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Thermally-fair Demand Response for District Heating andCooling (DHC) Networks
Saptarshi Bhattacharya
1
⇤, Vikas Chandan
2
, Vijay Arya
2
and Koushik Kar
1
1
Rensselaer Polytechnic Institute,
2
IBM Research, India
[email protected],[email protected], [email protected],
ABSTRACTDistrict heating and cooling networks (DHCs) are complexthermal grids wherein a centrally heated/cooled fluid is cir-culated through a network of pipes and heat exchangers tomeet the heating/cooling needs of residential and commer-cial buildings. Several factors can hinder e�ciencies andimpartial distribution of energy among customers in thesenetworks. These include varying levels of building insula-tion, distance of individual buildings from the central en-ergy source, and thermal losses in network pipes. Moreover,shortage of energy at the central energy source and extremeweather conditions can exacerbate these issues, leading todi↵ering levels of thermal comfort and customer disgruntle-ment in the long run.
In this paper, we propose and study a demand responsescheme that attempts to ensure thermal fairness among end-use energy consumers in modern thermal grids. We developoptimization formulations based on thermodynamic modelsof DHCs, which determine optimal heat inflow/thermostatsettings for individual buildings in order to achieve targetedthermal fairness across the network. Our experimental re-sults using physics based models for DHC networks showthat it is possible to achieve targeted thermal fairness basedsocial welfare objectives in the DHC network by controllingnetwork parameters such as mass flow rates of water to theconsumer premises and the supply water temperature.
CCS Concepts•Information Systems Applications!Miscellaneous;
KeywordsThermal fairness, District Heating and Cooling (DHC) net-works, Demand Response (DR)
1. INTRODUCTION⇤The work was done during the author’s internship at IBMResearch, India.
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DOI: http://dx.doi.org/10.1145/2934328.2934343
In the backdrop of increasing renewable energy generationfor catering to the global energy needs, District Heating andCooling (DHC) networks are fast emerging as a major com-ponent of sustainable energy systems worldwide [14], [13],[16]. These DHC networks (also known as thermal grids)act as sustainable means of providing for the space heatingand hot water requirements of buildings. They are partic-ularly common in European countries (especially countrieslike Sweden, Finland and Denmark) where almost 62 millionconsumers are served through them, totaling about 12% ofthe entire population [1]. The main advantages of such net-works lie in their ability to use waste heat from industrialprocesses, reduce emission levels, increase e�ciency and flex-ibility of operation and integration with renewable energysystems.
Although sustainable to a large degree in general, an im-portant limitation of these systems is the deficit of adequateenergy available at the central energy source. For example,under extreme ambient conditions (very cold temperatures),if the energy available at the central energy source is notenough to su�ciently cater to the heating requirements ofthe entire network, then the network manager may be forcedto procure energy from uneconomical fossil fuel based plantsfor satisfying the space heating requirements of all buildingsin the network. Also, absence of centralized coordinationand control of network parameters may often hinder fair en-ergy distribution to end-use consumers within the thermalgrid. This may lead to consumer disgruntlement in the longrun. For example, uncoordinated operation may lead tolesser energy distribution in buildings located far away fromthe energy source leading to preferential treatment amongconsumers. Thus, it is imperative for grid managers to en-sure fair distribution of available energy through centralizedmechanisms so that the notion of thermal fairness is pre-served within the network among the consumers.
The District Heating (DH) network which we model inthis work has a centralized energy source that heats up theincoming cold water to be sent across a network of pipesto individual buildings, which then employ the hot waterfor their space heating purposes. We establish through nu-merical experiments that in scenarios of energy inadequacy,one possible way of avoiding (or reducing) the use of fossilfuel based energy is by implementing demand-response (DR)
algorithms in the network. Through these DR algorithms,some buildings (consumers) are chosen to reduce their heat-ing needs by some margin (thus becoming DR facilitators)and the conserved energy is e↵ectively redistributed to other
buildings to achieve a greater degree of comfort (thus becom-ing DR beneficiaries). We then develop concrete optimiza-tion formulations to achieve targeted thermal fairness basedsocial objectives through centralized control of network pa-rameters such as mass flow rate of water to the buildingsand the supply water temperature.
Note that the DR facilitators must be su�ciently com-pensated by the network manager/utility for accepting amargin of discomfort. The compensation structure and theeconomic aspects of the DR algorithm are an important partof the overall mechanism, but is outside the scope of thiswork. In this work, we mostly focus on the modeling aspectsof a thermal grid and study appropriate DR frameworks inthat network under various network objectives pertainingto thermal fairness. The rest of the paper is organized asfollows. In Section 2, we describe the overall system modelfor a district heating network. We then empirically evalu-ate the potential for demand response in such a network inSection 3. In Section 4, we simplify the complex detailedsystem model in Section 2 by considering ideal conditions ofoperation and utilize the ideal model to develop the demandresponse optimization framework to achieve thermal fair-ness in Section 4.1. In Section 5, we extend our analysis tothe case where the network is non-ideal with inherent ther-mal losses, and extend our optimization frameworks to suchpractical networks. We evaluate our algorithms through nu-merical studies in Section 6 and summarize our key findingsalong with future directions in Section 7.
