# Market Integration of Excess Heat

Linde Frölke∗, Ida-Marie Palm, Jalal Kazempour
.
Technical University of Denmark
Department of Wind and Energy Systems
∗ Corresponding Author Email<EMAIL_ADDRESS>
###### Abstract

Excess heat will be an important heat source in future carbon-neutral district
heating systems. A barrier to excess heat integration is the lack of
appropriate scheduling and pricing systems for these producers, which
generally have small capacity and limited flexibility. In this work, we
formulate and analyze two methods for scheduling and pricing excess heat
producers: self-scheduling and market participation. In the former, a price
signal is sent to excess heat producers, based on which they determine their
optimal schedule. The latter approach allows excess heat producers to
participate in a market clearing. In a realistic case study of the Copenhagen
district heating system, we investigate market outcomes for the two excess
heat integration paradigms under increasing excess heat penetration. An
important conclusion is that in systems of high excess heat penetration,
simple price signal methods will not suffice, and more sophisticated price
signals or coordinated dispatch become a necessity.

Keywords: Heat market, open district heating, excess heat, pricing.

##  1 Introduction

District heating is expected to play an important role in future carbon-
neutral energy systems, especially in urban areas [1]. District heating
networks facilitate the decarbonization of heat generation, for instance by
allowing for distribution of excess heat to consumers. Examples of excess heat
producers include energy intensive industries such as metal and cement
factories, sources in the service sector that produce heat as a by-product of
their refrigeration systems [2], and data centers that produce heat from
cooling their servers [3]. In many cities, excess heat has the potential to
cover a large share of total heat demand. For example, [4] finds that excess
heat could cover over half of Greater Copenhagen’s heat demand in 2050. Excess
heat could also cover over 80% of demand in several other large Danish
district heating networks [5]. Current district heating systems typically rely
on few large generators, such as Combined Heat and Power (CHP) plants and
waste incinerators. Compared to these conventional sources, excess heat
producers are generally of smaller capacity and lower flexibility. Integration
of excess heat producers would therefore result in a more distributed heating
system, with a large number of small heat sources. In such a system,
coordination and scheduling of heat generation becomes more challenging.

A major barrier to excess heat injection in district heating networks is the
lack of suitable methods for scheduling and pricing excess heat. Most
potential excess heat sources are untapped, even though it has been shown that
their integration decreases both fuel usage and operational cost of the system
[6], and the integration of excess heat has proven feasible in simulation
studies and in practice [7, 8]. It remains an open question how heat
scheduling and pricing systems can be designed to optimally integrate excess
heat producers.

Most existing district heating systems do not have a liberalized market. In
Greater Copenhagen, the daily heat dispatch is determined by Varmelast.dk,
which is a regulated heat market that dispatches generators based on submitted
price-quantity bids. While the scheduled quantities are determined by this
market, the prices that could be derived from the market are not used.
Instead, the price of heat is fixed in advance in contracts between suppliers
of district heating and distributors/transmitters. Excess heat providers
currently do not have the possibility to participate in this dispatch
procedure. See e.g. [9] for more details on Varmelast.dk.

Only few existing works have studied excess heat producers in a market
setting. In [6], the impact of excess heat producers on the heat market in
Espoo, Finland, is investigated. References [10] and [11] study the potential
effect of dynamic pricing on marginal cost of district heating systems, for a
system including excess heat producers. In both works, it is assumed that the
excess heat production profile is constant over each month or over the entire
year, and that this profile is fully inflexible. Furthermore, the price
bidding behaviour of CHPs is not modeled accurately, either disregarding
electricity price dependence [11], or disregarding the dependence on
opportunity cost in the electricity market [10]. Reference [12] applies
marginal-cost pricing to a case study in the Netherlands to assess whether
different producers can recover their fixed costs from market revenues. Their
market clearing consists of a combined unit commitment and economic dispatch.
Excess heat from industrial processes is included, but its flexibility is not
modeled. None of these works investigate the effect of increasing excess heat
penetration.

In this work, we investigate whether price signals to be disseminated by the
heat market operator can suffice for market integration of excess heat
producers. To the best of our knowledge, this is the first paper in the
literature that explores the integration of excess heat producers through
price signals. Given a price signal, excess heat producers self-schedule their
production. We evaluate this self-scheduling model by comparison to an ideal
benchmark, namely direct market participation of excess heat producers. This
comparison is done by performing a realistic case study of the Copenhagen
district heating system, which currently includes 13 CHP plants. We evaluate
the success of the self-scheduling method by adding an increasing number of
excess heat producers to the Copenhagen system. We aim to show the
consequences of integrating cooling-based excess heat producers under these
two paradigms, including how the suboptimality of the self-scheduling model
evolves under increased penetration of excess heat. Our main finding is that
price signals can be used as an alternative for market participation of excess
heat producers, but their success depends highly on the quality of the signal,
as well as the penetration of excess heat. As long as the installed excess
heat capacity is sufficiently low, a simple price signal may suffice, but as
excess heat penetration increases there may be significant downsides to this
approach.

