# HetuMoE: An Efficient Trillion-scale Mixture-of-Expert Distributed Training
System

Xiaonan Nie, Pinxue Zhao, Xupeng Miao, Tong Zhao, Bin Cui
School of Computer Science,
Peking University, Beijing, China
{xiaonan.nie, 1800017766, xupeng.miao, zhaotong<EMAIL_ADDRESS>

###### Abstract

As giant dense models advance quality but require large amounts of GPU budgets
for training, the sparsely gated Mixture-of-Experts (MoE), a kind of
conditional computation architecture, is proposed to scale models while
keeping their computation constant. Specifically, the input tokens are routed
by the gate network and only activates part of the expert network. Existing
MoE training systems only support part of mainstream MoE models (e.g. Top k)
training under expensive high-bandwidth GPU clusters. In this paper, we
present HetuMoE, a high-performance large-scale sparse MoE training system
built on Hetu. HetuMoE provides multiple gating strategies and efficient GPU
kernel implementations. To further improve the training efficiency on
commodity GPU clusters (e.g, with only 1 NiC), we introduce the hierarchical
AllToAll communication that combines hierarchical networks and aggregating
messages. Compared with existing state-of-the-art MoE systems, HetuMoE obtains
at least $15\%$ speedup. Specifically, HetuMoE outperforms DeepSpeed-MoE up to
$8.1\times$ under the switch gate with a batch size of 32. Our code is
available at: https://github.com/PKU-DAIR/Hetu.

##  1 Introduction

Models tend to perform better with increasing data size, parameter size in
many fields, such as natural language processing (NLP) and computer vision
(CV) [9, 2, 14, 15, 3]. However, these large models always require huge
amounts of GPU resources and take weeks or even months to train. For example,
it takes approximately 288 years with a single V100 NVIDIA GPU to train the
GPT-3 model with 175 billion parameters [1, 12]. The sparsely gated Mixture of
Experts (MoE), as a kind of conditional computation architecture, has been
proved to be an effective way to expand model size without significantly
increasing the computation [19, 5, 10, 17, 4]. Specifically, an MoE model
contains a gating network and a pool of experts. During training, a token will
be assigned to a small number of experts for computation. This kind of
sparsely-activated nature enables MoE to significantly expand the model size
while keeping the amount of computation almost constant. The sparsely-
activated training paradigm necessitates new systems’ support. However,
existing MoE training systems, including DeepSpeed-MoE [16], Tutel [7], and
FastMoE [6], are still facing some limitations in both usability and
efficiency. First, they only support parts of mainstream MoE models and gate
networks (e.g. Top k). Second, most of them assume “hyperclusters” that have
expensive, high-speed interconnects such as NVLink or Infiniband. They are
suffering from a severe communication bottleneck for common clusters. The
above shortcomings have restricted their usage and limited the exploration of
MoE models in real applications.

In this paper, we first briefly introduce the common training process of MoE
models and then propose HetuMoE, a high-performance distributed MoE training
system built on Hetu, which supports a variety of mainstream gating strategies
and achieves state-of-the-art training speed compared to existing baseline
systems. HetuMoE adopts several customized gating kernels’ implementation and
utilizes the hierarchical All-To-All communication optimization. Experiments
show that HetuMoE outperforms existing MoE systems at least a $15\%$ speedup
under different settings. Meanwhile, HetuMoE outperforms DeepSpeed-MoE up to
$8.1\times$ under the switch gate with the batch size of 32.

##  2 MoE Training Process

An MoE model consists of a gating network and a pool of sub-networks (e.g.
Feed-Forward Networks in Transformer), which are called experts. We formulate
the common training process of MoE models in Algorithm 1.

  1. (1)

The input tokens are first processed by the gate network to know the target
experts.

  2. (2)

The layout transform operation is conducted on each device to put tokens with
the same target experts in a continuous memory buffer for the incoming
communication. (Line 4)

  3. (3)

An AllToAll communication operation is incurred to dispatch tokens to their
corresponding experts. (Line 6)

  4. (4)

Then each expert process its tokens respectively. (Line 8-11)

  5. (5)

The processed tokens are dispatched back to their GPUs. (Line 13)

  6. (6)

The reverse layout transform operation is conducted to put tokens back to
their original position in the training batch for future computation. (Line
15)

Data: $x_{S}$: a group of tokens of size $S$,

$Gate$: gate network

$E$: expert network

Result: $y_{S}$: tokens after precessing

1 //Step 1: Get the target experts and corresponding weights. ;

2 $\mathcal{W}_{(S,\ E)},id_{S}=Gate(x_{S})$ ;

3 //Step 2: Layout Transform on Input Data ;

4 $x_{S}^{\prime}=Layout\\_Transform(x_{S},id_{S})$ ;

5 //Step 3: AllToAll Communication ;

6 $x_{S}^{\prime}=AllToAll(x_{S}^{\prime})$ ;

7 //Step 4: Expert Processing ;

8 for  _$i\leftarrow 1\ to\ S$_ do

9 $y_{s}=0$ ;

10 for  _$idx\in id_{i}$_ do

11 $y_{s}=y_{s}+w_{(i,\ idx)}*e_{idx}(x_{i})$ ;

12

13//Step 5: AllToAll Communication ;

14 $y_{S}=AllToAll(y_{S})$ ;

15 //Step 6: Reverse Layout Transform on Output Data ;

16 $y_{S}=Reverse\\_Layout\\_Transform(y_{S},id_{S})$ ;

Algorithm 1 General MoE Training Process Figure 1: Time consumption of MoE
layer of DeepSpeed-MoE under 8 A100 GPUs in a single node

We then evaluate DeepSpeed-MoE and profile its time costs of MoE layer under 8
A100 GPUs in a single node and the result is shown in Figure 1. As we can see,
the computation of gate network, including layout transform and its reverse
operation, and AllToAll communication account for more than 50% of training
time totally. When scaling training into distributed scenarios across multiple
nodes, AllToAll communication overheads would deteriorate the entire training
process, which takes around 99% of training time under a 100 Gbps network.

Figure 2: HetuMoE compared with other MoE systems

##  3 HetuMoE

In order to solve above problems, we develop HetuMoE based on Hetu
111https://github.com/PKU-DAIR/Hetu, a high-performance distributed deep
learning system. HetuMoE supports various gating strategies proposed by recent
MoE approaches, such as Switch [5], GShard [10], M6 [21], BASE Layer [11] ,
Hash Layer [18], SAM [8], and Dense-to-Sparse [13]. In addition, we implement
hierarchical AllToAll [20], which greatly improves network bandwidth
utilization in the case of multi-node distributed training under commodity
network conditions, compared with vanilla AllToAll. The feature supported in
HetuMoE is compared with other MoE systems in Figure 2.

###  3.1 Gating Strategy in HetuMoE

#### Top1/Top2/Topk

Shazeer et al., 2017 [19] proposed to utilize Topk activated MoE layer in LSTM
increase the model capacity up to $1000\times$ with only minor losses in
efficiency. The Topk gate is formulated in Equation 1, where $x$ is the $N$
input tokens, $W$ is the gate’s weight, E experts $e_{i}(i\in{1..E})$ and $y$
is the output tokens. GShard [10] and Switch [5] propose to simplify the gate
to Top2 and Top1, and utilize a capacity factor $C$ to force the max received
tokens by each expert. As the $K$ increases, the MoE layer tends to perform
better while leading to more computation. It’s a trade-off on $K$ considering
computation efficiency and model performance.

$\displaystyle g=softmax(TopK(x\cdot W,K))$ (1) $\displaystyle
y=\sum_{i=1}^{E}g(x)_{i}\cdot e_{i}(x)$

(a) Different Expert Number

(b) Different Token Number

Figure 3: Topk kernel performance comparison with PyTorch

#### kTop1

Inspired by the observation that $K$ and expert capacity $C$ in Topk routing
can significantly make a difference in model performance, M6-T [21] propose
the KTop1 gating strategy, where experts are divided into $k$ prototypes and
each token is assigned to the highest scored expert in each prototype.
Finally, the outputs of different prototypes are summed together as for the
same input token.

#### Hierarchical Topk(H Topk)

As increasing the number of activated experts can boost the model performance
with a higher sparse ratio, SAM [8] (Switch and Mixture) proposed an efficient
hierarchical routing mechanism that divides the experts into different groups
according to their devices and activates multiple experts in the same group to
avoid the remote communication cost between devices. Specifically, the Switch
Router first selects one group and then the Mixture Router selects multiple
experts in the same group for each token.

