def testNEdges(N):
for M in range(3):
(D, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
D = (M + 1) * D / 3
B = loss(N)
n = 2**(N - 1)
elements = np.arange(0, N)
Limits = np.vstack((Pmin, Pmax))
N_edges = 0
for k in range(N):
index = np.append(elements[:k], elements[k + 1:])
for i in range(n):
binary = bin(i)
coord = list(binary)[2:] #get rid of '0b'
coord = np.array([int(i) for i in coord])
seq = np.zeros(N - 1, dtype=int)
seq[N - 1 - len(coord):] = coord
V_sup = np.zeros(N)
V_sup[index] = Limits[seq, index]
V_sup[k] = Pmax[k]
D_sup = sum(V_sup) - V_sup @ B @ V_sup
V_inf = np.zeros(N)
V_inf[index] = Limits[seq, index]
V_inf[k] = Pmin[k]
D_inf = sum(V_inf) - V_inf @ B @ V_sup
if (D_inf <= D and D_sup >= D):
N_edges = N_edges + 1
print(N_edges, 'edges for problem of size ', N, ' and ', D, 'demand')
def RelaxTime(N, Demand):
(Unused, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
B = loss(N)
price = SimplePriceFun(Pmin, Pmax, a, b, c, alpha, beta, gamma, Demand)
Time = np.zeros(2)
problems = ["Convex relaxation", "Linear Relax"]
n_meth = 0
for method in problems:
m = gp.Model(method)
m.setParam('OutputFlag', False)
P = m.addVars(range(N), lb=Pmin, ub=Pmax, name='P')
PL = m.addVar()
x = m.addVars(N)
y = m.addVars(N)
for i in range(N):
m.addConstr(x[i] == delta[i] * P[i])
m.addGenConstrExp(x[i], y[i])
if (method == "Convex relaxation"):
m.addQConstr(PL >= sum(
sum(P[i] * P[j] * B[i, j] for j in range(N))
for i in range(N)))
m.addConstr(P.sum() - PL == Demand, name='Demand')
elif (method == "Linear Relax"):
[opt, P0] = Solve(N, price, 1, Demand)
n = np.ones(N) - 2 * B @ P0
k = -np.dot(n, P0)
m.addConstr(sum(P[i] * n[i] for i in range(N)) + k >= 0)
t0 = time.time()
Cost = sum(a[k] + b[k] * P[k] + c[k] * P[k] * P[k] for k in range(N))
Emission = sum(alpha[k] + beta[k] * P[k] + gamma[k] * P[k] * P[k] +
eta[k] * y[k] for k in range(N))
obj = price * Emission + Cost
m.setObjective(obj)
m.optimize()
t1 = time.time()
Time[n_meth] = t1 - t0
n_meth = n_meth + 1
return (Time)
def LinearRelaxation(N, D):
(Unused, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
B = loss(N)
#Upper bound
(n, k_upper, p) = LinUpperB(Pmin, Pmax, B, D)
#Lower bound
model = gp.Model('Lower bound of demand')
P = model.addVars(range(N), lb=Pmin, ub=Pmax, name='P')
PL = model.addVar()
model.addQConstr(PL >= sum(
sum(P[i] * P[j] * B[i, j] for j in range(N)) for i in range(N)))
model.addConstr(P.sum() - PL == D, name='Demand')
obj = sum(P[i] * n[i] for i in range(N))
model.setObjective(obj)
model.setParam('OutputFlag', False)
model.optimize()
k_lower = -obj.getValue()
return (n, k_upper, k_lower)
def Solve(N,w_E,w_C,D, method='trust-const'):
(Unused,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
B=loss(N)
bnds=np.transpose(np.vstack((Pmin,Pmax)))
P0=Pmin.copy()
def objective(P):
Cost = sum(a[i]+b[i]*P[i]+c[i]*P[i]*P[i] for i in range(N))
Emission = sum(alpha[i]+beta[i]*P[i]+gamma[i]*P[i]*P[i] +eta[i]*np.exp(P[i]*delta[i]) for i in range(N))
return (w_E*Emission+w_C*Cost)
def Gradient(P):
GradC=b+2*c*P
GradE= beta+2*gamma*P+delta*eta*np.exp(delta*P)
Grad=w_C*GradC+w_E*GradE
return(Grad)
def Hessian(P):
Hess= 2*w_C*c+w_E*(2*gamma+delta*delta*eta*np.exp(delta*P))
H=Hess*np.eye(N)
return(H)
def cons_f(P):
PL=sum(sum(P[i]*P[j]*B[i,j] for j in range(N)) for i in range(N))
sum_eq=sum(P)-PL-D
return (sum_eq)
def cons_J(P):
Jac=np.ones(N)-2*P@B
return(Jac)
def cons_H(P,v):
return(-2*v*B)
if (method=='SLSQP'):
const=[{'type': 'eq', 'fun': cons_f}]
solution = minimize(objective,P0, method='SLSQP',jac=Gradient, bounds=bnds,constraints=const)
else:
NL_const = NonlinearConstraint(cons_f, 0, 0, jac=cons_J, hess=cons_H)
solution = minimize(objective,P0, method='trust-constr',jac=Gradient,
hess=Hessian,constraints=NL_const, bounds=bnds)
P = solution.x
return(objective(P),P)
def SolveGurobi(N,w_E,w_C, Demand, method="ConvexRelax"):
(Unused,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
B=loss(N)
m = gp.Model('Static Nonlinear EED with transmission losses')
Pow = m.addVars(range(N),lb=Pmin,ub=Pmax, name='P')
PLoss = m.addVar()
x = m.addVars(N)
y = m.addVars(N)
for n in range(N):
m.addConstr(x[n]==delta[n]*Pow[n])
m.addGenConstrExp(x[n], y[n])
if (method=="NonConvex"):
m.setParam('NonConvex', 2)
m.addQConstr(PLoss== sum( sum(Pow[i]*Pow[j]*B[i][j] for j in range(N))for i in range(N)))
m.addConstr(Pow.sum() == Demand+PLoss)
elif (method=="ConvexRelax"):
m.addQConstr(PLoss>= sum( sum(Pow[i]*Pow[j]*B[i][j] for j in range(N))for i in range(N)))
m.addConstr(Pow.sum() == Demand+PLoss)
elif (method=="LinRelax"):
t=time.time()
(n,k_upper, k_lower) = LinearRelaxation(N,Demand)
print(time.time()-t,' sec to add for computing the extreme points')
m.addConstr( sum(Pow[i]*n[i] for i in range(N))+k_upper<=0)
m.addConstr( sum(Pow[i]*n[i] for i in range(N))+k_lower>=0)
Cost = sum(a[k]+b[k]*Pow[k]+c[k]*Pow[k]*Pow[k] for k in range(N))
Emission = sum(alpha[k]+beta[k]*Pow[k]+gamma[k]*Pow[k]*Pow[k]+eta[k]*y[k] for k in range(N))
obj= w_E*Emission + w_C*Cost
m.setObjective(obj)
m.setParam( 'OutputFlag', False )
m.optimize()
opt=obj.getValue()
P=np.zeros(N)
for i in range(N):
P[i] = Pow[i].x
return(opt,P)
def figures(N):
plt.close("all")
(d,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
B=loss(N)
M=np.zeros([N+1,N+1])
M[:-1,:-1]=B
M[-1,:-1]=-np.ones(N)*0.5
M[:-1,-1] = -np.ones(N)*0.5
M[-1,-1] = d
[l,v]=np.linalg.eig(B)
samples=10**(5-N)
xc=-np.linalg.inv(B).dot(M[:-1,-1])
center=np.transpose(np.repeat([xc],samples, axis=0))
detB=np.linalg.det(B)
detM=np.linalg.det(M)
rad=np.zeros(N)
for i in range(N):