The remainder of this article is structured as follows. In Section 2, we
discuss the self-scheduling and market participation models in more detail. We
provide model formulations, and outline the bidding behavior of different
market participants. The real case study of the Copenhagen district heating
system is presented in Section 3, including numerical results. In Section 4 we
conclude with further discussions regarding the implications of both methods
and provide several recommendations for future work.

##  2 Market participation versus self-scheduling

One natural way of integrating excess heat producers in heat markets, is by
direct market participation. For example in the district heating system of
Copenhagen, a marginal-cost based heat market is operated by
Varmelast111www.varmelast.dk/. This paradigm will further be referred to as
market participation. Another option is for heat market operators to publish a
time-varying price signal for excess heat producers, which optimally self-
schedule their heat generation accordingly, and share the resulting schedule
with the market operator. We further refer to this paradigm as self-
scheduling. In the Open District Heating system of Stockholm, small excess
heat producers are successfully integrated in this way using an ambient
temperature-dependent price signal222www.opendistrictheating.com.

We first discuss the advantages and benefits of each model in Section 2.1, and
then present the model formulations in Sections 2.2 and 2.3. Finally, Section
2.4 outlines CHP and excess heat provider bidding behaviour.

###  2.1 Comparison of market integration and self-scheduling

We compare the two paradigms in Table 1. First of all, they differ in the
formation of the price received by excess heat participants. Under the market
participation scheme, the market-clearing price follows from the bids
submitted by market participants (including excess heat producers), while the
price for excess heat is set exogenously by the market operator in the self-
scheduling case. As also indicated in Table 1, market participation would
provide incentives to optimally schedule excess heat production, as it
minimizes total generation cost. If the price signal for self-scheduling is
designed perfectly, the resulting schedules may be the same as in a market
setting. Otherwise, the schedule resulting from price signals will be
suboptimal. This suboptimality may increase with the penetration of excess
heat producers.

Although market participation of excess heat producers would be optimal from a
cost minimization perspective, it has some drawbacks in practice for a heat
system with many small (excess) heat producers. For small excess heat
producers, it may be difficult to decide on market bids, and they may
therefore prefer to receive a price signal. For the market operator, the
market participation of many small excess heat producers poses a
communicational challenge: the operator receives bids from many participants,
clears a more complex market, and then needs to send the individual schedules
to each small market participant. Therefore, the self-scheduling paradigm may
be preferred in practice, as it is a rather simple and computationally cheap
way of scheduling and pricing excess heat, with lower IT requirements. We
consider the market participation scheme as ideal benchmark, and explore the
success of self-scheduling scheme in comparison to this ideal benchmark. Both
the market participation and the self-scheduling model are Linear Programs, in
which no binary variables are used.

###  2.2 Model formulation for market participation

| Market Participation | Self-scheduling
---|---|---
Price formation | Endogenous | Exogenous
Optimal scheduling | ✓ | ✓ / $\times$
Relevance for small producers | $\times$ | ✓
Problem type | Linear program | Linear program
Table 1: Comparison of the two paradigms for scheduling and pricing excess
heat

In the market participation scheme, excess heat producers participate in a
market clearing. We formulate a heat market clearing without network
constraints as a linear optimization problem. The market clearing results in a
uniform market price and scheduled quantities for all participants, including
excess heat producers. All market participants submit price-quantity bids. The
market then dispatches generators to minimize total generation cost, i.e.,
according to the merit order. Following EU electricity market design, we do
not consider unit commitment constraints. This implies that unit commitment
constraints should be internalized into the bids. We also choose not to
enforce network constraints in the optimal dispatch, because the current
optimal dispatch mechanism in Copenhagen does not include such constraints
either. Instead, hydraulic conditions are checked after clearing the market
[9].

The objective of the market clearing is to minimize total heat generation
cost, given by the function

$\displaystyle
f(\Gamma^{\mathrm{mp}})=\sum_{t\in\mathcal{T}}\biggl{(}\sum_{e\in\mathcal{E}}c_{et}(G_{et}^{\mathrm{H}}-W_{et})+\sum_{i\in\mathcal{G}}c_{it}G^{\mathrm{H}}_{it}+c^{\mathrm{U}}U_{t}\biggr{)}\,,$
(1)

where $\Gamma^{\mathrm{mp}}$ is the set of optimization variables for the
market participation model, $c_{it}$ is the bid price of CHP $i$ at time $t$,
$c_{et}$ is the bid price of excess heat producer $e$, $G^{\mathrm{H}}_{it}$
is the generated heat by a CHP, and $c^{\mathrm{U}}$ represents the (constant)
penalty cost per unit of unsupplied load $U_{t}$. The excess heat production
$G_{et}^{\mathrm{H}}$ may in some cases exceed the load, so that an amount of
$W_{et}$ will have to be wasted, i.e., vented to the outside air. The wasted
excess heat must be non-negative and cannot exceed the produced excess heat:

$\displaystyle G_{et}\geq W_{et}\geq 0$ $\displaystyle\forall e,t\,.$ (2)

A power balance must hold between the predicted heat load
$\hat{L}^{\mathrm{H}}_{t}$ and the scheduled generation. If the load cannot be
supplied due to insufficient installed capacity, the load can be curtailed by
an amount $U_{t}$ of unsupplied load:

$\displaystyle\sum_{e\in\mathcal{E}}G_{et}^{\mathrm{H}}-W_{et}+\sum_{i\in\mathcal{I}}G_{it}^{\mathrm{H}}=\hat{L}^{\mathrm{H}}_{t}-U_{t}\quad:\lambda^{\mathrm{H}}_{t}$
$\displaystyle\forall t\,.$ (3)

The uniform market price $\lambda^{\mathrm{H}}_{t}$ is given by the dual
variable corresponding to constraint (3), which is equal to the marginal price
bid of the most expensive scheduled generator.