#### BASE Layer

BASELayer [11] formulates token-to-expert allocation as a linear assignment
problem to improve efficiency, where balanced loads are guaranteed among each
expert. Because no new parameters or auxiliary expert-balanced losses are
introduced, the training process is simplified. The problem is formulated in
Equation 2, where $N$ tokens with representations $x_{i}$ and $E$ experts with
embeddings $w_{e}$, we assign each token to an expert via the assignment index
$a_{i}\in 0..E$:

$\displaystyle max\sum_{i=1}^{N}x_{i}\cdot w_{a_{t}}$ (2) $\displaystyle s.t.\
\forall{e}\sum_{i=1}^{N}1_{a_{i}=e}=\frac{N}{E}$

#### Hash Layer

Hash Layer [18] utilizes various hash functions to map tokens to experts, such
as Random Hashes, Balanced assignment, and Clustered Hashes. Specifically, in
Hash Layer, tokens are placed in the corresponding buckets according to the
used hash function and each bucket refers to an expert.

(a) Transformation

(b) Reverse transformation

(c) Performance comparison with Tutel

Figure 4: Data layout transformation illustration and performance comparison

#### Dense-To-Sparse

Dense-To-Sparse Gate [13] considers that current approaches of jointly
training experts and the sparse gate introduce a negative impact on model
accuracy, diminishing the efficiency of expensive large-scale model training
[5]. It proposes to begin as a dense gate that routes tokens to all experts,
then gradually and adaptively become sparser while routing to fewer experts.
Specifically, it utilizes the Gumbel Softmax and decreases the temperature
during training.

###  3.2 HetuMoE Optimization

As shown in Algorithm 1, Gate, Layout Transform and AllToAll are three key
components related to MoE models’ training, where the module of expert
networks also exists in common models and thus is not our target in MoE-
specific optimization. The optimizations in HetuMoE about these three modules
are detailed as follows.

#### Gate Optimization

Various gating strategies are supported in HetuMoE and we mainly optimize the
$topk$ operator as it is widely adopted in mainstream models [5, 10, 17]. The
operator’s input is a 2-D tensor which has shape
$(num\\_tokens,num\\_experts)$ and executes as finding the top-k values and
the corresponding index for each row. The Topk operator implemented in PyTorch
or Tensorflow for arbitrary K. We conduct algorithmic optimizations for useful
$k$ in MoE, such as 1 and 2. We vary $num\\_experts$ and $num\\_tokens$ to
conduct comparison with PyTorch and the experimental results are shown in
Figure 3. We outperform PyTorch Topk by $25\%$ speed improvement averagely.

#### Layout Transform Optimization

Data layout transformation is another important step in the MoE training
process. After the gating network decides the token-to-expert mapping, tokens
assigned to the same expert need to be put in physically continuous memory
locations. We undertake kernel level optimization to accelerate this process,
and we achieve more than 26% improvement compared with state-of-art
implementation.

#### All-To-All Optimization

In AllToAll operation, each GPU sends its data to all GPUs (one-for-all) and
receives data sent by all GPUs (all-for-one), where each data will be divided
equally into n parts, illustrated in Figure 5. Current AllToAll operations
implemented in NCCL and MPI may suffer from low utilization of network
bandwidth because of the small message size. Specifically, if we have N nodes,
each with G GPUs, and each GPU has data size B, then the size of data
transferred between 2 GPUs is $\frac{B}{GN}$. A common setting is $N=8$,
$G=8$, and $B=16$MB.

Figure 5: Illustration of NCCL AllToAll. Figure 6: Illustration of
hierarchical AllToAll

(a) 4 nodes

(b) 8 nodes

Figure 7: Hierarchical AllToAll performance

In HetuMoE, we utilize Hierarchical AllToAll, which combines hierarchical
networks (intra-node and inter-node) and aggregates messages, to optimize the
communication between multi-nodes equipped with one NIC. Jesper et al., 2014
[20] also utilize hierarchy to optimize alltoall communication for MPI.
Illustrated as Figure 6, it first gathers the data of all GPUs inside one node
into one GPU. Then, a data layout transformation is undertaken to place the
token assigned to the same node in physically continuous memory. Afterwards,
it launches All2All communication between nodes. After AllToAll is done, it
conducts the corresponding data layout transformation and scatter operation to
put each token to its corresponding expert. In this way, the size of data
transferred between nodes becomes $\frac{BG}{N}$, which is $G^{2}$ times
larger than before. Meanwhile, this two-level decoupled AllToAll also fully
utilizes the intra-node (NvLink or PCIe) and inter-node bandwidth (Infiniband
or Ethernet). Experiments show that Hierarchical AllToAll achieves
$1.66\times$ speedup in $4\times 8$ GPUs setting and $2\times$ speedup in
$8\times 8$ GPUs setting.

(a) Switch Gate

(b) GShard Gate

Figure 8: Overall performance comparison with DeepSpeed-MoE, FastMoE, and
Tutel.

#### Overall Performance

Experiments are conducted on GPU Clusters, where each node is equipped with 8
TITAN RTX GPUs and these eight GPUs are connected through PCIe. We compared
HetuMoE with existing MoE systems, including Tutel, FastMoE and DeepSpeed-MoE.
Our test model is a 16-experts MoE layer, where each expert represents a
FeedForward Network and the hidden size is 2048, The sequence length and
embedding dimension of input data is 1024 and 2048 respectively. We vary a
batch size to compare the performance of each system.

As shown in Figure 8, HetuMoE achieves the state-of-art training performance
in both Switch gate and GShard gate. Specifically, compared with FastMoE, we
achieve $18\%$ speed-up in Switch Gate and $15\%$ speed-up in GShard gate.
Meanwhile, HetuMoE outperforms DeepSpeed-MoE up to $8.1\times$ under the
switch gate with a batch size of 32.

##  4 Conclusion

Existing MoE training systems lack various gating strategy support and perform
badly in distributed training. In this paper, we propose HetuMoE, a high-
performance distributed MoE training system built on Hetu, which supports a
variety of mainstream gating strategies and achieves state-of-the-art training
speed compared to existing baseline systems. HetuMoE adopts several customized
gating kernels’ implementation and utilizes the hierarchical All-To-All to
optimize distributed communication by combining hierarchical network (intra-
node and inter-node) and aggregating small messages.

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# A Dynamically Similar Lab-Scale District Heating Network via Dimensional
Analysis

Audrey Blizard<EMAIL_ADDRESS>Stephanie Stockar

###### Abstract

Strict user demands and large variability in external disturbances, along with
limited richness in the data collected on the daily operating conditions of
district heating networks makes the design and testing of novel energy-
reducing control algorithms for district heating networks challenging. This
paper presents the development of a dynamically similar lab-scale district
heating network that can be used as a test bench for such control algorithms.
This test bench is developed using the Buckingham $\pi$ theorem to the match
the lab-scale components to the full-scale. By retaining the relative
thermodynamics and fluid dynamics of a full-scale network in the lab-scale
system, the experimental setup allows for repeatability of the experiments
being performed and flexibility in the testing conditions. Moreover, the down-
scaling of the experiment is leveraged to accelerate testing, allowing for the
recreation of operating periods of weeks and months in hours and days. A PID
controller is implemented on the lab-scale test bench to validate its response
against literature data. Results show 63% efficiency during heating operations
compared to 70% efficiency for a similar full-scale system, with comparable
pressure losses across the system.

###### keywords:

district heating network, thermal fluid systems, dynamic similitude,
optimization, control verification

††journal: Energy Conversion and Management

##  1 Introduction

Reducing energy consumption is one of the main tools available to combat
climate change and lessen humans’ impact on the environment. A large source of
energy demand is the heating and cooling of buildings. For instance, in 2015
these processes account for 35% of the energy consumed by buildings in the
United States [1]. Increasing the process efficiency will decrease the carbon
footprint of commercial and residential buildings and reduce their operating
costs.

In this context, district heating networks (DHNs) are a promising method to
efficiently deliver heat to buildings in an urban environment. Rather than
implementing individual heating systems in each building, DHNs centralize the
process, relying on economies of scale to increase efficiency and reduce
carbon emissions. Additionally, because the heat is generated at a centralized
plant, DHNs also simplify the process of integrating of a variety of advanced
technologies, including combined heating and power and bio-fuels, along with
the use and storage of energy generated by intermittently-available renewable
energy sources, such as solar [2]. While DHNs offer inherent performance
benefit over traditional heating methods, the system performance can be
greatly improved through advanced control strategies. Existing controllers
rely on limited information, mainly the supply and return temperatures and the
network pressure differential to set the supply temperature and initial flow
rate [3]. While simple to implement, these controllers are ineffective in
meeting the heat demands of users especially in large and highly
interconnected DHNs due to the lack of granular data on the network’s current
operating conditions and limited knowledge of the network’s future demands
[3]. Conventional control strategies also lack the flexibility needed to take
full advantage of inconsistent renewable energy sources [4]. Optimized control
algorithms enable existing DHNs to be operated more efficiently, reducing
energy consumption and providing a more comfortable user experience at a lower
operational cost. Preliminary studies of model-based control algorithms for
DHNs, which are able to account for predicted demands and disturbances, have
been shown to reduce the energy consumption by up 34% [5].