1.1 Related workExisting research has identified the need for controlling
the system parameters within the thermal grids for optimiz-ing social welfare objectives. For example, authors in [10]show that improvements in energy e�ciency in a districtheating network can be brought about by controlling massflow rates of water to individual buildings. They also showthat the topology of the network has an e↵ect on the energye�ciency as well. Mass flow rate control for the optimal op-eration (minimized pumping loss and heat loss) of districtheating networks has also been studied in [6]. Authors in[5] have developed optimal control strategies to minimizefossil fuel consumption in an integrated district heating sys-tem with availability of renewable energy sources like solarenergy and wind energy. An approach for system cost mini-mization along with minimization of energy consumption foran entire year in district heating networks through a casestudy has been presented in [15]. Similar studies for mod-eling and optimization of operational cost for thermal gridshave been proposed in [7]. A detailed modeling of a thermalgrid along with a mass flow control methodology to achieveappreciable temperature cooling has been proposed in [9].Several researchers have also identified the need for opti-mizing the supply temperature of water in district heatingsystems [3], [18], [11]. Authors in [3] propose a predictivecontrol scheme using fuzzy Direct Matrix Control (DMC)for optimizing supply water temperature in district heatingnetworks. Authors in [17] highlight the importance of prop-erly selecting the control period for controlling the supplytemperature of water. Authors in [2] discuss a methodol-ogy where mass flow rates and supply water temperatureare optimized to minimize heat loss rate in thermal grids.
While the need for optimizing system cost, heat lossesand supply water temperature within thermal grids has been
well established [6], [2], there is still a need for developingconcrete demand response control strategies for optimizingsocial welfare objectives such as thermal fairness. Our keycontribution in this work is that we consider the notion ofthermal fairness and explicitly show how the centralized con-trol of network parameters through demand response canlead to social welfare maximization within the thermal grid.Another contribution is that our physics based thermody-namic models for the DHC network are reasonably realisticand yet are amenable to analysis for gaining useful insightsinto the operation of the system. Additionally, our modelingallows us to frame our DR optimization problem as a simpleconvex optimization problem with linear constraints whichis computationally viable for network managers (utilities).
2. MODELING THE DISTRICT HEATINGNETWORK
We consider a district heating network as shown in Fig-ure 1. Let Nb = {1, 2, . . . , n} represent the set of all build-ings in the DH network. We assume a simple parallel topol-ogy to define the connectivity of the buildings in this net-work. Each building i 2 Nb is assumed to have a thermalcapacity Ci and an e↵ective thermal insulation given by Ri.We assume that the energy distribution in buildings are sub-jected to thermal loss, which typically increases with theirdistances from the heat source. The heat source is assumedto be receiving cold return water at a temperature of TR,and using the input power Qin is able to heat the water upto a supply temperature of TS .
The heated water is networked through the connection ofpipes to individual buildings as shown in Figure 1. Duringthis transport, the thermal losses incurred along the waycauses the e↵ective supply temperature available (T i
MS) atthe mains (primary side) of the heat exchange circuit inbuilding i to be less than TS , i.e. T
iMS = TS+wi where wi <
0 is the thermal loss (loss in temperature of supply water)incurred for building i. The e↵ective incoming heat energyfrom the water is transferred to the house side (secondaryside) of the heat exchange circuit which then has a supplytemperature of T i
HS . It must be noted that T iHS T i
MS .For our case, we assume that T i
MS � T iHS = �, 8i 2 Nb,
where � is the temperature di↵erential between T iMS and
T iHS . The hot water in the secondary side is now circulated
through the radiators of the building (HVAC equipment) i toprovide for space heating and ideally maintain the ambientzone temperature T i
z of that building at a preferred set-pointT isp. The return cold water temperature in the secondary,
i.e. T iHR, determines the return temperature in the mains i.e.
T iMR. Subsequently the loss-adjusted return temperatures of
the buildings i.e. the T iR, 8i 2 Nb, determine the e↵ective
return temperature TR of the water at the heat source.
2.1 Modeling of individual building thermo-dynamics
In this section, we attempt to capture the detailed ther-modynamics of each individual building i 2 Nb, dependingon several factors including its e↵ective equivalent thermalparameters (Ri and Ci), its heat exchanger e↵ectiveness ✏i,the respective supply and return temperatures in the pri-mary and secondary of the heat exchanger circuit of thebuilding, its radiator characteristics (including its thermalconductivity Ui and e↵ective surface area Ai), and the ambi-
TS
TR
TR,1 TR,2 TR,3 TR,4
,4sm,3sm,2sm,1sm
b1 b2 b3 b4inQ
Hot water
Cold water
Heat exchanger
HE1
1MST
1MRT
1HST
1HRT
,1sm
TS1loss
1zT
Energy source
Building b1
RadiatorNetwork manager
Power availability information
(DR signal)
Indoor zone temperature relayed to network manager
Observed indoor temperature of
buildings 1loss
,1Hm
TR,1
District heating (DH) network
Heat exchanger
HE2
Figure 1: Schematic diagram of a district heating network where buildings are connected in parallel topology.