The market participants’ bid includes a description of the feasible region
$\mathcal{F}$ of their heat generation. These feasible regions are respected
in the market clearing:

$\displaystyle G^{\mathrm{H}}_{it}$ $\displaystyle\in\mathcal{F}_{it}$
$\displaystyle\forall i,t$ (4) $\displaystyle G^{\mathrm{H}}_{et}$
$\displaystyle\in\mathcal{F}_{et}$ $\displaystyle\forall e,t\,.$ (5)

We will define these feasible regions for CHPs and excess heat producers in
Section 2.4, and in more detail in Appendices A and B. The set of optimization
variables in the market-clearing optimization problem is given by
$\Gamma^{\mathrm{mp}}=\\{G_{it}^{\mathrm{H}},G^{\mathrm{H}}_{et},W_{et}^{\mathrm{H}},U_{t}^{\mathrm{H}}\\}$.
Fig. 1 shows a graphical overview of the market participation model. The
forecasted electricity price $\lambda_{t}^{\mathrm{E}}$ is an input to the
marginal cost model of both CHPs and excess heat producers, which will be
discussed in Section 2.4.

Bid submission $\forall e,i,t$$\lambda^{\mathrm{E}}_{t}$MO clears
market$G^{\mathrm{H}}_{it}$, $G^{\mathrm{H}}_{et}$,
$\lambda^{\mathrm{H}}_{t}$$c_{it},\mathcal{F}_{it}$$c_{et},\mathcal{F}_{et}$
Figure 1: Overview diagram of market participation model. Market Operator is
abbreviated to MO.

###  2.3 Model formulation for self-scheduling

In the self-scheduling model, the market operator broadcasts a price signal
$\mu_{t}^{\mathrm{H}}$ for each market period $t$ to all excess heat
producers, who self-schedule their production accordingly. The resulting
schedule is submitted to the market operator, who uses the total excess heat
production as a fixed input to the market clearing with conventional
generators only. This market clearing is as described previously in Section
2.2, except that $G^{\mathrm{H}}_{et}$ is now a parameter instead of a
variable for all $e,t$. As a result, the total self-scheduled heat generation
by the excess heat producers is prioritized in the heat market, and the CHPs
may supply any remaining unsupplied load. If the self-scheduled excess heat
exceeds the heat load at certain hours, some of the excess heat is wasted,
i.e., vented to the air.

We assume an ambient-temperature-dependent price signal for excess heat
producers, inspired by the Stockholm Open District Heating pricing system. In
particular, the received price decreases with the ambient temperature, as heat
demand often decreases with ambient temperature too. Under the self-scheduling
scheme, the excess heat producers are paid as in the price signal for each
unit generated, regardless of whether (part of) the produced excess heat
exceeds the supplied load and needs to be wasted. Clearly, this is an
undesirable effect of the self-scheduling paradigm. The CHPs are still paid at
the uniform marginal price resulting from the market clearing.

During self-scheduling, excess heat producers aim to minimize total cost,
given by the difference between costs for electricity used by the heat pump
$L^{\mathrm{E}}_{et}$ bought at the (forecasted) electricity spot price
$\lambda^{\mathrm{E}}$, and the income from selling the generated heat
$G^{\mathrm{H}}_{et}$:

$\displaystyle
C^{\mathrm{H}}_{e}=\sum_{t\in\mathcal{T}}\left(\lambda^{\mathrm{E}}_{t}L^{\mathrm{E}}_{et}-\mu^{\mathrm{H}}_{t}G^{\mathrm{H}}_{et}\right)\,.$
(6)

It is assumed that all excess heat producers and CHPs use the same forecasted
electricity prices $\lambda_{t}^{\mathrm{E}}$. The excess heat producers must
schedule an amount of heat that respects their physical constraints:

$\displaystyle G^{\mathrm{H}}_{et}\in\mathcal{F}_{et}$ $\displaystyle\forall
e,t\,.$ (7)

The feasible region $\mathcal{F}_{et}$ will be defined in the next Section
2.4, and more details can be found in Appendix B. Fig. 2 shows a graphical
overview of the self-scheduling model.

MO clears market$G^{\mathrm{H}}_{it}$, $\lambda^{\mathrm{H}}_{t}$Submission of
CHP bids $\forall i,t$$c_{it},\mathcal{F}_{it}$Excess heat self-scheduling
$\forall
e,t$$G^{\mathrm{H}}_{et}$$\lambda^{\mathrm{E}}_{t}$$\mu^{\mathrm{H}}_{t}$
Figure 2: Overview diagram of self-scheduling model and the following market
clearing. Market Operator is abbreviated to MO.

