A Constraint on Chern Classes of Strictly Pseudoconvex CR Manifolds

A Constraint on Chern Classes
of Strictly Pseudoconvex CR Manifolds

Yuya TAKEUCHI

Y. Takeuchi

Department of Mathematics, Graduate School of Science, Osaka University,
1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan yu-
<EMAIL_ADDRESS><EMAIL_ADDRESS>https://sites.google.com/view/yuya-takeuchi-english/home

Received July 25, 2019, in final form January 18, 2020; Published online
January 21, 2020

This short paper gives a constraint on Chern classes of closed strictly
pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable
contact manifolds) of dimension at least five. We also see that our result is
“optimal” through some examples.

strictly pseudoconvex CR manifold; holomorphically fillable; Chern class

32V15; 32T15; 32V05; 57R17

##  1 Introduction

Let $\big{(}M,T^{1,0}M\big{)}$ be a closed connected strictly pseudoconvex CR
manifold of dimension $2n+1\allowbreak\geq 5$. It is known that $M$ can be
realized as the boundary of a strictly pseudoconvex domain in a complex
manifold [1, 6, 8]. This fact gives some restrictions to its topology. For
example, Bungart [2] has proved that the cup product

$\displaystyle H^{i_{1}}(M,\mathbb{C})\otimes\dots\otimes
H^{i_{m}}(M,\mathbb{C})\to H^{\absolutevalue{I}}(M,\mathbb{C})$

vanishes for any multi-index $I=(i_{1},\dots,i_{m})\in\mathbb{N}_{+}^{m}$
satisfying $i_{l}\leq n-1$ and $\absolutevalue{I}=i_{1}+\dots+i_{m}\geq n+2$;
see also [12]. As noted in the last paragraph of [2], this result follows also
from a result of $L^{2}$ Hodge theory on strictly pseudoconvex domains by
Ohsawa [10, 11].

In this paper, we will point out that Ohsawa’s result gives also a constraint
on Chern classes of strictly pseudoconvex CR manifolds. For a complex vector
bundle $E$ and a multi-index $K=(k_{1},\dots,k_{m})\in\mathbb{N}_{+}^{m}$, we
denote by $c_{K}(E)$ the cohomology class $c_{k_{1}}(E)\dotsm c_{k_{m}}(E)$.

######  Theorem 1.1.

Let $\big{(}M,T^{1,0}M\big{)}$ be a closed strictly pseudoconvex CR manifold
of dimension $2n+1\geq 5$. Then $c_{K}\big{(}T^{1,0}M\big{)}$ vanishes in
$H^{2\absolutevalue{K}}(M,\mathbb{C})$ for any multi-index $K$ with
$2\absolutevalue{K}\geq n+2$.

Note that sharper results than Theorem 1.1 hold for some particular cases. For
instance, Theorem 1.1 holds for $\mathbb{Z}$-coefficients if $M$ can be
realized as a real hypersurface in a Stein manifold. On the other hand, we can
relax the degree condition to $2\absolutevalue{K}\geq n+1$ if $M$ is the link
of an isolated singularity; see [7] for example. However, we will see in
Section 4 that Theorem 1.1 is “optimal” for a general closed strictly
pseudoconvex CR manifold. More precisely, we will give an example of a closed
strictly pseudoconvex CR manifold $\big{(}M,T^{1,0}M\big{)}$ of dimension
$2n+1$ with $c_{K}\big{(}T^{1,0}M\big{)}\neq 0$ in
$H^{2\absolutevalue{K}}(M,\mathbb{Z})$ for any multi-index $K$ with
$1\leq\absolutevalue{K}\leq n$ (Proposition 4.1). We will also construct a
$(2n+1)$-dimensional closed strictly pseudoconvex CR manifold
$\big{(}M,T^{1,0}M\big{)}$ with $c_{K}\big{(}T^{1,0}M\big{)}\neq 0$ in
$H^{2\absolutevalue{K}}(M,\mathbb{C})$ for any multi-index $K$ with $1\leq
2\absolutevalue{K}\leq n+1$ (Proposition 4.2).

Theorem 1.1 in particular implies that $c_{2}\big{(}T^{1,0}M\big{)}=0$ in
$H^{4}(M,\mathbb{C})$ for any closed strictly pseudoconvex CR manifold
$\big{(}M,T^{1,0}M\big{)}$ of dimension five. Hence the assumption on the
second Chern class in [4, Proposition 8.8] automatically holds for the
strictly pseudoconvex case, which is compatible with [9, Theorem 4.6]. It
would be an interesting problem whether Theorem 1.1 remains true for non-
degenerate CR manifolds or non-compact strictly pseudoconvex CR manifolds.

We also remark a relation between Theorem 1.1 and contact topology. Let $M$ be
a closed oriented $(2n+1)$-dimensional manifold with a positive co-oriented
contact structure $\xi$. Then we can define the $k$-th Chern class
$c_{k}(\xi)\in H^{2k}(M,\mathbb{Z})$ of $\xi$ by using an adapted almost
complex structure on $\xi$. A contact structure $\xi$ is said to be
_holomorphically fillable_ if $M$ can be realized as the boundary of a
strictly pseudoconvex domain and $\xi=\real T^{1,0}M$ holds. In this case,
$c_{k}(\xi)$ coincides with $c_{k}\big{(}T^{1,0}M\big{)}$. Therefore Theorem
1.1 implies the following

######  Corollary 1.2.

Let $(M,\xi)$ be a holomorphically fillable contact manifold of dimension
$2n+1\geq 5$. Then $c_{K}(\xi)$ is a torsion class in
$H^{2\absolutevalue{K}}(M,\mathbb{Z})$ for any multi-index $K$ satisfying
$2\absolutevalue{K}\geq n+2$.

This result is no longer true for a general contact structure. As far as the
author knows, such a constraint on Chern classes has not been obtained yet for
a contact structure satisfying other fillability conditions.

This paper is organized as follows. In Section 2, we recall some basic notions
related to strictly pseudoconvex domains and CR manifolds. Section 3 provides
a proof of Theorem 1.1. In Section 4, we will see that, by constructing some
examples, Theorem 1.1 does not hold for $\mathbb{Z}$-coefficients in general,
and the degree condition is optimal.

##  2 Preliminaries

Let $X$ be a complex manifold of complex dimension $n+1$. A real-valued smooth
function $\varphi$ on $X$ is called an _exhaustion function_ if
$\varphi^{-1}((-\infty,c])$ is compact in $X$ for any $c\in\mathbb{R}$. We say
$\varphi$ to be _strictly plurisubharmonic_ if its complex Hessian
$\pqty{\partialderivative*{\varphi}{z^{i}}{\overline{z}^{j}}}$ is a positive-
definite Hermitian matrix for any holomorphic local coordinate
$\big{(}z^{1},\dots,z^{n+1}\big{)}$. A _Stein manifold_ is a complex manifold
admitting a strictly plurisubharmonic exhaustion function.

