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Optimal DC Power Flow
The optimal power flow problem minimizes the overall operating cost in a electricity network while subject to a number of generation and transmission constraints. The DC model is a widely used linearized approximation of the actual nonlinear power flow model, and it can be written as
\[\begin{align} \min~&\sum\limits_{i=1}^m (a_ig_i^2 + b_ig_i + c_i) \\ \text{s.t.}~&\pmb{B}_{\text{bus}} \pmb{v} + \pmb{I}_{\text{bg}} \pmb{g} = \pmb{L} \\ &|\pmb{B}_{\text{f}} \pmb{v}| \leq \pmb{R} \\ &P_i^{\min} \leq g_i \leq P_i^{\max} &\forall i = 1, 2, ..., m \\ &v_{ref} = 0, \end{align}\]where \(n\) is the number of buses, and \(m\) is the number of generators. The decision variables:
- \(v_j\): the voltage angle of bus \(j\), with \(j=1, 2, …, n\)
- \(g_i\): the output of the \(i\)th generator, with \(i=1, 2, …, m\)
and parameters:
- \(\pmb{B}_{\text{bus}}\): the matrix of reciprocals of reactance between buses
- \(\pmb{I}_{\text{bg}}\): the matrix that places generations to their corresponding buses
- \(\pmb{L}\): the array of electricity load at each bus
- \(\pmb{B}_{\text{f}}\): the matrix for calculating the transmitted power
- \(P_i^{\min}\) and \(P_i^{\max}\): the minimum and maximum output of the \(i\)th generator
- \(\pmb{R}\): the array of transmission line capacity ratings
In this case study, we consider the IEEE RTS-1996, and the dataset is generated according to the one used in MATPOWER. You may find the dataset as an Excel file from here. The following code is used to import system parameters from the file.
# Load data from the Excel file
import pandas as pd
Bbus_df = pd.read_excel('ieee_rts.xlsx', sheet_name='Bbus')
Bf_df = pd.read_excel('ieee_rts.xlsx', sheet_name='Bf')
Gen_df = pd.read_excel('ieee_rts.xlsx', sheet_name='Gen')
Rates_df = pd.read_excel('ieee_rts.xlsx', sheet_name='Rates')
Load_df = pd.read_excel('ieee_rts.xlsx', sheet_name='Load')
Bbus = Bbus_df.values # Bbus array
Bf = Bf_df.values # Bf array
R = Rates_df['Rate_A'].values # line capacity ratings
GBus = Gen_df['GBus'].values # generator buses
Pmin = Gen_df['Pmin'].values # minimum outputs of generators
Pmax = Gen_df['Pmax'].values # maximum outputs of generators
Coeff = Gen_df.loc[:, 'Cost_a':'Cost_c'].values # cost coefficients of generators
L = Load_df['Load'].values # electricity loads
n = Bf.shape[1] # n: Number of buses
m = len(GBus) # m: Number of generators
The DC optimal power flow problem can be solved by the following Python code.
from rsome import ro
from rsome import grb_solver as grb
import rsome as rso
import numpy as np
model = ro.Model()
v = model.dvar(n) # decision variable as the voltage angle
g = model.dvar(m) # decision variable as the generation output
model.min(rso.sumsqr(Coeff[:, 0]**0.5 * g) +
Coeff[:, 1]@g +
Coeff[:, 2].sum()) # quadratic objective function
I_bg = np.zeros((n, m))
I_bg[GBus-1, range(m)] = 1 # array that places generators to their buses
model.st(Bbus@v + I_bg@g == L) # power balance equation
model.st(abs(Bf@v) <= R) # transmission line capacities
model.st(g >= Pmin, g <= Pmax) # output capacities of generators
Ref = 12 # index of the reference bus
model.st(v[Ref] == 0) # set the reference bus
model.solve(grb) # solve the problem by Gurobi
print('\nOptimal generation cost: {0:0.2f}'.format(model.get()))
Being solved by Gurobi...
Solution status: 2
Running time: 0.0045s
Optimal generation cost: 61000.92