Multi-Period Distributed Optimal Power Flow

Optimization for Balanced Three-Phase Power Distribution Networks with Renewables and Storage in MATLAB.

Naive Brute Force Multi-Period OPF. A spatially decomposed, temporally brute-forced MPOPF has been implemented.

Objectives currently covered:

• Loss Minimization

Description of the Modelling of the Radial Power Distribution System

Description of State Variables

Variable Notation	Variable Description	Number of Variables	Nature of Constraint
$P^{t}_{ij}$	Real Power flowing in branch	$m$	Nonlinear
$Q^{t}_{ij}$	Reactive Power flowing in branch	$m$	Nonlinear
$l^{t}_{ij}$	Square of Magnitude of branch Current	$m$	Nonlinear
$v^{t}_{j}$	Square of Magnitude of node Voltage	$N$	Nonlinear
$B^{t}_{j}$	Battery State of Charge	$n_{B}$	Linear

Description of Control Variables

Variable Notation	Variable Description	Number of Variables	Nature of Constraint
$q^{t}_{D_j}$	Reactive Power of DER (via inverter)	$n_{D}$	Linear1
$P^{t}_{c_j}$	Charging Power of Battery	$n_{B}$	Linear
$P^{t}_{d_j}$	Discharging Power of Battery	$n_{B}$	Linear
$q^{t}_{B_j}$	Reactive Power of Battery (via inverter)	$n_{B}$	Linear1

Description of Independent Variables

Variable Notation	Variable Description	Number of Variables	Nature of Constraint
$P^{t}_{L_j}$	Real Power Demand	$N$	Linear
$Q^{t}_{L_j}$	Reactive Power Demand	$N$	Linear
$P^{t}_{D_j}$	Real Power of DER	$n_{D}$	Linear1
$B^{0}_{j}$	Battery Initial State of Charge	$n_{B}$	Linear

Miscellaneous Notation

Variable Notation	Variable Description	Cardinality
$\mathbb{N}$	Set of all the nodes	$N$
$\mathbb{L}$	Set containing all the branches	$m$
$\mathbb{D}$	Set containing all the nodes containing DERs. $\mathbb{D} \subset \mathbb{N}$	$n_{D}$
$\mathbb{B}$	Set containing all the nodes containing Batteries. $\mathbb{B} \subset \mathbb{N}$	$n_{B}$
$\mathbb{T}$	Set containing all the time-periods	$T$
$j$	Denotes a node. $j \in \mathbb{N}$
$(i, j)$	Denotes a branch connecting nodes $i$ and $j$. $(i, j) \in \mathbb{L}$
$t$	Denotes a time-period2. $t \in \mathbb{T}$

Notes

1. Current modelling. Future modelling will incorporate reactive power as a non-linear function wrt maximum apparent power and real power. ↩
2. Except when used as a superscript in denoting Battery SOC $B^{t}_j$, $t$ refers to the average value of the variable within the time-period $t$. For Battery SOC, $B^{t}_j$ refers to the value of SOC at the end of time-period $t$. ↩

Related: You may also check out the Greedy Single Time Period Sequential OPF Model repo here. Temporal decomposition will be applied there later, after algorithm development.