###  2.4 Excess heat and CHP models

To simulate the two paradigms, the bidding behavior of different market
participants needs to be modeled. In this work, we consider CHPs and cooling-
based excess heat producers only. The derivation of their bidding behavior and
feasible regions is presented in more detail in Appendices A and B. We assume
that the heat market is cleared daily before the electricity market is
cleared, as is currently the case in Copenhagen. The resulting feasible region
for CHP $i$ at time $t$ is given by

$\displaystyle\mathcal{F}_{it}=\left\\{G^{\mathrm{H}}_{it}\quad\middle|\quad
0\leq
G^{\mathrm{H}}_{it}\leq\min\left(\overline{G}^{\mathrm{H}}_{i},\frac{\overline{F}_{i}}{\rho^{\mathrm{H}}_{i}+r_{i}\,\rho^{\mathrm{E}}_{i}}\right)\right\\}\,,$
(8)

where $\overline{G}^{\mathrm{H}}_{i}$ is the maximum heat generation,
$\rho^{\mathrm{E}}_{i}$ and $\rho^{\mathrm{H}}_{i}$ are the fuel efficiency
for electricity and heat, respectively, $r_{i}$ is the minimum power-to-heat
ratio, and $\overline{F}_{i}$ is the maximum fuel consumption. Note that these
parameters are here considered time-invariant, but this could easily be
adapted.

Reference [13] derives the optimal heat bid $c^{\mathrm{H}}_{it}$ for CHPs in
a sequential heat and electricity market setting. The marginal cost of heat
depends on the (forecasted) electricity price $\lambda^{\mathrm{E}}$ and
(constant) fuel price $\alpha_{i}$ as follows:

$\displaystyle
c^{\mathrm{H}}_{it}=\begin{cases}\alpha_{i}\,(\rho^{\mathrm{E}}_{i}\,r_{i}+\rho^{\mathrm{H}}_{i})-\lambda^{\mathrm{E}}_{t}\,r_{i}&\text{if
}\lambda^{\mathrm{E}}_{t}\leq\alpha_{i}\rho_{i}^{\mathrm{E}}\\\
\lambda^{\mathrm{E}}_{t}\,\frac{\rho^{\mathrm{H}}_{i}}{\rho^{\mathrm{E}}_{i}}&\text{if
}\lambda^{\mathrm{E}}_{t}>\alpha_{i}\rho^{\mathrm{E}}_{i}\,.\end{cases}$ (9)

This bidding function assumes that in case the forecasted electricity price is
lower than or equal to a certain threshold, CPHs bid their fuel cost minus the
income from electricity sale, as represented by the first case in (9).
Otherwise, if the forecasted electricity price is relatively high, CHPs bid
the lost opportunity cost from selling heat instead of electricity, as in the
second case in (9).

The feasible region for excess heat producers is derived in Appendix B. The
excess heat producers’ flexibility in heat production is represented by
modeling cooling cabinet heat dynamics as a set of linear constraints as given
in Appendix B through Eqs. (21)-(26). We define the feasible region for excess
heat production as the set of production setpoints that satisfy the
aforementioned constraints:

$\displaystyle\mathcal{F}_{et}=\left\\{G^{\mathrm{H}}_{et}\quad\middle|\quad\eqref{eq:ex1}-\eqref{eq:exend}\right\\}\,.$
(10)

The price bid of excess heat producers in the market participation model is
assumed to be at zero333Bidding at zero is reasonable if any sold excess heat
is considered extra income for these producers. However, the electricity
consumption cost of a given excess heat production profile may be greater than
the electricity cost of the production profile that minimizes these costs. One
may therefore choose to define the price bid of a certain production profile
as the difference between the electricity cost of the given profile and the
minimum electricity cost this producer could obtain if it would minimize
electricity cost only. , i.e.,

$\displaystyle c_{et}=0$ $\displaystyle\forall e,t\,.$ (11)

##  3 Copenhagen case study

We analyze the application of the market participation and self-scheduling
models in the district heating system of Copenhagen, Denmark.

###  3.1 Case study description

The district heating system of Copenhagen consists of 13 CHPs. In our case
study, we vary the level of excess heat capacity added to this system from 0
to 2100 MW. The excess heat is assumed to be produced as a by-product of
cooling. In particular, we assume excess heat producers cool refrigeration
cabinets using a local heat pump. We simulate in an hourly time resolution for
a full year. In our online appendix, a detailed description of all used
parameters can be found, as well as the code used to generate our
results444www.github.com/linde-fr/excess-heat-in-market.

Several time series are needed as inputs for the market participation and/or
self-scheduling models. The forecasted ambient temperature is an input used to
model the Coefficient Of Performance (COP) of the heat pumps, and also to
determine the price of waste heat in the self-scheduling model. We use hourly
temperature measurements from 2019 from the Danish Meteorological Institute
[14]. For the month March we use measurements from 2020, due to many missing
measurements for March 2019. As the electricity price forecast, we use Nord
Pool historical electricity prices for DK2 [15]. For the forecasted heat load,
we use the hourly heat load in the entire Copenhagen district heating area for
2019, provided by Varmelast.

