -1.0 + 12.0 / nopts * matrix(list(range(nopts)))
w = exp(a * pts + b)
pylab.plot(u, y, 'o', pts, div(w, 1 + w), '-')
pylab.title('Logistic regression (fig. 7.1)')
pylab.axis([-1, 11, -0.1, 1.1])
pylab.xlabel('u')
pylab.ylabel('Prob(y=1)')
pylab.show()
def _solve(self):
# An alias to the internal problem instance.
p = self.int
# This class doubles as an implementation for the SMCP solver.
solver = self.ext.options['solver'].lower()
assert solver in ("cvxopt", "smcp")
isGP = self._is_geometric_program()
# Check if the problem type is supported.
# TODO: Check if general-obj problems are properly handled.
if solver == 'smcp':
# SMCP is a standalone solver made available through the SMCPSolver
# class. However, since SMCP both depends on CVXOPT and has a
# similar interface, its logic is implemented inside CVXOPTSolver,
# and SMCPSolver subclasses it. Ensure that self is indeed a
# SMCPSolver instance.
assert self.__class__.__name__ == "SMCPSolver", \
"SMCP needs to be used through the SMCPSolver class."
import smcp
# Clear all options set previously. This is necessary because CVXOPT
# options are global, and might be changed even by another problem.
cvx.solvers.options.clear()
# Handle "verbose" option.
cvx.solvers.options['show_progress'] = (self.verbosity() >= 1)
# Handle "tol", "feastol", "abstol" and "reltol" options.
feastol = self.ext.options['feastol']
if feastol is None:
feastol = self.ext.options['tol']
abstol = self.ext.options['abstol']
if abstol is None:
abstol = self.ext.options['tol']
reltol = self.ext.options['reltol']
if reltol is None:
reltol = 10 * self.ext.options['tol']
cvx.solvers.options['feastol'] = feastol
cvx.solvers.options['abstol'] = abstol
cvx.solvers.options['reltol'] = reltol
# Handle "maxit" option.
maxit = self.ext.options['maxit']
if maxit is None:
maxit = int(1e6)
cvx.solvers.options['maxiters'] = maxit
# Allow solving GPs that are not strictly feasible.
# TODO: Turn this into a proper option?
cvx.solvers.options['kktreg'] = 1e-6
# Handle unsupported options.
self._handle_unsupported_options('lp_root_method', 'lp_node_method',
'timelimit', 'treememory', 'nbsol',
'hotstart')
# TODO: Add CVXOPT-sepcific options. Candidates are:
# - refinement
# Set options for SMCP.
if solver == 'smcp':
# Restore default options.
smcp.solvers.options = self._smcp_default_options.copy()
# Copy options also used by CVXOPT.
smcp.solvers.options.update(cvx.solvers.options)
# Further handle "verbose" option.
smcp.solvers.options['debug'] = (self.verbosity() >= 2)
# TODO: Add SMCP-sepcific options.
if self._debug():
from pprint import pformat
if solver == 'smcp':
options = smcp.solvers.options
else:
options = cvx.solvers.options
self._debug("Setting options:\n{}\n".format(pformat(options)))
# Print a header.
if solver == 'smcp':
subsolverText = None
else:
if isGP:
subsolverText = 'internal GP solver'
else:
subsolverText = 'internal CONELP solver'
# Further prepare the problem for the CVXOPT/SMCP CONELP solvers.
# TODO: This should be done during import.
if not isGP:
# Retrieve the structure of the cone, which is a cartesian product
# of the non-negative orthant of dimension l, a number of second
# order cones with dimensions in q and a number of positive
# semidefinite cones with dimensions in s.
dims = {
'l': p['Gl'].size[0],
'q': [Gqi.size[0] for Gqi in p['Gq']],
's': [int(numpy.sqrt(Gsi.size[0])) for Gsi in p['Gs']]
