PowSysMod

The PowSysMod environment enables the construction of polynomial optimization problems in complex variables for power systems. It uses the PolynomialOptim environment. The environment can adapt to several input formats (Matpower, GOC, IIDM) if the user defines the proper functions (a read function and elements of the power network along with their parameters and constraints). Once the optimization problem is built, the problem can be exported in real numbers and in text files (to be treated by AMPL for example). It is also possible to convert the problem in a JuMP model in real numbers (cartesian or polar form).

PolynomialOptim

The PolynomialOptim environment provides a structure and methods for working with complex polynomial optimization problems subject to polynomial constraints. It is based on a polynomial environment that allows to work on polynomial objects with natural operations (+, -, *, conj, |.|).

The base type is Variable, from which Exponents, Monomial, and Polynomial can be constructed by calling the respective constructors or with algebraic operations (+, -, *, conj, |.|). The Point type holds the variables at which polynomials can be evaluated.

Available functions on Polynomial, Mononomial, Variable types:

• isconst, isone
• evaluate
• abs2, conj
• is_homogeneous: tests if p(exp(iϕ)X) = p(X) ∀X∈R^n, Φ∈R (phase invariant equation)
• cplx2real: convert the provided object to a tuple of real and imaginary part, expressed with real and imaginary part variables.

include(joinpath("src_PolynomialOptim", "PolynomialOptim.jl"))

a = Variable("a", Complex)
b = Variable("b", Real)
p = a*conj(a) + b + 2

print(p)
# (2.0) + b + conj(a) * a

pt = Point([a, b], [1+2im, 1+im])
print(pt)
# a 1 + 2im
# b 1

pt = pt - Point([a], [1])
print(pt)
# a 0 + 2im
# b 1

val = evaluate(p, pt) # 7 + 0im

p_real, p_imag = cplx2real(p)
pt_r = cplx2real(pt)
val_real = evaluate(p_real, pt_r) # 7 + 0im
val_imag = evaluate(p_imag, pt_r) # 0

Polynomial problems

A Constraint structure holding a Polynomial and complex upper and lower bounds is defined to build the Problem type made of:

• several Variables
• a Polynomial objective,
• several constraints names along with their corresponding Constraint

Implemented methods

• get_objective, set_objective!
• get_variables, get_variabletype, has_variable, add_variable!
• get_constraint, get_constraints, has_constraint, constraint_type, add_constraint!, rm_constraint!
• get_slacks, get_minslack
• pb_cplx2real: converts the problem variables, objective and constraints to real expressions function of real and imaginary part of the original problem variables.

include(joinpath("src_PolynomialOptim", "PolynomialOptim.jl"))

a = Variable("a", Complex)
b = Variable("b", Real)
p_obj = abs2(a) + abs2(b) + 2
p_cstr1 = 3a + b + 2
p_cstr2 = abs2(b) + 5a*b + 2

pb = Problem()

add_variable!(pb, a); add_variable!(pb, b)

set_objective!(pb, p_obj)

add_constraint!(pb, "Cstr 1", p_cstr1 << 3+5im) # -∞ ≤ (2.0) + b^2 + (5.0) * a * b ≤ 2 + 6im
add_constraint!(pb, "Cstr 2", 2-im << p_cstr2 << 3+7im) # -∞ ≤ (2.0) + b^2 + (5.0) * a * b ≤ 2 + 6im
add_constraint!(pb, "Cstr 3", p_cstr2 == 0) # -∞ ≤ (2.0) + b^2 + (5.0) * a * b ≤ 2 + 6im

print(pb)
# ▶ variables: a b
# ▶ objective: (2.0) + conj(a) * a + b^2
# ▶ constraints:
#     Cstr 1: -∞ < (2.0) + (3.0) * a + b < 3 + 5im
#     Cstr 2: 2 - 1im < (2.0) + (5.0) * a * b + b^2 < 3 + 7im
#     Cstr 3: (2.0) + (5.0) * a * b + b^2 = 0

pt_sol = Point([a, b], [1, 1+2im])
print("slack by constraint at given point:\n", get_slacks(pb, pt_sol))
# slack by constraint at given point:
# Cstr 1 -3.0 + 0.0im
# Cstr 2 -5 + 1im
# Cstr 3 -8 + 0im