For the self-scheduling model, the excess heat price signal needs to be given.
The self-scheduling pricing signal used here is inspired by Stockholm’s Open
District Heating Spot Prima price. We approximate their ambient-temperature
dependent price function using an exponential regression on data available
from their website. The resulting price signal $\mu^{\mathrm{H}}_{t}(\cdot)$
is defined as follows:

$\displaystyle\mu^{\mathrm{H}}_{t}(T^{\mathrm{A}}_{t})=\begin{cases}380\cdot
0.92^{T^{\mathrm{A}}_{t}}&\text{for }T^{\mathrm{A}}_{t}<17.5\\\ 0&\text{for
}T^{\mathrm{A}}_{t}\geq 17.5\,.\end{cases}$ (12)

Note that the price is decreasing with the ambient temperature, as the base of
the exponent $0.92$ is non-negative and below 1.

We further require input parameters for CHP and excess heat producer models.
For the CHPs, most input parameters have been obtained from [16]. The minimum
power-to-heat ratio was not given there, and therefore a default value of
$r=0.45$ has been taken from [17].

The excess heat producers are assumed to have the same input parameters. The
heat dynamics parameters are $A=0.1$ and $B=\frac{1}{21}$. The temperature in
the cooled room has to be within 2-8 , while the average temperature every 6
hours has to be within 4-5 . The indoor temperature at the excess heat
producers is assumed to be constant at 25 . The heat pumps’ maximum generation
capacity is $\overline{G}^{\mathrm{H}}=30$ $\mathrm{kW}$. Heat pump ramping
limits are set to $0.25$ of its maximum generation capacity. It is assumed
that the COP of the excess heat producers’ heat pumps varies with the ambient
temperature. The approximate ambient temperature dependence of the COP was
obtained using a more detailed heat pump simulation model, under several
simplifying assumptions, including a linear dependence of supply and return
temperatures on ambient temperature.

To validate our CHP bidding model, we have compared our resulting heat market
prices to Varmelast heat market prices for a given electricity price signal.
The results showed satisfactory correspondence between our and Varmelast’s
heat prices.555Exact results cannot be shared here due to confidentiality of
Varmelast pricing data.

###  3.2 Results

The total heat load in 2019 in Copenhagen was around
$8.3$\mathrm{TW}\text{\,}\mathrm{h}$$ or $30$\mathrm{PJ}$$. For the different
maximum capacity levels of the excess heat producers of 300, 1200, and 2100
$\mathrm{MW}$ participating in the heat market, we find that the excess heat
providers are scheduled for $1.8$, $5.8$, and
$8.0$\mathrm{TW}\text{\,}\mathrm{h}$$, respectively. Considering that [4]
reports that excess heat could cover over $50\%$ of the Copenhagen heat load,
the 1200MW case could be realistic for the Copenhagen system in 2050.

#### Suboptimality

When the excess heat producers are self-scheduling and the market for CHPs is
cleared afterwards, the total generation cost for CHPs will be greater than it
is in the case the market is cleared with excess heat producers integrated.
Here we consider suboptimality in terms of the total CHP generation cost,
where we treat this total cost similarly as in the objective function of the
market clearing. That is, generation cost of each scheduled CHP is computed as
its scheduled quantity multiplied by its price bid, and the total generation
cost is obtained by summing over all CHPs. In Fig. 3, we visualize how the
total suboptimality over a full year depends on the penetration of excess heat
producers. In the left-hand Fig. 3(a), we observe that both schemes experience
a steep decrease of the total generation cost with the installation of the
first 500 MW of excess heat capacity, but this decrease flattens out
afterwards. Fig. 3(b) shows that the absolute suboptimality grows almost
linearly with the installed excess heat capacity. The curve is slightly
steeper when the latest installed excess heat is replacing a CHP that is
expensive compared to the CHPs that bid a price just below it.

(a) Total generation cost

(b) Difference in generation cost

Figure 3: Full year suboptimality of self-scheduling compared to market
integration in terms of total CHP generation cost. Note that this does not
include excess heat generation cost.

Next, we investigate how this suboptimality in total CHP generation cost is
distributed over the year, by zooming in on three levels of excess heat
penetration in Fig. 4. The left-hand Fig. 4(a) shows that the generation cost
is unequally distributed over the year, as monthly heat load varies
significantly over the year. As seen in the right Fig. 4(b), the suboptimality
is quite equally distributed over the year for a low capacity of excess heat
at 300 MW (blue). For higher excess heat penetration, the suboptimality is
increasingly shifted to the colder months. In the warmest summer months, i.e.,
from June to August, the suboptimality of the 300 MW case (blue) is relatively
high compared to the 1200 MW and 2100 MW cases (orange and green). The reason
for this is that from a certain level of excess heat capacity, the
overcapacity in summer is very high, so that the total demand is (almost)
always completely supplied by excess heat, both under self-scheduling and
market participation. Therefore, suboptimality will be low in summer for a
higher penetration of excess heat.