v[i]=v[i]/np.linalg.norm(v[i])
rad[i]=np.sqrt(-detM/(l[i]*detB))
if N==2:
"""Plot the entire ellipse """
theta=np.linspace(0,2*PI,num=samples)
ellipse=np.array([rad[0]*np.sin(theta), rad[1]*np.cos(theta)])
P=v.dot(ellipse)+center
plt.figure()
plt.plot(P[0],P[1])
plt.xlabel('$P_1$')
plt.ylabel('$P_2$')
plt.title('Domain')
plt.grid()
""" end """
theta=np.linspace(101*PI/80,51*PI/40,num=samples)
ellipse=np.array([rad[0]*np.sin(theta), rad[1]*np.cos(theta)])
P=v.dot(ellipse)+center
plt.figure()
plt.plot(P[0],P[1], label='Ellipsoid')
plt.vlines(x=Pmin[0], ymin=Pmin[1], ymax=Pmax[1], linewidth=2.0)
plt.vlines(x=Pmax[0], ymin=Pmin[1], ymax=Pmax[1], linewidth=2.0)
plt.hlines(y=Pmin[1], xmin=Pmin[0], xmax=Pmax[0], linewidth=2.0)
plt.hlines(y=Pmax[1], xmin=Pmin[0], xmax=Pmax[0], linewidth=2.0)
plt.xlabel('$P_1$')
plt.ylabel('$P_2$')
plt.title('Domain')
plt.grid()
plt.figure(3)
plt.xlabel('Emission [lb]')
plt.ylabel('Cost [$]')
plt.title('Achievable domain of the objective')
plt.grid()
box=np.zeros(N)
index=0
for i in range(samples):
const=np.zeros(2*N)
const[:N]=P[:,i]-Pmin
const[N:]=Pmax-P[:,i]
if all(const>=0):
Cost = sum(a[k]+b[k]*P[k,i]+c[k]*P[k,i]*P[k,i] for k in range(N))
Emission = sum(alpha[k]+beta[k]*P[k,i]+gamma[k]*P[k,i]*P[k,i] for k in range(N))
plt.figure(3)
plt.plot(Emission,Cost,'b.',label=str(i))
if index==0:
box= P[:,i-1]
index=i
box=np.vstack((box,P[:,i]))
box=np.vstack((box,P[:,index+len(box)]))
box=box.T
(n,k,p)=LinUpperB(Pmin,Pmax,B,d)
x=np.linspace(min(p[:,0]),Pmax[0])
y=-(n[0]*x+k)/n[1]
it=10
P_relax=np.zeros([N,it])
price= SimplePriceFun(Pmin,Pmax,a,b,c,alpha,beta,gamma,d)
w= np.linspace(0, 1 , num=it)
for i in range(it) :
w_E=price*w[i]
w_C=1-w[i]
[opt,P_opt]=SolveGurobi(N,w_E,w_C, d, method="ConvexRelax")
P_relax[:,i]=P_opt.copy()
plt.figure()
plt.plot(box[0],box[1],label="Demand constraint")
plt.plot(x,y, label = "Linear adjustment")
plt.plot(P_relax[0],P_relax[1],'ro', label='Relaxed Solutions')
plt.xlabel('$P_1$')
plt.ylabel('$P_2$')
plt.title('Relaxed convex set')
plt.grid()
plt.legend()
elif N==3:
theta=np.linspace(2.32,2.34,num=samples)
phi=np.linspace(4.92,4.935,num=samples)
ellipse=np.zeros([N,samples,samples])
ellipse[0]= rad[0]*np.outer(np.sin(theta),np.sin(phi))
ellipse[1]= rad[1]*np.outer(np.cos(theta),np.sin(phi))
ellipse[2]= rad[2]*np.outer(np.ones(samples),np.cos(phi))
P=np.zeros([N,samples,samples])
for i in range(samples):
P[:,:,i]=v.dot(ellipse[:,:,i])+center
plt.figure(1)
plt.xlabel('Emission [lb]')
plt.ylabel('Cost [$]')
plt.title('Achievable domain of the objective')
plt.grid()
box=np.zeros([N,samples,samples])
for i in range(samples):
for j in range(samples):
const=np.zeros(2*N)
const[:N]=P[:,i,j]-Pmin
const[N:]=Pmax-P[:,i,j]
if all(const>=0):
Cost = sum(a[k]+b[k]*P[k,i,j]+c[k]*P[k,i,j]*P[k,i,j] for k in range(N))
Emission = sum(alpha[k]+beta[k]*P[k,i,j]+gamma[k]*P[k,i,j]*P[k,i,j] for k in range(N))
plt.figure(1)
plt.plot(Emission,Cost,'b.')
box[:,i,j]= P[:,i,j]
P[:,i,j]=np.zeros(N) #In order to display in red the box points
fig=plt.figure()
ax = fig.add_subplot(111, projection='3d')
x=np.linspace(Pmin[0],Pmax[0])
y=np.linspace(Pmin[1],Pmax[1])
z=np.linspace(Pmin[2],Pmax[2])
[Yz,Zy]=np.meshgrid(y,z)
[Xz,Zx]=np.meshgrid(x,z)
[Xy,Yx]=np.meshgrid(x,y)
m=50
Xmin=Pmin[0]*np.ones([m,m])
Xmax=Pmax[0]*np.ones([m,m])
Ymin=Pmin[1]*np.ones([m,m])
Ymax=Pmax[1]*np.ones([m,m])
Zmin=Pmin[2]*np.ones([m,m])
Zmax=Pmax[2]*np.ones([m,m])
ax.plot_wireframe(Xmax,Yz,Zy,rstride=5, cstride=5)
ax.plot_wireframe(Xmin,Yz,Zy,rstride=5, cstride=5)
ax.plot_wireframe(Xz,Ymax,Zx,rstride=5, cstride=5)
ax.plot_wireframe(Xy,Yx,Zmax,rstride=5, cstride=5)
box=box[np.nonzero(box)]
box=box.reshape(N,int(len(box)/N))
ax.scatter(P[0,:,:], P[1,:,:], P[2,:,:],c='b', marker='o', s=0.5)
ax.scatter(box[0],box[1], box[2], c='r',s=2)
ax.set_xlabel('$P_1$')
ax.set_ylabel('$P_2$')
ax.set_zlabel('$P_3$')
plt.grid()
plt.title('Domain')
""" Linear approximation """
(n,k,p)=LinUpperB(Pmin,Pmax,B,d)
xx=np.linspace(min(p[:,0]),Pmax[0])
yy=np.linspace(min(p[:,1]),Pmax[1])
[Xx,Yy]=np.meshgrid(xx,yy)
Z=-(n[0]*Xx+n[1]*Yy+k)/n[2]
fig=plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(Xx,Yy, Z, label="Linear approximation")
ax.scatter(box[0],box[1], box[2], c='r',s=2, label= 'Domain of definition')
plt.title('Linear Approximation of the domain')
ax.set_xlabel('$P_1$')
ax.set_ylabel('$P_2$')
ax.set_zlabel('$P_3$')
plt.grid()
Dist=np.zeros(len(box[0]))
for i in range(len(box[0])):
Dist[i]=abs(box[:,i].dot(n)+k)/np.linalg.norm(n)
fig=plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(box[0],box[1], Dist, c='b')
plt.title('Distance of the Linear Approximation')
ax.set_xlabel('$P_1$')
ax.set_ylabel('$P_2$')
ax.set_zlabel('$Error$')
""" Get the maximum error """
model = gp.Model('Max distance')
P = model.addVars(range(N),lb=Pmin,ub=Pmax, name='P')
model.setParam('NonConvex', 2)
PL = model.addVar()
model.addQConstr(PL== sum( sum(P[i]*P[j]*B[i,j] for j in range(N))for i in range(N)))
model.addConstr(P.sum()-PL == d, name='Demand')
dist=-(sum(P[i]*n[i] for i in range(N))+k)/np.linalg.norm(n)
model.setObjective(dist, GRB.MAXIMIZE)
model.setParam( 'OutputFlag', False )
model.optimize()
print(dist.getValue()) #0.15661626890427272
def DEED_PL(N,method):
(StaticD,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
B=loss(N)
T=24
t1=int(T/3)
theta= np.linspace(-PI/2, PI/2, num=t1)
theta2= np.linspace(-PI, 3*PI/2, num=T-t1)
Demand=np.zeros(T)
phi= 0.15
Demand[:t1]= 0.75*StaticD*(1+phi*np.sin(theta))
for i in np.arange(0,T-t1):
#Modified demand otherwise the problem is not feasible anymore...