Let $\Omega$ be a relatively compact domain in $X$ with smooth boundary
$M=\partial\Omega$. Then there exists a smooth function $\rho$ on $X$ such
that

$\displaystyle\Omega=\rho^{-1}((-\infty,0)),\qquad M=\rho^{-1}(0),\qquad{\rm
d}\rho\neq 0\ \text{on}\ M;$

such a $\rho$ is called a _defining function_ of $\Omega$. A domain $\Omega$
is said to be _strictly pseudoconvex_ if we can take a defining function
$\rho$ of $\Omega$ that is strictly plurisubharmonic near $M$. We call
$\Omega$ a _Stein domain_ if $\Omega$ admits a defining function $\rho$ that
is strictly plurisubharmonic on a neighborhood of the closure of $\Omega$.
Note that a Stein domain is a Stein manifold; this is because $-1/\rho$
defines a strictly plurisubharmonic exhaustion function on $\Omega$.

We next recall some notions related to CR manifolds. Let $M$ be a
$(2n+1)$-dimensional manifold without boundary. A _CR structure_ is an
$n$-dimensional complex subbundle $T^{1,0}M$ of the complexified tangent
bundle $TM\otimes\mathbb{C}$ satisfying the following two conditions:

$\displaystyle
T^{1,0}M\cap\overline{T^{1,0}M}=0,\qquad\big{[}\Gamma\big{(}T^{1,0}M\big{)},\Gamma\big{(}T^{1,0}M\big{)}\big{]}\subset\Gamma\big{(}T^{1,0}M\big{)}.$

A typical example of a CR manifold is a real hypersurface $M$ in a complex
manifold $X$; it has the natural CR structure

$\displaystyle T^{1,0}M=T^{1,0}X|_{M}\cap(TM\otimes\mathbb{C}).$

A CR structure $T^{1,0}M$ is said to be _strictly pseudoconvex_ if there
exists a nowhere-vanishing real one-form $\theta$ on $M$ such that $\theta$
annihilates $T^{1,0}M$ and

$\displaystyle-\sqrt{-1}{\rm d}\theta\big{(}Z,\overline{Z}\big{)}>0,\qquad
0\neq Z\in T^{1,0}M;$

we call such a one-form a _contact form_. The natural CR structure on the
boundary of a strictly pseudoconvex domain is strictly pseudoconvex.
Conversely, any closed connected strictly pseudoconvex CR manifold of
dimension at least five can be realized as the boundary of a strictly
pseudoconvex domain.

##  3 Proof of Theorem 1.1

Let $\big{(}M,T^{1,0}M\big{)}$ be a closed strictly pseudoconvex CR manifold
of dimension $2n+1\geq 5$. Without loss of generality, we may assume that $M$
is connected. Then it can be realized as the boundary of a strictly
pseudoconvex domain $\Omega$ in an $(n+1)$-dimensional complex projective
manifold $X$ [8, Theorem 8.1]. The vector bundle $T^{1,0}X|_{M}$ is decomposed
into the direct sum of $T^{1,0}M$ and a trivial complex line bundle, and
consequently $c_{K}\big{(}T^{1,0}M\big{)}=c_{K}\big{(}T^{1,0}X|_{M}\big{)}$ is
in the image of the restriction morphism
$H^{2\absolutevalue{K}}(\Omega,\mathbb{C})\to
H^{2\absolutevalue{K}}(M,\mathbb{C})$. On the other hand, the natural morphism
$H^{i}_{c}(\Omega,\mathbb{C})\to H^{i}(\Omega,\mathbb{C})$ is surjective for
$i\geq n+2$ according to [11, Corollary 4]; see [2, Lemma] for another proof.
From the exact sequence

$\displaystyle H^{i}_{c}(\Omega,\mathbb{C})\to H^{i}(\Omega,\mathbb{C})\to
H^{i}(M,\mathbb{C}),$

it follows that $H^{i}(\Omega,\mathbb{C})\to H^{i}(M,\mathbb{C})$ is
identically zero for $i\geq n+2$. This completes the proof of Theorem 1.1.

##  4 Examples

In this section, we treat some examples related to Theorem 1.1. We first show
that Theorem 1.1 does not hold for $\mathbb{Z}$-coefficients in general.

######  Proposition 4.1.

For each positive integer $n$, there exists a closed strictly pseudoconvex CR
manifold $\big{(}M,T^{1,0}M\big{)}$ of dimension $2n+1$ such that, for every
multi-index $K$ with $1\leq\absolutevalue{K}\leq n$, the $K$-th Chern class
$c_{K}\big{(}T^{1,0}M\big{)}$ is not equal to zero in
$H^{2\absolutevalue{K}}(M,\mathbb{Z})$.

###### Proof.

Let $\mathbb{C}P^{n}$ be the $n$-dimensional complex projective space and
$\mathcal{O}(-d)$ be the holomorphic line bundle over $\mathbb{C}P^{n}$ of
degree $-d<0$. Fix a Hermitian metric $h$ on $\mathcal{O}(-d)$ with negative
curvature, and consider

$\displaystyle M=\big{\\{}v\in\mathcal{O}(-d)\,|\,h(v,v)=1\big{\\}},$

which is a principal $S^{1}$-bundle over $\mathbb{C}P^{n}$. This manifold has
a strictly pseudoconvex CR structure $T^{1,0}M$ induced from the total space
of $\mathcal{O}(-d)$, and $T^{1,0}M$ is isomorphic to
$\pi^{*}T^{1,0}\mathbb{C}P^{n}$ as a complex vector bundle, where $\pi\colon
M\to\mathbb{C}P^{n}$ is the canonical projection. Consider the Gysin exact
sequence

$\displaystyle
H^{i-2}\big{(}\mathbb{C}P^{n},\mathbb{Z}\big{)}\xrightarrow{-d\cdot\alpha}H^{i}\big{(}\mathbb{C}P^{n},\mathbb{Z}\big{)}\xrightarrow{\pi^{*}}H^{i}(M,\mathbb{Z})\rightarrow
H^{i-1}\big{(}\mathbb{C}P^{n},\mathbb{Z}\big{)},$

where $\alpha=c_{1}(\mathcal{O}(1))$ is a generator of
$H^{2}\big{(}\mathbb{C}P^{n},\mathbb{Z}\big{)}\cong\mathbb{Z}$. This gives
that