(a) Total generation cost

(b) Difference in generation cost

Figure 4: Monthly suboptimality of self-scheduling compared to market
integration in terms of total CHP generation cost.

#### Scheduled and wasted excess heat

The scheduled excess heat varies over the year. There can be a difference
between generated and scheduled excess heat, as some excess heat may have to
be wasted in case it exceeds the heat load. We compare the monthly scheduled
excess heat for the self-scheduling and market participation under different
levels of excess heat penetration in Fig. 5(a), and do the same for monthly
wasted excess heat in Fig. 5(b). Under the market participation model, a
capacity of 2100 MW (green) is enough to supply the load fully in all but the
coldest months, i.e., December to March. This is not the case for the self-
scheduling model, which is due to a greater mismatch between supplied excess
heat and heat load. This manifests itself in the consistently greater amount
of wasted heat for the self-scheduling model. In general, differences in the
self-scheduling and market participation model in total scheduled excess heat
are greatest in months where excess heat is the marginal supplier in some
hours, but not in all hours. With increasing excess heat capacity, the months
where this is the case shift more towards the winter months with greater heat
load. Note that the scheduled volume in the summer months is almost equal for
the 1200 MW (orange) and 2100 MW (green) cases, which is the reason that
suboptimality in summer months is similar for these cases, as we observed
previously in Fig. 4(b).

Finally, we highlight that the total wasted excess heat increases steadily
with increasing excess heat penetration, for both scheduling paradigms. This
is due to the limited flexibility of these producers, combined with the fact
that these producers have a minimum heat generation that may exceed the load.
To decrease the wasted excess heat and supply a higher share of the load, it
would be beneficial to install heat storage as the penetration of excess heat
increases.

(a) Scheduled volume

(b) Wasted volume

Figure 5: Monthly values for excess heat scheduled volume and wasted volume
for self-scheduling compared to market integration

#### Market prices

We compare monthly average market prices resulting from the market clearing in
both the self-scheduling and market participation models in Fig. 6. Recall
that we also clear the market in the self-scheduling model, but in this case
the excess heat schedule is a fixed input to the market. Under a low excess
heat penetration of 300 MW (blue), the average market price only differs for
the summer months. Counter-intuitively, the average market price for the
market participation model is higher in this case than that in the self-
scheduling model. The explanation of this is that the market participation
model schedules excess heat when it can reduce total generation cost as much
as possible, which means that it will spread out the excess heat schedule to
avoid scheduling of more expensive generators. The effect of this spread can
be that on average, the marginal generator is more expensive than in the self-
scheduling case, as is the case for our case study. In other words, the
minimization of total generation cost does not necessarily lead to the lowest
average market prices. This is also seen for the month January for the 2100 MW
case (green). However, in most cases and months the market participation model
does lead to lower average market prices compared to the self-scheduling
model.

As expected, the average market price decreases under increasing excess heat
penetration. With a relatively high installed excess heat capacity of 2100 MW
(green), the market price for the market participation model is close to zero
from April up to and including October. In the case of high installed
capacity, the average marginal price difference between the two models is most
pronounced.

Figure 6: Average market prices per month for self-scheduling compared to
market participation, under various levels of excess heat penetration.

##  4 Conclusions and Future Perspectives

We have investigated the consequences of integrating excess heat in district
heating systems under two different scheduling and pricing paradigms: self-
scheduling and market participation. The self-scheduling is attractive due to
its simplicity for both market operator and excess heat producers, but may
lead to suboptimal scheduling. Our main conclusion is that at higher excess
heat penetration, a simple price signal is no longer adequate, and more
sophisticated pricing signals and/or other market setups may be needed. In our
case study, we have shown that the disadvantages of using a price signal under
high excess heat penetration include:

  1. 1.

Expensive scheduling: Excess heat is scheduled in hours where CHPs can produce
relatively cheap heat instead of where CHPs produce more expensive heat. As a
result, total CHP generation cost is higher under self-scheduling than the
market participation.

  2. 2.

Wasted excess heat: Excess heat production is not matched to heat load, so
that a greater amount of excess heat is wasted.

  3. 3.

High market prices: Even though market participation may lead to higher
average market prices in some cases, market prices are most of the time higher
in the self-scheduling model, especially under high excess heat penetration.

###  4.1 Discussion

Under increasing penetration of excess heat, market prices decrease under both
paradigms, and most drastically under market participation. Market prices get
close to zero during the summer time already for intermediate penetration of
excess heat. This may affect the recovery of fixed cost for generators. For
example, in [12] it is shown that market revenues can be insufficient for
investment cost recovery for most heat producers in a case study in the
Netherlands. This problem has also been encountered in electricity systems
with high shares of solar and wind energy [18]. For this case, it has been
suggested that marginal-cost-based market clearing is not necessarily a proper
solution for systems of generators with high fixed and low marginal costs, and
a rethinking of power markets is needed [18]. This problem can be expected to
arise in excess heat based heat systems too.