Demand[i+t1]= max(min(Demand[i+t1-1]+0.9*sum(UR)*np.sin(theta2[i]), 0.8*sum(Pmax)), sum(Pmin))
Demand[-1]=Demand[0]
price = SimplePriceFun(Pmin,Pmax,a,b,c,alpha,beta,gamma, np.mean(Demand))
SR=0.05*Demand
SR2=SR/3
it=10
C= np.zeros(it)
E= np.zeros(it)
w= np.linspace(0.001, 0.999 , num=it)
LB= np.repeat([Pmin],T, axis=0)
UB= np.repeat([Pmax],T, axis=0)
model = gp.Model('Non-Convex Problem')
P = model.addVars(T,N,lb=LB, ub=UB, name='P')
PL = model.addVars(T)
x = model.addVars(T,N)
y = model.addVars(T,N)
z = model.addVars(T,N)
m = model.addVars(T,N)
m2 = model.addVars(T,N)
model.Params.NonConvex = 2
for t in range(T):
"""Losses on the transfers """
model.addQConstr(PL[t]<= sum( sum(P[t,i]*P[t,j]*B[i,j] for j in range(N))for i in range(N)))
model.addQConstr(PL[t]>= sum( sum(P[t,i]*P[t,j]*B[i,j] for j in range(N))for i in range(N)))
model.addConstr(sum(P[t,k] for k in range(N))-PL[t] == Demand[t], name='Demand at '+ str(t))
for n in range(0,N):
if t>0:
""" Ramp rate limits """
model.addConstr(P[t,n] <= P[t-1,n] + UR[n], name='Maxramp'+ str(t) + str(n))
model.addConstr(P[t,n] >= P[t-1,n] - DR[n], name='Minramp'+ str(t) + str(n))
""" Reserve constraints"""
model.addConstr(z[t,n]==Pmax[n]-P[t,n])
model.addConstr(m[t,n]==min_([z[t,n],UR[n]]))
model.addConstr(m2[t,n]==min_([z[t,n],UR[n]/6]))
""" Exponential term of Emission objective """
model.addConstr(x[t,n]==delta[n]*P[t,n])
model.addGenConstrExp(x[t,n], y[t,n])
model.addConstr(sum(m[t,i]for i in range(N))>=SR[t])
model.addConstr(sum(m2[t,i]for i in range(N))>=SR2[t])
Cost = sum(sum(a[k]+b[k]*P[t,k]+c[k]*P[t,k]*P[t,k] for k in range(N)) for t in range(T))
Emission = sum(sum(alpha[k]+beta[k]*P[t,k]+gamma[k]*P[t,k]*P[t,k]+eta[k]*y[t,k] for k in range(N))for t in range(T))
if N==6:
LimE=np.linspace(14700,17000,num=it)
LimF=np.linspace(1010000, 1050000,num=it)
elif N==10:
LimE=np.linspace(57000,63000,num=it)
LimF=np.linspace(1795000,1875000,num=it)
for i in range(it) :
if (method== "scal"):
obj= price*w[i]*Emission+ (1-w[i])*Cost
elif (method=="LimE"):
obj= Cost
if i>0:
model.remove(const)
const=model.addQConstr(Emission<=LimE[i])
elif (method=="LimF"):
obj=Emission
if i>0:
model.remove(const)
const=model.addQConstr(Cost<=LimF[it-i-1])
model.setObjective(obj)
model.setParam( 'OutputFlag', False )
model.Params.timeLimit = 500
model.update()
tm=1
model.optimize()
# In case the problem takes too much time
while (model.status!=2):
model.update()
model.optimize()
tm=tm+1
print(model.SolCount)
print(tm, 'Time periods')
print(model.status)
C[i]=Cost.getValue()
E[i]=Emission.getValue()
return(E,C,price,w,T)
def RelaxErrors(N, Demand):
(Unused, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
B = loss(N)
price = SimplePriceFun(Pmin, Pmax, a, b, c, alpha, beta, gamma, Demand)
it = 10
C = np.zeros(it)
E = np.zeros(it)
problems = [
"Convex relaxation", "Linear lower bound", "Linear upper bound",
"Linear lower bound LVH", "Linear upper bound LVH"
]
n_meth = 0
wErrror = np.zeros([len(problems), 2])
plt.figure()
prop_cycle = plt.rcParams['axes.prop_cycle']
colors = prop_cycle.by_key()['color']
for method in problems:
m = gp.Model(method)
m.setParam('OutputFlag', False)
P = m.addVars(range(N), lb=Pmin, ub=Pmax, name='P')
PL = m.addVar()
x = m.addVars(N)
y = m.addVars(N)
for i in range(N):
m.addConstr(x[i] == delta[i] * P[i])
m.addGenConstrExp(x[i], y[i])
if (method == "Non-convex problem"):
m.setParam('NonConvex', 2)
m.addQConstr(PL == sum(
sum(P[i] * P[j] * B[i, j] for j in range(N))
for i in range(N)))
m.addConstr(P.sum() == Demand + PL)
methname = method
line = '-'
elif (method == "Convex relaxation"):
m.addQConstr(PL >= sum(
sum(P[i] * P[j] * B[i, j] for j in range(N))
for i in range(N)))
m.addConstr(P.sum() - PL == Demand, name='Demand')
methname = method
line = '-'
elif (method == "Linear lower bound"):
E_truth = E.copy()
C_truth = C.copy()
(n, k_upper, k_lower) = LinearRelaxation(N, Demand)
m.addConstr(sum(P[i] * n[i]
for i in range(N)) + k_lower >= 0) #Lowerbound
n_meth = n_meth + 1
methname = "Linear lower and upper bound"
line = '--'
elif (method == "Linear upper bound"):
E_linL = E.copy()
C_linL = C.copy()
m.addConstr(sum(P[i] * n[i]
for i in range(N)) + k_upper >= 0) #Upperbound
methname = ''
elif (method == "Linear lower bound LVH"):
E_linU = E.copy()
C_linU = C.copy()
model = gp.Model('Find P0')
model.setParam('OutputFlag', False)
model.setParam('NonConvex', 2)
Pow = model.addVars(range(N), lb=Pmin, ub=Pmax, name='P')
PLoss = model.addVar()
model.addQConstr(PLoss == sum(
sum(Pow[i] * Pow[j] * B[i, j] for j in range(N))
for i in range(N)))
model.addConstr(Pow.sum() - PLoss == Demand, name='Demand')
model.setObjective(0)
model.optimize()
P0 = np.zeros(N)
for i in range(N):
P0[i] = Pow[i].x
(n, k_lower, k_upper) = get_approx_planes(P0, B, Demand, Pmin,
Pmax)
m.addConstr(sum(P[i] * n[i]
for i in range(N)) + k_lower >= 0) #Lowerbound
n_meth = n_meth + 1
methname = "Linear lower and upper bound LVH"
line = ':'
elif (method == "Linear upper bound LVH"):
E_LVHL = E.copy()
C_LVHL = C.copy()
m.addConstr(sum(P[i] * n[i]
for i in range(N)) + k_upper >= 0) #Upperbound
methname = ''
index = 0
while (method != problems[index]):
index = index + 1
Cost = sum(a[k] + b[k] * P[k] + c[k] * P[k] * P[k] for k in range(N))
Emission = sum(alpha[k] + beta[k] * P[k] + gamma[k] * P[k] * P[k] +
eta[k] * y[k] for k in range(N))
w = np.linspace(0.001, 0.999, num=it)
for i in range(it):
obj = price * w[i] * Emission + (1 - w[i]) * Cost
m.setObjective(obj)
m.optimize()
C[i] = Cost.getValue()
E[i] = Emission.getValue()
if (n_meth > 0):
E_error = abs(E[i] - E_truth[i])
C_error = abs(C[i] - C_truth[i])
wErrror[index, 0] = max(wErrror[index, 0],
round(E_error / E_truth[i], 5))
wErrror[index, 1] = max(wErrror[index, 1],
round(C_error / C_truth[i], 5))
plt.plot(E, C, '.', color=colors[n_meth], label=methname)
