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lp_relaxation.py
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lp_relaxation.py
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# -*- coding: utf-8 -*-
# lp_relaxation.py: Linear Programmaing relaxation for SUC
# authors: Antoine Passemiers, Cedric Simar
from instance import SUPInstance
from utils import SUCLpProblem
from variables import init_variables
import numpy as np
import pulp
def create_formulation(instance, variables=None, relax=True):
""" Create instance of the original problem, with relaxed integrality
constraints, if requested.
Args:
instance (SUPInstance):
stores constants and indices that are part of the problem instance
variables (tuple, optional):
Tuple of LpArray objects. If not provided, new PuLP variables
will be initialized.
relax (bool, optional):
Whether to relax integrality constraints
"""
# Get set sizes
(G, n_scenarios, T, L, N, n_import_groups) = instance.get_sizes()
n_generators, n_periods, n_lines, n_nodes = G, T, L, N
# Get indices arrays
(Gs, Gf, Gn, LIn, LOn, IG, LI_indices, LO_indices, \
L_node_indices) = instance.get_indices()
# Get problem constants
(PI, K, S, C, D, P_plus, P_minus, R_plus, R_minus, \
UT, DT, T_req, F_req, B, TC, FR, IC, GAMMA) = instance.get_constants()
# Initialize PuLP variables if not provided
if variables is None:
variables = init_variables(
Gs, Gf, n_scenarios, T, N, L, n_import_groups, relax=relax)
(u, v, p, theta, w, z, e) = variables
problem = SUCLpProblem("SUC", pulp.LpMinimize)
# Define objective function:
# sum_g sum_s sum_t PI[s] * (K[g]*u[g, s, t] + S[g]*v[g, s, t] + C[g]*p[g, s, t])
obj = np.sum(PI * np.swapaxes(
K * np.swapaxes(u, 0, 2) + \
S * np.swapaxes(v, 0, 2) + \
C * np.swapaxes(p, 0, 2), 1, 2))
problem += obj
# Define constraints group 3.21
# Market-clearing constraint: uncertainty in demand
# and production of renewable resources for each node
# sum_LIn e[l, s, t] + sum_g p[g, s, t] == D[n, s, t] + sum_LOn e[l, s, t]
problem.set_constraint_group("3.21")
for n in range(N):
LIn_ids = LI_indices[n][LI_indices[n] != SUPInstance.NO_LINE]
LOn_ids = LO_indices[n][LO_indices[n] != SUPInstance.NO_LINE]
sum_g = np.sum(p[Gn[n], :, :], axis=0) if len(Gn[n]) > 1 else p[Gn[n][0], :, :]
problem += (np.sum(e[LIn_ids, :, :], axis=0) + sum_g == \
D[n, :, :] + np.sum(e[LOn_ids, :, :], axis=0))
# Define constraints group 3.22
# e[l, s, t] == B[l, s] * (theta[n, s, t] - theta[m, s, t])
problem.set_constraint_group("3.22")
for l in range(L):
m, n = L_node_indices[l]
problem += (e[l, :, :] == B[l, :][..., np.newaxis] * \
(theta[n, :, :] - theta[m, :, :]))
# Define constraints group 3.23
# e[l, s, t] <= TC[l]
problem.set_constraint_group("3.23")
problem += (np.swapaxes(e, 0, 2) <= TC)
# Define constraints group 3.24
# -TC[l] <= e[l, s, t]
problem.set_constraint_group("3.24")
problem += (-TC <= np.swapaxes(e, 0, 2))
# Define constraints group 3.25
# Generator contingencies: Maximum generator capicity limits
# p[g, s, t] <= P_plus[g, s] * u[g, s, t]
problem.set_constraint_group("3.25")
problem += (np.transpose(p, (2, 0, 1)) <= P_plus * np.transpose(u, (2, 0, 1)))