We have concluded that more sophisticated pricing signals than the Stockholm
price are needed in systems with high excess heat penetration. However, the
Stockholm ambient-temperature dependent pricing signal has two attractive
properties: transparency and interpretability. These properties should be
considered when designing new methods for generating pricing signals.

Finally, we note that cooling-based excess heat producers provide most excess
heat during the summer months, which is a mismatch with the load that is
minimal in this period. This relation indicates that a seasonal storage may be
a suitable supplement in systems dominated by cooling-based excess heat
producers.

###  4.2 Recommendations for future work

In this work, we have designed a model of cooling-based excess heat producers
with the aim was to mimic general dynamics of such producers in a convex
manner. The model has been formulated after discussion with experts in more
detailed heat pump modeling. However, the model has not been verified using
real data of excess heat producers. This should be done in future work.

Furthermore, the model could be extended and made more realistic in several
ways. In our model, flexibility of heat producers was limited using an energy
budget to be respected over every six hours. This is a stylized representation
of excess heat flexibility. Future reformulations of the model could focus on
improving the representation of this (limited) flexibility. On the
conventional generator side, the market considered here assumes that unit
commitment constraints are included in the price-quantity bids. However, we
have ignored unit commitment considerations in the construction of supply
bids. Our work could be extended with block bids to represent bidding behavior
of CHPs more realistically.

It could be argued that a scaling is needed to adapt the Stockholm heat price
signal to the Copenhagen case. A sensitivity analysis for the value of scaling
factor could show how this would affect our results.

We have addressed effects of excess heat integration on the day-ahead market,
without considering potential sources of uncertainty. Future work could
investigate how uncertainty of heat load, as well as uncertainty in excess
heat production, could affect the scheduling and pricing of excess heat. We
have also disregarded network considerations. Inclusion of such constraints
could change our results quantitatively, as there would be a delayed arrival
of CHP heat, while local excess heat would be delivered close to real time. In
addition, heat transported from a distance would be accompanied by greater
heat losses. Our work could be extended to include network constraints, for
example using the linear formulation in [19].

Our analyses could also be extended by adding different types of market
participants. For example, future work could investigate the effect of adding
flexible loads to the system. It is furthermore likely that future district
heating systems will include other excess heat producers that are not cooling
based, such as energy intensive industries.

## Acknowledgment

This work is partly supported by EMB3Rs (EU H2020 grant no. 847121). The
authors would like to thank Wiebke Meesenburg and Torben Schmidt Ommen from
DTU Mechanical Engineering for discussion of convex heat pump modeling, and
for the supply of temperature dependent COP profiles. We thank Tore Gad Kjeld
of HOFOR for a constructive correspondence and kind provision of data, and
Pierre Pinson for his inputs on an early version of this work. Finally, we
thank two anonymous reviewers for their valuable feedback.

We derive the price bids and feasible regions $\mathcal{F}$ for CHPs and
excess heat producers in Appendices A and B, respectively.

##  Appendix A CHPs

Our model of CHP plants, including their bidding behavior, is identical to the
sequential decoupled formulation in [13]. As it is the current practice in
Copenhagen, we assume sequential heat and electricity markets, where the heat
market is cleared first.

The fuel intake $F_{it}$ of CHP $i$ is equal to the fuel used for electricity
generation $G^{\mathrm{E}}_{it}$ and heat generation $G^{\mathrm{H}}_{it}$.
The fuel generation is upper bounded as

$\displaystyle
F_{it}=\rho_{i}^{\mathrm{E}}G^{\mathrm{E}}_{it}+\rho_{i}^{\mathrm{H}}G^{\mathrm{H}}_{it}\leq\overline{F}_{i}$
$\displaystyle\forall i,t\,.$ (13)

Recall that $\rho^{\mathrm{E}}_{i}$ and $\rho_{i}^{\mathrm{H}}$ represent the
fuel efficiency for electricity and heat, respectively. A minimum power-to-
heat ratio $r_{i}$ relates heat and electricity production as

$\displaystyle G^{\mathrm{E}}_{it}\geq r_{i}G^{\mathrm{H}}_{it}$
$\displaystyle\forall i,t\,.$ (14)

Both heat and electricity generation must be non-negative, but due to (14)
limiting heat generation suffices.

$\displaystyle 0\leq G^{\mathrm{H}}_{it}\,.$ (15)

If drawn in a diagram with $G^{\mathrm{E}}_{it}$ on the y-axis and
$G^{\mathrm{H}}_{it}$ on the x-axis, constraints (13)-(15) form a triangle
with the y-axis as a base. At the tip of the triangle, the amount of generated
heat is

$\displaystyle
G^{\mathrm{H,*}}=\frac{\overline{F}_{i}}{\rho^{\mathrm{H}}_{i}+r_{i}\,\rho^{\mathrm{E}}_{i}}$
$\displaystyle\forall i\,.$ (16)

Additionally, the heat generation of a CHP may be upper bounded as

$\displaystyle G^{\mathrm{H}}_{it}\leq\overline{G}^{\mathrm{H}}_{i}$
$\displaystyle\forall i,t\,.$ (17)

As the heat market is cleared before the electricity market, constraints
(13)-(17) can be replaced by the following bounds on the generated heat
quantity:

$\displaystyle 0\leq
G^{\mathrm{H}}_{it}\leq\min\left\\{\overline{G}^{\mathrm{H}}_{i},\frac{\overline{F}_{i}}{\rho^{\mathrm{H}}_{i}+r_{i}\,\rho^{\mathrm{E}}_{i}}\right\\}$
$\displaystyle\forall i,t\,.$ (18)

Recall that we have already provided the feasible space for CHPS resulting
from this constraint in Section 2.4 in Equation (8).