[contE, s] = Hermite(E, C, price * w, 1 - w)
plt.plot(contE, s, color=colors[n_meth], linestyle=line)
plt.xlabel('Emission [lb]')
plt.ylabel('Cost [$]')
plt.title('Comparison of the relaxations of ' + str(N) +
' generators problem and demand = ' + str(int(Demand)) + ' MW')
plt.grid()
plt.legend()
Error1 = np.zeros([len(problems) - 1, 2])
[E_error, C_error] = HermiteError(E_truth, C_truth, E_linL, C_linL,
price * w, 1 - w)
Error1[0, 0] = E_error
Error1[0, 1] = C_error
[E_error, C_error] = HermiteError(E_truth, C_truth, E_LVHL, C_LVHL,
price * w, 1 - w)
Error1[2, 0] = E_error
Error1[2, 1] = C_error
Error2 = np.zeros([len(problems) - 1, 2])
[E_error, C_error] = SmartHermiteError(E_truth, C_truth, E_linL, C_linL,
price * w, 1 - w)
Error2[0, 0] = E_error
Error2[0, 1] = C_error
[E_error, C_error] = SmartHermiteError(E_truth, C_truth, E_LVHL, C_LVHL,
price * w, 1 - w)
Error2[2, 0] = E_error
Error2[2, 1] = C_error
return (Error1, Error2, wErrror)
def SQP(N,w_E,w_C,D,Pl=0,Pu=0, figures='False'):
(Unused,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
B=loss(N)
if (type(Pl)!=int):
Pmin=Pl.copy()
Pmax=Pu.copy()
t0=time.time()
"""Computing P0"""
bnds=np.transpose(np.vstack((Pmin,Pmax)))
P0=Pmin.copy()
def objective(P):
return (0)
def Gradient(P):
return(np.zeros(N))
def Hessian(P):
return(np.zeros([N,N]))
def cons_f(P):
PL=sum(sum(P[i]*P[j]*B[i,j] for j in range(N)) for i in range(N))
sum_eq=sum(P)-PL-D
return (sum_eq)
if (N<=10):
const=[{'type': 'eq', 'fun': cons_f}]
solution = minimize(objective ,P0, method='SLSQP',jac=Gradient, bounds=bnds,constraints=const)
else:
def cons_J(P):
Jac=np.ones(N)-2*P@B
return(Jac)
def cons_H(P,v):
return(-2*v*B)
NL_const = NonlinearConstraint(cons_f, 0, 0, jac=cons_J, hess=cons_H)
solution = minimize(objective ,P0, method='trust-constr',jac=Gradient,
hess=Hessian,constraints=NL_const, bounds=bnds)
P0 = solution.x
tol=1e-2
Maxiter=25
Obj=np.zeros(Maxiter)
C = sum(a[k]+b[k]*P0[k]+c[k]*P0[k]*P0[k] for k in range(N))
E = sum(alpha[k]+beta[k]*P0[k]+gamma[k]*P0[k]*P0[k]+eta[k]*np.exp(delta[k]*P0[k]) for k in range(N))
Obj[0]=w_C*C+w_E*E
Pk=P0.copy()
it=1
stepsize=1
while (it<Maxiter and stepsize>tol): #and tol<Obj[it-1]-opt
model=gp.Model('SQP Step')
model.setParam( 'OutputFlag', False )
DeltaP = model.addVars(range(N),lb=Pmin-Pk,ub=Pmax-Pk)
Surplus=sum(Pk)-Pk@B@Pk-D
model.addConstr(Surplus+sum(DeltaP[k]*(1-2*Pk@B[k]) for k in range(N))==0)
GradC=b+c*Pk*2
GradE= beta+gamma*Pk*2+delta*eta*np.exp(delta*Pk)
Grad=w_C*GradC+w_E*GradE
Hessian= w_C*2*c+w_E*(2*gamma+delta*delta*eta*np.exp(delta*Pk))
Lagr=sum(DeltaP[k]*DeltaP[k]*Hessian[k] for k in range(N))
objective = sum(Grad[k]*DeltaP[k] for k in range(N)) + 0.5*Lagr
model.setObjective(objective)
model.optimize()
Prev=Pk.copy()
for i in range(N):
Pk[i] = Pk[i] + DeltaP[i].x
stepsize=np.linalg.norm(Prev-Pk)
C = sum(a[k]+b[k]*Pk[k]+c[k]*Pk[k]*Pk[k] for k in range(N))
E = sum(alpha[k]+beta[k]*Pk[k]+gamma[k]*Pk[k]*Pk[k]+eta[k]*np.exp(delta[k]*Pk[k]) for k in range(N))
Obj[it]=w_C*C+w_E*E
if( (it % 10)==0):
print(it, " of ", Maxiter)
it=it+1
if (figures==True):
t1=time.time()
[opt,P_opt]=Solve(N,w_E,w_C,D)
t2=time.time()
plt.figure()
Pos=Obj[:it]-np.ones(it)*opt
Neg=-Pos.copy()
Pos=(Obj[:it]-np.ones(it)*opt>0)*Pos
Neg=(Obj[:it]-np.ones(it)*opt<0)*Neg
plt.plot(range(it),Pos, label='Positive Part ')
plt.plot(range(it),Neg, label='Negative Part ')
plt.xlabel('Iterations')
plt.ylabel('$f_k-f*$')
plt.title("Rate of convergence of SQP method ")
plt.legend()
plt.grid()
print(t1-t0, "sec for SQP ")
print(t2-t1, "sec for Scipy ")
print('\007')
return(E,C,Pk)
def GradMethod(N=10, method='ConvexRelax', solver='Gurobi'):
plt.close("all")
(Demand,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
B=loss(N)
price = SimplePriceFun(Pmin,Pmax,a,b,c,alpha,beta,gamma,Demand)
model=gp.Model('Projection Model')
model.setParam( 'OutputFlag', False )
P = model.addVars(range(N),lb=Pmin,ub=Pmax)
PL = model.addVar()
if (method=="NonConvex"):
model.setParam('NonConvex', 2)
model.addQConstr(PL== sum( sum(P[i]*P[j]*B[i][j] for j in range(N))for i in range(N)))
model.addConstr(P.sum() == Demand+PL)
else:
model.addQConstr(PL>= sum( sum(P[i]*P[j]*B[i,j] for j in range(N))for i in range(N)))
model.addConstr(P.sum()-PL == Demand, name='Demand')
if (solver=='Gurobi'):
t0=time.time()
[opt,P_opt]=SolveGurobi(N,price,1,Demand, method)
t1=time.time()
print(t1-t0 ,'sec for Gurobi')
"""Computing P0"""
model.setObjective(0)
model.optimize()
t2=time.time()
print(t2-t1 ,'P0')
P0=np.zeros(N)
for i in range(N):
P0[i] = P[i].x
else:
t0=time.time()
[opt,P_opt]=Solve(N,price,1,Demand)
t1=time.time()
print(t1-t0 ,'sec for Scipy')
"""Computing P0"""
bnds=np.transpose(np.vstack((Pmin,Pmax)))
P0=Pmin.copy()
def objective(P):
return (0)
def Gradient(P):
return(np.zeros(N))
def Hessian(P):
return(np.zeros([N,N]))
def cons_f(P):
PL=sum(sum(P[i]*P[j]*B[i,j] for j in range(N)) for i in range(N))
sum_eq=sum(P)-PL-Demand
return (sum_eq)
if (N<=10):
const=[{'type': 'eq', 'fun': cons_f}]
solution = minimize(objective ,P0, method='SLSQP',jac=Gradient, bounds=bnds,constraints=const)
else:
def cons_J(P):
Jac=np.ones(N)-2*P@B
return(Jac)
def cons_H(P,v):
return(-2*v*B)
NL_const = NonlinearConstraint(cons_f, 0, 0, jac=cons_J, hess=cons_H)
solution = minimize(objective ,P0, method='trust-constr',jac=Gradient,
hess=Hessian,constraints=NL_const, bounds=bnds)
P0 = solution.x
t2=time.time()
print(t2-t1 ,'P0')
print()
print("Gradient Method")
tol=1e-2
L=max(2*c+price*(2*gamma+delta*delta*eta*np.exp(delta*Pmax)))
mu=min(2*c+price*(2*gamma+delta*delta*eta*np.exp(delta*Pmin)))
Maxiter=int(0.25*(1+L/mu)*np.log(L*np.linalg.norm(P0-P_opt)**2/(2*tol)))+1
Maxiter=min(Maxiter,50) #Otherwise too large vector of iterates
Obj=np.zeros(Maxiter)
C = sum(a[k]+b[k]*P0[k]+c[k]*P0[k]*P0[k] for k in range(N))