$\displaystyle H^{2k}(M,\mathbb{Z})\cong\begin{cases}\mathbb{Z},&k=0,\\\
\mathbb{Z}/d\mathbb{Z},&1\leq k\leq n,\\\ 0,&\text{otherwise},\end{cases}$

and $\pi^{*}\alpha^{k}$ is a generator of $H^{2k}(M,\mathbb{Z})$. On the other
hand,

$\displaystyle
c_{K}\big{(}T^{1,0}M\big{)}=\pi^{*}c_{K}\big{(}T^{1,0}\mathbb{C}P^{n}\big{)}=\bqty{\prod_{l=1}^{m}\binom{n+1}{k_{l}}}\pi^{*}\alpha^{\absolutevalue{K}}.$

Hence, for a multi-index $K$ with $1\leq\absolutevalue{K}\leq n$, the $K$-th
Chern class $c_{K}\big{(}T^{1,0}M\big{)}$ is equal to zero in
$H^{2\absolutevalue{K}}(M,\mathbb{Z})$ if and only if
$\prod\limits_{l=1}^{m}\binom{n+1}{k_{l}}\in d\mathbb{Z}$. In particular if we
choose $d$ as a prime integer greater than $n+1$, then
$c_{K}\big{(}T^{1,0}M\big{)}$ does not vanish for any $K$ with
$1\leq\absolutevalue{K}\leq n$. ∎

We next see that the degree condition in Theorem 1.1 is optimal.

######  Proposition 4.2.

For each positive integer $n$, there exists an $(n+1)$-dimensional Stein
domain $\Omega$ such that its boundary $M=\partial\Omega$ satisfies
$c_{K}\big{(}T^{1,0}M\big{)}\neq 0$ in $H^{2\absolutevalue{K}}(M,\mathbb{C})$
for any multi-index $K$ with $1\leq 2\absolutevalue{K}\leq n+1$.

###### Proof.

Let $\Omega_{0}$ be a Stein domain in a two-dimensional complex manifold
$X_{0}$ such that its boundary $M_{0}=\partial\Omega_{0}$ satisfies
$c_{1}\big{(}T^{1,0}M_{0}\big{)}\neq 0$ in $H^{2}(M_{0},\mathbb{C})$; see [5,
Theorem 6.2] for an example of such $\Omega_{0}$. Take a defining function
$\rho$ of $\Omega_{0}$ that is strictly plurisubharmonic near the closure of
$\Omega_{0}$. Without loss of generality, we may assume that $\rho$ is an
exhaustion function on $X_{0}$. Then, for sufficiently small $\epsilon$, there
exists a diffeomorphism $\chi\colon(-\epsilon,\epsilon)\times
M_{0}\to\rho^{-1}((-\epsilon,\epsilon))$ satisfying $\chi(0,p)=p$ and
$\rho\circ\chi(t,p)=t$. The function $\psi_{0}=-1/\rho$ gives a strictly
plurisubharmonic exhaustion function on $\Omega_{0}$.

We first show the statement for the case of $n=2k-1$. Consider the domain

$\displaystyle\Omega=\big{\\{}(p_{1},\dots,p_{k})\in(\Omega_{0})^{k}\,|\,\psi_{0}(p_{1})+\dots+\psi_{0}(p_{k})<2k/\epsilon\big{\\}}.$

The function $\psi(p_{1},\dots,p_{k})=\psi_{0}(p_{1})+\dots+\psi_{0}(p_{k})$
is a strictly plurisubharmonic exhaustion function on $(\Omega_{0})^{k}$, and
$d\psi\neq 0$ on $M=\partial\Omega$. Hence $\Omega$ is a Stein domain in
$(\Omega_{0})^{k}\subset(X_{0})^{k}$. As noted in the proof of Theorem 1.1,
the cohomology class $c_{K}\big{(}T^{1,0}M\big{)}$ coincides with
$c_{K}\big{(}T^{1,0}(X_{0})^{k}|_{M}\big{)}$. Consider the map
$\iota\colon(M_{0})^{k}\to M\big{(}{\subset}(X_{0})^{k}\big{)}$ defined by

$\displaystyle\iota(p_{1},\dots,p_{k})=(\chi(-\epsilon/2,p_{1}),\dots,\chi(-\epsilon/2,p_{k})).$

Since this map is homotopic to the natural embedding
$(M_{0})^{k}\hookrightarrow(X_{0})^{k}$,

$\displaystyle\iota^{*}c_{K}\big{(}T^{1,0}M\big{)}=c_{K}\big{(}\iota^{*}T^{1,0}(X_{0})^{k}\big{)}=c_{K}\big{(}T^{1,0}(X_{0})^{k}|_{(M_{0})^{k}}\big{)}=c_{K}\big{(}\big{(}T^{1,0}M_{0}\big{)}^{k}\big{)}.$

From the assumption on $\Omega_{0}$, it follows that
$c_{K}\big{(}\big{(}T^{1,0}M_{0}\big{)}^{k}\big{)}\neq 0$ in
$H^{2\absolutevalue{K}}\big{(}(M_{0})^{k},\mathbb{C}\big{)}$ for any multi-
index $K$ with $1\leq\absolutevalue{K}\leq k$, and consequently
$c_{K}\big{(}T^{1,0}M\big{)}\neq 0$ in $H^{2\absolutevalue{K}}(M,\mathbb{C})$.

We next treat the case of $n=2k$. Consider the domain

$\displaystyle\Omega=\big{\\{}(p_{1},\dots,p_{k},z)\in(\Omega_{0})^{k}\times\mathbb{C}\,\big{|}\,\psi_{0}(p_{1})+\dots+\psi_{0}(p_{k})+|z|^{2}<2k/\epsilon\big{\\}}.$

This $\Omega$ is a Stein domain in
$(\Omega_{0})^{k}\times\mathbb{C}\subset(X_{0})^{k}\times\mathbb{C}$. Consider
the map $\iota\colon(M_{0})^{k}\to M=\partial\Omega$ given by

$\displaystyle\iota(p_{1},\dots,p_{k})=(\chi(-\epsilon/2,p_{1}),\dots,\chi(-\epsilon/2,p_{k}),0).$

Then we obtain

$\displaystyle\iota^{*}c_{K}\big{(}T^{1,0}M\big{)}=c_{K}\big{(}\big{(}T^{1,0}M_{0}\big{)}^{k}\big{)}\neq
0$

in $H^{2\absolutevalue{K}}\big{(}(M_{0})^{k},\mathbb{C}\big{)}$ for any $K$
satisfying $1\leq\absolutevalue{K}\leq k$ in a similar way to the case of odd
$n$. This proves the statement. ∎

######  Remark 4.3.

Cao and Chang have stated that, for any Stein domain $\Omega$, the first Chern
class $c_{1}\big{(}T^{1,0}\Omega\big{)}$ vanishes in
$H^{2}(\Omega,\mathbb{C})$; see Proposition 3.2(1) and the proof of
Proposition 2.4 in [3]. This statement contradicts the above example since
$c_{1}\big{(}T^{1,0}M\big{)}$ is the image of
$c_{1}\big{(}T^{1,0}\Omega\big{)}$ by the restriction morphism. Unfortunately,
their proof contains an error. They have claimed that the vanishing of
$H^{1,1}(\Omega)$ would imply that of the first Chern class. However, this is
not true since $H^{2}(\Omega,\mathbb{C})$ has no Hodge decomposition like on
closed Kähler manifolds. In particular, we need to add some assumptions on the
first Chern class to Main Theorems (1) and (2) in [3] at least.