# Define constraints group 3.26
# Generator contingencies: Minimum generator capacity limits
# P_minus[g, s]* u[g, s, t] <= p[g, s, t]
problem.set_constraint_group("3.26")
problem += (P_minus * np.transpose(u, (2, 0, 1)) <= np.transpose(p, (2, 0, 1)))
# Define constraints group 3.27
# p[g, s, t] - p[g, s, t-1] <= R_plus[g]
problem.set_constraint_group("3.27")
problem += (np.swapaxes(p[:, :, 1:] - p[:, :, :-1], 0, 2) <= R_plus)
# Define constraints group 3.28
# p[g, s, t-1] - p[g, s, t] <= R_minus[g]
problem.set_constraint_group("3.28")
problem += (np.swapaxes(p[:, :, :-1] - p[:, :, 1:], 0, 2) <= R_minus)
# Define constraints group 3.29
# sum_{t-UT[g]+1}^t z[g, q] <= w[g, t]
# t >= UT[g]
problem.set_constraint_group("3.29")
for g in Gs:
UTg = int(UT[g])
for t in range(UTg, T):
problem += (np.sum(z[g, t-UTg+1:t+1]) <= w[g, t])
# Define constraints group 3.30
# sum_{t+1}^{t+DT[g]} z[g, q] <= 1 - w[g, t]
# t <= N - DT[g]
problem.set_constraint_group("3.30")
for g in Gs:
DTg = int(DT[g])
# Number of periods in horizon = T
for t in range(0, T-DTg+1):
if t + 1 < T:
problem += (np.sum(z[g, t+1:t+DTg+1]) <= 1 - w[g, t])
# Define contraints group 3.31
# sum_{t-UT[g]+1}^t v[g, s, q] <= u[g, s, t]
# t >= UT[g]
problem.set_constraint_group("3.31")
for g in Gf:
UTg = int(UT[g])
for t in range(UTg, T):
problem += (np.sum(v[g, :, t-UTg+1:t+1], axis=1) <= u[g, :, t])
# Define constraints group 3.32
# sum_{t+1}^{t+DT[g]} v[g, s, q] <= 1 - u[g, s, t]
# t <= N - DT[g]
problem.set_constraint_group("3.32")
for g in Gf:
DTg = int(DT[g])
# Number of periods in horizon = T
for t in range(0, T-DTg-1):
if t + 1 < T:
problem += (np.sum(v[g, :, t+1:t+DTg+1], axis=1) <= 1 - u[g, :, t])
# Define constraints group 3.33
# z[g, t] <= 1 for slow generators
# Those constraints have been added during variables initialization
# Define constraints group 3.34
# v[g, s, t] <= 1 for slow generators
# Those constraints have been added during variables initialization
# Define constraints group 3.35
# z[g, t] >= w[g, t] - w[g, t-1] for slow generators
problem.set_constraint_group("3.35")
problem += (z[Gs, 1:] >= w[Gs, 1:] - w[Gs, :-1])
problem += z[Gs, 0] >= 0
problem += w[Gs, 0] >= 0
# Define constraints group 3.36
# v[g, s, t] >= u[g, s, t] - u[g, s, t-1]
if len(Gf) > 0:
problem.set_constraint_group("3.36")
problem += (v[Gf, :, 1:] >= u[Gf, :, 1:] - u[Gf, :, :-1])
# Define constraints group 3.37
# PI[s] * u[g, s, t] == PI[s] * w[g, t]
problem.set_constraint_group("3.37")
problem += (np.swapaxes(u[Gs, :, :], 0, 1) == w[Gs, :])
# Define constraints group 3.38
# PI[s] * v[g, s, t] == PI[s] * z[g, t]
problem.set_constraint_group("3.38")
problem += (np.swapaxes(v[Gs, :, :], 0, 1) == z[Gs, :])
# Define constraints group 3.39
# For all generators:
# p[g, s, t] >= 0
# v[g, s, t] >= 0
# 0 <= u[g, s, t] <= 1
# Those constraints have been added during variables initialization
# Define constraints group 3.40
# For slow generators:
# z[g, t] >= 0
# 0 <= w[g, t] <= 1
# Those constraints have been added during variables initialization
return problem, variables