Optimal Scheduling of Heat Pumps for Power Peak Shaving and

Customers Thermal Comfort

Jochen L. Cremer, Marco Pau, Ferdinanda Ponci and Antonello Monti

Institute for Automation of Complex Power Systems, E.ON Energy Research Center - RWTH Aachen University,

Mathieustrasse 10, 52074 Aachen, Germany

Keywords:

Demand Side Management, Heat Pump S cheduling, Power Peak Shaving, Load Flexibility, Load Balancing,

Mixed Integer Linear Programming.

Abstract:

Final customers are expected to play an active role in the Smart Grid scenario by offering their ﬂexibility to

allow a more efﬁcient and reliable operation of the electric grid. Among the household appliances, heat pumps

used for space heating are commonly recognized as ﬂexible loads that can be suitably handled to gain beneﬁt

in the Smart Grid context. This paper proposes an optimization algorithm, based on a Mixed-Integer Linear

Programming approach, designed to achieve power peak shaving in the distr ibution grid while providing at

the same time the required t hermal comfort to the end-users. The developed model allows considering a

continuous operation mode of the heat pumps and different comfort requirements deﬁned by the users over the

day. Performed simulations prove the proper operation of the proposed algorithm and the technical beneﬁts

potentially achievable t hrough the devised management of the heating devices.

1 INTRODUCTION

With the evolution towards the Smart Grid (SG) para-

digm, new technologies and applications will be put

in place to obtain a more efﬁcient, reliable and sustai-

nable utilization of the electric system assets. Some of

the most important changes concern the distribution

grid, where the p enetration of Distributed Generation

(DG) and other D istributed Energy Resourc es (DERs)

requires novel management tools to deal with the in-

creasing complexity of the network (Fan and Borlase,

2009). Differently from the past, end-users are also

expected to play an active role in the SG scen a rio.

Many customers already evolved into the so-called

prosumers, thanks to the installation of photovoltaic

panels or small wind turbines in their household pre-

mises. From on e side, this goes in the direction of

a more environmentally friendly system, on the other

hand it also enables a better use of the network infra-

structure if these resour c es are suitably managed.

Customers’ role, however, is not only limited to

the in stallation of generation units based on renewa-

ble energy sources, but also includes the possibility

to support the grid operation by offering ﬂexibility

in the power demand. Th e exploitation of the ﬂexi-

bility available on the custome r side has been a hot

research topic in the last years. Several Demand Re-

sponse (DR) and Demand Side Management (DSM)

models have been designed to achieve economic be-

neﬁts o r speciﬁc technical goals through the control of

different appliances (Balijepalli et al., 2011; Cap rino

et al., 2014; Klaa ssen et al., 2016a). Even though

many challenges still prevent a wide diffusion of DR

and DSM (such as the lack o f a suitable regulato ry

framework, or the absence of the metering and com-

munication infrastructure ), the beneﬁts d eriving from

the application of these schemes are well recognized

(Strbac, 2008). As a consequence, it is foreseeable

that such applications will play a relevant role in fu-

ture SGs.

Nowadays, DR schemes are already deployed and

well established in the U.S. (US DoE, 2006). From a

market perspective the existing programs can be divi-

ded in two main categories:

• Price-based pro grams: customers are motivated to

change their d e mand pattern in r esponse to day

ahead or real-time price signals. Acc ording to this

model, utilities or energy agg regators can not di-

rectly act on end- users appliances but they mo-

tivate people to change their power consumption

habits u sually by offering higher prices in peak

hours and lower prices during off-peak hou rs.

• Incentive-based prog rams: customers provide to

utilities or energy aggregators the possibility to di-

Cremer, J., Pau, M., Ponci, F. and Monti, A.

Optimal Scheduling of Heat Pumps for Power Peak Shaving and Customers Thermal Comfort.

DOI: 10.5220/0006305800230034

In Proceedings of the 6th International Conference on Smart Cities and Green ICT Systems (SMARTGREENS 2017), pages 23-34

ISBN: 978-989-758-241-7

rectly co ntrol or schedule some of their appliances

and are rewarded f or this ser vice through speciﬁc

incentives in th e tariff scheme. In this case, thu s,

the DR program provider can manage the ﬂexible

loads allowed by the customer following his own

needs, wh ile fulﬁlling some customer comfort re-

quirements if this is speciﬁed in th e agreemen t.

According to (FERC, 2011), DR programs de-

ployed in U.S. unlock a potential power peak re-

duction larger than 53 GW. More than 8 0% of

this peak reduction comes from incentive-based pro-

grams. This solution, despite being more invasive

with respect to the price-based alternatives, allows

an optimum management of the load ﬂexibility lea-

ding to the certain achievement of the de sired targets.

Given the invasiveness of these sch e mes, incentive-

based DR is usually implemented to control not cr iti-

cal shiftable or interruptible loads, such as heating de-

vices, air conditioners and water heaters. In E urope,

DR and DSM p rograms are still at an early stage. This

is mainly because of the heterogeneity of the regu-

latory framework in the different countries and, so-

metimes, also within the same co untry. Nevertheless,

these services are r ecently being proposed more in-

sistently and DR is regarded as a key tool to achieve

the targets of at least 27% for renewable energy and

energy savings by 2030 (SEDC, 201 4).

This paper proposes an optimization algorithm

conceived to exploit the ﬂexibility provided by he-

ating devices, like heat pumps. Electro-thermal de-

vices are in fact becoming more and more used for

space heating, also thanks to th e support of recent re-

gulations aimed at improving the energy efﬁciency in

the residential sector. Thanks to the relatively slow

dynamics of thermal phenomena, electric heat pumps

can be o perated ﬂexibly, thus offering a great poten-

tial for the de ployment of DSM an d DR schemes de-

signed for their management (Arteconi et al., 2013).

The goa l of the optimization algorithm here presented

is twofold. The main objec tive is to minimize th e po-

wer peaks on the grid, but the hea t pumps scheduling

is performed also in order to gua rantee the thermal

comfort required by the end-users.