ent temperature T1. Note that there are two heat exchangecircuits: (a) HE
1
which captures heat exchange between thenetwork pipes (primary/mains) and the house pipes (sec-ondary), and (b) HE
2
which captures the heat exchangebetween the house pipes and the indoor house space throughthe radiator coils. We begin with the indoor zone tempera-ture evolution. We assume that the indoor zone temperature(T i
z) of a building i evolves according to,
T iz =
1RiCi
⇣T1 � T i
z
⌘+
1Ci
Qia +
1Ci
QiHV AC . (1)
Here, Qia is the power generated by active elements (such
as humans, lighting devices etc.) within building i andQi
HV AC is the HVAC power available through the radiatorof building i for indoor space heating. Note that
QiHV AC = mH,iCp(T
iHS � T i
HR), (2)
where mH,i, TiHS and T i
HR are the mass flow rate, supplyand return temperatures of the water in the secondary ofHE
1
circuit respectively. Cp is the specific heat capacityof water. The e↵ectiveness of heat exchanger in the HE
1
circuit is modeled according to,
T iMS � T i
MR
T iMS � T i
HR
= ✏i, (3)
where ✏i is the heat exchanger e↵ectiveness in building i.Note that 0 ✏i 1, 8i. The radiator heat exchangedynamics in HE
2
circuit is governed by the equation,
QiHV AC = mH,iCp(T
iHS � T i
HR) = UiAi(�T ieff ), (4)
where �T ieff is the temperature di↵erence between the radi-
ator coils and the indoor ambient in building i. The �T ieff
can be approximated as �T ieff ⇡
⇣T iHS+T i
HR2
�T iz
⌘for prac-
tical purposes. Finally, due to conservation of energy, we canalso write,
ms,iCp(TiMS � T i
MR) = mH,iCp(TiHS � T i
HR) (5)
where ms,i is the mass flow rate of water in the primary/mainsof HE
1
circuit.
2.2 Modeling of network thermodynamics
Assume that Qin is the total rate of heat energy avail-able at the central source for district heating needs. Fromconservation of energy in the overall district heating net-
work, we can write Qin =X
i2Nb
ms,iCp(TS � T iR). Noting
that Qin = MCp(TS � TR), where M =X
i2Nb
ms,i, we can
express the return temperature of water (TR) in the networkas,
TR =
X
i2Nb
ms,iTiR
X
i2Nb
ms,i
=
X
i2Nb
ms,iTiR
M. (6)
The network manager uses the available energy at centralenergy source Qin to heat up the cold return water (at TR)to TS as captured by the following equation.
TS =Qin
MCp
+ TR. (7)
Note that thermal grid managers usually have an operatingchart to determine the maximum TS that can be employedgiven a certain T1. Such considerations impose an upperbound TS,sat on the temperature to which supply water canbe heated to during operation.
3. MOTIVATION FOR DR IN A DHC NET-WORK
In thermal grids, several factors may hinder the fair dis-tribution of available energy among end-use consumers, asdiscussed in Section 1. In this section, we seek to evaluatethe feasibility of ensuring fair distribution of energy amonggrid consumers by centrally controlling network parameters.
To this end, we consider a test district heating network of10 buildings (small residential units) connected in a paralleltopology as shown in Figure 1. Assume that the buildingshave similar thermal capacities (i.e. Ci values) but havedi↵ering levels of thermal insulation parameters (Ri). Thebuilding nearest to the heat source is assumed to have thebest insulation value, and insulation values are assumed toprogressively decrease as buildings move farther away fromcentral heat source. Note that in general buildings will have
arbitrary thermal parameters independent of location withinthe grid, but our assumptions relating to the nature of vari-ation of building insulation parameters have been made forease of exposition of our arguments. In general, however,our arguments hold true for any network.
We assume that the e↵ective supply temperature availableto any building i also depends upon its location with respectto the central energy source. Specifically, the e↵ective sup-ply temperature available at the mains of HE
1
circuit inany house, i.e. T i
MS , is assumed to linearly decrease withincrease in distance of that building from the central heatsource. Thus the building farthest from heat source (whichhas worst insulation in this case) would have the least e↵ec-tive supply water temperature. The radiator characteristicsof all buildings are assumed to be similar: UiAi = 0.2, 8i,heat exchanger e�ciency ✏i = 0.9 8i, and the mass flow rateof water in the house side (mH,i) is assumed to be upperbounded by mub = 0.044 kg/sec for all buildings (valve lim-its). We also assume that the preferred temperature setpointin these buildings is 22�C. The subsequent experiments inthis section are carried out over a 24 hour interval duringwhich ambient temperature evolves as in Figure 5 (see Sec-tion 6). Note that for these experiments, we assume thateach building i has a proportional controller controlling itsmH,i with an aim of reducing the temperature di↵erentialT isp � T i
z .Under this setting, we evaluate the feasibility of demand
response by centrally altering network parameters to achievethermal fairness in the district heating network. Note thatin general, the grid manager can control several parameters.These include the individual mass flow rates to buildings orthe thermostat set points in buildings. In these experiments,we choose the change in thermostat set point, i.e. �T i
sp asour demand response signal. This means that the e↵ectiveset point of building i is T i
sp ��T isp. A case of no demand
response for building i is realized by setting �T isp = 0. In
later sections, we also analyze the case where direct controlof mass flow rates to buildings are used to optimize thermalfairness.In this work, we are considering a district heating net-
work in an energy constrained environment where the netthermal discomfort incurred over a time t is given as di(t),where di(t) = max(0, T i
sp � T iz(t)). Intuitively, whenever
the indoor space temperature of building i fails to reach itspreferred set-point T i
sp, the consumer in building i incursdiscomfort. A method for quantifying the social welfare inthe network in time period [0, T ] is through observation ofthe variance of the term Di where Di =
Rt2[0,T ]
di(t) dt. A
lower variance would indicate that the spread of the net dis-comfort faced by the buildings is narrow and thus buildingsare subjected to similar discomfort levels in the network. Ahigher variance indicates the possibility of some buildingssu↵ering more discomfort than others.As a baseline reference, consider the case when there is no
demand response, i.e. �T isp = 0, 8i. This is when the vari-
ance of Di is maximum: let this value be var(Di)max
. Thenormalized variance of Di under a DR scenario is definedas var(Di)norm = var(Di)
var(Di)max
, where var(Di) is the vari-ance in Di when there is demand response in the network.We define the time-avergaed total discomfort in the entire
DH network as⇣ R T
t=0
X
i2Nb
di(t)dt⌘/(
Z T
t=0
dt). This gives us
Input power at energy source (kW)42 43 44 45 46 47
Tota
l tim
e av
erag
ed
disc
omfo
rt (o C
)
0
10
20
30
40
50
Selected for ourempiricalexperiments
Figure 2: E↵ect of change in the the input source
power availability on the social welfare.