### Acknowledgements

This paper is a part of the author’s Ph.D. Thesis. He is grateful to his
supervisor Kengo Hirachi for various valuable suggestions. He also would like
to thank Masanori Adachi, Tomohiro Asano, and Takeo Ohsawa for helpful
comments. This work was supported by JSPS Research Fellowship for Young
Scientists, JSPS KAKENHI Grant Numbers JP16J04653 and JP19J00063, and the
Program for Leading Graduate Schools, MEXT, Japan.

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# Quantifying national space heating flexibility potential at high spatial
resolution with heating consumption data

Claire Halloran  Jesus Lizana  Malcolm McCulloch

###### Abstract

Decarbonizing the building stock in cold countries by replacing fossil fuel
boilers with heat pumps is expected to drastically increase electricity
demand. While heating flexibility could reduce the impact of additional demand
from heat pumps on the power system, characterizing the national spatial
distribution of heating flexibility capacity to incorporate into sophisticated
power system models is challenging. This paper introduces a novel method for
quantifying at large scale and high spatial resolution the energy capacity and
duration of heating flexibility in existing building stock based on historical
heating consumption data and temperature data. This method can reflect the
geographic diversity of the national building stock in sophisticated power
system models. The proposed heating consumption-based method was tested in
Britain using national residential gas data and was validated by comparing it
with an Energy Performance Certificate-based method and an indoor temperature-
based method. The results demonstrate the potential of this approach to
characterize the heterogeneous distribution of heating flexibility capacity at
the national scale. Assuming a 3ºC temperature flexibility window, a total
thermal energy storage capacity of 500 GWhth is identified in the British
residential housing stock. The electricity storage equivalent of this figure
depends on the temperature-dependent coefficient of performance (COP) of the
heat pumps: for an illustrative cold weather COP value of 2.5, this thermal
energy storage capacity is equivalent to 200 GWh of electricity storage.
Regarding heating flexibility duration, gas-heated homes have a median of 5.9
heat-free hours for 20th percentile regional daily winter temperatures from
2010 to 2022. However, extreme cold days nearly halve flexibility duration to
a median of 3.6 heat-free hours. These high spatial resolution energy capacity
and self-discharge parameters can account for geographic diversity at the
national scale and provide a new data-based layer of information for
sophisticated power system models to support energy transition.

###### keywords:

Demand response , Demand side management , Thermal energy storage , Heat
electrification

[Oxford]organization=Department of Engineering Science, University of
Oxford,addressline=Parks Road, city=Oxford, postcode=OX1 3PJ, country=UK

##  1 Introduction

This study addresses the research question: _What is the potential energy
capacity and duration of heating flexibility from building fabric heat
capacity?_ A data-driven method for quantifying this heating flexibility
duration at high spatial resolution is introduced and validated using Britain
as a case study.

Residential heating accounts for a significant share of greenhouse gas
emissions for countries with cold climates: in the United Kingdom (UK),
heating homes caused 17% of emissions in 2019 [1]. Replacing gas boilers with
electric heat pumps is expected to be the dominant strategy for heating
decarbonization in the UK [2]. Based on the 2018 and 2019 generation mix in
Britain, making this switch reduces greenhouse gas emissions by at least 65%
[3].

Despite the carbon reductions that heat pumps offer, they pose a significant
challenge to the power system. Electrifying residential heating is expected to
significantly increase electricity demand, particularly at peak times. Heating
all homes in Britain with heat pumps in 2019 would have added 42.3 GW of peak
demand, increasing total peak electricity demand by 78% [4], and in a cold
year in the 2020s, this figure could increase to 78 GW [5].

For this reason, there has been increasing interest in the potential for
heating flexibility, that is, shifting the consumption of heat pumps to reduce
peak demand or achieve other benefits for the power system. Watson et al. [5]
show that if 100% of hot water demand and 20% of space heating demand were
flexible, peak heat pump demand could be reduced by 16%. Despite reported
benefits, the large-scale quantification and management of heating flexibility
capacity and duration, that is, how long homes in different regions turn their
heating off before they become too cold, is a crucial challenge.

Different approaches have been used to model the interaction between heating
flexibility and the wider power system. They can be classified into three
groups: detailed transient simulation models, reduced-order models, and
steady-state models [6]. Each group has a different trade-off between building
system model detail, energy system model detail, and the diversity of
buildings represented.

Transient simulation models (also known as dynamic simulation models or white-
box models) are “fundamentally based on conservation of energy, mass, and
momentum” [6] and include detailed information about building structure,
occupancy schedules, and heating systems. These models can represent a variety
of building types, from a single archetype of a detached Irish house [7] to
144 configurations of Danish single-family homes [8]. However, transient
simulation models tend to be too computationally intensive to couple with
energy system models. A few studies [9, 8] use dynamic electricity prices to
account for building interactions with the power system. Similarly, Langevin
et al. [10] use net load (hourly load minus renewable generation) in 22
regions in the United States as a proxy for marginal electricity costs and
quantify change in annual electricity use and daily net peak load from energy
efficiency and demand response measures. More commonly, transient simulation
models are used to quantify flexibility metrics of interest such as solar
photovoltaic self-consumption [7, 11]; peak heating demand [11]; flexibility
energy capacity [7, 9, 12, 13], power capacity [12, 13], and duration [9, 12,
13]; flexibility recovery time [13]; and thermal energy storage efficiency [7,
12].

To enable integration with power system models, reduced-order models (also
called lumped-parameter, resistance-capacitance (RC), or gray-box models)
represent only the main building components, decreasing computational
complexity compared to transient simulation models [6]. These reduced-order
models represent the temperature of a small number of components, ranging from
2 to 3 [14], 4 [15], and 5 thermal masses [16, 17]. These studies trade off
between the details of the power system model and the diversity of building
types considered. At one extreme, Hedegaard and Balyk [14] represent 10
archetypes of Danish single-family homes in a linear energy system investment
optimization model considering five representative weeks at hourly temporal
resolution. In contrast, Heinen et al. [15] consider only 2 building
archetypes (new and existing detached houses in Dublin, Ireland), and include
hourly weather data for an entire year at a time in their electricity system
planning model. Similarly, Patteeuw et al. [16] only include a single building
archetype due to the computational complexity of the mixed-integer linear unit
commitment and economic dispatch problem power system model. Patteeuw et al.
[17] later consider 36 building types and 3 heating systems in this model, but
each combination is considered individually due to computing constraints.