In the following, Section 2 shows how the ﬂexibi-

lity given by heat pumps can be used for DSM purpo-

ses and points out the differences between the prop o-

sed approach and those already available in literatu re.

In Section 3, the designed optimization algorithm is

presented and the constraints taken into account in the

used model are described. Section 4 presents the ap-

plication of the proposed optimization algorithm in

different case studies, high lighting the technical be-

neﬁts potentially achievable through the devised heat

pumps management. Sectio n 5 ﬁnally sum marizes the

obtained resu lts a nd con cludes the pape r.

2 USE OF HEAT PUMPS FOR

DEMAND SIDE MANAGEMENT

The ﬂexibility provided by heating systems has been

studied and evaluated in several works, proving that a

large p otential exists for the application of DR sche-

mes based on the management of electro-thermal de-

vices (Klaassen et al., 2016b; Chapman et al., 2016).

As a consequence, large efforts h ave been focused

on this research ﬁeld, dealing with different aspects

like the modelling of th e the rmal system (Good et al.,

2013; Akmal and Fox, 2016) or the estimation of the

heating demand (Kouzelis et al., 2015) in order to de-

sign tailored DR schemes. Many of the DSM and DR

programs proposed in the literature refer to the p rice-

based model a nd aim at minimizing the costs incu r-

red by the ﬁnal customer. Th erefore, the developed

models are usually conceived as a service to the cu -

stomer, while the utilities can address their needs (in

terms of grid management) by sending different price

signals over the time an d relying on the response of

the users to the varying prices.

In (Molitor et al., 2011), different price schemes

are used as input to an optimization algorithm run-

ning at the end-user premises for the scheduling of

heat pum ps. Results show that the optimal schedu-

ling leads to a reduction of the energy consumption of

the customer, but this is obtained at the expense of a

thermal discom fort. In (Lo esch et al., 2014), an evo-

lutionary algo rithm is proposed to schedule the he a t

pump so to minimize the costs for the user, given the

price of the energy in the spot market. Here utilities

can also deﬁne power limitations in speciﬁc periods

of the day for solving possible contingencies in their

grid and charge penalties to the custom er if such li-

mitations are not respected. The algorithm is able to

exploit the ﬂexibility provided b y the heating system

and to min imize the user costs, but a direct link to the

thermal comf ort delivered to the customer is missing.

The proposal in (Bhattarai et al. , 2014) also tries to

combine the objective of minimizing the costs for the

customer with a service that is orie nted to the distribu-

tion grid management. A two-step optimization pro-

cess is presented, where the ﬁrst step gives the sche-

duling of the he a t pumps ( minimizing the costs) while

the second step checks possible voltage problems in

the grid a nd, in case, shifts the he at pumps operation

to the following time slots. Again, customer discom-

fort is in general possible in case of realloca tion of

the heat pumps oper ation. Hea t pump ﬂexibility is

directly used to improve the operation of distribution

SMARTGREENS 2017 - 6th International Conference on Smart Cities and Green ICT Systems

grids in (Csetvei et al., 2011). A method to deﬁne lo-

cal price signals for the end-users is presented, where

additional costs are added to the spot mar ket prices if

overload conditions exist. The price signals are then

used to determine the set point temperature of the heat

pumps. The method allows eliminating the overloads,

but customer d iscomfort can still arise during over-

load periods.

To avoid thermal discomfort for the end-user,

some proposals include in the optimization model

constraints on the indoor temperature provided to the

customer. In (De Angelis et al., 2013), temperature

boundaries are considered in a h ome energy ma nage-

ment system which is used to schedule th e operation

of ﬂexible loads (including heat pumps) and possible

storage systems. The objective is to reduce the costs

for the customer, so utilities can pursue their goa ls

only by settin g different pr ic e signals over the time. In

(Nielsen et a l., 20 12), in stead, the price-based scheme

is comp a red to two d ifferent DR approaches where

the power consumption or the temperature set point

of the heat pump are directly controlled by the DR

provider. The objective is in this case to minimize the

costs for the energy aggregator ( which is providing

the DR program), while minimizing the discomfort

for the custom e rs by keeping their home temperature

between the consid ered boundaries. Similarly to the

case of th e heat pump s, (Li et al., 2017) propose an

algorithm to manage air conditioners by acting on the

temperature set points in order both to reduce energy

consumption and to provide the interruptibility of the

load as DR service. In this case, no ﬁxed temperature

boundaries are used, but the control scheme was tes-

ted in the ﬁeld and tuned ac cording to the custome rs

feedback in order to minimize their thermal discom-

fort.

All these approaches, while proposing solutions to

make DR and DSM programs more attractive for the

ﬁnal customer, do not a llow to fully exploit the availa-

ble ﬂexibility for enhancing the efﬁciency of the elec-

tric system operation. A s described in (Strbac, 2008)

and reported in (US DoE, 2006), one of the main be-

neﬁts for the system would be the power peak mi-

nimization. By minimizing the power peaks in the

grid, utilities can min imize power losses, improve the

voltage proﬁle in the grid, red uce the risk of contin-

gencies and postpone network reinforcement in ar eas

with increasing connected power. At system level,

this also leads to avoid the use of expe nsive genera-

tion units du ring peak hours and to reduce the needed

spinning reserve, thus minimizing the overall costs.

For this reason, d ifferently from the o ther propo-

sals available in the literature, the DSM model here

presented perfor ms an optimal day ahead scheduling

Figure 1: Example of user-deﬁned comfort requirement.

of the heat pumps for minimizing the power peaks on

the grid over the da y. The minimization is performed

by taking into account user-deﬁned requirements in

terms of thermal comfo rt, so that both utilities and

end users can take advantage from the proposed DSM

program. Further reward to the ﬁnal cu stomers could

be also deﬁned, in ter ms of incentives in their tariff,

to make the DSM scheme more a ppealing, depen ding

on the savings th e utilities estimate to achieve from

the application of this o ptimization on a large scale.

