Computational Optimization and Applications

, Volume 71, Issue 2, pp 553–608

# Mathematical programming methods for microgrid design and operations: a survey on deterministic and stochastic approaches

• Guanglei Wang
• Hassan Hijazi
Article

## Abstract

Planning and operating a power grid is a nontrivial exercise due to conflicting objectives, nonlinear constraints and uncertainties at multiple decision levels. Considerable research work has been dedicated to independently solve different aspects of the overall problem. This survey provides a detailed review of state-of-the-art techniques in mathematical optimization trying to address challenges in this area. We also provide a set of open problems and research perspectives.

## Keywords

Global optimization Convex relaxation Uncertainty Power systems

## Sets

$$\mathbb {N}$$

Set of natural numbers

$$\mathbb {R}$$

Set of real numbers

$$\mathbb {C}$$

Set of complex numbers

N

Set of buses in the network

$$N_S$$

Bus set of distributed storage units

$$N_T$$

Bus set of transformers

$$N_G$$

Bus set of dispatchable generators

$$N_R$$

Set of buses having renewable generators connected to them

$$N_F$$

Set of fault buses

E

Set of lines (ij) in a power network network where i is the from node

$$E^R$$

Set of lines (ij) in a power network network where i is the to node

$$\mathcal {T}$$

Time horizon (e.g., 24 h).

## Power flow parameters

$$\varvec{Z} = r+ \varvec{j}x$$

Line impedance, where j is the imaginary unit.

$$\varvec{Y} = g+ \varvec{j}b$$

Line admittance, $$g = \frac{r}{r^2 + x^2}, b = \frac{-x}{r^2 + x^2}$$

$$\varvec{b}^c$$

Line charging

$$\varvec{T} = \tau \angle \theta ^{s}$$

Transformer whose tap ratio has magnitude $$\tau$$ and phase shifter angle $$\theta ^{s}$$

$$\varvec{\theta }^M$$

Phase angle difference limit

$$\varvec{Y^{s}} = g^s + \varvec{j} b^s$$

$$\varvec{S^d} = p^d + \varvec{j} q^d$$

AC power demand

$$\varvec{c}_0, \varvec{c}_1,\varvec{c}_2$$

Generation cost coefficients

$$\mathfrak {R}(\cdot )$$

Real component of a complex number

$$\mathfrak {I}(\cdot )$$

Imaginary component of a complex number

$$(\cdot )^*$$

Conjugate of a complex number

$$|\cdot |$$

Magnitude of a complex number, $$l^2$$- norm

$$\angle$$

Angle of a complex number

$$x^u$$

Upper bound of variable x

$$x^l$$

Lower bound of variable x

$$\mathbf {x}$$

A constant value

## Scheduling parameters

$$\varvec{p}^r_t$$

Global real power reserve requirement at time $$t \in \mathcal {T}$$

$$\varvec{u}^{su}_i, \varvec{u}^{sd}_i$$

Startup and shutdown cost of unit $$i\in N_G$$ at time t

$$\varvec{\tau }^{tu}, \varvec{\tau }^{td}$$

Minimum up and minimum down time

## Storage parameters

$$\varvec{\rho }^c_i, \varvec{\rho }^d_i$$

Charging and discharging efficiencies for a storage unit i

## Facility location parameters

$$\varvec{\kappa }_i$$

Capacity of facility constructed at node i

$$\varvec{c}^f_i$$

Cost of constructing a facility at node i

$$\varvec{c}^l_{ij}$$

Cost of constructing link $$(i, j) \in E$$

$$\varvec{c}^s_{ij}$$

Unit shipping cost on link $$(i, j)\in E$$

$$\varvec{d}_i$$

Demand at node i

## Power flow variables

I

AC current

$$V_i = e_i + \varvec{j} f_i$$

AC voltage at bus $$i \in N$$ in rectangular form

$$V = {\left| {V}\right| } \angle \theta$$

AC voltage in polar form

W

Product of two AC voltages

$$S^g = p^g + \varvec{j} q^g$$

AC power generation

## Switching and restoration variables

$$\lambda _{ij} \in \left\{ 0, 1\right\}$$

1 if the state of line (ij) is on; 0 otherwise

$$\beta _{i} \in \left\{ 0, 1\right\}$$

1 if bus i is fed; 0 otherwise

## Storage sizing variables

$$S^s = p^s + \varvec{j} q^s$$

AC power injection at storage device.

$$\gamma _i$$

Capacity of power rating for storage unit i to be determined

$$\eta _i$$

Capacity of energy for storage unit i to be determined

$${\varDelta }_{it}$$

Storage change at time $$t \in \mathcal {T}$$

## Scheduling variables

$$\delta ^{su}_{it}, ~\delta ^{sd}_{it}~\in \left\{ 0, 1\right\}$$

Startup or shutdown generator i at time $$t\in \mathcal {T}$$

$$\sigma ^g_{i} \in \left\{ 0, 1\right\}$$

1 if the state of generator i is on; 0 otherwise

$$p_{ijt}, q_{ijt}$$

Active, reactive power flow of branch (ij) at hour t

$$l_{ijt}$$

The square of current magnitude of branch (ij) at hour t

$$p^{ga}_{it}$$

Maximum real power available from generator i at time $$t \in \mathcal {T}$$

## Facility location variables

$$y_{i} \in \left\{ 0, 1\right\}$$

1 if a facility is located at bus i, 0 otherwise

$$z_{ij} \in \left\{ 0, 1\right\}$$

1 if line (ij) is constructed; 0 otherwise

$$f_{ij}$$

$$p_i$$

Total production of the facility at node i

## Notes

### Acknowledgements

The authors are grateful to the editors and anonymous referees for their helpful suggestions that greatly improved the quality of the paper. This research was partly funded by the Australia Indonesia Centre.

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