}
# Construct G and h to contain all conic inequalities, starting with
# those with respect to the non-negative orthant.
G = p['Gl']
h = p['hl']
# SMCP's ConeLP solver does not handle (linear) equalities, so cast
# them as inequalities.
if solver == 'smcp':
if p['A'].size[0] > 0:
G = cvx.sparse([G, p['A']])
G = cvx.sparse([G, -p['A']])
h = cvx.matrix([h, p['b']])
h = cvx.matrix([h, -p['b']])
dims['l'] += (2 * p['A'].size[0])
# Add second-order cone inequalities.
for i in range(len(dims['q'])):
G = cvx.sparse([G, p['Gq'][i]])
h = cvx.matrix([h, p['hq'][i]])
# Add semidefinite cone inequalities.
for i in range(len(dims['s'])):
G = cvx.sparse([G, p['Gs'][i]])
h = cvx.matrix([h, p['hs'][i]])
# Remove zero lines from linear equality constraint matrix, as
# CVXOPT expects this matrix to have full row rank.
JP = list(set(p['A'].I))
IP = range(len(JP))
VP = [1] * len(JP)
if any([b for (i, b) in enumerate(p['b']) if i not in JP]):
raise SolverError(
'Infeasible constraint of the form 0 = a with a != 0.')
P = spmatrix(VP, IP, JP, (len(IP), p['A'].size[0]))
A = P * p['A']
b = P * p['b']
# Attempt to solve the problem.
with self._header(subsolverText), self._stopwatch():
if solver == 'smcp':
if self._debug():
self._debug(
"Calling smcp.solvers.conelp(c, G, h, dims) "
"with\nc:\n{}\nG:\n{}\nh:\n{}\ndims:\n{}\n".format(
p['c'], G, h, dims))
try:
result = smcp.solvers.conelp(p['c'], G, h, dims)
except TypeError:
# HACK: Work around "'NoneType' object is not subscriptable"
# exception with infeasible/unbounded problems.
result = None
else:
if isGP:
result = cvx.solvers.gp(p['K'],
p['F'],
p['g'],
p['Gl'],
p['hl'],
p['A'],
p['b'],
kktsolver='ldl')
else:
result = cvx.solvers.conelp(p['c'], G, h, dims, A, b)
# Retrieve primals.
if self.ext.options["noprimals"]:
primals = None
elif result is None or result["x"] is None:
primals = None
else:
primals = {}
# Retrieve values for all but LSE auxiliary variables.
for var in self.ext.variables.values():
# Auxiliary variables for LSE metaconstraints have no
# meaningful value at this point.
if isinstance(var.origin, LSEConstraint):
continue
# HACK: Cast from numpy array to list so that _retrieve_matrix
# will respect the target matrix size.
# TODO: Refactor _retrieve_matrix to handle different input
# types consistently.
value = list(result["x"][var.startIndex:var.endIndex])
primals[var.name] = value
# Give auxiliary variables added for an LSE metaconstraint a
# solution value.
for var in self.ext.variables.values():
if isinstance(var.origin, LSEConstraint):
value = []
for i, exponent in enumerate(var.origin.le0Exponents):
value.append(
cvx.matrix(exponent.constant if exponent.
constant else 0.0))
for v, factor in exponent.factors.items():
value[i] += factor * cvx.matrix(primals[v.name])
value[i] = cvx.exp(value[i])[0, 0]
primals[var.name] = cvx.matrix(value, var.size)
# Retrieve duals.
if self.ext.options["noduals"]:
duals = None
elif result is None:
duals = None
else:
duals = []
(indy, indzl, indzq, indznl, indzs) = (0, 0, 0, 0, 0)
if isGP:
zkey = 'zl'
zqkey = 'zq'
zskey = 'zs'
else:
zkey = 'z'
zqkey = 'z'
zskey = 'z'
indzq = dims['l']
indzs = dims['l'] + _bsum(dims['q'])
if solver == 'smcp':
# Equality constraints were cast as two inequalities.
ieq = p['Gl'].size[0]
neq = (dims['l'] - ieq) // 2
soleq = result['z'][ieq:ieq + neq]
soleq -= result['z'][ieq + neq:ieq + 2 * neq]
else:
soleq = result['y']
seenLSEConstraints = []
for constraint in self.ext.constraints:
dual = None
consSz = len(constraint)
if isinstance(constraint.origin, LSEConstraint):
