Increase in power demand guarantee: a bilevel approach

(1)

HAL Id: hal-01971748

https://hal.inria.fr/hal-01971748

Submitted on 7 Jan 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Increase in power demand guarantee: a bilevel approach

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

To cite this version:

Mathieu Besançon, Luce Brotcorne, Miguel Anjos, Juan Gomez-Herrera. Increase in power demand

guarantee: a bilevel approach. Optimization days, May 2018, Montréal, Canada. �hal-01971748�

(2)

Increase in power demand guarantee: a bilevel approach

Mathieu Besançon

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

8

^th

May 2018

Polytechnique Montréal - GERAD - INRIA INOCS

(3)

Outline

Introduction

Time-and-Level-of-Use tariff Bilevel formulation

Solution concept Numerical results

Further work and extensions

(4)

Introduction

(5)

Introduction

Changing context for power grids:

I

Integration of Distributed Renewable Sources

I

Easier, automated data exchange

I

Contribution of energy users to better grid operations Multi-objective, bilevel optimization:

I

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

I

Choose trade-off upfront or after building front?

(6)

Problem context, design goals

Context:

I

Generation planning requires information on demand

I

More than forecast, guarantee on consumption

I

Demand Response still not targeting residential users Design goals:

I

Offer users cost reductions for this guarantee on demand

I

Maintain privacy and user-side flexibility by design

I

Minimal information exchange and user actions

(7)

Time-and-Level-of-Use tariff

(8)

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:

I

K, booking fee

I

π

^L

, lower tariff decreasing with capacity

I

π

^H

, higher tariff increasing with capacity

User cost = Booking cost + Expected energy cost

User cost(c) = K · c + Expected energy cost

(9)

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.

(10)

Bilevel formulation

(11)

Decision variables

Supplier chooses a price setting:

I

K ≥ 0

I

π

^L

∈ R

^|π

L|

≥ 0

I

π

^H

∈ R

^|π

H|

≥ 0

π

^L

(0) = π

^H

(0): the baseline Time-of-Use price.

User:

I

Booked capacity for the time frame c ≥ 0

(12)

User and supplier objectives

User minimizes their total expected cost:

min

c

K · c + X

ω∈Ω⁻(c)

π

^L

(c) · p

ω

· x

ω

+ X

ω∈Ω⁺(c)

π

^H

(c) · p

ω

· x

ω

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

L

^F

≡ X

ω∈Ω

p

ω

·x

ω

·π

^H0

−



K · c + X

ω∈Ω⁻(c)

π

^L

(c) · p

ω

· x

ω

+ X

ω∈Ω⁺(c)

π

^H

(c) · p

ω

· x

ω





L

^F

: conflicting user and supplier, min-max problem:

min

K,π^L,π^H

max

c

L

^F

(K, π

^L

, π

^H

, c)

(13)

User and supplier objectives

Also maximizes a guarantee on the consumption.

I

Monotonous increasing

I

Linked to generation or demand min

K,π^L,π^H

L

^G

(c) Guarantee loss function: increasing the predictability.

⇒ forecast still as good, but error compensated by users

(14)

Solution concept

(15)

User cost for one price setting

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Booked capacity 0.25

0.50 0.75 1.00 1.25 1.50 1.75 2.00

Relative price $/ kWh

Price curve below capacity Price curve above capacity

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Booked capacity 0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0

User cost ($)

Cost at 7hr Cost at 8hr Cost at 10hr Cost at 12hr Cost at 14hr

(16)

Elements of bilevel and vector optimization

I

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

I

Bilevel: optimal value transformation [1] to turn the lower level problem

into a set of constraints

(17)

Sub-problem definition

I

Finite set of capacities potential optima

I

Independent of price settings with few conditions

I

These candidates are known from both supplier and user S

c

= {0} ∪ C

^L

∪ {x

i

, i ∈ I } Supplier sub-problem at k-th candidate:

min

K,π^L,π^H

L

^F

(K , π

^L

, π

^H

, c

k

) (1)

s.t. L

^F

(c ) ≥ L

^F

(c ) + δ (c ) ∀m ∈ S \{k} (2)

(18)

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

(19)

Numerical results

(20)

Residential power consumption dataset

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

I

Aggregation per hour

I

Probability discretization using Kernel Density Estimate

0 5 10 15 20

Hour of the day (0-23) 0.4

0.6 0.8 1.0 1.2 1.4 1.6 1.8

Power (kW)

Figure: Average consumption

0 1 2 3 4 5

Hourly consumption (kW*h) 0.0

0.2 0.4 0.6 0.8 1.0

Density

Discrete levels KDE fX(x)

Figure: Discretization at 7AM

(21)

Numerical resolution: Pareto-optimal solutions

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

0 1 2 3 4 5

c (kW.h) 0.5

1.0 1.5 2.0 2.5 3.0

($/kW.h)

Tariff curves c18= 4.51kW. h K = 0.1531$/kW.h

booked capacity

0 1 2 3 4 5

capacity (kW.h) 1.8

2.0 2.2 2.4 2.6

cost ($)

User utility c18= 4.51kW. h candidates optimum equilibrium limit

2.5 3.0

Tariff curves c15= 3.9kW. h K = 0.1964$/kW.h

booked capacity 2.4

2.6 User utility c15= 3.9kW. h candidates optimum equilibrium limit

(22)

Resolution: influence of δ and β

0 1 2 3 4 5

c (kW.h) 0.000

0.001 0.002 0.003 0.004 0.005 0.006 0.007

LF ($)

Pareto-optimum candidate Utopia point

Figure: Discrete Pareto front and sub-problem candidates

0 1 2 3 4

c (kW h) 0.000

0.002 0.004 0.006 0.008 0.010

LF ($)

= 0.0005, = 0.0

= 0.001, = 0.0

= 0.0, = 0.01

= 0.0001, = 0.005

= 0.001, = 0.005

Figure: Pareto front variation with (δ, β)

(23)

Further work and extensions

(24)

Further work and extensions

I

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

I

Load shifting & reduction possibility for users

I

Match L

^G

to generation costs

I

Handle continuous load probability distributions

(25)

Conclusion

(26)

Conclusion

I

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

I

Simple framework with minimal user action required

I

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

I

For supplier at higher level: guarantees when planning Unit Commitment

I

Forecasts can be wrong, but users contribute

(27)

References

(28)