3 MODEL FORMULATION

This section presents the formulation used for the p ro-

posed DSM model. First, the thermal model, con-

sisting of comfort c onstraints, boundary constraints,

energy balance equations and heat pump equati-

ons/constraints, is described. Then, the optimization

algorithm de signed to perform the day a head schedu-

ling of the heat pumps in the con sidered grid is pre-

sented.

3.1 Thermal Model

Comfort constraints They are used in the model to

guaran tee the comfort requ irements o f the residents

living in each house. In the proposed DSM scheme,

users choose the referenc e temperature they want to

have (it can also vary during the day ) and pr ovide

a certain boundary around su ch referen c e tempe ra-

ture. Figure 1 shows an example of possible tempe-

rature requirement for a customer. The comfort con-

straints are thus deﬁned so that, for every time perio d

t, the indoor temperature is always within the permit-

ted range. Indicating w ith Γ

h,t

and Γ

h,t

the lower a nd

the upper bound, respectively, of the temper a ture in

house h at tim e t, the following holds:

h,t

≥ Γ

h,t

∀h, t (1a)

h,t

≤ Γ

h,t

∀h, t (1b)

Optimal Scheduling of Heat Pumps for Power Peak Shaving and Customers Thermal Comfort

where T

h,t

is the variable associated to the indoor

temperature of house h at time t.

Boundary constraints They are added to deﬁne the

initial and ﬁnal states of th e temperature for the daily

optimization. Given a starting temperature Γ

INI

, the

temperature at time t = 0 is:

h,0

= Γ

INI

∀h

(2)

while at the ﬁnal time period f , the indo or tempera-

ture is bounde d with th e inequality constraint:

h,f

≥ Γ

REF

h,f

∀h,

(3)

where Γ

REF

h,f

+Γ

h,f

is the reference temperature

of house h at t = f. Such a choice is done in order

not to have a ﬁnal temperature too close to th e lower

bound, since this would force to turn on the heat pump

at the beginning of the following day (thus removing

any ﬂexibility for the ﬁrst time steps o f the subseq uent

day ah ead scheduling ).

Energy balance The energy b alance e quation deﬁ-

nes how the ind oor temperature changes over the time

due to the heat p rovided by the heat pump and the heat

loss to th e outdoor environment. The used equation is

based on the model described in (De Angelis et a l.,

2013) and it is:

h,t

= T

h,t−1

∆t



h,t

− Q

h,t



∀h, t

(4)

where ∆t is the duration of the tim e period between

two consecutive discrete time steps, µ

and γ

are

speciﬁc parame te rs, na mely the house in door a ir mass

and the air heat capacity, and Q

h,t

and Q

h,t

are varia-

bles indicating the heat ﬂow given by the heat pu mp

and the heat loss, respectively.

The indoor air mass µ

is a parameter that de-

pends on the size and geometrical characteristics of

the ho use (see (De Angelis et al., 2013) for more de-

tails) an d, combined with the air heat capacity γ

, ap-

pears as a thermal energy storage for the house, thus

affecting the dynamics of the thermal phenomena .

The heat losses are instead deﬁned through the fol-

lowing relationship:

h,t

= κ

h,t−1

− Γ

h,t−1

) ∀h, t (5)

Such losses depend on a heat loss facto r κ

and on

the temper a ture difference between the indoor and the

outdoor temperature Γ

h,t−1

As for Q

h,t

, more details will be provided in the

following p aragraph where the used heat pump model

is fully described.

500

1000

1500

2000

400 500 600 700 800 900

Power [W]

Air mass flow [kg h

-1

]

Continuous HP operation Binary HP operation

Figure 2: Power demand of the heat pump i n binary or con-

tinuous mode.

Heat pump model This model has to link the deli-

vered heat Q

h,t

to the electrical power P

h,t

required

to produce such heat, and has to accoun t for all the

possible constraints present in the heat pump opera-

tion. In the literature, heat pumps are often conside-

red to work at a ﬁxed power and thus a simple binary

variable is adopted to deﬁne if their status is on or off.

In some papers, a multi-op eration mode is instead de-

ﬁned by considering different discr ete air mass ﬂows

to which different electrical powers are consequently

associated. In this case, binary variables are intro-

duced for each d iscrete operation mode, hence deter-

mining an inc reasing complexity of the optimization

problem. In this p a per, a continuous operation mode

of the heat pump is considered. T his mean s that the

heat pump can generate any value of air mass ﬂow

included in the range be tween a minimum and a max-

imum lim it. The elec trical power needed to generate

the output a ir mass ﬂow can be described through a

function, whic h can be in ﬁrst approximation lineari-

sed through a given number of lin ear segments. Fi-

gure 2 shows an example of linearised curve map-

ping the air mass ﬂow to the r e quired electrical power,

which has been obtained using heat pump data given

in (De Angelis et al., 2013).

In Figure 2, it is possible to observe that th ree ope-

ration modes are deﬁned: the ﬁrst one, named m0, is a

discrete value corresponding to the minimum air mass

ﬂow of the heat pump; the second one, m1, is associ-

ated to the ﬁrst segment o f the curve; the la st o ne,

called m2, is linked to the upper segment o f the curve

and ar rives till the maximum air mass ﬂow for the heat

pump. As it will be shown in the following, such a

solution can be implemented in the optimization al-

gorithm by using integer variables for each operating

mode, while just one bin ary value is used to deter-

mine the status (on or off) of the heat pump. Figure 2

also shows the possible limits present in the deﬁnition

of a simple binary operation mode for the heat pump.

In fact, in su c h a case a single operating point of the

heat pump has to be decided, which does not reﬂect

the actual operation mode of many heat pumps.