However, in most cases, the validation of novel control algorithms for DHNs
has been focused on tests performed in simulation environments [6, 7, 8].
Without validation of the control strategies on the physical system, it is
difficult to guarantee that similar performance improvements observed in
simulations will be obtained in real-world DHNs. There are some examples of
physical tests benches for DHNs [9, 10], but none are sufficient for the
testing of novel control algorithms due to their narrow focus on individual
components of the DHN network. Additionally, there are examples of control
algorithms being tested on real world DHNs [11, 12]. However, these tests
required significant modifications to the communication infrastructure to
enable the implementation this controller on an operational DHN. Furthermore,
when performing tests on full-scale DHNs, there is unpredictability in the
external disturbances, such as ambient temperature, solar irradiation, and
occupancy times. The performance characteristics of DHNs are highly seasonal,
and a validation test of new controllers across the variety of operating
conditions experienced by DHNs can take weeks or even months. For example, one
of the preliminary tests discussed above were conducted over an entire winter
season [11].

A dynamically similar lab-scale test bench will eliminate challenges
associated with the validation of novel control strategies, allowing for the
repeatability in the experiments being performed, flexibility in the
conditions being explored, and reducing the time needed to perform tests,
while ensuring the applicability of the control algorithm to real-world DHNs.
Existing literature supports the idea of using dynamically scaled experimental
setups in the design of novel controllers, with the Buckingham $\pi$ Theorem
being the most common method of obtaining dynamically similar setups. For
example, this technique has been show effective in matching airborne wind
energy systems dynamics [13] and vehicle dynamics during high speed maneuvers
[14]. The Buckingham $\pi$ Theorem is a formalization of the procedure of
dimensional analysis which works by identifying the relevant relationships
within a system and systematically scaling them while retaining their relative
proportions [15]. This technique is commonly used to create smaller and
economical experimental test benches in a variety of fields, and its use in
the dynamic scaling of thermo-fluid systems is well known. For example, the
Buckingham $\pi$ Theorem has been applied in the analysis of the heat equation
[16] and more specifically, has been used in the design of the thermal
characteristics of spacecrafts [17]. The Buckingham $\pi$ theorem has been
used in DHNs to analyze individual components of the network, but has never
been used to simultaneously scale all components of the network. For example,
a reduce parameter model of a building’s temperature dynamics was obtained
through the correlation of different heating and cooling modes into related
nondimensional groups [18, 19]. Similarly, applications to the
characterization of the heat transfer rate in heat exchangers was developed
based on their geometry [20, 21]. A model of a DHN pipe network was created
using the Buckingham $\pi$ Theorem with the goal of reducing the number of
network characteristics needed to accurately predicting the heat losses [22].
Finally, this approach has been used to perform a detailed analysis of the
pressure losses of a liquid flow in a single length of pipe [23]. While the
Buckingham $\pi$ Theorem is an established method for scaling the static
responses of systems and analyzing individual components of DHNs, this
technique has never been used in the scaling the dynamic response of an entire
district heating network with the intended application of model validation and
control testing. Using the Buckingham $\pi$ Theorem, this paper will present
the design of a dynamically similar lab-scale test bench that will allow for
rapid testing of new control algorithms on physical hardware, and the
nondimensional framework developed in this paper provides a method to compare
the results obtained on the scaled network to that of a full-sized DHN. As
DHNs operate in both the thermodynamic and fluid domains, the characterization
of both has to be considered in the scaling procedure.

The remainder of this paper is organized as follows. First, Section 2 provides
a description of the components of the lab-scale DHN, along with the equations
used to model the network’s dynamics. Section 3 describes the application of
the Buckingham $\pi$ theorem to establish the nondimensionalization parameters
used to characterize the system. Then, Section 4 presents ranges for the size
of components in a full-scale DHN found in literature and describes the
desired parameters for the equivalent lab-scale components using the
established $\pi$ groups. In Section 5, the lab-scale system is validated
against simulation results from an equivalent full-scale system. Finally,
conclusions are presented in Section 6.

##  2 District Heating Networks

###  2.1 Experimental Setup

The lab-scale setup considered in this paper is representative of a two-user
DHN, however the sizing methodology presented is applicable to any network
size. The two buildings in this network are represented by fluid-filled
acrylic boxes. These thermal masses contain a submerged winding of copper pipe
acting as the heat exchangers in a full-scale network to transfer heat from
the distribution network into the buildings. Submerged impellers are used to
distribute the heat throughout the thermal masses and ensure the fluid is
well-mixed.

The distribution network, made from 1/2” PEX pipes, circulates water from the
heating plant to the thermal masses. This network consists of a main supply
line that splits into two loops, one for each user. Each loop has a
characterized control valve that divides the water between the user and a
bypass. The user branch sends water to the heat exchanger, while the bypass
branch diverts water from the heat exchanger. This division allows the
operator to control the mass flow rate in the heat exchanger, adjusting the
heat supplied to the thermal mass to control its temperature to a desired
value, $T_{set}$. The flows from each heat exchanger and bypass are mixed at
return nodes 1 and 2 and the user loops rejoin at the main return node. From
here, the water is sent back to the centralized plant through the main return
line to be reheated. The network is connected to a residential 30-gal water
heater and a fixed-speed 3/4 HP pump, which reproduce the connection with the
heating plant in a full-scale network. The detailed discussion of the
experimental setup and its construction is presented in Krieger et al. [24]. A
diagram of the described system is shown in Fig. 1, while a picture of the
physical setup is shown in Fig. 2.

The component sizing in the original design were performed based on a static
similarity analysis. However, an exact and systematic method to ensure dynamic
similarity is needed to guarantee the performance of the lab-scale system is
representative of a full-scale DHN.

Figure 1: Layout of the two-user lab-scale DHN with components labeled. Figure
2: Photograph of experimental setup with relevant components labeled.

###  2.2 Dynamic System Model of a DHN

The derivation of a physics-based model of a DHN, applicable to both the full-
scale and lab-scale network is presented in this section. The derivation of
these fundamental equations allows for the identification of the variables
relevant to the system’s behavior. Additionally, these equations will be used
to simplify the dimensional analysis performed in Section 3 as these equations
describe the relationship between the variables and parameters in the system.

The system is divided into three major components: the water supply, the
distribution pipes, and the buildings, all of which exhibit both temperature
and fluid dynamics. The lab-scale system is supplied by a water heater and
pump that deliver water with a set supply temperature $T_{s}$ and initial mass
flow rate $\dot{m}_{I}$ to the network. These values serve as the inlet values
for the first pipe in the distribution network. The distribution network is
then used to circulate this heated water to the buildings. Each pipe in the
distribution network must be modeled in both the thermal and fluid domains.
The bulk temperature $\left(T_{p}\right)$, which is the relevant temperature
in each pipe segment, is modeled using the conservation of energy equation

$\frac{dT_{p}}{dt}=\frac{\dot{m}}{\rho V}(T_{p_{in}}-T_{p})-\frac{hA_{s}}{\rho
c_{p}V}(T_{p}-T_{a})$ (1)

where ${\dot{m}}$ is the mass flow rate of water through the segment, $T_{a}$
is the ambient temperature, $\rho$ is the fluid density, $hA_{s}$ is the
conductive heat transfer coefficient of the pipe segment, $c_{p}$ is the
specific heat of water at constant pressure, assumed constant, and $V$ is the
volume of each pipe segment. Finally, $T_{p_{in}}$ is the inlet temperature
obtained from the conservation of energy equation of the previous pipe. Note
that for the supply line, $T_{p_{in}}=T_{s}$. The pipe’s volume is given by

$V=\frac{\pi}{4}D^{2}l$ (2)

where $D$ is the pipes internal diameter and $l$ is the length of the pipe
segment.

