Distribution System Planning with Representation of Uncertainties Based on Interval Analysis

This work presents an approach for optimal distribution system planning (DSP) to minimize the total costs of expansion and operation with the representation of uncertainties in the load demand and in the wind-based distributed generation (WDG). The proposed approach, called interval distribution system planning (I-DSP), is based on an interval power flow (IPF) and the metaheuristic artificial immune system (AIS). The IPF is used to obtain an interval of the total cost that reflects the uncertainties over load and generation. The interval cost is the merit function of the optimization algorithm. The network constraints as the limits of current, voltage and power from substations, in addition to the radiality and connectivity are taken into account. Well-known test systems are used to assess the impact of the uncertainties representation in the DSP problem.

a :

Index for cable type

e :

Index for existing cable type

d :

Index for deterministic variables

i :

Index for interval variables

f :

Index for system buses

j :

Index for buses directly connected to f

fj :

Index for system branches

n :

Index for substations

lo, up:

Index for lower and upper limits for any variable

NBP :

Set of candidate branches

NC :

Set of cable types

NBE :

Set of existing branches

NDG :

Set of buses candidate for receiving WDG

NSP :

Set of proposed substations, i.e., substations candidate for building

NSE :

Set of existing substations

NST :

Set of existing and candidate substations

NBT :

Set of existing and candidate branches

NBU :

Set of buses

Ω _f :

Set of buses directly connected to f

P :

Set of initial candidate solutions

CI _a :

Cost for building a new branch with cable type ‘a’ (US$/km)

CR _ea :

Cost for retrofitting an existing branch of cable type ‘e’ by replacing it by cable type ‘a’ (US$/km)

CWT _f :

Cost for installation of wind turbine at bus f (US$)

CB _n :

Cost for building a proposed substation ‘n’ (US$)

CE _n :

Cost for expanding the capacity of an existing substation (US$)

C _op :

Substation operation cost (US$/kVA² h)

C _l :

Energy loss cost (US$/kWh)

C _wg :

Operating and maintenance cost of the wind turbine (US$/kWh)

C _pe :

Energy purchase cost (US$/kWh)

l _fj :

Length of branch fj (km)

VP :

Conversion of any cost to its present value

g _fj :

Conductance of branch fj

α :

Number of hours in 1 year

φ_S, φ_l :

Loss factors for substations and branches

τ :

Annual interest rate

T :

Planning horizon in years

\( Pd^{i}_{f} ,Qd^{i}_{f} \) :

Interval active and reactive loads

\( Pd^{d}_{f} ,Qd^{d}_{f} \) :

Deterministic active and reactive loads

α_pkl, α_pku, α_qkl, α_qku :

Active and reactive load percent variations

\( Pwt^{\text{lo}}_{f} ,Pwt^{\text{up}}_{f} \) :

Limits of the active power from a wind generator

\( Qwt^{\text{lo}}_{f} ,Qwt^{\text{up}}_{f} \) :

Limits of the reactive power from a wind generator

\( V^{\text{lo}}_{f} ,V^{\text{up}}_{f} \) :

Limits for the interval voltage variable

\( I^{\text{lo}}_{fj} ,I^{\text{up}}_{fj} \) :

Limits of the interval current variable

\( Ss^{\text{lo}}_{n} ,Ss^{\text{up}}_{n} \) :

Limits of the apparent power supplied by substation

V^min, V^max :

Minimum and maximum operational voltage levels

\( I^{\hbox{max} }_{fj} \) :

Maximum current at branch fj

\( Ss^{0}_{n} \) :

Maximum apparent power from existing substation

\( Ss^{\text{EX}}_{n} \) :

Predefined value of apparent power considered as an expansion for existing substation

\( Ss^{\text{NS}}_{n} \) :

Maximum apparent power from new substation

v^lo, v^up :

Wind speed limits

α_wl, α_wu :

Percent variations of the wind speed

v_in, v_n, v_ou :

Input, nominal and output speeds of a wind generator

P _n :

Nominal WDG active power

ap, bp :

Coefficients that relate wind speed to the active power from a wind generator

cq, dq :

Coefficients that relate wind speed to the reactive power from a wind generator

X_p, Y_p, Z_p, W_p :

Rows of the inverse Jacobian matrix at the solution point of deterministic power flow

Jacⁱ, Jac^d :

Interval and deterministic Jacobian matrix

C :

Preconditioning matrix given by the inverse of the Jacobian matrix at the solution point of deterministic power flow

Id :

Identity matrix

m_x, r_X :

Midpoint and radius of interval X = [x₁; x₂]

round(.) :

Rounding operator

β :

Cloning parameter of the CLONR algorithm

nb :

Number of candidate solutions for cloning

db :

Number of candidate solutions randomly generated for the receptor editing process

f^*(i):

Normalized fitness of a candidate solution i

f(i):