SMARTGREENS 2017 - 6th International Conference on Smart Cities and Green ICT Systems

Relying on the described continuous operation,

the generated heat is deﬁned as:

h,t

= γ

∑

∆F

h,m,t



− Γ

h,t−1



∀h, t

(6)

where Γ

is th e output temperature of the heat pump

(assumed as constan t) and Γ

h,t−1

is th e reference tem-

perature of the house h at the time step t − 1. It is

worth noting that a rigorous deﬁnition of the genera-

ted heat Q

h,t

would require the use of the actual ind-

oor temperature T

h,t−1

in ( 6) in place of the refer e nce

temperature Γ

h,t−1

. However, su c h a solution would

lead to a nonlinear relationship and for th is r eason it is

here approximated by usin g the constant value g iven

by Γ

h,t−1

. This approximatio n is considered accepta-

ble since the indoor temperature is constrained to be

close to the refe rence temperature due to the comfort

constraints p reviously deﬁned.

The other term appearing in (6), namely ∆F

h,m,t

, is

the additional air ma ss ﬂow of mode m with respect

to the upper bound a ir mass ﬂow of mode m − 1. As

for the ﬁrst operating mode m0, the air mass ﬂow is

constrained b y the following equality constraint:

∆F

h,m0,t

= y

h,t

h,m0

∀h, t. (7)

where Φ

h,m0

is the minimal air m ass ﬂow o f the h eat

pump. The air mass ﬂow of mode m0 is thus either

0 or Φ

h,m0

depending on th e binary decision variable

h,t

. The additional air mass ﬂows of all other opera-

ting modes m are in stead constrained by the following

inequality constraints:

∆F

h,m,t

≤ y

h,t

∆Φ

h,m

∀h, m /∈ {m0}, t

(8)

where ∆Φ

h,m

is the upper bound of the additiona l

air mass ﬂow of the linearised segment associated to

mode m.

Given these deﬁnitions of the additional air mass

ﬂows, the required power of the heat pump is directly

mapped to the air mass ﬂow by means of the follo-

wing equation:

h,t

∑

∆F

h,m,t

∀h, t

(9)

where the parameter β

is the power per air mass ﬂow

associated to each mode m. The total power of the

heat pump is thus determined by taking the sum of all

the additional air m ass ﬂows ∆F

h,m,t

multiplied by the

respective pa rameter β

over all the operating modes

m. The propo sed formulatio n works properly for in-

creasing values of β

(β

≤ β

...) as in Fi-

gure 2. In fact, since the following optimization o pe-

rates to minimize the used powers, this ensures that

the modes will be automatically selected by the sol-

ver in the order m = {m0,m1,m2... } .

A further aspect considered in heat pump model is

the possible presence of time constraints. These con-

straints acco unt for the minimum (or maximum) ti-

mes the equipment has to operate or have to be turned

off since, usually, m any operational switches result

in inefﬁciency and mechanical stress. In (Hedm an

et al., 2009) , several different methods to account for

time constrain ts in another scheduling problem (the

unit commitment problem) were proposed and exa-

mined. Results of such work are here adapted to the

heat pump scheduling problem. In this case, only a

minimum number of time periods τ, during which the

heat pump has to be turne d on, is implemented (e.g.,

no minimum turn-off time) by the two following con-

straints:

h,t

≥ y

h,t

− y

h,t−1

∀h, t

(10)

h,t

≤y

h,τ

∀h, t, τ ∈ {t,··· ,min(t + τ

MIN

− 1, f)}.

(11)

Note, th e switching variable 0 ≤ Z

h,t

≤ 1 is a boun-

ded continuo us variable. Therefore by using these

time constrain ts, the intr oduction of new binary va-

riables is not required. More binary variables would

result in a larger branch and bound tree and thus in a

more complicated problem. As a result, the complex-

ity of the problem decreases by using the inequality

constraints proposed in (10) and (1 1).

3.2 The Optimization Algorithm

As discussed in Section 2, the objective of the DSM

here proposed is to minimize the power peaks in the

grid. To achieve this target, Q uadratic Programming

(QP) could be used to m inimize the squared power

resulting on the monitored network over all the time

periods. However, if binary variables are included in

the problem, QP approaches lead to very high compu-

tational burden and execution times. For this reason,

in the proposed approach, the objective function has

been linearised as pr esented in the following. This,

together with the linear constraints deﬁned in Sectio n

3.1, allows obtaining a linear problem that can be sol-

ved more easily through a Mixed Integer L inear Pro-

gramming (MILP) formulatio n. In this way, execu-

tion time s can be reduced, which is an essential as-

pect when d e aling with large optimiza tion p roblems

(in this scenario, when optim izing the heat pump ope-

ration of a large numbe r of houses).

The basic idea used here to linearise the objective

function is to discretize the power consumption a t

time t through a given number of blocks b and to as-

sign increasing weights to blocks associated to hig-

her levels of power (see Figure 3); in this way, the

minimization of the weighted blocks lead s to avoid

Optimal Scheduling of Heat Pumps for Power Peak Shaving and Customers Thermal Comfort

Figure 3: The weight α

of the energy ∆E

b,t

in box b.

the allocation of ﬂexible load consumption in periods

where power peaks are occurring. These blocks can

be interpreted as boxes tha t can be ﬁlled with energy

up to their respective capacity ε

. Each energy box

b is th us a continuous variable (indicated in the follo-

wing with ∆E

b,t

) that is lower bounded by 0 and upper

bounded through this inequality constraint:

∆E

b,t

≤ ε

∀t

(12)

where ε

is the maximum capacity o f the energy

box, which, in general, can be different for each block

For each time step t, the su m of all the energy

boxes is related to the power consumption in that pe-

riod b y means of:

∑

∆E

b,t

≥

∑

∆tP

h,t

+ ε

∀t

(13)

where ε

is the energy consumptio n at time t given

by all the non-scheduled loads in the gr id and P

h,t

the already m e ntioned power consumption of the heat

pumps for each house h.