The simplest heating flexibility models are thermal storage models based on a
steady-state approach. These models limit the charging and discharging power
of thermal storage as well as its state of charge, usually represented as an
internal temperature within a range of comfortable temperatures. Self-
discharge heat losses to the environment are commonly included based on a
thermal time constant that includes a single thermal mass and a single thermal
resistance [18, 19, 20, 21, 22]. As power system models increase in complexity
[23], considering higher spatial and temporal resolution, thermal storage
models offer an opportunity to include the variety of the building stock in
these models without compromising computational tractability. Despite the
opportunity to represent a large diversity of buildings, many of these models
do not account for heterogeneity in the building stock in large geographies
[20, 21, 22] while others only consider a few building parameter variations
[19, 24]. Considering 7 building archetypes to represent the German building
stock, Papaefthymiou et al. [25] use a transient simulation model to derive
the parameters of a thermal storage model. Mathieu et al. [18] consider
regional weather differences to quantify flexibility potential from heat pumps
and other thermostatic loads for 13 climate zones in California, but they
assume a uniform distribution of thermal parameters for the entire state.

Despite the variety of approaches, none accurately represents the diversity of
a national building stock at high spatial resolution and its flexibility
duration in complex power system models. To overcome this challenge, a recent
study explores the use of energy performance certificate (EPC) data. Canet and
Qadrdan [26] use a thermal storage model to reflect the spatial diversity in
heating flexibility potential across geographies using an EPC-based method
using national data at high spatial resolution in England and Wales. Because
of its basis in EPCs, this method is likely to underestimate heating
flexibility duration in the current residential building stock. Previous works
have shown that EPCs are based on standard occupancy procedures that
overestimate energy consumption [27, 28] and assume default heating losses
based on the year of construction that fail to capture retrofit improvements
to the building fabric [29]. Moreover, comparing EPC-modeled energy use with
smart meter-measured gas use in British households, Few et al. [30] find that
differences between the two for lower bands persist even in homes where
occupant behavior matches EPC assumptions. These differences are likely to be
particularly pronounced in rural areas, where Canet et al. [31] found that an
EPC-based method overestimated heating demand compared to gas consumption by a
median of 61% in the most rural LSOAs in England and Wales. Thus, although
many methods have been applied to quantify heating flexibility, none
accurately represent the geographic diversity of a national building stock
with sufficiently low computational requirements to be incorporated with
complex power system models (research gap 1). Moreover, the only study that
represents the variety of flexibility duration potential in British homes is
likely underestimating that potential because of its basis in EPCs [26], which
are known to overestimate heating demand (research gap 2).

This paper presents a novel heating consumption-based method for quantifying
heating flexibility duration in the current building stock at high spatial
resolution. The approach integrates high spatial resolution heating
consumption data, temperature, and building size data to capture the
geographic diversity of the national building stock. The method was tested in
Britain and validated by comparing the results with the EPC-based method from
Canet and Qadrdan [26] and a temperature-based method using heat pump trial
data from 742 homes [32].

While this method can be applied to buildings in different sectors with
different heating fuels, the scope of this case study only includes the
residential sector. Because of the availability of high spatial resolution
data on heating fuel consumption in Britain, this case study quantifies the
flexibility duration potential of gas-heated homes. Therefore, the flexibility
duration of homes in regions that are not on the gas grid is not quantified.

This study has two primary research contributions:

  * 1.

Introducing a method for quantifying national heating flexibility potential at
high spatial resolution based on annual heating consumption data and
historical temperature data (addressing research gap 1). The software for this
method is available under an MIT license on GitHub as GeoHeatFlex.

  * 2.

Characterizing Britain’s residential heating flexibility potential at high
spatial resolution to effectively support power system planning (addressing
research gap 2). This dataset is openly available on the Oxford Research
Archive [33].

This paper is structured as follows. First, the methods and case study data
are described in Section 2. Second, the results are divided into Section 3.1
describing the spatial variation in heat capacity and heat loss rate, the two
most important variables in the model; Section 3.2 discussing the model
validation; and Section 3.3 presenting the results for the case study of gas-
heated British homes. Finally, the conclusions are drawn in Section 4,
highlighting the practical implications of this approach.

##  2 Data and methods

Figure 1 summarizes the methods and data used in this paper. Sections 2.1
through 2.5 introduce the method. Section 2.6 introduces the British case
study and demonstration input data, and Section 2.7 discusses the validation
of this heating consumption-based method.

Figure 1: Summary of methods and case study data used in this paper. Outputs
for each step are shown in orange boxes, and intermediate outputs for each
step are shown in bold.

###  2.1 Heating degree days

Heating degree days (HDDs) are calculated based on gridded historical minimum
and maximum temperature observations. Following Kendon et al. [34], the mean
daily temperature for each point is calculated based on the maximum and
minimum temperature observations:

$T^{mean}=\frac{T^{max}+T^{min}}{2}$ (1)

HDDs are calculated based on the resulting daily mean temperature $T^{mean}$
assuming a base temperature of 15.5ºC. Gridded annual HDDs are assigned to
each region based on the location of the representative point of each region.
If the representative point is not within the gridded range, the mean number
of HDDs among all neighboring regions is assigned.

###  2.2 Heat loss rate

The average heat loss rate for residential buildings in each region $r$ is
calculated using a top-down methodology from Chen et al. [20] based on annual
heating consumption $Q_{r}$ and HDDs $HDD_{r}$:

$H_{r}=\frac{Q_{r}}{24\sum HDD_{r}}$ (2)

HDDs are multiplied by 24 hours to give heat loss rate $H_{r}$ in kW/ºC.

###  2.3 Heat capacity

The average heat capacity $C_{r}$ in each region is calculated based on the
average number of rooms per household. This value is converted to an estimated
floor area and finally to heat capacity using a heat capacity per floor area
value.

