Distribution System Expansion Planning

Abstract

The widespread growth of distributed generation (DG), mainly due to its numerous operational and planning benefits and to the penetration of renewable energy, inevitably requires the inclusion of this kind of generation in distribution planning models. This chapter addresses the multistage expansion planning problem of a distribution system where investments in the distribution network and in DG are jointly considered. The optimal expansion plan identifies the best alternative, location, and installation time for the candidate assets. The incorporation of DG in distribution system expansion planning drastically increases the complexity of the optimization process. In order to shed light on the modeling difficulties associated with the co-optimized planning problem, a deterministic model is presented first. The model is driven by the minimization of the net present value of the total cost including the costs related to investment, maintenance, production, losses, and unserved energy. As a relevant feature, radiality conditions are specifically tailored to accommodate the presence of DG in order to avoid the islanding of distributed generators and the issues associated with transfer nodes. Since a large portion of DG relies on non-dispatchable renewable-based technologies, the uncertainty associated with the high variability of the corresponding energy sources needs to be properly characterized in the planning models. Based on the previous deterministic model, uncertainty is incorporated using a stochastic programming framework. Within such a context, the uncertainty featured by renewable-based generation and demand is characterized through a set of scenarios that explicitly capture the correlation between uncertainty sources. The resulting stochastic program is driven by the minimization of the total expected cost. Both deterministic and stochastic optimization problems are formulated as mixed-integer linear programs for which finite convergence to optimality is guaranteed and efficient off-the-shelf software is available. Numerical results illustrate the effective performance of the approaches presented in this chapter.

Appendix

The symbols used throughout this chapter are listed below:

1.1.1 Indices

b :

Index for time blocks.

h :

Index for the blocks used in the piecewise linearization of energy losses.

i,j:

Indices for nodes.

$k,\kappa$ :

Indices for available investment alternatives.

l :

Index for branch types.

p :

Index for generator types.

s :

Index for segments of the cumulative distribution functions.

$t,\tau$ :

Indices for time stages.

tr :

Index for transformer types.

$\omega$ :

Index for scenarios.

1.1.2 Sets

$\mathcal{B}$ :

Index set of time blocks.

$K^{l}$ :

Index set of available alternatives for branches of type l.

$K^{p}$ :

Index set of available alternatives for generators of type p.

$K^{tr}$ :

Index set of available alternatives for transformers of type tr.

$\mathcal{L}$ :

Set of branch types. $\mathcal{L} = \left\{ {EFB} \right.$, $ERB$, $NRB$, $\left. {NAB} \right\}$ where EFB, ERB, NRB, and NAB denote existing fixed branch, existing replaceable branch, new replacement branch, and newly added branch, respectively.

$\mathcal{P}$ :

Set of generator types. $\mathcal{P} = \left\{ C \right.$, $W$, $\left. {\Theta} \right\}$ where C, W, and ${\Theta}$ stand for conventional, wind power, and photovoltaic generation, respectively.

$\mathcal{T}$ :

Index set of time stages.

TR :

Set of transformer types. $TR = \left\{ {ET} \right.$, $\left. {NT} \right\}$ where ET and NT denote existing transformer and new transformer, respectively.

${\Xi}^{DT}$ :

Set of variables associated with the deterministic model.

${\Xi}^{ST}$ :

Set of variables associated with the stochastic model.

${\Upsilon}^{l}$ :

Index set of branches of type l.

${\Upsilon}^{SW,l}$ :

Subset of ${\Upsilon}^{l}$ comprising those branches that are switchable under normal operation.

${\Psi}_{i}^{l}$ :

Index set of nodes connected to node i by a branch of type l.

${\Psi}_{t}^{LN}$ :

Index set of load nodes at stage t.

${\Psi}^{N}$ :

Index set of system nodes.

${\Psi}^{p}$ :

Index set of candidate nodes for the installation of generators of type p.

${\Psi}^{SS}$ :

Index set of substation nodes.

${\Omega}_{b}$ :

Set of scenarios for time block b.