Number of buildings2 3 4 5 6 7 8 9
Nor
mal
ized
var
ianc
e of
Di
0
0.2
0.4
0.6
Optimum numberof buildings = 7
Figure 3: E↵ect of change in the number of buildings
controlled for DR on social welfare.
a measure of the net system wide discomfort subject to agiven ambient temperature profile and a given power avail-ability at the input source. From Figure 2, we understandthat the time-averaged total discomfort is dependent on theamount of power available at input source. In a practicalscenario, the power availability is going to be variable acrosstime slots, with underlying uncertainty. Here for the sakeof study, we assume that the input power is constant duringthe 24 hour interval chosen for the study. We find that ina setting where Qin = 44 kW, 8t 2 [0, T ], all buildings facesome quantity of discomfort when there is no DR. Note thatthe choice of Qin is made entirely for illustration purposes;the experiment can be done under any other conditions ofinput power availability at the central heat source.
Temperature deviation from setpoint (oC)0 0.5 1 1.5 2 2.5 3
Nor
mal
ized
var
ianc
e of
Di
0
0.5
1
1.5
2
"T $sp = 1:5/C
Figure 4: E↵ect of change in the T isp of buildings
controlled for DR on social welfare.
We now consider a varying number of buildings by settingthe �T i
sp = 1�C for the buildings and evaluate the extent
of such control on the social welfare in the network. We ob-serve from Figure 3 that the social welfare is optimized when7 buildings are selected for demand response. Note that forimparting demand response, we select buildings in increas-ing order of their distances from the central heat source. Ina subsequent study, we fix buildings 1 through 7 for impart-ing demand response by allowing change in their respectiveT isp, and vary �T i
sp, 8i 2 {1, 2, . . . , 7} in a range of valuesfrom 0�C to 3�C in intervals of 0.5�C. Note that in thisstudy we change the set-point of all selected buildings bysame amount. We observe from Figure 4 that under thissetting, the maximum improvement in social welfare is ob-served when the set-point change is 1.5�C.
From the above experiments, we infer that demand re-sponse through centralized control of network parameterscan indeed enhance thermal fairness within the network pro-vided the number of buildings on which DR is applied andthe extent to which DR is done is chosen carefully. Next,we will look to develop concrete algorithms through whichoptimal thermal fairness can be attained. We also demon-strate the distinction between such appropriate frameworkswithin ideal and non-ideal thermal grids.
4. ANALYSIS OF IDEAL DHC NETWORKSIn order to fully understand the energy distribution dy-
namics via fluid flow within an ideal district heating net-work, in this section, we make certain simplifying assump-tions:
• The heat exchanger e�ciency ✏i = 1 8i 2 Nb.
• � = 0, 8i 2 Nb, meaning T iMS = T i
HS , 8i 2 Nb.
• There are no thermal losses in the network, i.e. T iMS =
TS , 8i 2 Nb.
From (3), since ✏i = 1, we can write T iMR = T i
HR. It alsofollows from our assumptions, that (T i
MS � T iMR) = (T i
HS �T iHR) = (TS�T i
MR). Again, from (5), it follows that mH,i =ms,i. Let us denote mH,i = ms,i = xi. Assuming Qi
a =0 8i 2 Nb, and under steady state conditions when T i
z = 0,we can write directly from (1),
1RiCi
⇣T1 � T i
z
⌘+
1Ci
mH,iCp(TiHS � T i
HR) = 0. (8)
Applying our simplifying assumptions to (8), we can write,
1RiCi
⇣T1 � T i
z
⌘+
1Ci
xiCp(TS � T iMR) = 0. (9)
Again, from (4) and our simplifying assumptions, we canwrite,
xiCp(TS � T iMR) = UiAi
⇣TS + T iMR
2� T i
z
⌘. (10)
From (9) and (10), after rearranging the terms, we can write,
T iMR = �iT
iz � ↵iT1 � TS , (11)
where �i =⇣2 + 2
UiAiRi
⌘and ↵i = 2
UiAiRi. Equation (11)
allows us to express the steady state return temperature ofthe water from a building i in terms of the temperatureof supply water TS , the ambient temperature T1 and the
indoor zone temperature of building i i.e. T iz . Again from
equations (9), (10) and (11) we can write,
xi =1
Ri(T i
z � T1)
Cp(2TS � �iT iz + ↵iT1)
. (12)
Equation (12) allows us to express the mass flow rate ofbuilding i under steady state operation as a function of thetemperature of supply water TS , the ambient temperatureT1 and the indoor zone temperature of building i, T i
z . Us-ing the expressions for T i
MR and xi as obtained in (11) and(12) and the equations (6)-(7), and after simplification andrearrangement of terms, we can write,
Qin =X
i2Nb
1Ri
(T iz � T1). (13)
From the above equation, we clearly see that under steadystate operations in an ideal environment (such as the oneassumed), the energy available at the central heat sourceis used to cater to the energy requirements for maintainingzone temperature in the buildings of the network.
