Distributed Multi-Period Optimal Power Flow for Demand Response in Microgrids
ANU / NICTA
Paul Scott
Sylvie Thiebaux
Demand Response
- Reduce network peaks
- Balance renewable supply
- Provide network support
Multiagent Problem
Our Focus
A distributed demand response mechanism that incentivises participation and coordinates activity whilst satisfying network constraints.
Challenges and Contribution
In general the power flow equations are non-convex and many typical household loads require discrete decisions.
We compare the performance of a wide range of power flow equations using the distributed ADMM algorithm, and show that in practice the non-convex power flows converge to a near-optimal solution. We show the algorithm can also manage typical household discrete loads.
Optimisation Problem
Minimise total cost of serving power, whilst preserving network and participant constraints.
Receding Horizon Control (MPC)
Model
- \(C:\) set of components (houses, gens, lines, buses)
- \(T:\) set of terminals
- \(L:\) set of connections between terminals
Variables
- \(x_c:\) vector of variables for component \(c\), including any:
- \(y_i := [p_i,q_i,v_i,\theta_i]^\mathsf{T}:\) terminal \(i\) variables
Optimisation Problem
\begin{align} &\min_x\sum_{c \in C} f_c(x_c)\quad\color{red}{\text{costs}}\\ &\text{ s.t. } \forall c \in C: g_c(x_c) \leq 0\quad\color{red}{\text{constraints}}\\ &\phantom{\text{ s.t. }} \forall (i, j) \in L: h(y_i, y_j) = 0\quad\color{red}{\text{connections}} \end{align}
where \(h(y, y') := y + Ay'\)
ADMM
Iterative algorithm that allows the problem to be decomposed across linear constraints. Proven to converge when the problem is convex, and linear constraints take on particular form.
ADMM
- Kim and Baldick 2000
- Kraning et al. 2014
- Peng and Low 2014
Augmented Lagrangian
\begin{align} \mathcal{L}_A(y, y', \lambda, \rho) &:= \sum_{c \in C} f_c(x_c)\\ &+ \sum_{(i, j) \in L} \left[\frac{\rho}{2} \|h(y_i, y'_j)\|_2^2 + \lambda_{i,j}^\mathsf{T} h(y_i, y'_j) \right] \end{align}
Split Into Two
Split Into Two
Split Into Two
ADMM
\begin{align} (1)\quad&\forall c \in \color{red}{C_1}: x_c^{(k)} = \argmin_{x_c} \mathcal{L}_A(y, \color{red}{y^{(k-1)}}, \lambda^{(k-1)}, \rho^{(k)})\label{eq:admm_st}\\ &\quad\text{ s.t. }\quad g_c(x_{c}) \leq 0\\ (2)\quad&\forall c \in \color{red}{C_2}: x_c^{(k)} = \argmin_{x_c} \mathcal{L}_A(\color{red}{y^{(k)}}, y, \lambda^{(k-1)}, \rho^{(k)})\\ &\quad\text{ s.t. }\quad g_c(x_{c}) \leq 0\\ (3)\quad&\forall (i, j) \in L: \lambda_{i,j}^{(k)} = \lambda_{i,j}^{(k-1)} + \rho^{(k)} h(y_i^{(k)}, y_j^{(k)})\label{eq:admm_en} \end{align}
Visualisation of Algorithm
Visualisation of Algorithm
Visualisation of Algorithm
Visualisation of Algorithm
Visualisation of Algorithm
Visualisation of Algorithm
Visualisation of Algorithm
Power Flow Models
| Equations | Variables | Accuracy | |
|---|---|---|---|
| AC | Non-convex | \(p,q,v,\theta\) | Exact |
| QC | SOCP | \(p,q,v,\theta\) | Relaxation |
| DF | SOCP | \(p,q,v\) | Relaxation |
| DC | Linear | \(p,\theta\) | Approximation |
| K | Quadratic | \(p,\theta\) | Approximation |
Convergence (Cold)
Timing (Cold)
- AC: 148s
- QC: 546s
- DF: 110s
- DC: 244s
- K: 15s
Warmstarting
For \(\sigma =\) 20%, relative number of iterations:
- Uncorrelated 11.4% (0.3%)
- Correlated 29% (18%)
Quality
-
QC:
-0.03%1% LB -
DF:
0.04%1% LB - DC: -3.5%
- K: 4.7%
Home Automation
Discrete Shiftable Load
- RP: Relax and Price
- RD: Relax and Decide
- UR: Unrelaxed
Discrete Shiftable Loads
-
RP-0:
0.25%1% -
RP-3:
0.24%1% -
RD:
0.23%1% -
UR:
0.01%1%
Conclusion
Get within 1% of the global optimal. Works with AC power flows. Works with household discrete loads. Only a couple of minutes to converge.