E = sum(alpha[k]+beta[k]*P0[k]+gamma[k]*P0[k]*P0[k]+eta[k]*np.exp(delta[k]*P0[k]) for k in range(N))
Obj[0]=C+price*E
#Used if method=ConvexRelax
GradRate=np.zeros(Maxiter)
normP0=np.linalg.norm(P0-P_opt)**2
GradRate[0]=L/2*normP0
Pk=P0.copy()
it=1
if (method=='NonConvex'):
print(L,mu)
h=1/L
else:
h=2/(mu+L)
while (it<Maxiter and tol<Obj[it-1]-opt):
GradC=b+c*Pk*2
GradE= beta+gamma*Pk*2+delta*eta*np.exp(delta*Pk)
Grad=GradC+price*GradE
Pk=Pk-h*Grad
projection= sum((P[i]-Pk[i])*(P[i]-Pk[i]) for i in range(N))
model.setObjective(projection)
model.optimize()
if model.Status!= GRB.OPTIMAL:
print('Optimization was stopped with status ' + str(model.Status))
for i in range(N):
Pk[i] = P[i].x
C = sum(a[k]+b[k]*Pk[k]+c[k]*Pk[k]*Pk[k] for k in range(N))
E = sum(alpha[k]+beta[k]*Pk[k]+gamma[k]*Pk[k]*Pk[k]+eta[k]*np.exp(delta[k]*Pk[k]) for k in range(N))
Obj[it]=C+price*E
GradRate[it]=L/2*((L-mu)/(L+mu))**(2*it)*normP0
if( (it % 10)==0):
print(it, " of ", Maxiter)
it=it+1
plt.figure()
if (method=='ConvexRelax'):
plt.plot(range(it),GradRate[:it],'b--', label='Gradient theoretical rate')
plt.plot(range(it),Obj[:it]-np.ones(it)*opt,'b', label='Gradient Method ')
plt.xlabel('Iterations')
plt.ylabel('$f_k-f*$')
plt.title('Rate of convergence of the Gradient method ')
plt.legend()
plt.grid(True)
t3=time.time()
print(t3-t1, "for gradient")
def SPG(N=10, solver="Gurobi"):
(Demand,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
if (N==40): # Default demand not suited for problem with Transmission Losses
Demand=7500
B=loss(N)
price = SimplePriceFun(Pmin,Pmax,a,b,c,alpha,beta,gamma,Demand)
t0=time.time()
if (solver=='Gurobi'):
[opt,P_opt]=SolveGurobi(N,price,1,Demand, 'ConvexRelax')
t1=time.time()
print(t1-t0 ,' sec for Gurobi')
print(opt)
#Gurobi does not provide an accurate solution
[opt,P_opt]=Solve(N,price,1,Demand)
print(opt)
else:
[opt,P_opt]=Solve(N,price,1,Demand)
t1=time.time()
print(t1-t0 ,' sec for scipy')
bnds=np.transpose(np.vstack((Pmin,Pmax)))
P0=Pmin.copy()
def objective(P): return (0)
def Gradient(P): return(np.zeros(N))
def Hessian(P): return(np.zeros([N,N]))
def cons_f(P):
PL=sum(sum(P[i]*P[j]*B[i,j] for j in range(N)) for i in range(N))
sum_eq=sum(P)-PL-Demand
return (sum_eq)
if (N<=10):
const=[{'type': 'eq', 'fun': cons_f}]
solution = minimize(objective ,P0, method='SLSQP',jac=Gradient, bounds=bnds,constraints=const)
else:
def cons_J(P):
Jac=np.ones(N)-2*P@B
return(Jac)
def cons_H(P,v):
return(-2*v*B)
NL_const = NonlinearConstraint(cons_f, 0, 0, jac=cons_J, hess=cons_H)
solution = minimize(objective ,P0, method='trust-constr',jac=Gradient,
hess=Hessian,constraints=NL_const, bounds=bnds)
P0 = solution.x
t2=time.time()
print(t2-t1 ,'P0')
tol=1e-2
L=max(2*c+price*(2*gamma+delta*delta*eta*np.exp(delta*Pmax)))
C = sum(a[k]+b[k]*P0[k]+c[k]*P0[k]*P0[k] for k in range(N))
E = sum(alpha[k]+beta[k]*P0[k]+gamma[k]*P0[k]*P0[k]+eta[k]*np.exp(delta[k]*P0[k]) for k in range(N))
Maxiter=50
Obj=np.zeros(Maxiter)
Obj[0]=C+price*E
print()
print("Spectral Projected Gradient")
model=gp.Model('Projection Model')
P = model.addVars(range(N),lb=Pmin,ub=Pmax)
PL = model.addVar()
model.addQConstr(PL>= sum( sum(P[i]*P[j]*B[i,j] for j in range(N))for i in range(N)))
model.addConstr(P.sum()-PL == Demand, name='Demand')
model.setParam( 'OutputFlag', False )
it=1
Pk=P0.copy()
dk=np.zeros([N])
stepmin=1e-10
stepmax=1e10
stepsize=1/L
sigma1=0.1
sigma2=0.9
g=1e-4
M=10
while (it<Maxiter and tol<Obj[it-1]-opt):
GradC=b+c*Pk*2
GradE= beta+gamma*Pk*2+delta*eta*np.exp(delta*Pk)
Grad=GradC+price*GradE
Prev=Pk.copy()
Pk=Pk-stepsize*Grad
projection= sum((P[i]-Pk[i])*(P[i]-Pk[i]) for i in range(N))
model.setObjective(projection)
model.optimize()
for i in range(N):
dk[i] = P[i].x - Prev[i]
coeff=1
index=max(0,it-M)
fmax=max(Obj[index:it])
Pk=Prev+coeff*dk
C = sum(a[k]+b[k]*Pk[k]+c[k]*Pk[k]*Pk[k] for k in range(N))
E = sum(alpha[k]+beta[k]*Pk[k]+gamma[k]*Pk[k]*Pk[k]+eta[k]*np.exp(delta[k]*Pk[k]) for k in range(N))
fk=C+price*E
while(fk>fmax+g*coeff*Grad@dk):
Num=-0.5*coeff**2*Grad@dk
Denom= fk-Obj[it-1]-coeff*Grad@dk
temp=Num/Denom
if (temp>=sigma1 and temp<=coeff*sigma2):
coeff=temp
else:
coeff=coeff/2
Pk=Prev+coeff*dk
C = sum(a[k]+b[k]*Pk[k]+c[k]*Pk[k]*Pk[k] for k in range(N))
E = sum(alpha[k]+beta[k]*Pk[k]+gamma[k]*Pk[k]*Pk[k]+eta[k]*np.exp(delta[k]*Pk[k]) for k in range(N))
fk=C+price*E
sk=Pk-Prev
yk=2*c*(Pk-Prev) + price*(2*gamma*(Pk-Prev)+ delta*eta*(np.exp(delta*Pk)-np.exp(delta*Prev)))
if sk@yk<=0: stepsize=stepmax
else: stepsize=max(stepmin,min(sk@sk/(sk@yk),stepmax))
Obj[it]=fk
if( (it % 10)==0):
print(it, " of ", Maxiter)
it=it+1
t3=time.time()
print(t3-t1, "sec for SPG")
plt.figure()
plt.plot(range(it),Obj[:it]-np.ones(it)*opt, label='SPG')
plt.title('Rate of convergence of the Spectral Projected Gradient')
plt.xlabel('Iterations')
plt.ylabel('$f_k-f*$')
plt.legend()
plt.grid(True)
def AccMethod(N=10, method='ConvexRelax', solver='Gurobi'):
(Demand,Pmax,Pmin,a,b,c,alpha,beta,gamma,delta,eta,UR,DR) = load(N)
B=loss(N)
price = SimplePriceFun(Pmin,Pmax,a,b,c,alpha,beta,gamma,Demand)
model=gp.Model('Projection Model')
model.setParam( 'OutputFlag', False )
P = model.addVars(range(N),lb=Pmin,ub=Pmax)
PL = model.addVar()
if (method=="NonConvex"):
model.setParam('NonConvex', 2)
model.addQConstr(PL== sum( sum(P[i]*P[j]*B[i][j] for j in range(N))for i in range(N)))
model.addConstr(P.sum() == Demand+PL)
elif (method=="ConvexRelax"):
model.addQConstr(PL>= sum( sum(P[i]*P[j]*B[i,j] for j in range(N))for i in range(N)))
model.addConstr(P.sum()-PL == Demand, name='Demand')
if (solver=='Gurobi'):
t0=time.time()
[opt,P_opt]=SolveGurobi(N,price,1,Demand, method)
t1=time.time()
print(t1-t0 ,'sec for Gurobi')
"""Computing P0"""
model.setObjective(0)
model.optimize()
t2=time.time()
print(t2-t1 ,'P0')
P0=np.zeros(N)
for i in range(N):
P0[i] = P[i].x
else:
t0=time.time()