# We are looking at an auxiliary constraint for a
# metaconstraint that was imported directly.
# We don't get any duals:
# - The affine constraint was imported for technical reasons
# and all variables used in it are irrelevant to the
# objective function. Hence the dual will always be 0.
# We don't store it since that could produce a bad result
# when querying the LSE metaconstraint for its dual.
# - The exponential cone constraints were never imported.
# Instead, a single LSE constraint was imported.
if isinstance(constraint, AffineConstraint):
assert constraint.is_inequality()
indzl += len(constraint) # = 1
else:
assert isinstance(constraint, ExpConeConstraint)
if constraint.origin not in seenLSEConstraints:
indznl += len(constraint.origin) # = 1
seenLSEConstraints.append(constraint.origin)
elif isinstance(constraint, AffineConstraint):
if constraint.is_equality():
if soleq is not None:
dual = (P.T * soleq)[indy:indy + consSz]
indy += consSz
else:
if result[zkey] is not None:
dual = result[zkey][indzl:indzl + consSz]
indzl += consSz
elif isinstance(constraint, SOCConstraint) \
or isinstance(constraint, RSOCConstraint):
if result[zqkey] is not None:
if isGP:
dual = result[zqkey][indzq]
dual[1:] = -dual[1:]
indzq += 1
else:
dual = result[zqkey][indzq:indzq + consSz]
# Swap the sign of the SOcone's Lagrange multiplier,
# as PICOS wants it the other way round.
# Note that this also affects RScone constraints
# which were cast as SOcone constraints on import.
dual[1:] = -dual[1:]
if isinstance(constraint, RSOCConstraint):
# RScone were cast as a SOcone on import, so
# transform the dual to a proper RScone dual.
alpha = dual[0] - dual[-1]
beta = dual[0] + dual[-1]
z = 2.0 * dual[1:-1]
dual = cvx.matrix([alpha, beta, z])
indzq += consSz
elif isinstance(constraint, LMIConstraint):
if result[zskey] is not None:
matSz = constraint.size[0]
if isGP:
dual = cvx.matrix(result[zskey][indzs],
(matSz, matSz))
indzs += 1
else:
dual = cvx.matrix(
result[zskey][indzs:indzs + consSz],
(matSz, matSz))
indzs += consSz
duals.append(dual)
# Retrieve objective value.
if result is None:
objectiveValue = None
elif isGP:
objectiveValue = 'toEval'
else:
p = result['primal objective']
d = result['dual objective']
if p is not None and d is not None:
objectiveValue = 0.5 * (p + d)
elif p is not None:
objectiveValue = p
elif d is not None:
objectiveValue = d
else:
objectiveValue = None
if self.ext.objective[0] == 'max' and objectiveValue is not None:
objectiveValue = -objectiveValue
# Retrieve solution metadata.
meta = {
'cvxopt_sol': result,
'status': result['status'] if result else "unavailable"
}
return (primals, duals, objectiveValue, meta)
def F(x=None, z=None):
""" Evaluates the objective and nonlinear constraint functions.
"""
if x is None:
# Include power mismatch constraint twice to force equality.
nln_N = self.om.nln_N + neqnln
return nln_N, x0
# Evaluate objective function -------------------------------------
Pgen = x[Pg.i1:Pg.iN + 1] # Active generation in p.u.
Qgen = x[Qg.i1:Qg.iN + 1] # Reactive generation in p.u.