In the fluid domain, the pressure drop $\left(\Delta P\right)$ across each
pipe segment is calculated using

$\Delta P=k_{tot}\left(\frac{\dot{m}}{A_{c}}\right)^{2}$ (3)

where $A_{c}$ is the pipe’s cross sectional area and $k_{tot}$ is a pressure
loss coefficient representative of both the distributed and concentrated
pressure losses [25]. In practice, DHNs are self balancing, meaning the mass
flow $\left(\dot{m}\right)$ is divided between the branches to equalize the
pressure losses throughout the network, resulting in a relative mass flow rate
that is inversely proportional to $k_{tot}$. Hence the mass flow rate in each
segment can be determined offline by solving the set of algebraic equations
characterizing the pressure losses in each segment. The total pressure loss
for each user loop $\Delta P_{Li},\ i=1,2$ can be calculated as a function of
mass flow rate by summing the $\Delta P$ values for each pipe segment. The
pressure balance in the network can be enforced using

$\Delta P_{L1}\left(\dot{m}_{1}\right)=\Delta P_{L2}\left(\dot{m}_{2}\right)$
(4)

where $\dot{m}_{i}$ is the mass flow in each user loop. Conservation of mass
ensures that

$\dot{m}_{1}+\dot{m}_{2}=\dot{m}_{I}$ (5)

Combining Eq. 4 and Eq. 5, the flow rate split between the branches can be
calculated.

The temperature dynamics of the thermal mass is modeled using the conservation
of energy equation, following the same principle as the pipe segments in Eq.
1:

$\left(\rho c_{p}V\right)_{ThM}\frac{dT_{ThM}}{dt}=\dot{Q}_{in}-\dot{Q}_{out}$
(6)

where $\left(\rho c_{p}V\right)_{ThM}$ is the heat capacity of the thermal
mass, $\dot{Q}_{out}$ is the heat lost by the thermal mass to the environment,
and $\dot{Q}_{in}$ is the heat provided to the thermal mass by the network
through the heat exchanger. The rate of heat transferred into the thermal mass
is given by

$\dot{Q}_{in}=\left(hA_{s}\right)_{HX}\left(T_{HX}-T_{ThM}\right)$ (7)

where $T_{HX}$ is the bulk temperature of the water in the heat exchanger, and
$(hA_{s})_{HX}$ is the convective heat transfer coefficient of the heat
exchanger. Similarly, the heat transferred from the thermal mass to the
environment is

$\dot{Q}_{out}=\left(hA_{s}\right)_{ThM}\left(T_{ThM}-T_{a}\right)$ (8)

where $(hA_{s})_{HX}$ is the convective heat transfer coefficient of the
thermal mass.

Equations 1, 2, 3, 4, 5, 6, 7 and 8 provide a model of the network that will
serve as the foundation to perform the scaling of the system. Having this
model simplifies the process of creating a similar lab-scale system by proving
an understanding of the underlying relationship between the relevant
variables.

##  3 Nondimensionalization of System Equations

To ensure applicability of tests performed on the lab-scale system, the
transient and steady state responses of the full-scale system must be scaled
to the lab-scale. The multi-domain nature of DHNs requires the lab-scale model
to match both the heat transfer characteristics and the fluid dynamics of the
original system. This section presents the extension of the Buckingham $\pi$
Theorem for the uniformly scaling of all components of a full-scale DHN. The
procedure established by the Buckingham $\pi$ Theorem is followed to identify
the relevant nondimensionalization parameters for the system and use them to
create dimensionless $\pi$ groups that describe the behavior of the system.
Finally, from these $\pi$ groups, the nondimensional form of the system
equations is found.

The first step of the Buckingham $\pi$ Theorem is to identify the variables
that are relevant to the system. From the equations presented in Section 2,
the 20 relevant variables associated with the system are

$f\left(\begin{gathered}t,\ T_{s},\ T_{a},\ T_{p},\ T_{HX},\ T_{ThM},\
\dot{m}_{I},\ \dot{m},\ \rho,\\\ c_{p},\ D,\ L,\ hAs,\ k_{tot},\ A_{c},\
\rho_{ThM},\\\ c_{p_{ThM}},\ V_{ThM},\ hAs_{ThM},\
hAs_{HX}\end{gathered}\right)=0$ (9)

However, due to design constraints, the available sizes of some components,
and lab space restrictions, there are a limited number of modifiable design
parameters. Due to the aforementioned constraints, matching all 20 variables
for every component of the system is impossible. Instead, using the system
model, it is possible to combine some of the variables into groups. Because
the equations governing the system’s dynamic responses and losses have been
established, the relationship between the some of the variables and these
variables can be grouped together during the nondimensionalization process.
This will reduce the number of dimensionless quantities to be matched, while
still providing a representative description of the system. Leveraging this
information, the variables listed in Eq. 9 are reduced to the following

$f\left(\begin{gathered}t,\ T_{s},\ T_{p},\ T_{HX},\ T_{ThM},\ \dot{m}_{I},\\\
D,\ \rho,\ \frac{\dot{m}}{\rho V},\ \frac{hA_{s}}{\rho
c_{p}V}\left(T_{p}-T_{a}\right),\\\ \Delta P,\ \left(\rho
c_{p}V\right)_{ThM},\ \ \dot{Q}_{in},\ \dot{Q}_{out}\end{gathered}\right)=0$
(10)

The next step in the Buckingham $\pi$ Theorem is to identify the fundamental
units of the variables in the system. Dimensional formulas can be used to show
the fundamental units associated with each physical quantity and to determine
what terms are needed to generate each nondimensional group. The dimensional
formulas for each variable group in the reduced list are

$t=[t]$ (11a) $T_{s},T_{p},T_{HX},T_{ThM}=[T]$ (11b)
$\dot{m}_{I}=\left[Mt^{-1}\right]$ (11c) $D=[L]$ (11d)
$\rho=\left[ML^{-3}\right]$ (11e) $\frac{\dot{m}}{\rho V}=\left[t^{-1}\right]$
(11f) $\frac{hA_{s}}{\rho c_{p}V}(T_{p}-T_{a})=\left[Tt^{-1}\right]$ (11g)
$\Delta P=\left[ML^{-1}t^{-2}\right]$ (11h) $\left(\rho
c_{p}V\right)_{ThM}=\left[ML^{2}t^{-2}T^{-1}\right]$ (11i)
$\dot{Q}_{in},\dot{Q}_{out}=\left[ML^{2}t^{-3}\right]$ (11j)

From these dimensional formulas, it can be seen that four fundamental units
appear in the variables used to describe the relevant system dynamics. These
units are length $\left(L\right)$, time $\left(t\right)$, mass
$\left(M\right)$, and temperature $\left(T\right)$. Therefore, four
fundamental quantities are needed to create the independent, nondimensional
variable groups. The quantities selected are fluid density, initial mass flow
rate, supply temperature, and internal pipe diameter and are summarized in
Table 1 . These values are selected because first, they best characterize the
scale of operation for each component in the system, and second, are the ones
subject to the most challenging constraints (dimensions, specifications) in
the lab-scale design.

Table 1: Nondimensionalization parameters selected to create the $\pi$ groups Parameter | Symbol | Unit
---|---|---
Density of operating fluid | $\rho$ | $ML^{-3}$
Initial mass flow rate | $\dot{m}_{I}$ | $Mt^{-1}$
Supply temperature | $T_{s}$ | $T$
Pipe internal diameter | $D$ | $L$

From the above steps, it can be seen there are fourteen relevant variable
groups and four fundamental units. Therefore, according to the Buckingham
$\pi$ Theorem, a total of ten independent $\pi$ groups can be found. The
process of finding the $\pi$ group for a single variable is

  1. 1.

Identify the fundamental units of the variable using its dimensional formula,
given in Eq. 11.

  2. 2.

Note the exponent for each of the fundamental units.

  3. 3.

Eliminate the units of the variable using the appropriate power of the
nondimensionalization parameters (Table 1) in order of $T_{s}$, $\dot{m}_{I}$,
$\rho$, and $D$.

  4. 4.

Write the final nondimensional group from the original variable and
nondimensionalization parameters raised to the appropriate powers.

This procedure is followed to nondimensionalize the variables listed in Eq.
10. The resulting $\pi$ groups are listed below.

Nondimensional time is found using

$t^{*}=t\cdot\frac{\dot{m}_{I}}{\rho D^{3}}$ (12)

The nondimensional temperatures of each pipe segment in the network can be
found by dividing the current temperature by the supply temperature.

