Uncertainty-aware Flexibility Envelope Prediction in Buildings with Controller-agnostic Battery Models Paul Scharnhorst12 Baptiste Schubnel1 Rafael E. Carrillo1 Pierre-Jean Alet1and Colin N. Jones2

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Uncertainty-aware Flexibility Envelope Prediction in Buildings with

Controller-agnostic Battery Models

Paul Scharnhorst1,2,∗, Baptiste Schubnel1, Rafael E. Carrillo1, Pierre-Jean Alet1and Colin N. Jones2

Abstract— Buildings are a promising source of ﬂexibility for

the application of demand response. In this work, we introduce

a novel battery model formulation to capture the state evolution

of a single building. Being fully data-driven, the battery model

identiﬁcation requires one dataset from a period of nominal

controller operation, and one from a period with ﬂexibility

requests, without making any assumptions on the underlying

controller structure. We consider parameter uncertainty in the

model formulation and show how to use risk measures to encode

risk preferences of the user in robust uncertainty sets. Finally,

we demonstrate the uncertainty-aware prediction of ﬂexibility

envelopes for a building simulation model from the Python

library Energym.

I. INTRODUCTION

Electriﬁcation, along with the increase of renewable en-

ergy production, serves as a means to mitigate climate

change [1]. However, increasing energy demand, together

with the intermittent behavior of renewable energy sources

like solar or wind power, creates new challenges for grid

operators in maintaining a balanced grid. A key tool to

achieve this balancing, is Demand Response (DR) [2], a way

for prosumers to adapt their consumption to available energy

generation.

Various studies have been undertaken on the topic of DR,

from reviews on DR deﬁnitions and metrics [3], to socio-

economic surveys on the acceptance of DR schemes [4], to

research on the probability and extent of consumer reaction

to price signals [5]. In this work, we will focus on direct

DR, a setting where participating prosumers are remunerated

for following explicit consumption signals from the grid

operator, while not providing ﬂexibility in a prespeciﬁed

range leads to penalties. This is different from indirect DR,

where participants are incentivized to adapt their consump-

tion behavior through varying price signals.

Buildings have been identiﬁed as promising assets to

provide ﬂexibility (e.g. [6]). Equipped with e.g. heat pumps,

photovoltaic systems, and electric vehicles, buildings have

the potential to signiﬁcantly change their consumption pat-

terns in the short term. An exact estimation of the ﬂexibility

is necessary to efﬁciently use the buildings’ capabilities for

DR.

This work received support from CSEM’s Data Program and the Swiss

National Science Foundation under the RISK project (Risk Aware Data

Driven Demand Response, grant number 200021 175627).

1Paul Scharnhorst, Baptiste Schubnel, Rafael E. Carrillo,

and Pierre-Jean Alet are with CSEM S.A., 2002 Neuchˆ

atel,

Switzerland, Email: {paul.scharnhorst, baptiste.schubnel,

rafael.carrillo, pierre-jean.alet}@csem.ch

2Paul Scharnhorst and Colin N. Jones are with LA, EPFL, 1015 Lau-

sanne, Switzerland, Email: colin.jones@epfl.ch

∗Corresponding author

We will focus on the single building ﬂexibility estimation

problem in this work. As a tool for this estimation, we will

use the concept of ﬂexibility envelopes. Flexibility envelopes,

as in [7], provide availability time predictions of power level

changes from a given baseline, depending on the time of

the day. Reference [8] uses ﬂexibility envelopes without

considering a baseline, by reporting availability times for dis-

cretized power levels. While being able to adapt the control

objectives, the approach relies on modeling the building and

its systems. A data-driven way of learning and approximating

ﬂexibility envelopes is presented in [9]. This approach is

capable of reducing the computational burden, but a training

set of precomputed ﬂexibility envelopes is needed.

Due to the potentially difﬁcult and complex modeling

of buildings and their equipment, virtual battery models

have been determined as an efﬁcient tool to model the

thermal capacity of a building. Reference [10] provides a set

description of the feasible power consumption proﬁles, with

the use of a data-driven battery model, for heating systems

with on/off behavior. Input tracking for constrained linear

discrete-time systems is considered in [11], with the aim of

certifying input trackability for a given reference set. In a

DR application, a ﬁxed-shape reference set is parameterized

by a battery model and optimized over to offer cost-optimal

ﬂexibility to the grid operator. Further usage of battery

models for aggregations of thermostatically controlled loads

is shown in e.g. [12].

Our contributions are threefold. Firstly, we introduce a

data-driven battery model for representing the state of a

building, without assuming a ﬁxed controller structure and

present a method for its parameter identiﬁcation. Secondly,

parameter uncertainty is considered for the resulting set of

feasible trajectories. Risk measures are used to formulate

robust uncertainty sets that take risk preferences of the user

into account. Thirdly, we use the battery model to compute

ﬂexibility envelopes for a simulated building from the Ener-

gym library [13] and compare the results for different risk

levels.

Notation: We denote the indicator function of the set {0}

by χ, so χq=(1,if q= 0

0,if q6= 0 . We use bold symbols for

vectors and trajectories, with trajectories written as r0:k−1=

[r0, . . . , rk−1]>, or simply rif the context is clear. The N-

dimensional probability simplex is given by ∆N={q∈

RN:qi≥0, i = 1, . . . , N, PN

i=1 qi= 1}.0and 1denote a

vector of appropriate size of zeros or ones, respectively. We

assume empty sums to be 0and R0={0}.

arXiv:2210.03604v1 [eess.SY] 7 Oct 2022

II. LEARNING BATTERY MODELS

A. Model Formulation

We assume that the state of a building at time t, in terms of

its thermal capacity, can be described by a scalar st, bounded

by smin and smax (w.l.o.g. smin = 0 and smax = 1). The

state is a measure of the energy stored in the system, and

its bounds depend on the thermal bounds which are related

to comfort or operational constraints in the building. This

means that st= 0 indicates that no energy can be extracted

from the building without violating constraints, and st= 1

indicates that no energy can be inserted. A possible deﬁnition

of this abstract state is given in Section IV-B.