The average heat capacity and the number of households in each region $A_{r}$
is used to calculate the total thermal energy storage in the building fabric
for each region. For a flexibility window $\Delta T^{flex}$, the total thermal
energy storage capacity $\bar{e_{r}}$ in each region is:

$\bar{e_{r}}=C_{r}A_{r}\Delta T^{flex}$ (3)

###  2.4 Thermal time constant

Spatial variation in the duration of heating flexibility is accounted for
using the average time constant $\tau_{r}$ for each region $r$. This lumped
thermal parameter reflects how quickly heat is lost from residential buildings
in each region, depending on the temperature difference between indoor and
outdoor air. The time constant is the ratio of the total heat capacity $C_{r}$
to the total heat losses $H_{r}$ for each region:

$\tau_{r}=\frac{C_{r}}{H_{r}}$ (4)

###  2.5 Comfortable heat-free hours

Newton’s law of cooling relates the time constant $\tau_{r}$ with the indoor
air temperature at a given time $t$ to the outdoor air temperature
$T^{out}_{r}$ (assumed constant in time) in each region $r$ and initial indoor
air temperature $T^{in}_{r,0}$:

$T^{in}_{r,t}=T^{out}_{r}+(T^{in}_{r,0}-T^{out}_{r})e^{-t/\tau_{r}}$ (5)

Assuming an indoor temperature range from 18 to 21ºC, the number of
comfortable heat-free hours $t^{c}_{r}$ in each region $r$ is calculated as:

$t^{c}_{r}=-\tau_{r}\ln{\frac{18-T^{out}_{r}}{21-T^{out}_{r}}}$ (6)

The number of heat-free hours in each region is calculated based on a variety
of outdoor temperatures. To provide an even comparison of the residential
building stock across Britain, comfortable heat-free hours are calculated
using a uniform outdoor temperature $T^{out}_{r}$ of 5ºC in every region $r$.
To account for regional variation in typical winter temperatures, the number
of heat-free hours is also calculated for different percentiles of historical
regional daily winter temperatures from 2010 to 2022. This time period was
selected to capture both changing winter patterns under climate change as well
as extreme cold periods in winter 2010/11.

###  2.6 Case study data

The heating consumption-based method for calculating heating flexibility
duration introduced in Sections 2.1 through 2.5 is applied to a case study of
gas-heated homes in Britain. For each step, values are calculated for each
2011 Lower Level Super Output Area (LSOA) in England and Wales and each 2011
Data Zone (DZ) in Scotland. LSOAs are regions with an average of 1,500
residents and a minimum of 1,000 residents, while DZs are slightly smaller,
with populations between 500 and 1,000 [35]. There are 34,753 LSOAs in England
and Wales and 6,505 DZs in Scotland. The boundaries for these regions are
available from Office for National Statistics [36] and Government of Scotland
[37].

Historical minimum and maximum temperature observations at 1 x 1 km resolution
from HadUK-Grid [38] are used to calculate HDDs and regional daily winter
temperature percentiles. Annual HDDs are calculated based on the gas heating
season the annual metered domestic gas demand data, which is typically from
May of each year to the following May [35].

Heating demand for each region is based on annual metered domestic gas
consumption from the Department for Energy Security and Net Zero (DESNZ) [39].
Because the gas data methodology changed in 2017, only data from 2017 to 2021
is used [35]. Total gas consumption is scaled to space heating demand in each
region based on the national share of domestic gas consumption used for space
heating for each year [40]. Heat losses are based on total space heating gas
consumption and HDDs from 2017 to 2021.

The average number of rooms in homes in each region is obtained from the 2011
censuses [41, 42]. Based on the mean number of rooms in English homes and the
mean floor area of 94 $m^{2}$ from the English Housing Survey [43], the
average number of rooms is converted to floor space assuming 17.6 $m^{2}$ of
floor area per room. The Standard Assessment Procedure medium heat capacity
value of 250 kJ/m²ºC is used to convert floor area to heat capacity [44].

###  2.7 Validation

To validate this heating consumption-based method, the regional thermal time
constants are compared with those obtained from 2 different data sources: the
EPC-based method from Canet and Qadrdan [26] and an indoor temperature-based
method using 2-minutely temperature and energy use data from a sample of 742
homes in the Electrification of Heat trial [32].

####  2.7.1 EPC-based method

Thermal building characteristics calculated based on the EPC-based method [26]
are available for each dwelling category (flats, terraced, semi-detached, and
detached) and heating system in each LSOA in England and Wales [45].
Considering only gas-heated households, the time constant for gas-heated homes
in each dwelling category in each LSOA is calculated based on the thermal
capacity and thermal losses using Equation 4. Since Canet and Qadrdan [26]
present thermal losses for both the current building stock and potential
retrofit, current and retrofit time constants are calculated.

####  2.7.2 Indoor temperature-based method

The Electrification of Heat trial includes 742 households in Southeast
England, Northeast England, and Scotland with heat pumps. A cleaned interim
dataset including 2-minutely internal temperature, external temperature, and
heat pump energy output from November 2020 to August 2022 for 740 of these
homes is available [32].

The time constants of the homes included in this trial were calculated using
an indoor temperature-based method. This approach uses an exponential fit of
the internal temperature decrease when the heat pump, as well as any boiler or
backup heater, was off. The internal temperature drop was fit using the
following equation, based on Equation 5:

$T^{in}(t)=T^{out}_{mean}+(T^{in}-T^{out}_{mean})e^{-t/\tau}$ (7)

Because Equation 5 assumes constant outdoor temperature, the mean outdoor
temperature is used to fit this equation.

Only data from time periods during the heating season (from October to April)
and with both internal and external temperature data availability are
considered to avoid data from warm days. Moreover, three data pre-processing
steps were applied. First, periods were selected when the heating was off for
at least 90 minutes, and second, time periods with sudden, large drops in
temperature (more than 5ºC in 2 minutes) were excluded based on the assumption
that an external door or window was opened. Third, because Equation 5 assumes
constant outdoor temperature, only time periods with less than 2ºC variation
in outdoor air temperature were considered. All homes have at least 1 time
period that fits these criteria. For homes with multiple time periods that fit
these criteria, the mean of the fitted $\tau$ values for each period is taken
as the time constant for that house.

##  3 Results and discussion

This section discusses the results and their implications for heating
flexibility in Britain. First, the regional variation in heat capacity and
heat loss rates is presented in Section 3.1. Then, Section 3.2 presents the
time constants calculated using the heating consumption-based approach and
validates them with two other methods: the EPC-based method from Canet and
Qadrdan [26] and a temperature-based method using heat pump trial data from
742 homes [32]. Finally, Section 3.3 explores the implications of these
findings for flexibility duration in Britain using uniform and spatially
varied outdoor temperatures.

###  3.1 Heat capacity and heat loss rate

Regional time constants are the ratio of heat capacity to heat loss rate.
Buildings with higher heat capacity (kWh/ºC) can store more thermal energy in
the building fabric and maintain the building temperature for longer while the
heating is off. In this study, heat capacity is proportional to building size
based on the mean number of rooms per household. Buildings with higher heat
loss rates (kW/ºC) are worse-insulated and lose more energy to their
surroundings when the outdoor temperature is below the indoor temperature. In
this study, higher heat loss rates reflect higher gas consumption per HDD. The
regional values of heat capacity and heat loss rate are illustrated in Figure
2, which reveals significant geographic variation.