$T_{p}^{*}=\frac{T_{p}}{T_{s}},\quad T_{HX}^{*}=\frac{T_{HX}}{T_{s}}$ (13)

Because the temperature of the fluid in the thermal mass can be modulated by
the bypass valve, it is independent from the network supply temperature
$T_{s}$, and the desired setpoint, $T_{set}$, can be decided arbitrarily,
provided that $T_{set}<T_{s}$. Therefore, the current thermal mass temperature
must be normalized by subtracting the setpoint before dividing by the supply
temperature

$T_{ThM}^{*}=\frac{T_{ThM}-T_{set}}{T_{s}}$ (14)

There are three $\pi$ groups needed to characterized the pipes, denoted as
$\pi_{1}-\pi_{3}$. The first two groups, $\pi_{1}$ and $\pi_{2}$ are from
conservation of energy equation for the pipe Eq. 1, where $\pi_{1}$ is the
nondimensional coefficient for the heat supplied into the pipe

$\pi_{1}=\frac{\dot{m}}{\rho V}\cdot\frac{D^{3}\rho}{\dot{m}_{I}}$ (15)

and $\pi_{2}$ comes from the term used to describe the heat lost to the
environment by the pipe.

$\pi_{2}=\frac{hA_{s}}{\rho
c_{p}V}(T_{p}-T_{a})\cdot\frac{D^{3}\rho}{\dot{m}_{I}T_{s}}$ (16)

The final group for the pipes, $\pi_{3}$, comes from the fluid dynamics model
of the pipe Eq. 3 and is used to ensure the pressure losses are consistent
between the network scales.

$\pi_{3}=\Delta P\cdot\frac{\rho D^{4}}{\dot{m}^{2}}$ (17)

The dynamics of the building components, given by Eq. 6 can be described using
$\pi_{4}-\pi_{6}$, where $\pi_{4}$ is the nondimensional heat capacity of the
building

$\pi_{4}=(\rho c_{p}V)_{ThM}\cdot\frac{\rho T_{s}D}{\dot{m}^{2}}$ (18)

The second group for the building dynamics, $\pi_{5}$, describes the energy
transferred into the thermal mass from the heat exchanger

$\pi_{5}=\dot{Q}_{in}\frac{\rho^{2}D^{4}}{\dot{m}^{3}}$ (19)

and the last group $\pi_{6}$ describes the heat lost by the thermal mass to
the environment

$\pi_{6}=\dot{Q}_{out}\frac{\rho^{2}D^{4}}{\dot{m}^{3}}$ (20)

Using the $\pi$ groups, the system equations are then written in their
dimensionless forms. Having the nondimensional form of these equations allows
for the modeling of the network behavior in the nondimensional space, which
makes comparison between the full-scale and lab-scale systems possible. The
temperature dynamics equation in the pipes in Equation 1 is reformulated as

$\frac{dT_{p}^{*}}{dt^{*}}=\pi_{1}\left(T_{p_{in}}^{*}-T_{p}^{*}\right)-\pi_{2}$
(21)

while the temperature dynamics equation for the thermal masses, Eq. 6, is
rewritten as

$\pi_{4}\frac{dT_{ThM}^{*}}{dt^{*}}=\pi_{5}-\pi_{6}$ (22)

##  4 Selection of the Lab-Scale System Parameters

Table 2: Average network component values for full-scale and lab-scale systems Nondimensionalization Parameters |  |
---|---|---
|  | Full-Scale | Lab-Scale |
Parameter | Symbol | Value | Unit | Value | Unit | Source
Density of network fluid | $\rho$ | 971 | $\left[kg/m^{3}\right]$ | 994 | $\left[kg/m^{3}\right]$ | [25]
Initial mass flow rate | $\dot{m}_{I}$ | 20 | $\left[kg/s\right]$ | 0.0862 | $\left[kg/s\right]$ | [26, 27]
Supply temperature | $T_{s}$ | 80 | $\left[C\right]$ | 36 | $\left[C\right]$ | [26, 28]
Pipe internal diameter | $\left[D\right]$ | 0.1 | $\left[m\right]$ | 12 | $\left[mm\right]$ | [28, 27]
Pipe Characteristics |  |
|  | Full-Scale | Lab-Scale |
Parameter | Symbol | Value | Unit | Value | Unit | Source
Length of segment | $l$ | $20-100$ | $\left[m\right]$ | $2.5-11$ | $\left[m\right]$ | [26, 29]
Pressure loss across segment | $\Delta P$ | 0.01-0.2 | $\left[MPa\right]$ | 1-10 | $\left[kPa\right]$ | [29, 30]
Conductive heat transfer coefficient | $hA_{s}$ | 5-90 | $\left[W/K\right]$ | 0.23-1.0 | $\left[W/K\right]$ | [31]
Thermal Mass Characteristics |  |
|  | Full-Scale | Lab-Scale |
Parameter | Symbol | Value | Unit | Value | Unit | Source
Heat capacity | $\left({\rho c_{p}V}\right)_{ThM}$ | $0.15-7$ | $\left[GJ/K\right]$ | $30-45$ | $\left[kJ/K\right]$ | [32, 18, 7]
Pressure loss across heat exchanger | $\Delta P_{HX}$ | 0.03 | $\left[MPa\right]$ | 2-3 | $\left[kPa\right]$ | [29]
Convective heat transfer coefficient | $\left(hA_{s}\right)_{HX}$ | 5-12 | $\left[kW/K\right]$ | 14.5-16 | $\left[W/K\right]$ | [33, 34]
of heat exchanger
Rate of heat lost by building | $\dot{Q}_{out}$ | 70-230 | $\left[kW\right]$ | 0-79 | $\left[W\right]$ | [32, 27]
Desired internal temperature of building | $T_{Set}$ | 20 | $\left[C\right]$ | 26 | $\left[C\right]$ | [33]

The physical parameters of the lab-scale system are selected to match the
nondimensional large-scale values for all the previously established $\pi$
groups. A literature review was conducted to find representative dimensional
values for the sizes of each component for a variety of real-world DHN
configurations. This information is then used to calculate the values of the
full-scale $\pi$ groups. The results of this literature review is summarized
in Table 2.

Then, the unmodifiable components of the lab-scale system were considered.
Specifically, the lab-scale network is supplied by a a fixed speed rotary vane
pump and 30 gallon residential water heater. These two components constrained
two of the nondimensionalization parameters, namely the size of the pump
determines $\dot{m}_{I}$, and the recovery rate of the water heater (the
amount of water that can be heated during a one hour period) dictates $T_{s}$.
Additionally, the network uses 1/2” PEX pipes for the supply network, fixing
the value for $D$. By modifying the original experimental setup, the rest of
the components in the lab-scale DHN are resized to ensure agreement between
the full-scale and lab-scale $\pi$ groups. The ranges of the lab-scale values
that result from matching the $\pi$ groups are presented in Table 2.

The lengths of the pipes are chosen to ensure agreement between the $\pi_{1}$
values. The pipes have been encased in two layers of R13 fiberglass insulation
to reduce the heat lost by the pipes to the environment, matching the
$\pi_{2}$ values. Additionally, the predicted pressure drops in the lab-scale
network are calculated based on the friction, length, and geometry changes in
the network and were found to be in the desired range to match the values of
$\pi_{3}$ for the lengths selected.

The thermal mass characteristics are also matched between the full-scale and
lab-scale. To decrease the volume required to match heat capacity of the
thermal mass and increase the rate of heat transfer from the heat exchanger,
the lab-scale thermal masses representing the buildings are filled with water,
rather than the air as would be in full-scale system. The volume of the
thermal masses is then chosen to match $\pi_{4}$. This results in the chosen
volumes of $7000\ cm^{3}$ and $10400\ cm^{3}$ respectively. The desired
steady-state temperature in the thermal mass is set to match the rate of heat
extracted from the network $\left(\pi_{5}\right)$ during steady state
operation. The desired building temperature is of particular relevance because
this variable is used to modulate the bypass valves supplying the heat
exchangers to meet the desired setpoint when each building is occupied.
Finally, because the lab-scale DHN is indoors, the difference between the set
point temperature $\left(T_{ThM}\right)$ and the ambient temperature
$\left(T_{a}\right)$ is less than for an equivalent outdoor system. Hence, the
required rate of heat transfer out of the buildings can not be met through
natural convection alone. Additionally, the ambient temperature can vary
widely throughout a day, and the ability to recreate these fluctuations is key
to performing realistic tests. To address these issues, Peltier junctions are
added to a wall of the thermal masses. An image of one of the thermal mass
with the embedded Peltier junction can be seen in Fig. 3. These Peltier
junctions remove heat from the thermal mass when a voltage is applied via the
thermoelectric cooling effect. The addition of the Peltier junction to the
thermal mass allows for the control of the rate of heat removed from each
thermal mass, allowing for more flexibility in the tests that can be
performed. The rate of heat lost by the thermal mass (Eq. 8) must be modified
to include the effects of the Peltier junction

$\dot{Q}_{out}=\left(hA_{s}\right)_{ThM}\left(T_{ThM}-T_{a}\right)+\dot{Q}_{pelt}$
(23)

where $\dot{Q}_{pelt}$ is the rate of heat removed by the Peltier junction,
which is set directly by modulating the power supplied to the Peltier
junctions. For example, to replicate an ambient temperature of -5 C in the
full-scale, the Peltier junctions must operate at approximately 40 W. The rate
of heat loss caused by the Peltier junctions can be converted to a simulated
ambient temperature $\left(T_{a_{sim}}\right)$ as

$T_{a_{sim}}=-\frac{hA_{s_{act}}}{hA_{s_{sim}}}\left(T_{ThM}-T_{a}\right)-\frac{\dot{Q}_{pelt}}{hA_{s_{sim}}}+T_{ThM}$
(24)

where $hA_{s_{act}}$ is the thermal mass’s heat transfer coefficient and
$hA_{s_{sim}}$ is the simulated heat transfer coefficient. The thermal masses’
heat transfer coefficients $hA_{s_{sim}}$ and the ambient temperature, both
components of $\pi_{5}$, are modulated by changing voltage supplied to the
Peltier junction to match the full-scale values.

