Distributed Residential Demand Response

ANU / NICTA

Paul Scott

Sylvie Thiebaux

Demand Response

Reduce network peaks
Balance renewable supply
Provide network support

Multiagent Problem

Home Automation

Retail Prices

Our Focus

A demand response mechanism that incentivises participation and coordinates activity whilst satisfying network constraints.

Contribution

We demonstrate that a particular distributed demand response algorithm can produce near-optimal results even with the non-convex power flows. Discrete loads typically found within households can also be managed by the algorithm. This brings the approach closer to where it can be deployed in a real system.

Optimisation Problem

Minimise total cost of serving power, whilst preserving network and participant constraints.

Model

\(C:\) set of components (houses, gens, lines, buses)
\(T:\) set of terminals
\(L:\) set of connections between terminals

Model

\(y_i = [p,q,v,\theta]^\mathsf{T}:\) variables for terminal \(i\)
\(x_c:\) variables for component \(c\) (inc. terminal vars)

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.

Split Into Two

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}

ADMM

\begin{align} (1):&~x_1^{(k)} = \argmin_{x_1} \mathcal{L}_A(y, \color{red}{y^{(k-1)}}, \lambda^{(k-1)}, \rho^k)\label{eq:admm_st}\\ &\quad\text{ s.t. } \forall c \in \color{red}{C_1}: g_c(x_{c}) \leq 0\\ (2):&~x_2^{(k)} = \argmin_{x_2} \mathcal{L}_A(\color{red}{y^{(k)}}, y, \lambda^{(k-1)}, \rho^k)\\ &\quad\text{ s.t. } \forall c \in \color{red}{C_2}: g_c(x_{c}) \leq 0\\ (3):&~\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}

House/Generator Payments

\(\lambda_pp + \lambda_qq\)

Visualisation of Algorithm

Receding Horizon Control (MPC)

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%

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.