4.1 Optimization framework for an ideal DHCnetwork
In this section, we propose and analyze optimization for-mulations for demand response in the district heating net-work which are targeted to achieve social welfare based ob-jectives. Note that in general, dynamic optimization for-mulations for an entire district heating network consider-ing temperature transients, system losses and other networknon-idealities over an extended duration of time (say 24hours) tend to become computationally challenging owingto the large number of variables. Also, it becomes di�cultto analytically examine the optimization formulation in suchcases. Therefore, to circumvent these problems, we granu-larize the entire time duration into suitably small contiguoustime windows. We then focus on optimization of mass flowrates in the ideal network within each time window when theambient temperature is assumed to remain constant (say 1hour) and considering steady state mode of operation. Theoverall dynamic control problem is therefore broken up intosmaller subproblems, one for each time window.
In our framework, we consider that the control variable forinvoking demand response in any building i is the mass flowrate xi. In other words, given a certain ambient temperatureT1, we try to determine the optimal mass flow rate of waterx⇤i that needs to be channelized to building i to achieve a
network level social welfare objective. Another variable wewant to determine is the optimal temperature T ⇤
S to whichthe supply water is to be heated.
4.1.1 Objective functionWe discuss two types of objective functions for seeking
thermal fairness within the district heating network.
1. Minimize the maximum thermal discomfort of
a consumer in the thermal grid: In this case,
Jfair =
Z T
t=0
minmaxi2Nb
(T isp � T i
z(t)), (14)
where T isp are the preferred temperature set point of
consumers in building i. The notion of thermal discom-fort in building i in this case, is given by the extent to
which the indoor zone temperature T iz is less than the
preferred set point T isp.
2. Maximize the overall utility in the thermal grid
Consider that consumer in building i has a utility func-tion Ui(T
iz) = ci � bi(T
isp � T i
z)2, where ci and bi are
positive scalar constants. Such utility functions arecommon for thermostatic loads as seen in [12]. In suchcases, a possible network objective for achieving ther-mal fairness would be,
Jfair =
Z T
t=0
maxX
i2Nb
Ui(Tiz(t)). (15)
Although we discuss the above two objectives in our work,it is worth noting that our proposed methodology can incor-porate any other objective functions intended at optimizingthermal fairness in the network, as long as they satisfy cer-tain convexity properties. As discussed before, we dividethis optimization problem over [0, T ] into smaller subprob-lems, each on a smaller time window (say 1 hour). Next,we define the decision variables and associated optimizationconstraints for a single such small time window in the sub-sequent sections.
4.1.2 Decision variablesIn our optimization framework, note that the decision
variables are the temperature of the supply water and indoorsteady state zone temperatures. Denote the decision vari-able vector as T = [TS , T
1
z , T2
z , . . . TNz ]. An optimal point of
operation corresponds to determining a set of optimal indoorzone temperatures for the N buildings i.e. T i,⇤
z 8i 2 Nb andthe optimal temperature of the supply water T ⇤
S . Note thatthe decision variable TS does not appear in the objectivefunctions we wish to analyze and hence needs to be deter-mined separately as we will see subsequently. Also note thatto achieve the optimal operating point, the requisite controlvariables in our formulation are the optimal mass flow ratesx⇤i derived from (12), as a function of T1, T ⇤
S and the cor-responding T i,⇤
z values.
4.1.3 Optimization constraintsThe first constraint for this optimization problem is the
energy balance constraint as derived in equation (13). Thenext set of constraints deal with setting the upper and lowerbounds for xi, 8i 2 Nb. Now, note that the mass flow ratesin all buildings must be a non-negative value so xi � 0, 8i 2Nb. For practical purposes, we can assume (T i
z � T1) �0, 8i 2 Nb. Therefore, using this information and (12), weobserve that to make xi � 0 for all buildings,
2TS � �iTiz + ↵iT1 � 0, 8i 2 Nb. (16)
Typically, mass flow rates of water in buildings will havean upper bound denoting the valve limits of the controller.Therefore,
xi =1
Ri(T i
z � T1)
Cp(2TS � �iT iz + ↵iT1)
xubi 8i 2 Nb. (17)
The above set of inequalities can be simplified to be writtenas,
T iz
⇣ 1Ri
+ xubi Cp�i
⌘� T1
⇣ 1Ri
+ xubi Cp↵i
⌘� 2xub
i CpTS 0,
(18)
for all i 2 Nb. We also identify that in general, TS shouldalso be selected such that TS TS,sat (saturation con-straint) where TS,sat is the maximum supply temperatureat T1 ambient temperature, as obtained from the operatingchart available to grid managers/operators. Let us denotethe set D be the set of all feasible vectors T, such that (16)and (18) along with the saturation constraint TS TS,sat
and energy balance constraint (13) hold. Therefore, the op-timization problem for demand response in the district heat-ing network can be written compactly as,
min Jfair(T), (19)
s.t. T 2 D. (20)
4.1.4 Selecting the optimal temperature of supply wa-ter (T ⇤
S)Note that once the optimal steady state indoor zone tem-
perature T i,⇤z are determined from (19)-(20), the grid man-
ager can rescale the optimal temperature to which the sup-ply water needs to be heated i.e. T ⇤
S without a↵ecting theT i,⇤z values. It may be in the best interest of the grid man-
ager to keep the T ⇤S as low as possible since it minimizes
thermal losses in the network and increases operational ef-ficiency. To do this and yet maintain the respective T i,⇤
z
and satisfy 0 x⇤i xub
i , 8i 2 Nb, the grid manager mustobserve the following:
• To ensure x⇤i � 0, 8i 2 Nb, from (16), the optimal
supply temperature must be such that,
T ⇤S � 1
2
⇣�iT
i,⇤z � ↵iT1
⌘8i 2 Nb. (21)
• To ensure x⇤i xub
i , from (17) the T ⇤S has to be such
that,
T ⇤S � T i,⇤
z
⇣ 1
2xubi CpRi
+�i
2
⌘� T1
⇣ 1
2xubi CpRi
+↵i
2
⌘,
(22)for all i 2 Nb.