Given the above deﬁnition of the energy boxes

and c onsidering all the constra ints introduced in the

problem, the optimization used to schedule the heat

pumps is a centralized algorithm with the following

objective function:

minimize

h,t

,∆F

h,m,t

∑

∆E

b,t

s.t. Eqs. (1) − (13).

where the optimiza tion decisions ar e the binary va-

riables y

h,t

and the continuous variables ∆F

h,m,t

. As

it can be observed, the designed alg orithm is thus a

centralized approach where the heat pumps of ea ch

house included in the problem are scheduled within

the same DSM optim iz ation procedure. Similarly

to the case of the additional air mass ﬂows, for the

Table 1: Parameters of the heat pump.

Mode m m0 m1 m2

(Whkg

−1

) 0.939 1.86 3.70

∆Φ

(kgh

−1

) 426 264 178

proper functioning of the method it is crucial that

the weight α

is increasing (α

≤ α

...).

In this case, indee d, the boxes will be selected (or

’ﬁlled with energy’) by the solver in the order b =

{b1,b2,b3... }. The box approach is rea sonable since

the target is only the cut of the hig hest peak. This

approa c h allows to be tailored to the considered sce-

nario. For example, the discretization in the energy

level can be modiﬁed, or any arbitrary strong functi-

ons (e.g. , exponential to the power x, etc.) can be

linearized by setting the values of the weights α

ac-

cordingly. Differently from other proposals available

in literature and, in general, from p rice-based DSM

schemes, the proposed centralized approach also al-

lows avoiding that possible high power peaks are sim-

ply shifted fro m a time to another due to th e similar

response of the customers to the DSM inputs.

4 TESTS AND RESULTS

4.1 Tests Setup

The proposed optimization algorithm has been tested

considerin g different scenarios whe re the DSM pro-

vider wants to minimize the power peak of the grid

using the ﬂexibility provided by 60 residential houses

endowed with electric heat pumps. Th e time horizon

for the scheduling is one day. The initial time of the

scheduling problem is midn ight a nd the day is separa-

ted in 96 time periods resulting in a discretization time

step o f ∆t = 15min. For the sake of simplicity, in the

simulation it is assumed that all houses have the same

heat pum p that can contin uously operate in 3 different

modes. However, the algorithm obviously allows for

the implementation of heat pump s with different cha-

racteristics for each house h. The paramete rs of the

heat pump model are stated in Table 1 and are derived

from (De Angelis et al., 2013). It is worth reminding

that mode m0 is the operating start point, while the

linear operating segments m1 an d m2 offers c ontinu-

ous operation of the heat pump as depicted in Fig ure

2. The output temperature of the heat p ump ha s bee n

chosen as Γ

= 30

◦

C and the minimal time period

the heat p ump has to run is τ = 2 (cor responding to a

minimal o peration time of 30min).

As shown in Figure 4, in the proposed DSM

scheme, the inputs needed for the optimization algo-

rithm are :

SMARTGREENS 2017 - 6th International Conference on Smart Cities and Green ICT Systems

• a foreca st of the inﬂexible load in th e grid

• a foreca st of the outdoor temperature

• the thermal comfort required by the customers, to-

gether with heat pumps and building characteris-

tics.

As for the inﬂexible load proﬁle in the grid, statis-

tical data are often available (for example at substa-

tion level) regarding the aggregated power co nsump-

tion in different periods of the year and for different

types of day (e.g. working or weekend day). In the

following simulatio ns, the aggregated proﬁles of re-

sidential houses have been taken from the standard

load proﬁle of 2012 (Bundesverband der Energie- und

Wasserwirtscha ft) using an average consu mption of

2000 kWh/year p e r customer. Two different perio ds

of the year, n amely a working day in May and one in

December, have been simulated, and the correspon-

ding load proﬁles have been assumed as inﬂexible

load for the residential customers. I n addition, the

presence of ind ustrial consumers has been also consi-

dered. This contributes to give the ﬁnal shape of the

forecast inﬂexible load, as it will be shown in the next

subsection when presenting the simulated scenarios.

For the fo recast of the outdoor temperature, the

actual temperature of a day in May is used in a ﬁrst

simulation, while the actual temperature of a day in

December is used to simulate a second scenario. The

used temperature proﬁles are presented in Figure 5.

The thermal comfort of the 60 houses differs from

house to house and individual paramete r sets (as des-

cribed in the previous section ) have to be taken into

account. For the case stu dies presented here, the re-

levant parameters are sampled ba sed on 5 different

temperature proﬁles and 12 different building charac-

teristics. Figure 5 shows the 5 different temp erature

proﬁles (the reference temperature is always the mean

Figure 4: Overall model of the designed DSM scheme.

0 2 4 6 8 10 12 14 16 18 20 22

Temperature [°C]

Time [h]

lower bound

outdoor temperature

upper bound

Figure 5: U pper and lower temperature bounds of the resi-

dents and outdoor temperature.

value of the upper and lower bound). As starting point

for the simulation, the initial indoor temp erature Γ

INI

is assum ed to be equal to the ref e rence tempe rature of

the ﬁrst time period. The 12 building types differ in

the indoor air mass and the heat loss factor (Figure 6 ).

The parameters are calculated based on the geometric

dimensions of the h ouse (De Angelis et al., 2013).

As an example, let us consider a house having the

length ξ

= 20m, ξ

= 20m, the height ξ

= 4m,

a roof pitch of σ

= 40

◦

and η

W I

= 6 windows,

each one with a n area of Λ

W I

= 1m

. The the r-

mal transmittanc e fo r walls and windows are assumed

= 0.15Wm

−2

−1

and ν

W I

= 1 Wm

−2

−1

, re-

spectively. The heat loss factor in this example house

is calculated as follows:

= ν



2 (ξ

+ ξ

) ξ

− η

W I



+ η

W I

= 191.16kJh

−1 ◦

(14)

By using the density of the air ρ

= 1.2041kgm

−3

at standard conditions, the total air mass is:

= ρ



+ 0.25 ξ





tan(σ

)



= 3946kg.