xx = matrix([Pgen, Qgen]) * base_mva
# Evaluate the objective function value.
if len(ipol) > 0:
# FIXME: Implement reactive power costs.
f0 = sum([g.total_cost(xx[i]) for i, g in enumerate(gn)])
else:
f0 = 0
# Piecewise linear cost of P and Q.
if ny:
y = self.om.get_var("y")
ccost = spmatrix(matrix(1.0, (ny, 1)),
range(y.i1, y.iN + 1),
matrix(0.0, (ny, 1)), size=(nxyz, 1)).T
f0 = f0 + ccost * x
else:
ccost = matrix(0.0, (1, nxyz))
# TODO: Generalised cost term.
# Evaluate cost gradient ------------------------------------------
iPg = range(Pg.i1, Pg.iN + 1)
iQg = range(Qg.i1, Qg.iN + 1)
# Polynomial cost of P and Q.
df_dPgQg = matrix(0.0, (2 * ng, 1)) # w.r.t p.u. Pg and Qg
for i in ipol:
df_dPgQg[i] = \
base_mva * polyval(polyder(list(gn[i].p_cost)), xx[i])
df0 = matrix(0.0, (nxyz, 1))
df0[iPg] = df_dPgQg[:ng]
df0[iQg] = df_dPgQg[ng:ng + ng]
# Piecewise linear cost of P and Q.
df0 = df0 + ccost.T
# TODO: Generalised cost term.
# Evaluate cost Hessian -------------------------------------------
# d2f = None
# Evaluate nonlinear equality constraints -------------------------
for i, g in enumerate(gn):
g.p = Pgen[i] * base_mva # active generation in MW
g.q = Qgen[i] * base_mva # reactive generation in MVAr
# Rebuild the net complex bus power injection vector in p.u.
Sbus = matrix(case.getSbus(bs))
Vang = x[Va.i1:Va.iN + 1]
Vmag = x[Vm.i1:Vm.iN + 1]
V = mul(Vmag, exp(1j * Vang))
# Evaluate the power flow equations.
mis = mul(V, conj(Ybus * V)) - Sbus
# Equality constraints (power flow).
g = matrix([mis.real(), # active power mismatch for all buses
mis.imag()]) # reactive power mismatch for all buses
# Evaluate nonlinear inequality constraints -----------------------
# Inequality constraints (branch flow limits).
# (line constraint is actually on square of limit)
flow_max = matrix([(l.rate_a / base_mva)**2 for l in ln])
# FIXME: There must be a more elegant way to do this.
for i, rate in enumerate(flow_max):
if rate == 0.0:
flow_max[i] = 1e5
if self.flow_lim == IFLOW:
If = Yf * V
It = Yt * V
# Branch current limits.
h = matrix([(If * conj(If)) - flow_max,
(If * conj(It)) - flow_max])
else:
i_fbus = [e.from_bus._i for e in ln]
i_tbus = [e.to_bus._i for e in ln]
# Complex power injected at "from" bus (p.u.).
Sf = mul(V[i_fbus], conj(Yf * V))
# Complex power injected at "to" bus (p.u.).
St = mul(V[i_tbus], conj(Yt * V))
if self.flow_lim == PFLOW: # active power limit, P (Pan Wei)
# Branch real power limits.
h = matrix([Sf.real()**2 - flow_max,
St.real()**2 - flow_max])
elif self.flow_lim == SFLOW: # apparent power limit, |S|
# Branch apparent power limits.
h = matrix([mul(Sf, conj(Sf)) - flow_max,
mul(St, conj(St)) - flow_max]).real()
else:
raise ValueError
# Evaluate partial derivatives of constraints ---------------------
iVa = matrix(range(Va.i1, Va.iN + 1))
iVm = matrix(range(Vm.i1, Vm.iN + 1))
iPg = matrix(range(Pg.i1, Pg.iN + 1))
iQg = matrix(range(Qg.i1, Qg.iN + 1))
iVaVmPgQg = matrix([iVa, iVm, iPg, iQg]).T
# Compute partials of injected bus powers.
dSbus_dVm, dSbus_dVa = dSbus_dV(Ybus, V)
i_gbus = [gen.bus._i for gen in gn]
neg_Cg = spmatrix(matrix(-1.0, (ng, 1)),
i_gbus,
range(ng), (nb, ng))
