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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.