We now proceed to the derivation of a battery like state

equation for the state st. In full generality, the state evolution

can be represented by the following difference equation:

st+1 −st=h(st,et, pt) + ωt,(1)

with the external weather conditions denoted by et, which

can comprise multiple measurements and past data, the

power injected pt, and a noise term ωt. The noise term ac-

counts for random disturbances in the system, e.g. occupants.

We assume that the controller operating the system leads

to the state following a speciﬁc pattern, here called “nominal

state” and denoted by sn,t. The evolution of this nominal state

can similarly be expressed as

sn,t+1 −sn,t =h(sn,t,et, pn,t) + ωt,(2)

with the baseline power injected given by pn,t.

Prediction of this baseline power is not the focus of this

work. An overview of data-driven methods to predict energy

consumption in buildings can be found in [14]. To intro-

duce uncertainty quantiﬁcation for consumption prediction,

methods like Gaussian Process (GP) regression [15], kernel

methods with error quantiﬁcation [16], or variational autoen-

coders [17] can be used. We will therefore assume to have

reasonably accurate baseline predictions where necessary.

Considering the difference of (1) and (2), we get

st+1 −st=sn,t+1 −sn,t +h(st,et, pt)−h(sn,t,et, pn,t).(3)

The goal of our work is to quantify the system behavior,

and therefore the evolution of st, in cases where the nominal

controller actions are augmented by speciﬁc requests.

Deﬁnition 1 (Relative Consumption Request): Given a

baseline power pn,t ∈R, we deﬁne a relative request as

rt∈Rsuch that the desired overall consumption of the

building at time tis pt=pn,t +rt.

We consider systems with controllers that drive the state

back to its nominal value after receiving requests (as typically

observed in thermal assets), therefore, we get two distinct

phases in the system operation:

1) The request phase where pt=pn,t +rt.

2) The recovery phase where stis driven towards sn,t, with

the injected power denoted by pcon,t.

Moreover, in the request phase, we can distinguish between

receiving positive or negative relative consumption requests,

due to equipment or controller characteristics.

We make the following assumption about the controller.

Assumption 1: For each st∈[0,1], the controller is able

to satisfy the comfort/operational constraints for all t0> t.

When receiving ﬂexibility requests, the controller follows

them as closely as possible, without violating constraints.

Furthermore, we assume to either receive state measurements

from the controller, or measurements from which we can

construct a state-like variable.

Note that Assumption 1 does not impose a ﬁxed controller,

and therefore, is very general in its application. The assump-

tion on fulﬁlling constraints is more an assumption on the

equipment than the controller since any decent controller

should be able to fulﬁll constraints with enough controlla-

bility. Lastly, not relying on a ﬁxed state deﬁnition further

increases generality, while still having the option to construct

a state from standard measurements (see Section IV-B).

Through Assumption 1, we have that the overall state

dynamics behave like a switched system, distinguishing the

cases where pt=pn,t +rtand pt=pcon,t. Assuming a

linear approximation of h, for simplicity, around the nominal

operation point, we get that

h(st,et, pt)−h(sn,t,et, pn,t)

≈









a+rtif rt>0

a−rtif rt<0

bf(st−sn,t)if rt= 0

.(4)

Due to the stochasticity of st, notice that a+,a−and bfare

in general stochastic.

We make a few further assumptions on the nominal state

evolution and the coefﬁcients a+,a−, and bfthat will ease

the rest of the analysis.

Assumption 2: In (4), we assume that

(a) The request-free nominal state evolution sn,t can be

well approximated by a function f:Rm→Rof

the current and recent past weather variables, denoted

hereafter by et:= [e>

1,t, ..., e>

n,t]>∈Rm,ei,t ∈Rη, i =

1, . . . , n, m =nη,

(b) bf∈Ris a constant,

probability space.

Assumption 2a) states that the request-free state evolution

can be well-captured by a deterministic function that only

depends on past and current weather variables. ndenotes the

number of measured variables, and ηdenotes the number of

considered time steps. Despite being strong, this modeling

assumption for thermal systems (in particular building assets)

often leads to good results in practice because errors do

not accumulate. Note that this assumption could be replaced

by modeling the nominal state with a GP instead to take

uncertainty into account, at the price of complicating fur-

ther the analysis. Assumption 2b) is justiﬁed by the fact

that the coefﬁcient bfhas little inﬂuence on the ﬂexibility

quantiﬁcation discussed here, see Sections II-B and IV. For a

reasonable choice of bfsee Section II-B. Finally, Assumption

2c) is useful to extract the distributions of a+and a−directly

from data. The random variable assumption also captures

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Uncertainty-awareFlexibilityEnvelopePredictioninBuildingswithController-agnosticBatteryModelsPaulScharnhorst1;2;,BaptisteSchubnel1,RafaelE.Carrillo1,Pierre-JeanAlet1andColinN.Jones2AbstractBuildingsareapromisingsourceofexibilityfortheapplicationofdemandresponse.Inthiswork,weintroduceanovelbattery...

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Uncertainty-aware Flexibility Envelope Prediction in Buildings with Controller-agnostic Battery Models Paul Scharnhorst12 Baptiste Schubnel1 Rafael E. Carrillo1 Pierre-Jean Alet1and Colin N. Jones2

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