The net heat production cost for a CHP is given by the difference of fuel cost
and revenue from electricity sale:

$\displaystyle
C^{\mathrm{H}}_{it}=\alpha_{i}\,(\rho_{i}^{\mathrm{E}}G^{\mathrm{E}}_{it}+\rho_{i}^{\mathrm{H}}G^{\mathrm{H}}_{it})-\lambda_{t}^{\mathrm{E}}G_{it}^{\mathrm{E}}$
$\displaystyle\forall i,t\,,$ (19)

where $\alpha$ is the fuel price. In [13], the optimal heat bid
$c^{\mathrm{H}}_{it}$ is derived for CHPs in a sequential heat and electricity
market setting. The price bid depends on the (forecasted) electricity price as

$\displaystyle
c^{\mathrm{H}}_{it}=\begin{cases}\alpha_{i}\,(\rho^{\mathrm{E}}_{i}\,r_{i}+\rho^{\mathrm{H}}_{i})-\lambda^{\mathrm{E}}_{t}\,r_{i}&\text{if
}\lambda^{\mathrm{E}}_{t}\leq\alpha_{i}\rho_{i}^{\mathrm{E}}\\\
\lambda^{\mathrm{E}}_{t}\,\frac{\rho^{\mathrm{H}}_{i}}{\rho^{\mathrm{E}}_{i}}&\text{if
}\lambda^{\mathrm{E}}_{t}>\alpha_{i}\rho^{\mathrm{E}}_{i}\,.\end{cases}$ (20)

##  Appendix B Excess heat producers

We assume that all excess heat is produced as a by-product of cooling. In
particular, we assume these agents cool their refrigeration cabinets using a
local heat pump. The heat pump’s heat output $G^{\mathrm{H}}_{et}$ relates to
its electricity load $L^{\mathrm{E}}_{et}$ as

$\displaystyle G^{\mathrm{H}}_{et}=\mathrm{COP}_{et}L^{\mathrm{E}}_{et}$
$\displaystyle\forall e,t\,,$ (21)

where the coefficient of performance $\mathrm{COP}_{et}$ is a time-varying
parameter in our model. This allows us to include its approximate ambient
temperature dependence. The heat output of the heat pump is subject to upper
and lower bounds:

$\displaystyle 0\leq G^{\mathrm{H}}_{et}\leq\overline{G}_{e}^{\mathrm{H}}$
$\displaystyle\forall e,t\,.$ (22)

The refrigeration cabinets are the heat source of the heat pump. This implies
that the available heat depends on the temperature dynamics in these cabinets.
We model the refrigerator temperature $T^{\mathrm{F}}_{et}$ using a linear
difference equation as

$\displaystyle
T_{et+1}^{\mathrm{F}}-T_{et}^{\mathrm{F}}=A_{e}\,(T_{et}^{\mathrm{I}}-T_{et}^{\mathrm{F}})-B_{e}\,(G_{et}^{\mathrm{H}}-L_{et}^{\mathrm{E}})$
$\displaystyle\forall e,t\,,$ (23)

where $T^{\mathrm{I}}$ is the indoor temperature in the supermarket. The
parameters $A_{e}$ and $B_{e}$ may differ per excess heat provider, depending
on the physical characteristics of the refrigerators. The refrigerator
temperature is subject to bounds:

$\displaystyle\underline{T}_{e}^{\mathrm{F}}\leq
T_{et}^{\mathrm{F}}\leq\overline{T}^{\mathrm{F}}$ $\displaystyle\forall
i,t\,.$ (24)

The average temperature of the refrigerator over chosen time periods
$\mathcal{P}$ must also stay within pre-set limits:

$\displaystyle
T^{\mathrm{F-}}_{e}\leq\frac{1}{|P|}\,\sum_{t\in{P}}T_{et}^{\mathrm{F}}\leq
T^{\mathrm{F+}}_{e}$ $\displaystyle\forall e,P\in\mathcal{P}\,,$ (25)

which ensures that the refrigerator temperature will not be on lower or upper
bounds for longer periods of time. The periods must be defined such that
$\bigcup_{P\in\mathcal{P}}P=\mathcal{T}$, so that all time steps are part of
at least one period. Finally, the heat output from the heat pump is subject to
ramping limits:

$\displaystyle\underline{R}_{e}\leq
G^{\mathrm{H}}_{et+1}-G^{\mathrm{H}}_{et}\leq\overline{R}_{e}$
$\displaystyle\forall e,t\,.$ (26)

Recall that we already provided the feasible space for excess heat producers
resulting from the previous constraints in Section 2.4 in Equation (10).

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