# Increase in power demand guarantee: a bilevel approach

Mathieu Besançon, Luce Brotcorne, Miguel F Anjos, Juan A Gomez-Herrera

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Mathieu Besançon, Luce Brotcorne, Miguel F Anjos, Juan A Gomez-Herrera. Increase in power demand guarantee: a bilevel approach. Optimization days, May 2018, Montréal, Canada. . hal-01971748

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POLYTECHNIQUE
MONTRÉAL

GERAD

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# Increase in power demand guarantee: a bilevel approach

Mathieu Besançon

Supervisors: Luce Brotcorne & Miguel F. Anjos
Juan A. Gomez-Herrera

8th May 2018

Polytechnique Montréal - GERAD - INRIA INOCS
# Outline

## Introduction

## Time-and-Level-of-Use tariff

## Bilevel formulation

## Solution concept

## Numerical results

## Further work and extensions

## Conclusion

## References
Introduction
# Introduction

Changing context for power grids:

* Integration of Distributed Renewable Sources

* Easier, automated data exchange

* Contribution of energy users to better grid operations

Multi-objective, bilevel optimization:

* NP-hard problem to solve or prove optimality in the general case [1]

* Choose trade-off upfront or after building front?
# Problem context, design goals

## Context:

* Generation planning requires information on demand

* More than forecast, **guarantee** on consumption

* Demand Response still not targeting residential users

## Design goals:

* Offer users cost reductions for this guarantee on demand

* Maintain privacy and user-side flexibility by design

* Minimal information exchange and user actions
Time-and-Level-of-Use tariff
Time-and-Level-of-Use tariff

Introduced in Gomez-Herrera & Anjos 2017 [2]

A capacity is booked by the user before consumption

Policy described by:

▶ K, booking fee

▶ $\pi^L$, lower tariff decreasing with capacity

▶ $\pi^H$, higher tariff increasing with capacity

$$ \text{User cost} = \text{Booking cost} + \text{Expected energy cost} $$

$$ \text{User cost}(c) = K \cdot c + \text{Expected energy cost} $$
TLOU operational steps

1. Supplier sends pricing

2. Consumers book a capacity

3. After consumption, total cost is computed and billed

Done for each time frame, at least day-ahead in the base case considered.
Bilevel formulation
Decision variables

Supplier chooses a price setting:

▶ $K \ge 0$

▶ $\pi^L \in \mathbb{R}^{|\pi^L|} \ge 0$

▶ $\pi^H \in \mathbb{R}^{|\pi^H|} \ge 0$

$\pi^L(0) = \pi^H(0)$: the baseline Time-of-Use price.

User:

▶ Booked capacity for the time frame $c \ge 0$
User and supplier objectives

User minimizes their total expected cost:

$$
\min_{c} K \cdot c + \sum_{\omega \in \Omega^{-}(c)} \pi^{L}(c) \cdot p_{\omega} \cdot x_{\omega} + \sum_{\omega \in \Omega^{+}(c)} \pi^{H}(c) \cdot p_{\omega} \cdot x_{\omega}
$$

Supplier minimizes the difference in profit between the baseline and current
solution:

$$
L^F \equiv \sum_{\omega \in \Omega} p_{\omega} \cdot x_{\omega} \cdot \pi_0^H - \left( K \cdot c + \sum_{\omega \in \Omega^{-}(c)} \pi^L(c) \cdot p_{\omega} \cdot x_{\omega} + \sum_{\omega \in \Omega^{+}(c)} \pi^H(c) \cdot p_{\omega} \cdot x_{\omega} \right)
$$

$L^F$: conflicting user and supplier, min-max problem:

$$
\min_{K, \pi^L, \pi^H} \max_c L^F(K, \pi^L, \pi^H, c)
$$
User and supplier objectives

Also maximizes a guarantee on the consumption.