[opt,P_opt]=Solve(N,price,1,Demand)
t1=time.time()
print(t1-t0 ,'sec for Scipy')
"""Computing P0"""
bnds=np.transpose(np.vstack((Pmin,Pmax)))
P0=Pmin.copy()
def objective(P):
return (0)
def Gradient(P):
return(np.zeros(N))
def Hessian(P):
return(np.zeros([N,N]))
def cons_f(P):
PL=sum(sum(P[i]*P[j]*B[i,j] for j in range(N)) for i in range(N))
sum_eq=sum(P)-PL-Demand
return (sum_eq)
if (N<=10):
const=[{'type': 'eq', 'fun': cons_f}]
solution = minimize(objective ,P0, method='SLSQP',jac=Gradient, bounds=bnds,constraints=const)
else:
def cons_J(P):
Jac=np.ones(N)-2*P@B
return(Jac)
def cons_H(P,v):
return(-2*v*B)
NL_const = NonlinearConstraint(cons_f, 0, 0, jac=cons_J, hess=cons_H)
solution = minimize(objective ,P0, method='trust-constr',jac=Gradient,
hess=Hessian,constraints=NL_const, bounds=bnds)
P0 = solution.x
t2=time.time()
print(t2-t1 ,'P0')
tol=1e-2
L=max(2*c+price*(2*gamma+delta*delta*eta*np.exp(delta*Pmax)))
mu=min(2*c+price*(2*gamma+delta*delta*eta*np.exp(delta*Pmin)))
C = sum(a[k]+b[k]*P0[k]+c[k]*P0[k]*P0[k] for k in range(N))
E = sum(alpha[k]+beta[k]*P0[k]+gamma[k]*P0[k]*P0[k]+eta[k]*np.exp(delta[k]*P0[k]) for k in range(N))
f0= C+price*E
Maxiter= int(np.sqrt(L/mu)*np.log(2*(f0-opt)/tol))+1
Maxiter=min(Maxiter,50)
Obj=np.zeros(Maxiter)
Obj[0]=f0
AccRate=np.zeros(Maxiter)
AccRate[0]=2*(Obj[0]-opt)
print()
print("Accelerated Gradient")
it=1
Pk=P0.copy()
yk=Pk.copy()
stepsize=(np.sqrt(L)-np.sqrt(mu))/(np.sqrt(L)+np.sqrt(mu))
while (it<Maxiter and tol<Obj[it-1]-opt):
GradC=b+c*yk*2
GradE= beta+gamma*yk*2+delta*eta*np.exp(delta*yk)
Grad=GradC+price*GradE
Prev=Pk.copy()
Pk=yk-Grad/L
projection= sum((P[i]-Pk[i])*(P[i]-Pk[i]) for i in range(N))
model.setObjective(projection)
model.optimize()
for i in range(N):
Pk[i] = P[i].x
yk=Pk+stepsize*(Pk-Prev)
C = sum(a[k]+b[k]*Pk[k]+c[k]*Pk[k]*Pk[k] for k in range(N))
E = sum(alpha[k]+beta[k]*Pk[k]+gamma[k]*Pk[k]*Pk[k]+eta[k]*np.exp(delta[k]*Pk[k]) for k in range(N))
Obj[it]=C+price*E
AccRate[it]=2*(1-np.sqrt(mu/L))**it*(Obj[0]-opt)
if( (it % 10)==0):
print(it, " of ", Maxiter)
it=it+1
t3=time.time()
print(t3-t1, "sec for Accelerated")
plt.figure(1)
if (method=='ConvexRelax'):
plt.plot(range(it),AccRate[:it], '--', color='orange', label= 'Accelerated gradient theoretical rate')
plt.plot(range(it),Obj[:it]-np.ones(it)*opt, color='orange', label='Accelerated Gradient Method')
plt.title('Rate of convergence of the two methods')
plt.xlabel('Iterations')
plt.ylabel('$f_k-f*$')
plt.legend()
plt.grid(True)
def TimeRelaxation():
Number = [100] #[2,3,6,10,40,100]
size = len(Number)
nRelax = 3
Computime = np.zeros([size, nRelax])
# Comparison of the 2 Linear Relaxations
for nPb in range(size):
N = Number[nPb]
(D, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
if (N == 40):
D = 7500
B = loss(N)
t1 = time.time()
""" Linear relaxation using N points """
LinearRelaxation(N, D)
t2 = time.time()
""" Linear relaxation using 1 feasible point and Gurobi"""
model = gp.Model('Find P0')
model.setParam('OutputFlag', False)
model.setParam('NonConvex', 2)
Pow = model.addVars(range(N), lb=Pmin, ub=Pmax, name='P')
PLoss = model.addVar()
model.addQConstr(PLoss == sum(
sum(Pow[i] * Pow[j] * B[i, j] for j in range(N))
for i in range(N)))
model.addConstr(Pow.sum() - PLoss == D, name='Demand')
model.setObjective(0)
model.optimize()
P0 = np.zeros(N)
for i in range(N):
P0[i] = Pow[i].x
get_approx_planes(P0, B, D, Pmin, Pmax, True)
t3 = time.time()
""" Linear relaxation using 1 feasible point and Scipy"""
bnds = np.transpose(np.vstack((Pmin, Pmax)))
P0 = Pmin.copy()
def objective(P):
return (0)
def Gradient(P):
return (np.zeros(N))
def Hessian(P):
return (np.zeros([N, N]))
def cons_f(P):
PL = sum(
sum(P[i] * P[j] * B[i, j] for j in range(N)) for i in range(N))
sum_eq = sum(P) - PL - D
return (sum_eq)
def cons_J(P):
Jac = np.ones(N) - 2 * P @ B
return (Jac)
def cons_H(P, v):
return (-2 * v * B)
if N <= 10:
const = [{'type': 'eq', 'fun': cons_f}]
solution = minimize(objective,
P0,
method='SLSQP',
jac=Gradient,
bounds=bnds,
constraints=const)
else:
NL_const = NonlinearConstraint(cons_f,
0,
0,
jac=cons_J,
hess=cons_H)
solution = minimize(objective,
P0,
method='trust-constr',
jac=Gradient,
hess=Hessian,
constraints=NL_const,
bounds=bnds)
P0 = solution.x
(n, k_lower, k_upper) = get_approx_planes(P0, B, D, Pmin, Pmax, True)
t4 = time.time()
Computime[nPb] = np.array([[t2 - t1, t3 - t2, t4 - t3]])
bw = 0.25
opacity = 0.8
prop_cycle = plt.rcParams['axes.prop_cycle']
colors = prop_cycle.by_key()['color']
plt.figure()
plt.bar(np.arange(size),
Computime[:, 0],
bw,
alpha=opacity,
color=colors[0],
label='Relaxation: N points')
plt.bar(np.arange(size) + bw,
Computime[:, 1],
bw,
alpha=opacity,
color=colors[1],
label='Relaxation: 1 point and Gurobi')
plt.bar(np.arange(size) + 2 * bw,
Computime[:, 2],
bw,
alpha=opacity,
color=colors[2],
label='Relaxation: 1 point and Scipy')
plt.xlabel('Problem size')
plt.ylabel('Time [s]')
plt.title(
'Computational time for the relaxations for different problem sizes')
plt.xticks(
np.arange(size) + bw, ('N=2', 'N=3', 'N=6', 'N=10', 'N=40', 'N=100'))
plt.legend()
plt.tight_layout()
plt.show()
#Comparison of Time for one Linear, quadratic, nonconvex problem
Computime2 = np.zeros([size, nRelax])
for nPb in range(size):
N = Number[nPb]
(D, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
if (N == 40):
D = 7500
B = loss(N)
t = RelaxTime(N, D)
Computime2[nPb, :-1] = t
t0 = time.time()
Solve(N, 1, 1, D)
t1 = time.time()
Computime2[nPb, -1] = t1 - t0
bw = 0.25
opacity = 0.8
prop_cycle = plt.rcParams['axes.prop_cycle']
colors = prop_cycle.by_key()['color']
plt.figure()
plt.bar(np.arange(size),
Computime2[:, 0],
bw,
alpha=opacity,
color=colors[0],
label='Convex Relaxation')
plt.bar(np.arange(size) + bw,
Computime2[:, 1],
bw,
alpha=opacity,
color=colors[1],
label='Linear Relaxation')