(15)

For all the 60 resid ential ho uses, the indoor air

masses and h eat loss factors are presen te d in Figure

0 100 200 300 400 500

Indoor air mass [10

kg]

Heat loss factor [kJ h

-1

°C

-1

]

Figure 6: Indoor air masses and heat loss factors of the re-

sidential houses.

Optimal Scheduling of Heat Pumps for Power Peak Shaving and Customers Thermal Comfort

6. The set of houses used in the simulation has been

obtained using all the possible combinations between

the 5 thermal comfort proﬁles (Fig. 5) and the 12 dif-

ferent house characteristics (Fig. 6 ).

During the presentation of the test re sults, the be-

neﬁts provided by the proposed DSM model are ana-

lyzed by comparing the results of the describ ed op-

timization algorithm to those of two different simu-

lations. In the ﬁrst case, the term of comparison is

given by a simulation where the target of the internal

control system of the heat pump is to keep the ind-

oor temperature as close as possible to the referen ce

temperature Γ

h,t

for all the time periods. This simu -

lation has been run separately for each HP h by using

a QP approach that minimizes the squ ared difference

of the indoor temperatur e of the house with respect to

the reference temperature selected by the custom e r,

accordin g to:

minimize

h,t

,∆F

h,m,t

∑



h,t

− Γ

h,t



∀h

s.t. Eqs. (1) − (11)

(16)

This c omparison aims at highlighting the advantages

offered by the proposed DSM scheme with respect to

a scenario in which no DSM is applied. In the follo-

wing, this opera tion mode of the HP will be referred

to as “internal HP control”.

In the second case, th e DSM model has been ap-

proxim ated by using the same model presented in

Section 3 but excluding multiple HP modes m. Thus,

HP operation is only valid in mode m = {m0} (by

using the equality constraint Equation (7)) and Equa-

tion (8) is not required any more in the optimiza-

tion. In this comparison, the binary operation of the

heat pump is selected to have an a ir mass ﬂow of

∆Phi

= 647kgh

−1

and a power per air mass ﬂow

= 1.25W h kg

−1

. Th is value is the mean air mass

ﬂow of the c ontinuous heat pump model. This sce-

nario allows showing the different results achievable

when considering a more realistic (continuous) ope -

ration m ode of the heat pump rathe r than a simpliﬁed

binary version.

4.2 Simulation Results

To assess the beneﬁts p rovided by the proposed DSM

scheme, a ﬁrst simu la tion scenario, using as input the

outdoor temperature of a day in May (see Fig. 5),

has been considered. In this test case, it is a ssumed

that the optimization has to be perf ormed in a portion

of a LV grid where all the 60 houses are equipped

with an elec tric heat pump. In addition, an indu s-

trial load is also taken into account, which operates

at {1 .5 kW, 8 kW} and switches with a p eriod of 4 h,

starting with 8 kW at midnight. This scena rio can be

representative, for example, of a distribution fee der

that supplies the simulated 60 houses.

At the household level, the results for an example

house are presented in Figure 7. The comparison of

the sche duled powers of the heat pump, for the case

of internal HP control and for the DSM with binary

and continuous HP operation mode, is presented in

the up per part of the ﬁgure, while the re spective ind-

oor temperatures are presented in the bottom part. In

the case of temperature minimization in the internal

control system of the heat pump, obviously the ind-

oor temperature follows c losely the reference tempe-

rature. It can be observed that mor e power is requi-

red in the morning, when the desired reference tem-

perature increases, and that the heat pump works re-

gardless of the loading c onditions of the grid. With

the DSM, since the optimization algorithm fo ster s the

power consumption in some time periods more than

in others, the full range of the speciﬁed temperature

bounds is used. However, it is possible to observe that

the te mperature always falls within the r ange accepted

by the customer. I n particular, mornin g hours (when

the loading of the grid is lower) are used to store ther-

mal energy in the house, while during peak hours the

operation of the heat pump is minimized in order not

to aggravate the situation in the grid (while providing

to the customer the required comfort).

The main differences between the binary and the

proposed contin uous HP oper ation mode are f rom an

energy consumption pe rspective. Indeed, it is possi-

ble to see that the continuous model leads to operate

the heat pump at lower power levels and for a longer

time during the day. This allows better modulating the

power before peak hours, when the storage of ther-

mal energy is needed, and during peak hours, when,

while re specting the customer thermal requirements,

the oper ation of the heat pump has to be minim ized.

In addition, operatin g the HP at its lower bound also

0 2 4 6 8 10 12 14 16 18 20 22

Temperature [°C]

Time [h]

DSM with binary HP DSM with continuous HP internal HP control

temperature bounds TUB

0.5

1.5

Power [kW]

Figure 7: Heat pump consumptions and temperature proﬁ-

les for one example house in the ﬁrst simulation scenario.

SMARTGREENS 2017 - 6th International Conference on Smart Cities and Green ICT Systems

Table 2: Results on the daily energy consumption, ﬁrst si-

mulation scenario.

case HP consumption increase

(kWh) (%)

Example household

DSM - continuous HP 4.19 -

DSM - binary HP 5.88 + 40.2

internal HP control 5.46 + 30.2

Overall scenario

DSM - continuous HP 333.6 -

DSM - binary HP 459.4 + 37.7

internal HP control 439.8 + 31.8

allows using the most efﬁcient operation points of

the HP, and this implies a signiﬁcant red uction in the

overall energy consumptio n for the end-user. Table 2

shows the r e sults related to th e energy consumption

for both the example household presented in Fig. 7

and for the overall scenario. It is possible to see that

a simpliﬁed binary mo del of the HP clearly leads to a

larger energy consumption, which may be not accep-

table for the ﬁnal cu stomers.