Figure 3: Photograph of thermal mass with embedded Peltier junction.

##  5 Validation of the Resulting Lab-Scale DHN

There is a limited amount of data available in literature on daily operating
conditions of real-world DHNs at the individual user level. More often, models
are developed for design and energy characterization purposes. For this
reason, in the paper, simulation results are used to validate the similarity
between the lab-scale and full-scale systems.

###  5.1 Data Acquisition System

The experimental setup is outfitted with temperature, pressure and mass flow
sensors to ensure all relevant dynamics are captured during operation. The
system has two separate but synchronized data acquisition systems (DAQs). The
first DAQ consists of three 8-channel USB data acquisition modules and is used
for collecting temperature data from the 17 thermistors located throughout the
network. The second system is a National Instruments PXIe1073 system, with a
PXIe4353 and a PXIe6363 installed, which is used for temperature, pressure and
flow rate data. The PXIe4353 provides thermocouple input channels, which are
used to measure the cold-side temperature of the Peltier junctions. The
PXIe6363 has 16 analog differential inputs, nine of which are used to collect
data from the pressure transducers located throughout the network.
Additionally, four analog input channels are used to collect data from the
mass flow sensors. The PXIe6363 also has 4 analog outputs, which supply
voltage to the two characterized control valves to modulate their positions
during the experiment. The location of the sensors in the network are shown in
Fig. 4. NI LabVIEW is used to interface with both DAQs to synchronize and
record the data, set the control valve positions, and display results in real
time.

Figure 4: Diagram of the two-user lab-scale DHN with sensor locations labeled.

###  5.2 Description of Experiment

Through the $\pi$ groups framework, any set of testing conditions can be
recreated in the lab. In this paper, the response of the lab-scale system is
evaluated against the simulation results presented in the paper by Saletti et
al. [27]. This test is representative of a typical day during winter operation
and shows a diverse response range for the two thermal mases. In the full-
scale experiment, a PID control algorithm was designed to act as a baseline
for comparison to a novel control algorithm. This PID controller was
implemented in a Model-in-the-Loop simulation to control two school buildings
to a desired set temperature during their hours of operation. In the full-
scale simulation, the maximum available supply mass flow rate for the two
buildings was 20 kg/s, while the maximum supply temperature was 80 C,
consistent with the ranges presented in Section 4.

A continuous 48 hour subset of the week-long simulation was selected to be
emulated in the lab-scale experiment. In the full-scale experiment, thermal
mass 1, associated with the smaller building (a school), started at its heated
temperature and was immediately cut off from the heat supply and allowed to
cool for 12 hours, before being heated again for an additional 12 hours. The
occupancy times of thermal mass 2, representative of the larger building (a
sports hall), were offset from that of thermal mass 1. The larger thermal mass
remained at its heated temperature for two hours and 45 minutes before being
set to cool for 11 hours. After its cooling period, thermal mass 2 was
reheated for the remaining 10 hours and 15 minutes. These occupancy cycles
were repeated twice during the 48 hour period. For the lab-scale experiment,
the timings of cycles were scaled using Eq. 12, resulting in the entire 48
hour period being recreated in just under 18 hours, a 63% reduction in time
needed to run the experiment. To recreate the thermal masses’ temperature
profiles, two PID controllers are implemented in the lab-scale network, where
the difference between the current thermal masses’ temperatures and their
desired setpoints are used to set the position of the bypass valves to drive
the thermal masses to the desired temperature. During hours of operation, the
setpoint was 28 C, while during cooling periods, the PID control setpoint was
0 C.

The ambient temperature profile used in the original simulation is show in
Fig. 5. This ambient temperature was recreated in the lab-scale experiment
using the Peltier junction. The power setpoint for each Peltier junction was
controlled using

$\dot{Q}_{pelt}=\left(hA_{s}\right)_{ThM}\bigl{(}\left(T_{a}-T_{set}\right)_{Lab}\\\
-k_{T}\left(T_{a}-T_{set}\right)_{Full}\bigr{)}$ (25)

where $k_{T}$ is the ratio of nondimensionalization parameters for the full-
scale and lab-scale temperatures:

$k_{T}=\frac{T_{s_{Lab}}}{T_{s_{Full}}}$ (26)

Each Peltier junction has a built-in PID controller that is used to track a
desired power setpoint.

###  5.3 Results

The temperature response of the thermal masses are shown in Figs. 5 and 6,
specifically the nondimensional temperature of the thermal masses, along with
the nondimensional simulated ambient temperatures for both the full-scale and
lab-scale systems are shown in Figs. 5a and 5b for thermal mass 1 and thermal
mass 2, respectively. The result of the direct comparison shows a consistent
dynamic response between the full-scale and the lab-scale system.
Additionally, it can be seen that the simulated ambient temperature is able to
effectively follow the desired full-scale temperature, with the RMS error
being 0.22% for both thermal masses.

Figure 6 shows the median nondimensional full-scale and lab-scale thermal mass
temperature along with the limits of 25th and 75th percentiles for the
temperature of both thermal masses. The mean and standard deviations of the
temperatures are listed in Table 3, along with the ratio of the mean lab-scale
temperatures to the mean full-scale temperatures. Consistently, the lab-scale
results have a larger spread in the nondimensional thermal mass temperature,
which is more evident in the smaller thermal mass ($ThM1$).

While the general trend between the full-scale and lab-scale thermal mass
temperatures are similar, there is a large variation between the extreme
values observed. This deviation can be attributed to the difference in values
of $\pi_{4}$ between the particular buildings being represented. Specifically,
$\pi_{4}$ describes the total heat capacity of the thermal mass, which is
mainly effected by the building’s volume. In the full-scale configuration, the
nondimensional heat capacities were approximately 8.6 times bigger for both
thermal mass 1 and thermal mass 2 in the full-scale simulation. These values
are consistent with the ratio of the mean temperature values between the two
scales. Moreover, when considering the differences in the heat transfer
coefficients for different sized buildings, this provides an explanation of
the variations observed. There was a large range of potential building volumes
presented in literature; in the lab-scale setup, the thermal masses are sized
more similarly to residential buildings, while the full-scale experiment used
commercial/industrial sized buildings. The effects of various building sizes
can be seen in Gambarotta et al. [35], where the temperature dynamics of 12
different buildings are simulated in one network. The building temperatures in
the lab-scale system will exhibit more rapid dynamic responses as compared to
larger buildings, but will still be valid as a proving ground for new modeling
and control techniques.

Table 3: Mean and standard deviation of nondimensional thermal mass temperatures | Scale | Mean | STD | Ratio
---|---|---|---|---
$T_{ThM1}$ | Lab | $-9.42\times 10^{-3}$ | $10.5\times 10^{-3}$ | 7.32
Full | $-1.29\times 10^{-3}$ | $2.15\times 10^{-3}$
$T_{ThM2}$ | Lab | $-6.81\times 10^{-3}$ | $8.17\times 10^{-3}$ | 2.11
Full | $-3.22\times 10^{-3}$ | $4.07\times 10^{-3}$

(a) Response of thermal mass 1 for full and lab-scale systems.

(b) Response of thermal mass 2 for full and lab-scale systems.

Figure 5: Comparison between the lab-scale and full-scale systems. Figure 6:
Median nondimensional thermal mass temperature with upper and lower quartiles
for full-scale and lab-scale thermal masses.