The grid manager then selects the minimum T ⇤S that sat-
isfies all the inequalities in (21) and (22). Selection of the T ⇤S
now allows determination of the optimum x⇤i for all buildings
using equation (17) where T iz = T i,⇤
z 8i 2 Nb and TS = T ⇤S .
Thus the key observation is that for lossless networks, theremay be a range over which the optimal supply water tem-perature T ⇤
S (the optimal x⇤i depend on the choice of T ⇤
S )can be heated to such that the optimal indoor temperaturesT i,⇤z (which are unique) may be attained. Equations (21)-
(22) define the minimum T ⇤S that can attain the optimal
solution.
5. ANALYSIS OF PRACTICAL LOSSY DHCNETWORKS
In Section 4, we were able to understand the energy dis-tribution dynamics of an ideal DHC network and use theknowledge to optimize key network parameters such as massflow rates to buildings and the supply water temperaturefor realizing thermal fairness based social welfare objectives.However, in reality a DHC network will include several non-idealities which must be considered while trying to tune ournetwork parameters with a view to realize similar social wel-fare objectives as in the last section.
Let us consider a lossy DHC network which has non-idealheat exchangers in buildings, i.e. ✏i 1, 8i 2 Nb and � > 0.The thermal losses in the network typically depend (increasewith) on the distance from the central energy source. Let wi
represent the thermal loss of the water supplied to buildingi. In other words, the e↵ective supply water temperatureat i is T i
MS = TS + wi where wi < 0. Also, the e↵ectivereturn water temperature of building i at the source is T i
R =T iMR+wi. As in Section 4, we consider steady state scenario
and express the steady state return temperature of water inthe buildings in terms of the ambient temperature T1, thesupply temperature TS , and the indoor zone temperatureT iz . A similar line of analysis as in Section 4 enables us to
write,
T iMR = �iT
iz � ↵iT1 � �iT
iMS + ✏i�, (23)
where �i = 2✏i +2✏i
UiAiRi, ↵i = 2✏i
UiAiRiand �i = (2✏i �
1). The steady state primary side mass flow rates can beexpressed as,
ms,i =1
Ri(T i
z � T1)
Cp(2✏iT iMS � �iT i
z + ↵iT1 � ✏i�). (24)
Using these above quantities as obtained in (23) and (24)in (7), the steady state energy balance equation for a prac-tical lossy network can be derived as,
Qin =X
i2Nb
1Ri
(T iz � T1)� 2
X
i2Nb
ms,iCpwi. (25)
Note that the second term in equation (25) accounts for thelosses in the network. This was absent in (13) where an idealand lossless DHC network was considered. Also note thatall the steady state balance equations obtained in Section 4can be retrieved from (23), (24) and (25) by putting ✏i = 1,� = 0 and wi = 0 for all buildings.
Remark: Note that for lossy networks, the presence of themass flow rates ms,i in the energy balance equation hindersthe suitable rescaling of the supply water temperature (andhence the mass flow rates) and yet maintain the optimalsteady state T i,⇤
z values as in the case of the lossless idealnetwork. Therefore, we put an additional constraint of up-per bounding TS by a suitable TS,ub within the optimizationframework to directly compute system optimum T
⇤ underlossy cases using suitable mass flow rates m⇤
s,i, 8i 2 Nb inthe primary circuit. We discuss this in details in the follow-ing subsection.
5.1 Optimization framework for practical DHCnetworks
The grid manager’s objective is to control the mass flowrates of primary circuit (note that we cannot consider a sin-gle mass flow rate xi as in the ideal network anymore due toinherent system non-idealities which leads to ms,i 6= mH,i
in general) to achieve social welfare objectives. The objec-tive functions and the decision variables for optimization areexactly same as in Section 4.1. Let Mub denote the upperbound on the mass flow rates. The first constraint is theenergy balance equation as given in (25). The rest of theconstraints in the lossy case can be summarized by the fol-lowing set of equations,
2✏iTiMS � �iT
iz + ↵iT1 � ✏i� � 0, 8i 2 Nb. (26)
Inequalities in (26) lower bound the mass flow rates to benon-negative. The constraints relating to the upper boundare given as,
�i1
T iz � �i
2
T1 � 2MubCp✏iTiMS +MubCp✏i� 0, 8i 2 Nb,
(27)
where �i1
=⇣
1
Ri+ Mub�iCp
⌘, �i
2
=⇣
1
Ri+ Mub↵iCp
⌘
are parametric constants of the system. The additional con-straint on TS can be written as,
TS TS,ub. (28)
In general, a good choice of TS,ub can be a few degrees lesserthan the corresponding saturation temperature for supplywater TS,sat at that respective T1. After solving the aboveoptimization problem, we can derive the optimal mass flowrates m⇤
s,i for each building i in order to maintain set pointtemperature at T i,⇤
z and implicitly tune the steady statesupply temperature to T ⇤
S . The above optimization problemis then solved for the subsequent time windows to determinethe optimal mass flow rates and supply water temperaturefor the entire duration of DR.
6. EXPERIMENTAL RESULTSIn this section, we evaluate the optimization algorithms
for demand response in the same network as was consideredin Section 3. Note that the demand response mechanismsare studied over a period of 24 hours. The ambient temper-ature during this time is assumed to vary in accordance tothe temperature of a typically extreme winter day in Lulea,Sweden; an area which caters to a sizable portion of localheating energy needs by district heating. The variation ofthe ambient temperature during this time is shown in Fig-ure 5. We assume that the input power available for heatingvaries as shown in Figure 6. The preferred set points of allbuildings throughout the optimization window is assumedto be 22�C.