plt.bar(np.arange(size) + 2 * bw,
Computime2[:, 2],
bw,
alpha=opacity,
color=colors[2],
label='Scipy on non-convex')
plt.xlabel('Problem size')
plt.ylabel('Time [s]')
plt.title(
'Computational time for solving one optimization problem for different problem sizes'
)
plt.xticks(
np.arange(size) + bw, ('N=2', 'N=3', 'N=6', 'N=10', 'N=40', 'N=100'))
plt.legend()
plt.tight_layout()
plt.show()
def eConstE_SQP(N, LimE, D):
(Unused, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
B = loss(N)
(Zones, Units) = zones(N)
"""Computing P0 without POZ"""
bnds = np.transpose(np.vstack((Pmin, Pmax)))
def P0_Obj(P):
return (0)
def P0_Grad(P):
return (np.zeros(N))
def Objective(P):
Obj = sum(a[i] + b[i] * P[i] + c[i] * P[i] * P[i] for i in range(N))
return (Obj)
def Gradient(P):
Grad = b + 2 * c * P
return (Grad)
def cons_f(P):
PL = sum(
sum(P[i] * P[j] * B[i, j] for j in range(N)) for i in range(N))
sum_eq = sum(P) - PL - D
return (sum_eq)
def cons_J(P):
Jac = np.ones(N) - 2 * P @ B
return (Jac)
def cons_C(P):
constraint = LimE - sum(alpha[k] + beta[k] * P[k] + gamma[k] * P[k] *
P[k] + eta[k] * np.exp(delta[k] * P[k])
for k in range(N))
return (constraint)
def cons_GradC(P):
Grad = -beta - 2 * gamma * P - delta * eta * np.exp(delta * P)
return (Grad)
const = [{'type': 'eq', 'fun': cons_f}, {'type': 'ineq', 'fun': cons_C}]
solution = minimize(P0_Obj,
Pmin,
method='SLSQP',
jac=P0_Grad,
bounds=bnds,
constraints=const)
P0 = solution.x
tol = 1e-2
Maxiter = 100
Obj = np.zeros(Maxiter)
Obj[0] = sum(a[k] + b[k] * P0[k] + c[k] * P0[k] * P0[k] for k in range(N))
Pk = P0.copy()
Prev = P0.copy()
it = 1
stepsize = 1
while (it < Maxiter and stepsize > tol):
model = gp.Model('SQP Step, MIQP')
model.setParam('OutputFlag', False)
DeltaP = model.addVars(range(N), lb=Pmin - Pk, ub=Pmax - Pk)
x = model.addVars(N)
y = model.addVars(N)
for i in range(N):
model.addConstr(x[i] == delta[i] * DeltaP[i])
model.addGenConstrExp(x[i], y[i])
Surplus = sum(Pk) - Pk @ B @ Pk - D
model.addConstr(Surplus + sum(DeltaP[k] * (1 - 2 * Pk @ B[k])
for k in range(N)) == 0)
model.addQConstr(
LimE -
sum(alpha[k] + beta[k] * (Pk[k] + DeltaP[k]) + gamma[k] *
(Pk[k] + DeltaP[k]) *
(Pk[k] + DeltaP[k]) + eta[k] * np.exp(delta[k] * Pk[k]) * y[k]
for k in range(N)) >= 0)
for i in range(N): # POZ
Zon_i = Zones[Units == i]
n_i = len(Zon_i)
if n_i >= 1:
bb = model.addVars(range(n_i + 1), vtype=GRB.BINARY)
model.addConstr(DeltaP[i] <= Zon_i[0, 0] * bb[0] +
(1 - bb[0]) * Pmax[i] - Pk[i])
for j in np.arange(1, n_i):
model.addConstr(
DeltaP[i] >= Zon_i[j - 1, 1] * bb[j] - Pk[i])
model.addConstr(DeltaP[i] <= Zon_i[j, 0] * bb[j] +
(1 - bb[j]) * Pmax[i] - Pk[i])
model.addConstr(DeltaP[i] >= Zon_i[-1, 1] * bb[n_i] - Pk[i])
model.addConstr(bb.sum() == 1)
Grad = b + c * Pk * 2
Hessian = 2 * c
Lagr = sum(DeltaP[k] * DeltaP[k] * Hessian[k] for k in range(N))
objec = sum(Grad[k] * DeltaP[k] for k in range(N)) + 0.5 * Lagr
model.setObjective(objec)
model.optimize()
PPrev = Prev.copy()
Prev = Pk.copy()
for i in range(N):
Pk[i] = Pk[i] + DeltaP[i].x
stepsize = np.linalg.norm(Prev - Pk)
if np.linalg.norm(PPrev - Pk) < tol:
res = sum(Pk) - Pk @ B @ Pk - D
resPrev = sum(Prev) - Prev @ B @ Prev - D
if (abs(res) <= 1e-4 or abs(resPrev) <= 1e-4):
if (resPrev == max(res, resPrev) and res <= 0):
Pk = Prev.copy()
break
(LowerB, UpperB) = InZone(Pk, Pmin, Pmax, Zones, Units)
Limits = np.transpose(np.vstack((LowerB, UpperB)))
sol = minimize(Objective,
Pk,
method='SLSQP',
jac=Gradient,
bounds=Limits,
constraints=const)
Pk = sol.x
obj = sum(a[k] + b[k] * Pk[k] + c[k] * Pk[k] * Pk[k]
for k in range(N))
(LowerB1, UpperB1) = InZone(Prev, Pmin, Pmax, Zones, Units)
obj1 = Obj[it - 1]
if np.prod(LowerB == LowerB1) == 0:
Limits = np.transpose(np.vstack((LowerB1, UpperB1)))
sol1 = minimize(Objective,
Prev,
method='SLSQP',
jac=Gradient,
bounds=Limits,
constraints=const)
Prev = sol1.x
obj1 = sum(a[k] + b[k] * Prev[k] + c[k] * Prev[k] * Prev[k]
for k in range(N))
if obj1 < obj:
Pk = Prev.copy()
stepsize = -1
Obj[it] = sum(a[k] + b[k] * Pk[k] + c[k] * Pk[k] * Pk[k]
for k in range(N))
if ((it % 10) == 0):
print(it, " of ", Maxiter)
it = it + 1
"""Figures"""
opt = Obj[it - 1] + 1e-6
plt.figure()
Pos = Obj[:it] - np.ones(it) * opt
Neg = -Pos.copy()
Pos = (Obj[:it] - np.ones(it) * opt > 0) * Pos
Neg = (Obj[:it] - np.ones(it) * opt < 0) * Neg
plt.plot(range(it), Pos, label='Positive Part ')
plt.plot(range(it), Neg, label='Negative Part ')
plt.xlabel('Iterations')
plt.ylabel('$f_k-f*$')
plt.title("Rate of convergence of eConstE ")
plt.legend()
plt.grid()
return (LimE - cons_C(Pk), Obj[it - 1], Pk)
def SQP_MINLP(N, w_E, w_C, D):
(Unused, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
B = loss(N)
(Zones, Units) = zones(N)
"""Computing P0 without POZ"""
bnds = np.transpose(np.vstack((Pmin, Pmax)))
P0 = Pmin.copy()
def objective(P):
return (0)
def Gradient(P):
return (np.zeros(N))
def Hessian(P):
return (np.zeros([N, N]))
def cons_f(P):
PL = sum(
sum(P[i] * P[j] * B[i, j] for j in range(N)) for i in range(N))
sum_eq = sum(P) - PL - D
return (sum_eq)
def cons_J(P):
Jac = np.ones(N) - 2 * P @ B
return (Jac)
if (N <= 10):
const = [{'type': 'eq', 'fun': cons_f, 'jac': cons_J}]
solution = minimize(objective,
P0,
method='SLSQP',
jac=Gradient,
bounds=bnds,
constraints=const)
else:
def cons_H(P, v):
return (-2 * v * B)
NL_const = NonlinearConstraint(cons_f, 0, 0, jac=cons_J, hess=cons_H)
solution = minimize(objective,
P0,
method='trust-constr',
jac=Gradient,
hess=Hessian,
constraints=NL_const,
bounds=bnds)
P0 = solution.x
tol = 1e-2
Maxiter = 100
Obj = np.zeros(Maxiter)
C = sum(a[k] + b[k] * P0[k] + c[k] * P0[k] * P0[k] for k in range(N))
E = sum(alpha[k] + beta[k] * P0[k] + gamma[k] * P0[k] * P0[k] +
eta[k] * np.exp(delta[k] * P0[k]) for k in range(N))
Obj[0] = w_C * C + w_E * E
Pk = P0.copy()
Prev = P0.copy()
it = 1
stepsize = 1
