Electric Distribution Network Planning Under Uncertainty

Abstract

This chapter presents a deterministic and an adaptive robust model for the short-term network expansion planning in electric distribution networks, considering siting and sizing of voltage regulators, capacitor banks, renewable energy generation, energy storage systems, and existing overloaded feeders reinforcement. The objective function to be minimized consists of investment and operation costs. Conventional expansion models in distribution networks are stated as a mixed-integer non-linear mathematical programs. In this chapter, we introduce the standard formulation and transform it into a mixed-integer linear programming form. This formulation is used to solve a deterministic short-term electric distribution network expansion planning case. Based on the deterministic formulation, we expand the formulation to a two-stage tri-level adaptive robust problem for considering load consumption and renewable-based DG uncertainties. By using Karush–Kuhn–Tucker conditions, this model is transformed into a two-stage bi-level adaptive robust optimization problem. A column and constraint generation framework is used to solve the problem. Computational results are obtained from a 123-node distribution system under different conditions to assess the performance of the proposed approach. Results show the effectiveness of the proposed methodology.

Appendix

The notations used throughout this chapter are listed below.

Acronym

AMPL:

A Modeling Language for Mathematical Programming.

Indexes

b :

Index of installed CB sizes.

c :

Index of conductor types.

k :

Index of buses of the system.

km :

Index of branches of the system.

t :

Index of periods of planning.

μ :

Index of uncertain parameters.

Sets

B :

Set of branches of the system.

C :

Set of conductor types.

CB :

Set of CBs.

E :

Set of candidate buses to install ESSs.

N :

Set of buses of the system.

P :

Set of periods.

R :

Set of candidate buses to install CBs.

SE :

Set of substations of the system.

U :

Set of uncertainties.

V :

Set of candidate buses to install PV plants.

VR :

Set of voltage regulators.

W :

Set of candidate buses to install WP plants.

Z :

Set of integer numbers.

Constants

ES _k,0 , ES _k,T :

Initial/final stored energy in the storage unit at node k.

$ES_{k}^{\min }, ES_{k}^{\max }$ :

Minimum/maximum storage capacity of the unit at node k.

L _km :

Length of circuit km.

N _ES :

Maximum number of storage units to be installed in the EDN.

N _PV , N _WT :

Maximum number of PV/WT plants to be installed in the EDN.

$PV_v^{\max }, WT_w^{\max }$ :

Active power capacity of PV module/WT unit.

$Pc^{\min }_{k}, Pc^{\max }_{k}$ :

Minimum/maximum charging power rate of the storage unit at node k.

$P^{\max }_{c}, Q^{\max }_{c}$ :

Minimum/maximum active/reactive power through conductor c.

$Pd^{\min }_{k}, Pd^{\max }_{k}$ :

Minimum/maximum discharging power rate of the storage unit at node k.

P _{D k}, Q _{D k} :

Active/reactive load demand in bus k.

$Q_b^{sp}$ :

Specified reactive power capacity of CB b.

R _c , X _c :

Resistance/reactance of conductor c.

$R^{\%}_{km}$ :

Regulation % of the VR to be installed in km.

r :

Discount rate.

$V_k^{\min }, V_k^{\max }$ :

Minimum/maximum voltage magnitude limit in bus k.

ηc _k , ηd _k :

Charging/discharging storage efficiency of the storage unit at node k.

Δ _t :

Time slot.

$\varGamma _{km}^{vr}$ :

Installation cost of VR at node km.

$\varGamma _{b}^{{cb}^{fx/sw}}$ :

Installation cost of FCB/SCB with capacity type b.

$\varGamma _{v/w}^{pv/wt}$ :

Installation cost of PV modules/WT units.

$\varGamma _{k}^{es}$ :

Installation cost of storage unit at node k.

$\varGamma _{km,c}^{cr}$ :

Replacement cost of overloaded conductor in branch km by a conductor c.

$\varPi _{k}^{es}$ :

Operation cost of storage unit at node k.

$\varPi _{v/w}^{pv/wt}$ :

Operation cost of PV modules/WT units.

$\varPi _{k}^{s}$ :

Cost of energy supply by the substation at node k.

$\underline {\phi }^S, \overline {\phi }^S$ :

Minimum/maximum power factor at substation.

$\underline {\varUpsilon }^D, \overline {\varUpsilon }^D$ :

Load demand uncertainty budget.

$\underline {\varUpsilon }^{pv}, \overline {\varUpsilon }^{pv}$ :

PVG uncertainty budget.

$\underline {\varUpsilon }^{wt}, \overline {\varUpsilon }^{wt}$ :

WTG uncertainty budget.

$\underline {\mu }^D_t, \overline {\mu }^D_t$ :

Minimum/maximum limits for load demand factor in period t.

$\underline {\mu }^{pv}_t, \overline {\mu }^{pv}_t$ :

Minimum/maximum limits for PVG factor in period t.

$\underline {\mu }^{wt}_t, \overline {\mu }^{wt}_t$ :

Minimum/maximum limits for WTG factor in period t.

Continuous Variables

p _km,c,t , q _km,c,t :

Active/reactive power flow through replaced circuit c in branch km in period t.

$p_{k,t}^S, q_{k,}^S$ :

Active/reactive power in substation k in period t.

$v_{k,t}, \tilde {v}_{k,t}$ :

Non regulated/regulated voltage magnitude in bus k in period t.

$q_{k,t}^{CB}$ :

Reactive capacitive power injected by the CB in bus k in period t.

$p_{k,t}^{ESS}$ :

Active power injected or absorbed to/from the system by storage unit k in period t.

pd _k,t , pc _k,t :

Charging/discharging active power of storage unit k in period t.

es _k,t :

Stored energy in storage unit k in period t.

$p_{k,t}^{PV}, p_{k,t}^{PV}$ :

PV/WT active power generation at node k in period t.

$\mu _t^D, \mu _t^{pv}, \mu _t^{wt}$ :

Uncertainty parameters for load demand, PV and WT power output in period t.

Binary and Integer Variables

$\alpha _{km}^{vr}$ :

Binary variable, $\alpha _{km}^{vr} = 1$ if the VR is installed in branch km in period t, $\alpha _{km}^{vr} = 0$ otherwise.

$\alpha _{k,b,t}^{fx/sw}$ :

Binary variable, $\alpha _{k,b,t}^{fx/sw} = 1$ if the FCB/SCB of capacity b is installed at node k in period t, $\alpha _{k,b,t}^{fx/sw} = 0$ otherwise.

αc _k,t :

Binary variable equal to 1 if the storage installed unit at node k is being charged in period t, and equal to 0 otherwise.

αd _k,t :

Binary variable equal to 1 if the storage installed unit at node k is being discharged in period t, and equal to 0 otherwise.

$\alpha _{k,t}^{es}$ :

Binary variable equal to 1 if storage unit is installed at node k in period t, and equal to 0 otherwise.

$\alpha _{k,v,t}^{pv}$ :

Binary variable equal to 1 if PV module v is installed at node k in period t, and equal to 0 otherwise.

$\alpha _{k,w,t}^{wt}$ :

Binary variable equal to 1 if WT unit w is installed at node k in period t, and equal to 0 otherwise.

$\alpha _{km,c,t}^{cr}$ :

Binary variable equal to 1 if conductor in branch km is replaced by conductor c in period t, and equal to 0 otherwise.

$n_{k,t}^{cb}$ :

Integer variable that define the CB modules installed at node k in period t.

$n_{k}^{\max }$ :

Integer variable that define the maximum CB modules to be installed at node k.