The response of the pipes in the distribution network is shown in Figs. 8, 7
and 9. Figure 7 presents the nondimensional pressure losses in the pipes. For
reference, the dimensional scale for pressure losses range from 0 to 25 kPa
and the pump provides a 75 kPa increase in pressure, which are in the
acceptable range for a similar full-scale system. The pressure losses across
the heat exhangers are 3.6 and 4.5 kPa for thermal mass 1 and 2 respectively.
Figure 8 shows the mass flow split between the network branches. This data is
divided into four time intervals, the average mass flow rate, the mass flow
rate during the cooling of both thermal masses, the mass flow rate during the
heating of both thermal masses, and the mass flow rate during the steady state
operation, where the PID controller is working to maintain the current
temperature in the thermal masses. The measured values correlate well with
relative pressure losses in the network . This data is used in the calculation
of $\pi_{1}$ and in the calculation of the heat losses in the network. The
values in these plots are validated against the data provided in Ancona et al.
[26].

The supply temperature, return temperature, and select pipe temperatures
throughout the network are presented in Fig. 9 for the portion of the
experiment where both thermal masses are being re-heated for the first time
and then maintained at the set temperature. This information was used to
validate the temperature dynamics in the pipes. The difference between the
supply and return temperature was used to calculate the energy losses
throughout the network. Additionally, there is a time delay between changes in
the supply temperature and those changes being seen in the return temperature.
This delay is caused by the time taken for the operating fluid to circulate
through the network. As the fluid dynamics of the system are much faster than
the temperature dynamics, this delay can be modeled as an algebraic offset.
The magnitude of the algebraic offset seen in the lab-scale data was validated
against the data collected from a CFD-based analysis of a DHN’s distribution
network presented in Zhao et al. [36]. This paper used the peak-valley method
to quantify the delay. In nondimensional time, the average delay in
temperature peaks and valleys for users a similar nondimensional distance away
from the supply is approximately 4,500, or about 90 seconds in the lab-scale
system, similar to the results seen in the lab-scale system. This time delay
can have a large impact on the energy consumption of a DHN and can
consequently greatly effect the efficiency of a predictive controller.
Therefore, accurately capturing this delay in the lab-scale system is critical
for control design.

The ratio of heat being lost to the environment compared to the heat being
extracted by the heat exchangers is presented in Fig. 11. The rate of heat
lost can be found using the change in enthalpy of the circulating water,
calculated according to

$\dot{Q}_{tot}=\dot{m}_{I}c_{p}\left(T_{s}-T_{r}\right)$ (27)
$\dot{Q}_{ThM_{i}}=\dot{m}_{HX_{i}}c_{p}\left(T_{HX_{i}in}-T_{HX_{i}out}\right)$
(28) $\dot{Q}_{tot}=\dot{Q}_{amb}+\dot{Q}_{ThM_{1}}+\dot{Q}_{ThM_{2}}$ (29)

where $T_{s}$ is collected by TM1, $T_{r}$ by TM2, $T_{HX_{i}in}$ by TM10 and
TM5, and $T_{HX_{i}out}$ by TM12 and TM17 respectively.

This efficiency measure is divided into the same time intervals as the mass
flow rates (overall, cooling, heating, and steady state). The reported average
efficiency of a full-scale district heating network is around 70% [37], while
the overall lab-scale is only around 28% efficient. There are many
explanations for the discrepancy between these values. The load diversity in a
DHN with many users is much higher than for just the two users being
represented here, leading to a higher percentage of the network’s operation
being in the heating phase, where the lab-scale DHN performs comparably to a
full-scale DHN. Additionally, the size of the thermal masses can also impact
this metric, as smaller residential buildings require less heat than larger
industrial buildings, thereby decreasing the time spent in the heating phase
of operation. Furthermore, in this experiment, no effort was made to reduce
the mass flow rate during the times when the buildings were not being heated.
A more optimal control algorithm would have reduced the mass flow rate during
the cooling periods, reducing the energy wasted and increasing efficiency.
Finally, the full-scale efficiency is calculated for an entire year of
operation, and heat losses vary drastically depending on the season and
ambient temperature.

The breakdown of the total heat lost to the environment by each of the
components of the system is presented in Fig. 11. Most of of the heat is being
lost by pipes, followed by the two thermal masses. The heat lost by the
thermal masses includes the heat lost through natural convection and the heat
lost due to the cooling of the Peltier junctions. The heat lost by the heater
is the smallest source of energy losses in the network. The relative energy
loss between the components is fairly consistent between the different phases
of operation.

Figure 7: Nondimensional pressure losses in each pipe segment. Figure 8: The
nondimensional mass flow rate through each pipe segment during different
operating periods. Figure 9: Nondimensional temperature in the pipes during
the first heating.

Figure 10: Breakdown of energy leaving the pipe network, separated into heat
lost to the ambient, and heat transferred to the two thermal masses during
different operating periods.

Figure 11: Breakdown of the energy losses in the network, separated into the
energy lost by pipe network, the water heater and the two buildings.

##  6 Conclusion

This paper describes the design and validation of a dynamically similar lab-
scale DHN. The equations used to model the system are presented, along with
the dimensionless groups that describe the relevant dynamics of the system.
Then, representative values for all components of a full-scale DHN based on
current literature are provided, along with the corresponding desired lab-
scale values. Finally, the scaled DHN is validated by recreating a simulated
two day period of operation and compares the results to desired values. The
data is presented in the nondimensional form to allow direct comparison to the
full-scale data. Future work will use the data collected from the lab-scale
DHN to validate modeling techniques developed for use in the design of novel
control algorithms.

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Homological monotonicity for configuration spaces of manifolds

by

Muhammad Yameen

###### Abstract

Consider the configuration spaces of manifolds. An influential theorem of
McDuff, Segal and Church shows that the (co)homology of the unordered
configuration space is independent of number of points in a range of degree
called the stable range. We study the another important (and general) property
of unordered configuration spaces of manifolds (not necessarily orientable,
and not necessarily admitting non-vanishing vector field) that is _homological
monotonicity_ in unstable part. We show that the homological dimension of
unordered configuration spaces of manifolds in each degree is monotonically
increasing. Our results show that the monotonicity property is not depend on
the differential structure and orientiability of manifold.

> Key Words: Configuration spaces, manifolds, homological monotonicity
>
>
> Mathematics Subject Classification: Primary 55R80, Secondary 55P62.

##  1 Introduction

For any manifold $M$, let

$F_{k}(M):=\\{(x_{1},\ldots,x_{k})\in M^{k}|x_{i}\neq x_{j}\,for\,i\neq j\\}$

be the configuration space of $k$ distinct ordered points in $M$ with induced
topology. The symmetric group $S_{k}$ acts on $F_{k}(M)$ by permuting the
coordinates. The quotient

$C_{k}(M):=F_{k}(M)/S_{k}$

is the unordered configuration space with quotient topology. All manifolds are
assumed to be finite type (Betti numbers are finite numbers). The boundary of
manifold $M$ is written $\partial M.$ Throughout the paper, $U$ is a closed
subset of $M$ such that $\partial M\cap U=\phi,$ and $M-U$ is connected.

An influential theorem of McDuff [7], Segal [9] and Church [2] shows that the
(co)homology of the unordered configuration space $C_{k}(M)$ is independent of
number of points $k$ in a range of degree called the stable range:

######  Theorem 1.

Let $M$ be a manifold. For each $i\geq 0,$ the map

$k\rightarrow\emph{dim}H_{i}(C_{k}(M);\mathbb{Q})$

is eventually constant.

Ellenberg and Wiltshire-Gordon [5] proved that the homological dimension of
unordered configuration spaces of orientable manifolds admitting non-vanishing
vector field is non-decreasing. We prove that the conditions of orientablility
and admitting vanishing vector field are not necessary.

######  Theorem 2.

Let $M$ be a non-orientable manifold. For each $i\geq 0,$ the map

$k\rightarrow\emph{dim}H_{i}(C_{k}(M);\mathbb{Q})$

is monotonically increasing.

######  Theorem 3.

Let $M$ be a orientable manifold and $\partial M\cup U\neq\phi.$ For each
$i\geq 0,$ the map

$k\rightarrow\emph{dim}H_{i}(C_{k}(M-U);\mathbb{Q})$

is monotonically increasing.

### General conventions

$\bullet$ We work throughout with finite dimensional graded vector spaces. The
degree of an element $v$ is written $deg(v)$.