The results obtained at the grid level are shown in

Figure 8. Whereas for all the cases the inﬂexible in-

dustrial and residential loads are the same, the ﬂexible

parts differ depending on the HP scheduling. In this

scenario, all the houses are equipped with heat pumps,

so a large amount of ﬂexible energy is available. As

a consequence, the ﬁnal curve of aggregated power

is mainly determined by the allocation of this ﬂexible

energy, rather than by the shape of the ﬁxed load. In

the case of temperature minimization using the inter-

nal control of th e HP, large power peaks are obtained.

The reason for these peaks is the presence of simi-

lar comfort proﬁles for many customer s (see Fig. 5),

which leads to the simultaneous operation of the heat

pumps. Even though these peaks are originated by the

particular thermal requirements used for the test, this

kind of problem is likely in a scenario with large pe-

netration of e lectric HPs managed in a decentralized

way. In fact, end-users can have same r e quiremen ts

100

0 2 4 6 8 10 12 14 16 18 20 22

Power [kW]

Time [h]

fixed power industrial fixed power residents internal HP control

DSM with binary HP DSM with continuous HP

Figure 8: Aggregated power in the grid for t he ﬁrst simula-

tion scenario.

Table 3: Technical beneﬁts with the D SM at peak times,

ﬁrst simulation scenario.

case max. peak power at 6:00 HP cut

(kW) (kW) (%)

internal HP control 95.6 95.6 -

DSM - binary HP 45.8 43.8 60.9

DSM - continuous HP 37.8 32.7 73.9

in som e periods of the day (e.g. due to similar wor-

king hours or consequently to the weather conditions)

or price-based DSM programs can lead to a similar

reaction of the customers. This would bring the si-

multaneous operation of the hea t pumps, thus d eter-

mining a signiﬁcant impact on the aggregated power

demand . The use of a centralized optimization appro-

ach leads signiﬁcant beneﬁts in this perspective, allo-

wing to achieve power peak shaving. Fig. 8 c le a rly

shows that a much ﬂatter demand proﬁle is obtained

thanks to the application of the DSM. Table 3 reports

the numeric results for the maximum power peaks ori-

ginated by each HP control. It is possible to see that a

reduction of the p ower peak larger than 60% is obtai-

ned for the DSM with continuous HP operation mode.

Since the actu al po te ntial of the DSM scheme is only

to manage the HP power, Table 3 also shows the re-

sults in terms of ﬂexible energy that is shifted thr ough

the DSM to avoid the power peaks. Considering the

power peak time for the case of internal HP control,

almost 74% of the ﬂexible power can be realloca te d

through the DSM scheme (this reduction is calcula-

ted considering only the part of the load associated

to the HP operation). The continuous HP operation

mode provides larger improvements due to its ﬂexibi-

lity in choosing the HP ope ration point and its better

efﬁciency with respect to the binary HP.

To evaluate the potential of the proposed DSM

scheme even when less ﬂexible energy is available,

a second test case has been run considering a scena-

rio with 240 resid e ntial houses, among whic h only 60

are endowed with electric HPs. This test case could

be representative, for example , of a MV/LV substa-

tion that subtends four d ifferent feeders. Due to the

assumed scenario, also th e industrial load has been

scaled up to consider four feeders, and p ower levels

equal to {6 kW, 32kW} have been assumed using the

same operation cycles as the previous te st. While the

same considerations as the previous case hold when

looking at the single household, different results can

be found when considering the aggregated power at

grid level. Fig. 9 shows the obtained power proﬁ-

les for th e d ifferent HP operation modes. In this case,

the level of the ﬁxed load is relevant with respect to

the ﬂexible power associated to the HPs, thus th e pro-

ﬁle of the aggregated power is strongly affected by

its shape. Non etheless, it is possible to observe that,

Optimal Scheduling of Heat Pumps for Power Peak Shaving and Customers Thermal Comfort

0 2 4 6 8 1012141618202224262830323436384042444648505254565860626466687072747678808284868890929496

160

0 2 4 6 8 10 12 14 16 18 20 22

Power [kW]

Time [h]

fixed power industrial fixed power residents internal HP control

DSM with binary HP DSM with continuous HP

Figure 9: Aggregated power in the grid for the second si-

mulation scenario.

when no DSM is applied, the th ree additional power

peaks brought by the customer thermal requirements

are still evident and give the largest power peaks over

the day. In case of DSM, a power pro ﬁle as ﬂat as

the one obtained in th e previous test scenario cannot

be found due to the relatively low amou nt of ﬂexible

energy. However, it is possible to note that the DSM

scheme accomplishes its task of power peak minimi-

zation by reducing the HP use at the peak hou rs and

scheduling the operation of the HPs during off-peak

periods. This behaviour is clearly depicted in Figure

10, which shows the distribution of the HPs o peration

over the day for the two DSM schem e s. It is pos-

sible to observe that, in the case of continuous HPs,

all the devices are activated in the period of lowest

power consumption (4:00 - 6:00), while only a mini-

mum set of HPs is scheduled to operate during peak

hours, like at 12:00 or at 20:00. Fig. 10 also permits

underlining once more the advantages of the continu-

ous HP mode with respect to the binary model. I n the

latter case, in fact, despite a generally lower use of

the HPs (because they generally operate at higher po-

wer), the same or a larger number of HPs is r unning

during peak periods, which implies a larger additio-

nal power due to the ﬁxed power chosen to represent

the binary behaviour. This is a lso reﬂected in Table

4, which shows the obtaine d values of power peak,

0 2 4 6 8 10 12 14 16 18 20 22

100

Time [h]

Operating Heat Pumps [%]

DSM with binary HP DSM with continuous HP

Figure 10: Distribution of the HP operation for the second

simulation scenario.