1.1.3 Parameters

$A_{kh}^{l}$ :

Width of block h of the piecewise linear energy losses for alternative k for branches of type l.

$A_{kh}^{tr}$ :

Width of block h of the piecewise linear energy losses for alternative k for transformers of type tr.

$C_{k}^{E,p}$ :

Cost coefficient for the energy supplied by alternative k for generators of type p.

$C_{k}^{I,l}$ :

Investment cost coefficient for alternative k for branches of type l.

$C_{k}^{I,NT}$ :

Investment cost coefficient for alternative k for new transformers.

$C_{k}^{I,p}$ :

Investment cost coefficient for alternative k for generators of type p.

$C_{i}^{I,SS}$ :

Investment cost coefficient for the substation at node i.

$C_{k}^{M,l}$ :

Maintenance cost coefficient for alternative k for branches of type l.

$C_{k}^{M,p}$ :

Maintenance cost coefficient for alternative k for generators of type p.

$C_{k}^{M,tr}$ :

Maintenance cost coefficient for alternative k for transformers of type tr.

$C_{b}^{SS}$ :

Cost coefficient for the energy supplied by substations for time block b.

$C^{U}$ :

Cost coefficient for unserved energy.

$D_{it}$ :

Actual peak demand at node i and stage t.

$\tilde{D}_{it}$ :

Fictitious peak demand at node i and stage t.

${\overline{F}}_{k}^{l}$ :

Upper limit for the actual current flow through alternative k for branches of type l.

${\overline{G}}_{k}^{p}$ :

Rated capacity for alternative k for generators of type p.

$\hat{G}_{ikb}^{p}$ :

Maximum power availability for alternative k for generators of type p at node i and time block b.

$\hat{G}_{ikb}^{p} \left( \omega \right)$ :

Maximum power availability for alternative k for generators of type p at node i, time block b, and scenario $\omega$.

${\overline{G}}_{k}^{tr}$ :

Rated capacity of alternative k for transformers of type tr.

$I$ :

Annual interest rate.

$IB_{t}$ :

Investment budget for stage t.

$J$ :

Sufficiently large positive constant.

$\ell_{ij}^{{}}$ :

Length of the branch connecting nodes i and j.

$M_{kh}^{l}$ :

Slope of block h of the piecewise linear energy losses for alternative k for branches of type l.

$M_{kh}^{tr}$ :

Slope of block h of the piecewise linear energy losses for alternative k for transformers of type tr.

$n_{B}$ :

Number of time blocks.

$n_{DG}$ :

Number of candidate nodes for installation of distributed generation.

$n_{H}$ :

Number of blocks of the piecewise linear energy losses.

$n_{T}$ :

Number of time stages.

$n_{{\Omega} }$ :

Number of scenarios per time block.

$n_{S}^{D}$ :

Number of segments for demand factors at each time block.

$n_{S}^{p}$ :

Number of segments for factors for generation of type p at each time block.

$pf$ :

System power factor.

$RR^{l}$ :

Capital recovery rate for investment in branches of type l.

$RR^{NT}$ :

Capital recovery rate for investment in new transformers.

$RR^{p}$ :

Capital recovery rate for investment in generators of type p.

$RR^{SS}$ :

Capital recovery rate for investment in substations.

$\underline{V}$ :

Lower bound for nodal voltages.

${\overline{V}}$ :

Upper bound for nodal voltages.

$Z_{k}^{l}$ :

Unitary impedance magnitude for alternative k for branches of type l.

$Z_{k}^{tr}$ :

Impedance magnitude for alternative k for transformers of type tr.

$\Delta_{b}$ :

Duration of time block b.

$\eta^{l}$ :

Lifetime of branches of type l.

$\eta^{NT}$ :

Lifetime of new transformers.

$\eta^{p}$ :

Lifetime of generators of type p.

$\eta^{SS}$ :

Lifetime of substation assets other than transformers.

$\mu_{b}^{D}$ :

Average demand factor of time block b.

$\mu_{sb}^{D}$ :

Average factor for demand in segment s of time block b.

$\mu_{sb}^{p}$ :

Average factor for generation of type p in segment s of time block b.