Midpoint fitness of a candidate solution i

f _av :

Average fitness for clone set of CLONR

δ* :

Standard deviation

p(ic):

Probability of clone ic to be mutated (p(ic) ∈ [0,1])

h :

Mutation parameter of the CLONR algorithm

h₁, h₂ :

Low- and high-mutation parameters of the CLONR algorithm

Nab :

Number of candidate solutions (antibodies) of set P

Nab_dist :

Number of unique individuals of the CLONR algorithm

gmax :

Maximum number of I-DSP algorithm iterations

gest :

Number of iterations in which the best solution of P remains unchanged

gimp :

Number of iterations in which the process stagnates

div, limd :

Solutions diversity and its inferior limit

δ_S, δ_l, δ_wt, δ_pe :

Auxiliary variables for the calculation of the following interval costs: operating of the substations, energy loss, wind power generation and the energy purchase cost, respectively

ε_vo, ε_cu, ε_ap, ε_acp, ε_lo, ε_op, ε_pe, ε_tc :

Maximum relative percent errors between IPF and MCS for voltages, currents, apparent power from substation, active power from substation, loss cost, operational cost, energy purchase cost and total cost, respectively

\( x^{bc}_{fj,a} \) :

Binary variable associated with building branch fj with cable type ‘a’

\( x^{br}_{fj,a} \) :

Binary variable associated with retrofitting branch fj

\( x^{wt}_{f} \) :

Binary variable associated with retrofitting branch fj

\( x^{sc}_{n} \) :

Binary variable associated with building the proposed substation

\( x^{sr}_{n} \) :

Binary variable associated with expanding the existing substation

\( x^{oc}_{n} \) :

Binary variable associated with operating substation

\( x^{cc}_{fj} \) :

Binary variable associated with building branch fj

vⁱ, v^d :

Interval and deterministic wind speed

\( Ss^{i}_{n} \) :

Apparent power supplied by substation ‘n’

\( L^{i}_{fj} ,L^{d}_{fj} \) :

Interval and deterministic power loss of branch fj

\( Pwt^{i}_{f} ,Qwt^{i}_{f} \) :

Active and reactive power from wind generator at bus f

\( Ps^{i}_{n} \) :

Active power supplied by substation n

\( \Delta L^{i}_{fj} \) :

Power loss increment of branch fj

\( V^{i}_{f} \) :

Interval voltage magnitude at bus f

\( V^{d}_{f} ,V^{d}_{j} \) :

Deterministic voltage magnitude at bus f and j

\( I^{i}_{fj} \) :

Interval current magnitude at branch fj

\( \theta^{d}_{fj} \) :

Deterministic phase angle between buses f and j

\( \theta^{d}_{f} \) :

Deterministic phase angle at bus f

\( Ps^{i}_{f} ,Qs^{i}_{f} \) :

Interval active and reactive powers supplied by substation at bus f

\( P^{i}_{fj} ,Q^{i}_{fj} \) :

Interval active and reactive power flows at branch fj

\( P^{i}_{f} ,Q^{i}_{f} \) :

Interval active and reactive powers injected at bus f

\( \Delta P^{i}_{f} ,\Delta Q^{i}_{f} \) :

Interval active and reactive power mismatches at bus f

Δθⁱ, ΔVⁱ :

Interval increments of phase angle and voltage magnitude

\( \theta^{i}_{f} \) :

Interval phase angle at bus f

X ^h :

Interval solution vector of the IPF updated at each iteration h

x ^h :

Vector given by the midpoints of the intervals contained in X^h

x, f(x):

State and power mismatch vectors, respectively

K(x^h,X^h), h(x^h):

Krawczyk operator and iteration counter

ε ⁱ :

Pre-specified tolerance used as IPF convergence criterion

OVⁱ, OV^d, ΔOVⁱ :

Interval, deterministic and interval increment for any output variable

μ :

Measure function used for comparing intervals

Nc(i):

Number of clones for a selected candidate solution i

The authors would like to thank the ‘Coordination for the Improvement of Higher Education Personnel’ (CAPES), ‘Foundation for Supporting Research in Minas Gerais’ (FAPEMIG), ‘Brazilian National Research Council’ (CNPq), ‘Electric Power National Institute’ (INERGE) and ‘Heuristic and Bioinspired Optimization Group’ (GOHB) for supporting this research.

Authors and Affiliations

1. Department of Electrical Engineering, Federal Center of Technological Education “Celso Suckow da Fonseca”, Angra dos Reis, RJ, Brazil

   Felipe da S. Seta

2. Department of Electrical Engineering, Federal University at Juiz de Fora (UFJF), Juiz de Fora, MG, Brazil

   Leonardo W. de Oliveira & Edimar J. de Oliveira

Corresponding author

Correspondence to Felipe da S. Seta.

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