▶ Monotonous increasing

▶ Linked to generation or demand

$$
\min_{K, \pi, L, \pi_H} L^G(c)
$$

Guarantee loss function: increasing the predictability.

⇒ forecast still as good, but error compensated by users
Solution concept
User cost for one price setting

<figure><img src="image_5.png" /><figcaption>Figure: Higher and lower tariff curves</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure: User cost vs booked capacity</figcaption></figure>
Elements of bilevel and vector optimization

▶ Multi-objective optimization: ε-constraint to obtain the Pareto set [4]

▶ Bilevel: optimal value transformation [1] to turn the lower level problem into a set of constraints
Sub-problem definition

► Finite set of capacities potential optima

► Independent of price settings with few conditions

► These candidates are known from both supplier and user

$$
S_c = \{0\} \cup C^L \cup \{x_i, i \in I\}
$$

Supplier sub-problem at k-th candidate:

$$
\min_{K, \pi^L, \pi^H} L^F(K, \pi^L, \pi^H, c_k) \qquad (1)
$$

$$
\text{s.t.} \quad L^F(c_k) \ge L^F(c_m) + \delta_u(c_k) \quad \forall m \in S_c \setminus \{k\} \qquad (2)
$$
Building the Pareto front

$L^G$ monotonous decreasing with c

1. Starting from highest possible candidate

2. Compute $L^F, L^G$

3. If $L^F$ not better than all previous points → dominated point

4. Continue for all points
Numerical results
# Residential power consumption dataset

Study on 47-month household consumption measurement data [3]

* Aggregation per hour

* Probability discretization using Kernel Density Estimate

<figure><img src="image_7.png" /><figcaption>Figure: Average consumption</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure: Discretization at 7AM</figcaption></figure>
# Numerical resolution: Pareto-optimal solutions

Resolution using Julia, JuMP and the Coin-OR LP solver Pareto-optimal solutions examples:

<figure><img src="image_9.png" /></figure>
Resolution: influence of δ and β

<figure><img src="image_10.png" /><figcaption>Figure: Discrete Pareto front and
sub-problem candidates</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure: Pareto front variation with (δ, β)</figcaption></figure>
Further work and extensions
Further work and extensions

* Booked capacity as maximal power in the time frame instead of energy.

* Load shifting & reduction possibility for users

* Match $L^G$ to generation costs

* Handle continuous load probability distributions
Conclusion
Conclusion

▶ Reduction of a bilevel, discontinuous, multi-objective problem into a set of LPs

▶ Simple framework with minimal user action required

▶ User always have to the option to opt-out for any time-frame (c = 0)

▶ For supplier at higher level: guarantees when planning Unit Commitment

▶ Forecasts can be wrong, but users contribute
References
References I

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Stephan Dempe, Vyacheslav Kalashnikov, Gerardo A. Prez-Valds, and
Nataliya Kalashnykova.

*Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks.*

Springer Publishing Company, Incorporated, 2015.

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Juan A. Gomez-Herrera and Miguel F. Anjos.

Optimization-based estimation of power capacity profiles for activity-based residential loads.

*International Journal of Electrical Power & Energy Systems, 104:664–672, jan 2019.*

<figure><img src="image_14.png" /></figure>

Georges Hébrail and Alice Bérard.

Individual household electric power consumption data set.

<figure><img src="image_15.png" /></figure>

Margaret M. Wiecek, Matthias Ehrgott, and Alexander Engau.

*Continuous Multiobjective Programming*, pages 739–815.

Springer New York, New York, NY, 2016.
Model formulation

$$
\min_{K, \pi^{L}, \pi^{H}} (L^{F}, L^{G}) \qquad (3)
$$

$$
L^G(c) = P_{\Omega}[X > \hat{c}] \quad (4)
$$

$$
(\pi^L, \pi^H) \in \Phi \qquad (5)
$$

$$
\hat{c} \in \arg \max_{c} L^{F}(c, K, \pi^{L}, \pi^{H}) \qquad (6)
$$