Time of day 12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
Ambi
ent t
empe
ratu
re (o C
)
-31
-30
-29
-28
-27
-26
Figure 5: Ambient temperature variation during pe-
riod of demand response.
The first experiment we conduct is to see how the networkparameters and the individual zone temperatures evolve un-der conditions of no demand response. In such cases, theindividual buildings are assumed to selfishly drive their in-door zone temperature towards their preferred set point by
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
Inpu
t pow
er a
vaila
ble
at e
nerg
y so
urce
(kW
)
35
40
45
50
Figure 6: Variation of input power available for dis-
trict heating at energy source during period of de-
mand response.
controlling the secondary mass flow rates through a suitableproportional controller as was considered in Section 3. Thesecondary mass flow rate upper bound was considered tobe 0.044 kg/sec as before. In this setting, we observe theevolution of the indoor zone temperature of the buildings.We then compare this with cases where demand response isinitiated by the grid manager to e↵ectively induce thermalfairness in the considered network. For better readability,in the remainder of this section, we only report the param-eters of 5 of the 10 buildings (buildings numbers 1, 3, 5, 7and 9) for studying the e↵ects of demand response. Notethat among these buildings, building 1 has the best ther-mal insulation and is subjected to least losses in the supplywater temperature. The building insulation progressivelydecreases and temperature loss progressively increases withincrease in building index. The thermal loss wi (in �C) inbuilding i located y meters away from heat source is mod-eled by wi = �0.0027y + 0.84. Building 1 is assumed tobe located 320 meters from the energy source, and subse-quent gaps between successive buildings are assumed to be50 meters.
When there is no demand response, the indoor tempera-ture profile of di↵erent buildings vary as shown in Figure 7.Clearly, buildings do not achieve desired set point since theQin available (see Figure 6) during the studied window isinsu�cient for meeting their energy needs. We also observethat when there is no DR, building 1 is su↵ering the leastdiscomfort and building 9 is su↵ering the maximum discom-fort. This is owing to the lesser insulation in building 9 ascompared to building 1 and greater thermal loss encounteredby building 9 due to being located farther down the networkfrom the central energy source. We now investigate the im-pact of demand response algorithms on the network. In ourstudies, motivated by the supply temperature curves in [4]we assume that TS,sat = 43 � 1.1T1. We also know thatideally the grid managers want to keep the supply tempera-ture lower than the saturation to minimize losses in network.Hence, the optimization upper bound of TS in our studies(which is also dependent on ambient temperature) is takento be TS,ub = 36� 1.1T1.
Also note that the demand response signals in our case arethe mass flow rates in the primary circuit of the individualbuildings as determined by the grid manager. Considering
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
Tem
pera
ture
(o C)
10
12
14
16
18
20
22
24
building 1building 3building 5building 7building 9Preferred setpoint
Figure 7: Variation of indoor zone temperature in
buildings without any demand response.
the typical mass flow rates of water in radiator systems ofresidential buildings in di↵erent seasons [8], the upper boundon the primary mass flow rate to all buildings is selected asMub = 0.044 kg/sec. In the first case we examine, the ther-mal fairness objective for the grid manager is to minimizethe maximum discomfort faced by buildings, as defined byequation (14).
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
_m$ s;
i(k
g/se
c)
0.01
0.015
0.02
0.025
0.03
0.035
0.04building 1building 3building 5building 7building 9
Figure 8: Variation of primary mass flow rates in
buildings during period of demand response.
In Figure 8, we report the optimal mass flow rates m⇤s,i for
each building i to realize the optimal set point temperatureT i,⇤z . We observe that amount of water to be channelized to
the building i increases with the increase in building index,i.e. grid manager channelizes more flow to less insulatedbuildings and buildings which face greater thermal loss inorder to maximize the social welfare. In Figure 9, we ob-serve that with these optimal mass flow rates, the indoorzone temperatures attained in the buildings are very closeto each other (note that the temperature curves in Figure 9are almost overlapping). Such network reconfiguration al-lows the consumers in the buildings to face similar levels ofdiscomfort, hence optimizing the social welfare metric. Wenow investigate the e↵ect of demand response on the supplywater temperature.
We observe from Figure 10 that the optimal supply tem-perature of water T ⇤
S during the demand response is suit-
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
Tem
pera
ture
(o C)
10
12
14
16
18
20
22
24
building 1building 3building 5building 7building 9Preferred setpointSteady state optimal setpoints
Figure 9: Variation of indoor zone temperature in
buildings during period of demand response.
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
Tem
pera
ture
(o C)
55
60
65
70
75
80
T $S
TS;sat
Figure 10: Variation of supply water temperature in
the network during period of demand response.
ably maintained. In conjunction with the optimal mass flowrates, they control the indoor zone temperature in the build-ings so as to achieve optimum thermal fairness. In order toquantify the improvement in thermal fairness across the en-tire network through demand response, we report the nor-malized discomforts, i.e. Di,norm = Di
maxi2NbDi,no�DR
(dis-
comfort of building i is Di =Rt2[0,T ]
di(t) dt) su↵ered by
each of the buildings under two scenarios: (a) without de-mand response (when decentralized/greedy control of massflow in buildings is allowed) and (b) with demand response.