$\bullet$ The symmetric algebra $Sym(V^{*})$ is the tensor product of a
polynomial algebra and an exterior algebra:

$Sym(V^{*})=\bigoplus_{k\geq 0}Sym^{k}(V^{*})=Poly(V^{even})\bigotimes
Ext(V^{odd}),$

where $Sym^{k}$ is generated by the monomials of length $k.$

$\bullet$ The $n$-th suspension of the graded vector space $V$ is the graded
vector space $V[n]$ with $V[n]_{i}=V_{i-n},$ and the element of $V[n]$
corresponding to $a\in V$ is denoted $s^{n}a;$ for example

$H^{*}(S^{d};\mathbb{Q})[n]=\begin{cases}\mathbb{Q},&\text{if
$*\in\\{n,n+d\\}$}\\\ 0,&\mbox{otherwise}.\\\ \end{cases}$

$\bullet$ We write $H_{-*}(M;\mathbb{Q})$ for the graded vector space whose
degree $-i$ part is the $i$-th homology group of $M;$ for example

$H^{-*}(S^{d};\mathbb{Q})=\begin{cases}\mathbb{Q},&\text{if
$*\in\\{-d,0.\\}$}\\\ 0,&\mbox{otherwise}.\\\ \end{cases}$

##  2 Chevalley–Eilenberg complex

Félix–Thomas [4] (see also [3]) constructed a Sullivan model for the rational
cohomology of configuration spaces of closed orientable even dimensional
manifolds. Furthermore, Knudsen [6] extended the result of Félix–Thomas for
general even dimensional manifolds.

Let us introduced some notations. Consider two graded vector spaces

$V^{*}=H^{-*}_{c}(M;\mathbb{Q}^{w})[d],\quad
W^{*}=H_{c}^{-*}(M;\mathbb{Q})[2d-1]:$

where

$V^{*}=\bigoplus_{i=0}^{d}V^{i},\quad W^{*}=\bigoplus_{j=d-1}^{2d-1}W^{j}.$

We choose bases in $V^{i}$ and $W^{j}$ as

$V^{i}=\mathbb{Q}\langle v_{i,1},v_{i,2},\ldots\rangle,\quad
W^{j}=\mathbb{Q}\langle w_{j,1},w_{j,2},\ldots\rangle$

(the degree of an element is marked by the first lower index; $x_{i}^{l}$
stands for the product $x_{i}\wedge x_{i}\wedge\ldots\wedge x_{i}$ of
$l$-factors). Always we take $V^{0}=\mathbb{Q}\langle v_{0}\rangle$. Now
consider the graded algebra

$\Omega^{*,*}_{k}(M)=\bigoplus_{i\geq
0}\bigoplus_{\omega=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\Omega^{i,\omega}_{k}(M)=\bigoplus_{\omega=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\,(Sym^{k-2\omega}(V^{*})\otimes
Sym^{\omega}(W^{*}))$

where the total degree $i$ is given by the grading of $V^{*}$ and $W^{*}$. We
called $\omega$ is a weight grading. The differential
$\partial:Sym^{2}(V^{*})\rightarrow W^{*}$ is defined as a coderivation by the
equation

$\partial(s^{d}a\wedge s^{d}b)=(-1)^{(d-1)|b|}s^{2d-1}(a\cup b),$

where

$\cup\,:H^{-*}_{c}(M;\mathbb{Q}^{w})^{\otimes 2}\rightarrow
H^{-*}_{c}(M;\mathbb{Q})$

(here $H^{-*}_{c}$ denotes compactly supported cohomology of $M$). The degree
of $\partial$ is $-1.$ It can be easily seen that $s^{d}a,\,s^{d}b\in V^{*}$
and $s^{2d-1}(a\cup b)\in W^{*}.$ The differential $\partial$ extends over
$\Omega^{*,*}_{k}(M)$ by co-Leibniz rule. By definition the elements in
$V^{*}$ have length 1 and weight 0 and the elements in $W^{*}$ have length 2
and weight 1. By definition of differential, we have

$\partial:\Omega^{*,*}_{k}(M)\longrightarrow\Omega^{*-1,*+1}_{k}(M).$

######  Theorem 4.

If $d$ is even, $H_{*}(C_{k}(M);\mathbb{Q})$ is isomorphic to the homology of
the complex

$(\Omega^{*,*}_{k}(M),\partial).$

##  3 Proofs of main results

In this section, we give the proofs of Theorems 2 and 3.

Proof of Theorems 2 and 3.

Case I. Let $d$ is odd. We have an isomorphism [1]:

$H_{*}(C_{k}(M);\mathbb{Q})\cong Sym^{k}(H_{*}(M;\mathbb{Q})).$

We can write the decomposition of algebras:

$Sym^{k}(V^{*})\cong(\langle v_{0}\rangle\otimes Sym^{k-1}(V^{*}))\oplus
Sym^{k}(V^{*})/\langle v_{0}\rangle,$

here we take $V^{*}\cong H_{*}(M;\mathbb{Q})$ and $v_{0}$ is a corresponding
element of $H_{0}(M;\mathbb{Q}).$ Moreover, we have an isomorphism :

$H_{*}(C_{k-1}(M);\mathbb{Q})\cong\langle v_{0}\rangle\otimes
Sym^{k-1}(V^{*}).$

As a consequence, we get

$H_{*}(C_{k}(M);\mathbb{Q})\cong H_{*}(C_{k-1}(M);\mathbb{Q})\oplus
Sym^{k}(V^{*})/\langle v_{0}\rangle.$

This implies that for each $i\geq 0,$ the map

$k\rightarrow\emph{dim}H_{i}(C_{k}(M-U);\mathbb{Q})$

is monotonically increasing.

Case II. Let $d$ is even and $M$ is not closed orientable. We have
decomposition of algebras:

$\Omega^{*,*}_{k}(M)\cong(\langle
v_{0}\rangle\otimes\Omega^{*,*}_{k-1}(M))\oplus\Omega^{*,*}_{k}(M)/\langle
v_{0}\rangle.$

The subspace $\Omega^{*,*}_{k}(M)/\langle v_{0}\rangle$ is
$\partial-$invariant:

$\partial(\Omega^{*,*}_{k}(M)/\langle
v_{0}\rangle)\subseteq\Omega^{*,*}_{k}(M)/\langle v_{0}\rangle.$

The degrees of elements of $V^{*}$ are less than $d.$ Also, the degree of
elements of $W^{*}$ are greater than and equal to $d-1.$ The differential
$\partial$ has degree $-1.$ Therefore, $\partial(\langle v_{0}\rangle\otimes
V^{*})=0.$ This implies that the subspace $\langle
v_{0}\rangle\otimes\Omega^{*,*}_{k-1}(M)$ is also $\partial-$invariant:

$\partial(\langle v_{0}\rangle\otimes\Omega^{*,*}_{k-1}(M))\subseteq\langle
v_{0}\rangle\otimes\Omega^{*,*}_{k-1}(M).$

As a consequence, we get the decomposition of complexes:

$(\Omega^{*,*}_{k}(M),\partial)\cong(\langle
v_{0}\rangle\otimes(\Omega^{*,*}_{k-1}(M),\partial)\oplus(\Omega^{*,*}_{k}(M)/\langle
v_{0}\rangle,\partial).$

As a consequence, we have an isomorphism :

$H_{*}(C_{k}(M);\mathbb{Q})\cong H_{*}(C_{k-1}(M);\mathbb{Q})\oplus
H_{*}(\Omega^{*,*}_{k}(M)/\langle v_{0}\rangle,\partial).$

This implies that for each $i\geq 0,$ the map

$k\rightarrow\emph{dim}H_{i}(C_{k}(M-U);\mathbb{Q})$

is monotonically increasing.

##  4 Final remark

Napolitano [8] computed the first integral homology group of 2-sphere:

$H_{1}(C_{k}(S^{2});\mathbb{Z}=\mathbb{Z}/(2k-2)\mathbb{Z}.$

The integral homological stability fails for 2-sphere. Similar behavior in
case of closed orientable manifolds is also observe for monotonicity. If $M$
is an orientable manifold (admitting non-vanishing vector field) and $\partial
M\cup U=\phi,$ then the monotonicity is also hold for $C_{k}(M)$ (see [5]).
However, this is not true in general for all closed orientable manifolds. For
example the map

$k\rightarrow\emph{dim}H_{2d}(C_{k}(S^{2d});\mathbb{Q})$

is not monotonically increasing (see [10]). The situation is unclear in this
case, and we ask the following.

Question. Let $M$ be an orientable closed manifold with $\chi(M)\neq 0.$ Then
under what condition the map

$k\rightarrow\emph{dim}H_{i}(C_{k}(M);\mathbb{Q})$

is monotonically increasing for each $i\geq 0?$

Acknowledgement. The author gratefully acknowledge the support from the ASSMS,
GC university Lahore. This research is partially supported by Higher Education
Commission of Pakistan.

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Abdus Salam School of Mathematical Sciences,
GC University Lahore, Pakistan.
E-mail<EMAIL_ADDRESS>