Figure 2: Maps of (a) mean heat capacity in each region and (b) mean heat loss
rate in houses heated with gas in each region. Inset maps display details for
the Greater London area. Domestic gas consumption data is unavailable for
regions shown in gray. Lower layer Super Output Area (LSOA) boundaries [36]
and Data Zone (DZ) boundaries [37] licensed under the Open Government Licence
v.3.0. LSOA boundaries contains OS data © Crown copyright and database right
2023. DZ boundaries copyright Scottish Government, contains Ordnance Survey
data © Crown copyright and database right 2021.

Figure 2a displays the mean heat capacity in each region. The median heat
capacity in each region is 6.5 kWh/ºC; however, heat capacity varies greatly
among regions, ranging from 10 kWh/ºC in rural regions and heat capacity to 3
kWh/ºC in densely populated urban regions. These figures reflect the diversity
in the size of dwellings in Britain. Assuming a 3ºC temperature flexibility
window, the total thermal energy storage capacity in the British housing
stock, including both gas- non-gas-heated homes, is 500 GWhth. This figure is
not directly comparable to electricity storage because the amount of delayed
electricity consumption depends on the temperature-dependent COP of the heat
pumps. For an illustrative cold temperature COP value of 2.5 [46], this
thermal energy storage capacity is equivalent 200 GWh of electricity storage.
Note that this figure is an order-of-magnitude estimate calculated assuming a
uniform room size and specific heat capacity across the housing stock.

Figure 2b displays the mean heat loss rate in gas-heated dwellings in each
region. Regions with too few houses on the gas grid to be included in the
metered demand data are in gray. Median heat losses per region are 0.21 kW/ºC,
but there is significant regional variation from 0.12 kW/ºC at the 1st
percentile to 0.41 kW/ºC at the 99th percentile. High heat losses in the outer
regions of Greater London stand out. Unlike heat capacity, there is not a
clear spatial pattern in heat losses that can be predicted from population
density. This underlines the importance of using the method presented in this
paper to analyze regional heat losses.

###  3.2 Model validation

The heating consumption-based method introduced in this paper calculates the
thermal time constant (h) for each region on the ratio of thermal capacity to
heat loss rate presented in Section 3.1. In this section, the heating
consumption-based thermal time constants are compared with time constants
obtained from the EPC-based approach and the indoor temperature-based method.
Figure 3 illustrates the distribution of time constant value for each
approach. The results demonstrate that time constants calculated based on the
heating consumption-based method are consistent with those calculated for the
British housing stock using other methods as follows.

The blue line shows the results from the heating consumption-based method
using national data at high spatial resolution presented in this paper. The
median time constant value is 30.6 hours.

Figure 3: Comparison of thermal time constants for the British housing stock
based on the heating consumption-based approach introduced in this paper, the
EPC-based method for current and retrofit housing stock, and an indoor
temperature-based method using data from heat pump trials in 742 homes.

The orange and green lines illustrate the distribution of time constants from
the EPC-based method for the current building stock in England and Wales as
well as the retrofit potential of the building stock. The similar magnitude of
time constants obtained from the heating consumption-based and EPC-based
methods verifies the soundness of the heating consumption-based results.
However, because EPCs overestimate heating losses [29, 30], the EPC-based
method for the current building stock results in a lower median time constant
of 22.8 hours than the heating consumption-based method. In contrast, the
distribution of time constants from the EPC-based method for retrofit
potential, with a median value of 33.8 hours, closely matches the distribution
of heating consumption-based time constants. This agreement may reflect
building retrofits that have taken place since the last EPC assessment, which
is updated every 10 years, as well as energy-saving occupant behaviors that
EPCs do not consider. Because the EPC-based method represents each dwelling
category in each region separately, there are smaller peaks in the EPC-based
time constant distributions at higher time constant values. These peaks
represent high heat capacity detached dwellings with low heat loss rates.

Finally, the purple line shows the distribution of time constants obtained
with the indoor temperature-based method using a sample of 742 homes in the
Electrification of Heat trial. These results further support the validity of
the consumption-based time constant values. The median time constant for these
homes is 44.4 hours, which is almost double the median heating consumption-
based regional time constant value of 30.6 hours. This difference can be
explained by the fact that compared to the national average, the
Electrification of Heat trial includes a higher proportion of detached and
semi-detached houses, which tend to have higher floor areas and higher thermal
capacities, and a lower proportion of flats, which have lower floor areas and
lower thermal capacities [47]. In addition, properties with a heat demand that
was too high were excluded from the trial, so this data is representative of a
well-insulated building stock with low heat losses.

###  3.3 Case study: British residential flexibility duration

Due to the geographical diversity in the current residential building stock
and outdoor temperature, flexibility duration varies considerably among
regions in Britain. Figure 4 displays the number of comfortable heat-free
hours in each region for an indoor temperature window from 18ºC to 21ºC and
various outdoor temperatures.

Figure 4: Maps of average comfortable heat-free hours in each region, for an
indoor window of 18ºC to 21ºC with (a) a uniform outdoor temperature of 5ºC
and (b) 40th percentile and (c) 5th percentile regional daily winter
temperature for 2010 to 2022. Map (a) displays the regional variation in
flexibility duration due to building stock differences, while maps (b) and (c)
demonstrate the effect of regional temperature differences on heat-free hours.
Domestic gas consumption data is unavailable for regions shown in gray. Lower
layer Super Output Area (LSOA) boundaries [36] and Data Zone (DZ) boundaries
[37] licensed under the Open Government Licence v.3.0. LSOA boundaries
contains OS data © Crown copyright and database right 2023. DZ boundaries
copyright Scottish Government, contains Ordnance Survey data © Crown copyright
and database right 2021.

Figure 4a displays the number of comfortable heat-free hours in each region
for an indoor temperature window from 18ºC to 21ºC with a uniform outdoor
temperature of 5ºC. The median number of heat-free hours is 6.4, with an
interquartile range of 5.5 to 7.0 hours. A few areas in central Wales,
northern England, and Scotland stand out with higher-than-average heat-free
hours. Despite regional differences in heat capacity and heat losses in
Greater London (as shown in the inset maps in Figure 2), the region has
uniformly low heat-free hours. Both small, well-insulated flats in central
London, with low heat capacity and low heat losses and large, poorly-insulated
detached houses with a high heat capacity and high heat losses in the
surrounding areas result in similarly low heating flexibility duration.
Notably, this work does not consider heating flexibility power, which
determines the total amount of energy shifted along with flexibility duration.
Regions with small, well-insulated flats are likely to have lower total heat
pump capacity than regions with large, poorly insulated detached houses.
Therefore, central London is expected to have lower flexibility power capacity
than the surrounding regions despite having similar flexibility duration.