We observe from Figure 11 that without any demand re-sponse, the discomfort su↵ered by buildings increase mono-tonically from building 1 to building 10. Through DR, thegrid manager is seen to have e↵ectively optimized the pri-mary mass flow rates so that the overall time-averaged totaldiscomfort in the network (as defined in Section 3) has de-creased from 34.46�C to 25.88�C. As seen in Figure 11, withDR, the individual discomforts of buildings 4 to 10 have de-creased. Therefore, these are classified as DR beneficiaries.However, discomforts in buildings 1 to 3 have slightly in-creased owing to DR. These buildings are thus classified asthe DR facilitators. In practice, the grid manager has toprovide incentives to the DR facilitators for facing greaterdiscomfort. However, the design of such incentive mecha-
Building index1 2 3 4 5 6 7 8 9 10
Di;nor
m
0
0.5
1
1.5Without DRPost DR
DR beneficiaries
DR facilitators
Figure 11: Discomforts faced by buildings before
and after DR.
nisms require an independent investigation considering thegrid manager’s other economic considerations and hence wedefer that to future work.
We repeat our experiment on the same test network un-der the same assumptions and settings with a di↵erent so-cial welfare objective: to maximize overall utility in thenetwork, as defined by (15). In this case, consumers inbuilding i are assumed to have a concave utility functionUi = ci � bi(T
isp � T i
z)2. We assume ci = 10 and bi = 0.05
for all buildings. Under such a setting, the time-averagedtotal discomfort in the network was observed to come downfrom 34.46�C to 25.30�C. The optimal mass flow rates tothe individual buildings, the resultant optimal supply tem-perature, the post DR indoor zone temperatures and thenormalized discomfort of consumers in individual buildingsbefore and after DR have been reported in Figures 12, 13,14 and 15 respectively. Note that under this objective, theindividual temperatures of the buildings are not as similarto each other as in the previous case. However, note thatthe reduction of time-averaged total discomfort in the net-work is slightly better under this setting. Thus, we infer thatthrough various network objectives, we can achieve di↵erentlevels of thermal fairness within the network by centralizedcontrol of network parameters.
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
_m$ s;
i(k
g/se
c)
0.01
0.015
0.02
0.025
0.03
0.035
0.04building 1building 3building 5building 7building 9
Figure 12: Variation of primary mass flow rates in
buildings during period of demand response when
optimization objective is utility maximization.
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
Tem
pera
ture
(o C)
10
12
14
16
18
20
22
24
building 1building 3building 5building 7building 9Preferred setpoint
Figure 13: Variation of indoor zone temperature in
buildings during period of demand response when
optimization objective is utility maximization.
Time of day12 am 4 am 8 am 12 pm 4 pm 8 pm 12 am
Tem
pera
ture
(o C)
55
60
65
70
75
80
T $S
TS;sat
Figure 14: Variation of supply water temperature in
the network during period of demand response when
optimization objective is utility maximization.
7. DISCUSSION AND FUTURE WORKIn this work, we have presented a DR mechanism for
thermal grids which optimizes network-wide thermal fair-ness based social welfare metrics. We modeled a thermalgrid through detailed thermodynamics of the componentbuildings and demonstrated the scope of demand responsein these networks to achieve thermal fairness. Throughdetailed analysis, we have proposed suitable optimizationframeworks where network parameters such as mass flow ofwater to individual buildings are controlled in a centralizedmanner by the grid manager to realize the optimized tem-peratures in the buildings of the network. In this context,we have covered the scenarios of both an ideal network and apractical network (with non-idealities such as losses) and dis-tinguished the di↵erences between them and how that a↵ectsthe DR mechanism. We demonstrate through experimentalstudies on a test network that such DR mechanisms are in-deed able to significantly improve the selected social welfaremetric when there is energy inadequacy in the central energysource. Finally, we note that proper incentive mechanismsmust be developed to implement such DR mechanisms bythe grid manager. This will require detailed investigationinto economic considerations of the grid manager and is de-ferred to future work.
Building index1 2 3 4 5 6 7 8 9 10
Di;nor
m
0
0.5
1
1.5Without DRPost DR
DR facilitators
DR beneficiaries
Figure 15: Discomforts faced by buildings before
and after DR when optimization objective is utility
maximization.
In this paper, while we have dealt with two objectivesfor improving thermal fairness, our framework can easily beextended to consider a broader set of social welfare met-rics relating to thermal fairness. A dynamic optimizationconsidering entire window of operation in a single shot maybecome computationally very resource intensive and hence,for large networks, may not be a viable solution to grid man-agers in the long run. Our optimization framework allowsus to build the optimal set point of network operation overan extended duration by breaking it into smaller time slots,determining the optimum solution for that smaller time slotand repeating it to include the entire window. Thus it isamenable for use in larger networks hence making our DRmechanism easily scalable for implementation.
In this work, we have considered a parallel topology of theDHC network. Practical DHC networks may have di↵erenttopologies like tree topology, ring topology, etc. Anotherpoint is that larger networks can also have multiple heatsources of di↵erent capacities within the network. We planto extend our analysis to cover those cases as well and de-vise suitable DR mechanisms under such settings. Through-out this paper, we assume that the radiator characteristics,thermal insulation parameters of buildings and the thermalcapacities of buildings are known to the grid manager. Inpractice, the thermal insulation parameters may change de-pending on the orientation of doors and windows within thebuildings. Also, radiator performances may not always beequal to their rated performances. In light of such possi-bilities, a DR algorithm which is agnostic to the explicitknowledge of network parameters and can learn and adaptto the changing network parameters is desirable. We planto investigate this aspect in our future work as well. Inthis context, with the advent of information and communi-cation technologies, a fully data driven framework for mod-eling and subsequent optimization may also be a possibility,thus allowing us to bypass the use of physics based modelingaltogether.
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