Table 4: Technical beneﬁts with the D SM at peak times,

second simulation scenario.

case max. peak HP power operating HPs

(kW) (kW)

internal HP control 145.1 47.2 40

DSM - binary HP 122.7 9.7 12

DSM - continuous HP 118.6 3.2 9

the cor responding quote brought by the HPs, and the

number of HPs operating at that time. Moreover, even

in this scen a rio, the binary HP model proves to be less

efﬁcient than the continuous one, with an increase in

the overall energy consumption (for all the 60 houses)

larger than 36%.

To further conﬁrm the results achieved until now,

the last scenar io has been simu la ted again conside-

ring as outdoor temperature a day in December (see

Fig. 5). The ﬁrst consideration in this test case con-

cerns the DSM with binary HPs: in these conditions

the optimization algorithm is una ble to ﬁnd a feasible

solution, b e cause with the considered operating po-

wer is n ot po ssible to fulﬁl the thermal comfort requi-

rements during the changes in the reference tempera -

ture. This outc ome highlights once again the possible

drawbacks associated to the introduction of this sim-

pliﬁcation in the HP model. Focusing on the other two

HP scheduling criteria, Figure 11 shows the results

obtained for th e aggr egated power at grid level. Com-

paring these results with those obtained in the same

scenario in May (Fig. 9), it is immediate to verify that

a larger amount of HP energy results on top of the

inﬂexible base load. This is a consequence of the c ol-

der outdoor temperature, which fo rces the HPs to run

more f requently in order to provide the required ther-

mal comfor t to the customers. L ooking at the schedu-

ling of the sing le ho useholds, in the case of DSM with

continuous HP, 5 hou ses out of 60 require to have th e

HP running at all the tim e steps, and 13 houses n e ed

an operating HP for at least 23 hours. These effects

are automatically propagated to the results of the ag-

0 2 4 6 8 1012141618202224262830323436384042444648505254565860626466687072747678808284868890929496

160

0 2 4 6 8 10 12 14 16 18 20 22

Power [kW]

Time [h]

fixed power industrial fixed power residents

internal HP control DSM with continuous HP

Figure 11: Aggregated power in the grid for the third simu-

lation scenario.

SMARTGREENS 2017 - 6th International Conference on Smart Cities and Green ICT Systems

Table 5: Technical beneﬁts with the D SM at peak times,

third si mulation scenario.

case max. peak HP power operating HPs

(kW) (kW)

internal HP control 149.8 52.5 43

DSM - continuous HP 136.5 14.9 28

gregated power. As shown in Table 5, in fact, the le-

vel of power and the number of operating HPs during

the peak time is signiﬁcantly larger than in the pre-

vious simulation scenario. Nonetheless, despite this

slight degradation of the DSM performa nce, it is still

possible to notice as the proposed DSM scheme al-

lows optimizing the scheduling of the HPs, reducing

as much as possible the operation at the peak time and

ﬁlling the valleys during off-peak ho urs. In compari-

son to the case of internal HP control, a reduction of

the power peak and of the overall energy consumption

larger than 9% and 20%, r e spectively, is obtained also

in this last scenario.

Finally, as for the com putational cost for the pro-

posed method, w e can state that it is r e la tively low.

The test cases were solved on a standard laptop using

CPLEX 12. 7.0.0 in GAM S 24.8 on an Intel i7(2.9

GHz) machine with 16 GB RAM. The termination

criteria fo r the DSM with binary and continuous ope-

ration mode was set to a computation time of 2400 s

and 1 200 s, respectively; thus, both optimizations

were not solved to global optimality. This is reaso-

nable since sub-optimal so lutions from the grid per-

spective are achieved q uickly. Table 6 shows the re-

sults r elated to the ﬁrst presented simulation scena-

rio. Note, before the relative gap is calculated all se-

parate parameters an d p roducts of multiple parame-

ters that arise in the objective function a re subtracted

from the objective function. Interestingly, the more

detailed the HP model, the smaller gets the relative

gap. This means that a more realistic model (continu-

ous HP operation) decreases the comp utational com-

plexity of the problem. As for the internal HP control

case, where the temperature differences are minimi-

zed, a QP optimization is solved f or each house (in

total 60). The termina tion criteria of each optimiza-

tion was set to a computation time of 300 s. The rela-

tive gap o f the 60 QPs varies much, but the majority

was solved to global optimality.

Table 6: Computational r esults.

case time relative gap

(s) (%)

internal HP control 10 922 differs

DSM with binary HP 2400 8.15

DSM with continuous HP 1200 1.33

5 CONCLUSIONS

This paper presented an optimization algorithm de-

signed to deﬁne the day ahe ad scheduling of heat

pumps for a chieving power peak shaving in the elec-

tric grid. The conceived approach exploits the ﬂex-

ibility given by the heating devices on the customer

side to o btain the minimization of the power peaks,

while providing the required thermal comfort to the ﬁ-

nal u sers. Performed tests prove that the proposed ap-

proach allows combining the beneﬁts for the utilities

with the service for the customer, wh ic h obtains the

required temperatur e over the day a nd a minimization

of the energy consumption. Moreover, the advantages

brought by the proposed continuou s operation model

of the heat pump, with respect to the simpliﬁed ca se of

binary operation of the heat pump, are presented. This

work will be u sed as a starting point for further deve-

lopments in th is ﬁeld. In particular, a deeper study on

the impact of the customer ﬂexibility on the ﬁnal re-

sults and the evaluation of the possible drawbacks led

by the unavoidable uncertainties p resent in the used

model (e.g. outdoor temperature, knowledge o f the

building parameters, e tc .) will be object of future stu-

dies. The possible use of dedicated ther mal storag e

will be also object of future work, since it can sig-

niﬁcantly increase the available ﬂexibility leading to

potential improvements in the ac hievable results and

in the design of the DSM scheme. The integration of

additional home appliances in the propo sed manage-

ment algorithm can b e a further step for the design of

a complete DSM program fully exploiting the ﬂexibi-

lity offered by residential customers.

ACKNOWLEDGEMENTS

This work was suppor ted by FLEXMETER, which is

an EU Horizon 2020 project under grant agreement

no. 646568.

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