Figure 4b displays heat-free hours for the 40th percentile daily winter
temperature in each region. The regional mean value of the 40th percentile
daily winter temperature is 4.4ºC. Comparing Figure 4b with Figure 4a, which
has a uniform outdoor temperature close to the mean temperature in (b),
demonstrates the effect of accounting for regional differences in outdoor
temperature on heat-free hours. In many central and northern regions, heat-
free hours decrease, and they increase heat-free hours in southern coastal
regions. Figure 4c demonstrates that for 5th percentile temperatures, flexible
heat-free hours reduce to less than 5 hours in the vast majority of regions.

Accounting for daily variation in winter temperatures demonstrates the impact
of outdoor temperature on heating flexibility duration. Figure 5 shows the
distribution of comfortable heat-free hours for each region at different
historical temperature percentiles during winters from 2010 to 2022.

Figure 5: Histogram of average comfortable heat-free hours in each region for
various outdoor temperature percentiles during winter 2010 to 2022. Mean
temperatures in each subfigure title are the mean regional temperature at each
percentile. Note that the colormap matches Figure 4.

At lower outdoor temperatures, the median number of heat-free hours in each
region drops: for the 80th percentile temperature, the median is 7.8 hours and
for the 5th percentile temperature, the median is 4.7 hours. The range of
regional heat-free hours also decreases as the outdoor temperature drops. For
the 80th percentile temperature, the interquartile range is 6.9 to 8.6 hours,
and for the 5th percentile temperature, the interquartile range is 4.1 to 5.1
hours. This range decrease means that regions with higher-than-median thermal
time constants will experience larger flexibility duration drops in cold
weather than regions with lower-than-median thermal time constants. This
increased sensitivity to outdoor temperature will limit the ability of heating
flexibility from regions with high thermal time constants to support the power
system during long cold snaps.

Although the current building stock offers significant flexibility duration
during typical winter days, flexibility duration drops significantly during
extreme cold events. For the lowest daily winter temperature from 2010 to
2022, with a mean value of -5.8ºC, the median flexibility duration was just
3.6 hours with an interquartile range of 3.2 to 4.0 hours. This decreased
duration will limit the ability of heating flexibility in the current building
stock to ensure power system reliability during extreme cold events, which are
predicted to be a major challenge for the electricity system in the future
with electrified heating. This finding suggests that longer-duration forms of
flexibility will be necessary to maintain power system reliability during
extreme cold events.

##  4 Conclusions

This paper aims to quantify the energy capacity and duration of heating
flexibility available from the existing building stock at high spatial
resolution to support power system planning. While many existing methods model
the interaction of heating flexibility with the power system, there is a need
to represent the geospatial diversity of the building stock in a heating
flexibility model that can be incorporated into complex power systems models.
To this end, a method for assessing heating flexibility potential based on
thermal time constants calculated from historical heating energy consumption,
temperature, and building size is introduced. This method is applied to a case
study of gas-heated homes in Britain and validated by comparing thermal time
constants with those obtained from energy performance certificates (EPCs) and
a heat pump trial. The number of comfortable heat-free hours without heating
was presented for a constant outdoor temperature of 5ºC as well as different
percentiles of regional winter temperatures. Based on the results, the
following conclusions can be drawn:

The proposed method produces high spatial resolution thermal time constant
values that are broadly consistent with those found using EPC-based and indoor
temperature-based methods. The heating consumption-based thermal time constant
values are higher than those obtained using current EPC values (median value
of 30.6 hours vs. 22.8 hours), which is expected because EPCs tend to
overestimate heating demand. Compared to heat pump trial data with a median of
44.4 hours, the values obtained from this paper’s method are slightly lower.
This difference is explained by the disproportionate representation of
detached and semi-detached houses in the trial, as well as the exclusion of
homes with high heating demand. Thus, the proposed consumption-based method
produces more accurate thermal time constant estimates for the current housing
than the EPC-based method and at higher spatial resolution than the indoor
temperature-based method.

In the case study of Britain, significant heating flexibility energy capacity
and duration are identified in the existing housing stock. The total thermal
energy storage capacity in residential buildings of both gas and non-gas-
heated homes is 500 GWhth for a 3ºC temperature flexibility window. This is
equivalent to 200 GWh of electricity storage for a cold weather COP value of
2.5. The housing stock also has significant heating flexibility duration
potential: assuming a uniform 5ºC outdoor temperature and indoor temperature
range from 18 to 21ºC, regions had a median of 6.4 comfortable heat-free hours
and an interquartile range of 5.5 to 7.0 heat-free hours. However, heating
flexibility duration varies significantly among regions. Due to low thermal
capacity in central London and high heating losses in the surrounding home
counties, there is relatively low flexibility duration in Greater London and
the Southeast. High flexibility duration is identified in some rural areas due
to higher heat capacity and lower heat losses.

Accounting for the impact of historical outdoor temperatures demonstrates that
extreme cold events can drastically decrease flexibility duration compared to
typical winter temperatures. For the 20th percentile of daily winter
temperatures, the median regional heat-free hours is 5.9, with an
interquartile range from 5.2 to 6.5 hours. At these temperatures, heating
flexibility can compete with the typical 4-hour duration of lithium-ion
batteries during the heating season. However, for the lowest daily winter
temperatures in each region from 2010 to 2022, this heating flexibility
duration nearly halved to a median of 3.6 hours with an interquartile range of
3.2 to 4.0 hours. This temperature-driven decrease in flexibility duration
will limit the ability of heating flexibility to contribute to power system
reliability during extreme cold events.

The results demonstrate how this novel consumption-based method can
characterize the heating flexibility potential of the current building stock
at high spatial resolution. This approach provides building stock data that
can be incorporated into sophisticated power system planning models. Future
work will evaluate the role of heating flexibility in generation, storage, and
transmission expansion planning for bulk power systems with high shares of
renewable generation. It should be noted that this study quantifies heating
flexibility duration and does not consider flexibility power, which depends on
the heat pump sizing and building efficiency. Because of the uncertainty about
the spatial distribution of these characteristics, this topic will be
addressed in future studies.

## CRediT authorship contribution statement

Claire Halloran: Conceptualization, Methodology, Software, Validation, Writing
- Original draft, Visualization. Jesus Lizana: Writing - Review & Editing.
Malcolm McCulloch: Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

C. Halloran acknowledges the support of the Rhodes Trust.

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