$\mu_{b}^{D} \left( \omega \right)$ :

Average demand factor of time block b and scenario $\omega$.

$\xi$ :

Penetration limit for distributed generation.

$\pi_{sb}^{D}$ :

Probability of the average factor for demand in segment s of time block b.

$\pi_{sb}^{p}$ :

Probability of the average factor for generation of type p in segment s of time block b.

$\pi_{b} \left( \omega \right)$ :

Probability of scenario $\omega$ of time block b.

1.1.4 Variables

$c_{t}^{E}$ :

Production cost at stage t.

$c_{t}^{I}$ :

Amortized investment cost at stage t.

$c_{t}^{M}$ :

Maintenance cost at stage t.

$c_{t}^{R}$ :

Energy losses cost at stage t.

$c_{t}^{U}$ :

Unserved energy cost at stage t.

$c^{TPV}$ :

Present value of the total cost.

$d_{itb}^{U}$ :

Unserved energy at node i, stage t, and time block b.

$d_{itb}^{U} \left( \omega \right)$ :

Unserved energy at node i, stage t, time block b, and scenario $\omega$.

$f_{ijktb}^{l}$ :

Actual current flow through alternative k for the branch of type l connecting nodes i and j at stage t and time block b.

$f_{ijktb}^{l} \left( \omega \right)$ :

Actual current flow through alternative k for the branch of type l connecting nodes i and j at stage t, time block b, and scenario $\omega$.

$\tilde{f}_{ijkt}^{l}$ :

Fictitious current flow through alternative k for the branch of type l connecting nodes i and j at stage t.

$g_{iktb}^{p}$ :

Current injection at node i for alternative k for the generator of type p at stage t and time block b.

$g_{iktb}^{p} \left( \omega \right)$ :

Current injection at node i for alternative k for the generator of type p at stage t, time block b, and scenario $\omega$.

$g_{iktb}^{tr}$ :

Actual current injection at substation node i for alternative k for the transformer of type tr at stage t and time block b.

$g_{iktb}^{tr} \left( \omega \right)$ :

Actual current injection at substation node i for alternative k for the transformer of type tr at stage t, time block b, and scenario $\omega$.

$\tilde{g}_{it}^{SS}$ :

Fictitious current injection at substation node i and stage t.

$v_{itb}$ :

Voltage magnitude at node i, stage t, and time block b.

$v_{itb} \left( \omega \right)$ :

Voltage magnitude at node i, stage t, time block b, and scenario $\omega$.

$x_{ijkt}^{l}$ :

Binary investment variable for alternative k for the branch of type l connecting nodes i and j at stage t.

$x_{ikt}^{NT}$ :

Binary investment variable for alternative k for the new transformer at substation node i and stage t.

$x_{ikt}^{p}$ :

Binary investment variable for alternative k for the generator of type p at node i and stage t.

$x_{it}^{SS}$ :

Binary investment variable for the substation at node i and stage t.

$y_{ijkt}^{l}$ :

Binary utilization variable for alternative k for the branch of type l connecting nodes i and j at stage t.

$y_{ikt}^{p}$ :

Binary utilization variable for alternative k for the generator of type p at node i and stage t.

$y_{ikt}^{tr}$ :

Binary utilization variable for alternative k for the transformer of type tr at substation node i and stage t.

$\delta_{ijktbh}^{l}$ :

Current in block h of the piecewise linear energy losses for alternative k for the branch of type l connecting nodes i and j at stage t and time block b.

$\delta_{ijktbh}^{l} \left( \omega \right)$ :

Current in block h of the piecewise linear energy losses for alternative k for the branch of type l connecting nodes i and j at stage t, time block b, and scenario $\omega$.

$\delta_{iktbh}^{tr}$ :

Current injection in block h of the piecewise linear energy losses for alternative k for the transformer of type tr at substation node i, stage t, and time block b.

$\delta_{iktbh}^{tr} \left( \omega \right)$ :

Current injection in block h of the piecewise linear energy losses for alternative k for the transformer of type tr at substation node i, stage t, time block b, and scenario $\omega.$