# Transposed Jacobian of the power balance equality constraints.
dg = spmatrix([], [], [], (nxyz, 2 * nb))
blank = spmatrix([], [], [], (nb, ng))
dg[iVaVmPgQg, :] = sparse([
[dSbus_dVa.real(), dSbus_dVa.imag()],
[dSbus_dVm.real(), dSbus_dVm.imag()],
[neg_Cg, blank],
[blank, neg_Cg]
]).T
# Compute partials of flows w.r.t V.
if self.flow_lim == IFLOW:
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dIbr_dV(Yf, Yt, V)
else:
dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft = \
dSbr_dV(Yf, Yt, V, bs, ln)
if self.flow_lim == PFLOW:
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 (complex power, 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 = spmatrix([], [], [], (nxyz, 2 * nl))
dh[matrix([iVa, iVm]).T, :] = sparse([[df_dVa, dt_dVa],
[df_dVm, dt_dVm]]).T
f = matrix([f0, g, h, -g])
df = matrix([[df0], [dg], [dh], [-dg]]).T
if z is None:
return f, df
# Evaluate cost Hessian -------------------------------------------
nxtra = nxyz - (2 * nb)
# Evaluate d2f ----------------------------------------------------
d2f_dPg2 = matrix(0.0, (ng, 1)) # w.r.t p.u. Pg
d2f_dQg2 = matrix(0.0, (ng, 1)) # w.r.t p.u. Qg
for i in ipol:
d2f_dPg2[i, 0] = polyval(polyder(list(gn[i].p_cost), 2),
Pg.v0[i] * base_mva) * base_mva**2
# TODO: Reactive power costs.
# for i in ipol:
# d2f_dQg2[i] = polyval(polyder(list(gn[i].q_cost), 2),
# Qg.v0[i] * base_mva) * base_mva**2
i = matrix([range(Pg.i1, Pg.iN + 1), range(Qg.i1, Qg.iN + 1)])
d2f = spmatrix(matrix([d2f_dPg2, d2f_dQg2]), i, i, (nxyz, nxyz))
# TODO: Generalised cost model.
d2f = d2f * self.opt["cost_mult"]
# Evaluate Hessian of power balance constraints -------------------
eqnonlin = z[1:neqnln + 1]
nlam = len(eqnonlin) / 2
lamP = eqnonlin[:nlam]
lamQ = eqnonlin[nlam:nlam + nlam]
Gpaa, Gpav, Gpva, Gpvv = d2Sbus_dV2(Ybus, V, lamP)
Gqaa, Gqav, Gqva, Gqvv = d2Sbus_dV2(Ybus, V, lamQ)
d2G_1 = sparse([[sparse([[Gpaa, Gpva], [Gpav, Gpvv]]).real() +
sparse([[Gqaa, Gqva], [Gqav, Gqvv]]).imag()],
[spmatrix([], [], [], (2 * nb, nxtra))]])
d2G_2 = spmatrix([], [], [], (nxtra, 2 * nb + nxtra))
d2G = sparse([d2G_1, d2G_2])
# Evaluate Hessian of flow constraints ---------------------------
ineqnonlin = z[1 + neqnln:1 + neqnln + niqnln]
nmu = len(ineqnonlin) / 2
muF = ineqnonlin[:nmu]
muT = ineqnonlin[nmu:nmu + nmu]
if self.flow_lim == IFLOW:
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:
fr = [e.from_bus._i for e in ln]
to = [e.to_bus._i for e in ln]
# Line-bus connection matrices.
Cf = spmatrix(1.0, range(nl), fr, (nl, nb))
Ct = spmatrix(1.0, range(nl), to, (nl, nb))
dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St = \
dSbr_dV(Yf, Yt, V, bs, ln)
if self.flow_lim == PFLOW:
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)
elif self.flow_lim == SFLOW:
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)
else:
raise ValueError
d2H_1 = sparse([[sparse([[Hfaa, Hfva], [Hfav, Hfvv]]) +
sparse([[Htaa, Htva], [Htav, Htvv]])],
[spmatrix([], [], [], (2 * nb, nxtra))]])
d2H_2 = spmatrix([], [], [], (nxtra, 2 * nb + nxtra))
d2H = sparse([d2H_1, d2H_2])
Lxx = d2f + d2G + d2H
return f, df, Lxx