while (it < Maxiter and stepsize > tol):
model = gp.Model('SQP Step, MIQP')
model.setParam('OutputFlag', False)
DeltaP = model.addVars(range(N), lb=Pmin - Pk, ub=Pmax - Pk)
Surplus = sum(Pk) - Pk @ B @ Pk - D
model.addConstr(Surplus + sum(DeltaP[k] * (1 - 2 * Pk @ B[k])
for k in range(N)) >= 0)
for i in range(N): # POZ
Zon_i = Zones[Units == i]
n_i = len(Zon_i)
if n_i >= 1:
bb = model.addVars(range(n_i + 1), vtype=GRB.BINARY)
model.addConstr(DeltaP[i] <= Zon_i[0, 0] * bb[0] +
(1 - bb[0]) * Pmax[i] - Pk[i])
for j in np.arange(1, n_i):
model.addConstr(
DeltaP[i] >= Zon_i[j - 1, 1] * bb[j] - Pk[i])
model.addConstr(DeltaP[i] <= Zon_i[j, 0] * bb[j] +
(1 - bb[j]) * Pmax[i] - Pk[i])
model.addConstr(DeltaP[i] >= Zon_i[-1, 1] * bb[n_i] - Pk[i])
model.addConstr(bb.sum() == 1)
GradC = b + c * Pk * 2
GradE = beta + gamma * Pk * 2 + delta * eta * np.exp(delta * Pk)
Grad = w_C * GradC + w_E * GradE
Hessian = w_C * 2 * c + w_E * (
2 * gamma + delta * delta * eta * np.exp(delta * Pk))
Lagr = sum(DeltaP[k] * DeltaP[k] * Hessian[k] for k in range(N))
objec = sum(Grad[k] * DeltaP[k] for k in range(N)) + 0.5 * Lagr
model.setObjective(objec)
model.optimize()
PPrev = Prev.copy()
Prev = Pk.copy()
for i in range(N):
Pk[i] = Pk[i] + DeltaP[i].x
stepsize = np.linalg.norm(Prev - Pk)
if np.linalg.norm(PPrev -
Pk) < tol: # The algorithm cycles between 2 zones
res = sum(Pk) - Pk @ B @ Pk - D
resPrev = sum(Prev) - Prev @ B @ Prev - D
if (resPrev == max(res, resPrev) and res <= 0):
Pk = Prev.copy()
(opt, p) = Solve(N, w_E, w_C, D)
stepsize = -1
C = sum(a[k] + b[k] * Pk[k] + c[k] * Pk[k] * Pk[k] for k in range(N))
E = sum(alpha[k] + beta[k] * Pk[k] + gamma[k] * Pk[k] * Pk[k] +
eta[k] * np.exp(delta[k] * Pk[k]) for k in range(N))
Obj[it] = w_C * C + w_E * E
if ((it % 10) == 0):
print(it, " of ", Maxiter)
it = it + 1
return (E, C, Pk)
def eConstF_SQP(N, LimF, D):
(Unused, Pmax, Pmin, a, b, c, alpha, beta, gamma, delta, eta, UR,
DR) = load(N)
B = loss(N)
(Zones, Units) = zones(N)
"""Computing P0 without POZ"""
bnds = np.transpose(np.vstack((Pmin, Pmax)))
def P0_Obj(P):
return (0)
def P0_Grad(P):
return (np.zeros(N))
def Objective(P):
Obj = sum(alpha[i] + beta[i] * P[i] + gamma[i] * P[i] * P[i] +
eta[i] * np.exp(P[i] * delta[i]) for i in range(N))
return (Obj)
def Gradient(P):
Grad = beta + 2 * gamma * P + delta * eta * np.exp(delta * P)
return (Grad)
def cons_f(P):
PL = sum(
sum(P[i] * P[j] * B[i, j] for j in range(N)) for i in range(N))
sum_eq = sum(P) - PL - D
return (sum_eq)
def cons_J(P):
Jac = np.ones(N) - 2 * P @ B
return (Jac)
def cons_C(P):
constraint = LimF - sum(a[k] + b[k] * P[k] + c[k] * P[k] * P[k]
for k in range(N))
return (constraint)
def cons_GradC(P):
Grad = -b - 2 * c * P
return (Grad)
const = [{'type': 'eq', 'fun': cons_f}, {'type': 'ineq', 'fun': cons_C}]
solution = minimize(P0_Obj,
Pmin,
method='SLSQP',
jac=P0_Grad,
bounds=bnds,
constraints=const)
P0 = solution.x
tol = 1e-2
Maxiter = 100
Obj = np.zeros(Maxiter)
Obj[0] = sum(alpha[k] + beta[k] * P0[k] + gamma[k] * P0[k] * P0[k] +
eta[k] * np.exp(delta[k] * P0[k]) for k in range(N))
Pk = P0.copy()
Prev = P0.copy()
it = 1
stepsize = 1
while (it < Maxiter and stepsize > tol):
model = gp.Model('SQP Step, MILP')
model.setParam('OutputFlag', False)
DeltaP = model.addVars(range(N), lb=Pmin - Pk, ub=Pmax - Pk)
Surplus = sum(Pk) - Pk @ B @ Pk - D
model.addConstr(Surplus + sum(DeltaP[k] * (1 - 2 * Pk @ B[k])
for k in range(N)) >= 0)
model.addQConstr(LimF - sum(a[k] + b[k] * (Pk[k] + DeltaP[k]) + c[k] *
(Pk[k] + DeltaP[k]) * (Pk[k] + DeltaP[k])
for k in range(N)) >= 0)
for i in range(N): # POZ
Zon_i = Zones[Units == i]
n_i = len(Zon_i)
if n_i >= 1:
bb = model.addVars(range(n_i + 1), vtype=GRB.BINARY)
model.addConstr(DeltaP[i] <= Zon_i[0, 0] * bb[0] +
(1 - bb[0]) * Pmax[i] - Pk[i])
for j in np.arange(1, n_i):
model.addConstr(
DeltaP[i] >= Zon_i[j - 1, 1] * bb[j] - Pk[i])
model.addConstr(DeltaP[i] <= Zon_i[j, 0] * bb[j] +
(1 - bb[j]) * Pmax[i] - Pk[i])
model.addConstr(DeltaP[i] >= Zon_i[-1, 1] * bb[n_i] - Pk[i])
model.addConstr(bb.sum() == 1)
Grad = beta + gamma * Pk * 2 + delta * eta * np.exp(delta * Pk)
Hessian = 2 * gamma + delta * delta * eta * np.exp(delta * Pk)
Lagr = sum(DeltaP[k] * DeltaP[k] * Hessian[k] for k in range(N))
objec = sum(Grad[k] * DeltaP[k] for k in range(N)) + 0.5 * Lagr
model.setObjective(objec)
model.optimize()
PPrev = Prev.copy()
Prev = Pk.copy()
for i in range(N):
Pk[i] = Pk[i] + DeltaP[i].x
stepsize = np.linalg.norm(Prev - Pk)
if np.linalg.norm(PPrev -
Pk) < tol: # The algorithm cycles between 2 zones
res = sum(Pk) - Pk @ B @ Pk - D
resPrev = sum(Prev) - Prev @ B @ Prev - D
if (abs(res) <= 1e-4 or abs(resPrev) <= 1e-4):
if (resPrev == max(res, resPrev) and res <= 0):
Pk = Prev.copy()
break
# Compares the best solution of the 2 zones
(LowerB, UpperB) = InZone(Pk, Pmin, Pmax, Zones,
Units) #Current zone
Limits = np.transpose(np.vstack((LowerB, UpperB)))
sol = minimize(Objective,
Pk,
method='SLSQP',
jac=Gradient,
bounds=Limits,
constraints=const)
Pk = sol.x
obj = sum(alpha[k] + beta[k] * Pk[k] + gamma[k] * Pk[k] * Pk[k] +
eta[k] * np.exp(delta[k] * Pk[k]) for k in range(N))
(LowerB1, UpperB1) = InZone(Prev, Pmin, Pmax, Zones,
Units) #Previous zone
obj1 = Obj[it - 1]
if np.prod(LowerB == LowerB1) == 0: #Different zone
Limits = np.transpose(np.vstack((LowerB1, UpperB1)))
sol1 = minimize(Objective,
Prev,
method='SLSQP',
jac=Gradient,
bounds=Limits,
constraints=const)
Prev = sol1.x
obj1 = sum(alpha[k] + beta[k] * Prev[k] +
gamma[k] * Prev[k] * Prev[k] +
eta[k] * np.exp(delta[k] * Prev[k])
for k in range(N))
if obj1 < obj:
Pk = Prev.copy()
stepsize = -1
Obj[it] = sum(alpha[k] + beta[k] * Pk[k] + gamma[k] * Pk[k] * Pk[k] +
eta[k] * np.exp(delta[k] * Pk[k]) for k in range(N))
if ((it % 10) == 0):
print(it, " of ", Maxiter)
it = it + 1
return (Obj[it - 1], LimF - cons_C(Pk), Pk)