# ILB: Graph Neural Network Enabled Emergency Demand Response Program For
Electricity

Sina Shaham  University of Southern California
Los Angeles, USA
<EMAIL_ADDRESS>Bhaskar Krishnamachari  University of Southern California
Los Angeles, USA
<EMAIL_ADDRESS>Matthew Kahn  University of Southern California
Los Angeles, USA
<EMAIL_ADDRESS>
###### Abstract

Demand Response (DR) programs have become a crucial component of smart
electricity grids as they shift the flexibility of electricity consumption
from supply to demand in response to the ever-growing demand for electricity.
In particular, in times of crisis, an emergency DR program is required to
manage unexpected spikes in energy demand. In this paper, we propose the
Incentive-Driven Load Balancer (ILB), a program designed to efficiently manage
demand and response during crisis situations. By offering incentives to
flexible households likely to reduce demand, the ILB facilitates effective
demand reduction and prepares them for unexpected events. To enable ILB, we
introduce a two-step machine learning-based framework for participant
selection, which employs a graph-based approach to identify households capable
of easily adjusting their electricity consumption. This framework utilizes two
Graph Neural Networks (GNNs): one for pattern recognition and another for
household selection. Through extensive experiments on household-level
electricity consumption in California, Michigan, and Texas, we demonstrate the
ILB program’s significant effectiveness in supporting communities during
emergencies.

###### Index Terms:

Demand Response Program, GNN

##  I Introduction

Demand Response (DR) programs have emerged as an integral component of smart
electricity grids. By shifting the flexibility of power consumption from the
supply side to the demand side, these programs manage the continually
increasing demand for electricity. Such programs incentivize households to
reduce their electricity usage, thereby enhancing the agility and
responsiveness of the network. The benefits of DR programs are manifold,
including, but not limited to, preventing the need for new power plants,
lowering electricity costs by avoiding high-priced energy purchases, enhancing
grid reliability to prevent blackouts, and mitigating environmental damage by
reducing the consumption of fossil fuels. An example of such a program is
Time-of-Use pricing (TOU), where the cost of electricity varies based on the
time of day, typically increasing during peak periods [1].

A more diligent and critical type of DR programs is designed for times of
crisis and is referred to as an emergency DR program. Such programs are
developed to manage unexpected spikes in energy demand during emergency
situations. Unfortunately, most of the existing emergency DR programs are only
activated after the accident and without prior provisions made for households,
such as during extreme weather events, unexpected equipment failures, or other
emergencies. Hence, they mostly work as mediators of the damage rather than
the prevention mechanism, which can have severe consequences for households
and businesses. For example, during a heat wave in California in August 2020,
the California Independent System Operator (CAISO) called for an emergency DR
program to reduce electricity usage and prevent blackouts [2]. This included
asking businesses to reduce their energy consumption during peak demand
periods, as well as calling on residential customers to save energy during the
hottest parts of the day. In another unfortunate incident during a winter
storm in Texas in February 2021, the Electric Reliability Council of Texas
(ERCOT) called for an emergency DR program to reduce electricity usage and
prevent blackouts [3]. This included asking businesses and residents to
conserve energy by reducing heating and hot water usage, as well as turning
off non-essential appliances and electronics.

The earlier examples illustrate the requirement for more advanced emergency DR
programs that can greatly ease the pressure on the power grid during crises
and avoid losses to communities. Nevertheless, there exist several obstacles
that have impeded the development of efficient DR programs, including the need
for real-time communication and monitoring of energy usage, precise prediction
of energy demand, and ensuring sufficient resources are available to meet the
heightened demand during emergencies. Thankfully, recent progress in Machine
Learning (ML) combined with the increased adoption of Advanced Metering
Infrastructure (AMI) in households has cleared the way for the implementation
of new DR programs [4]. AMI meters are capable of measuring and recording
electricity usage at least once per hour and transmitting this information to
both the utility and the customer at least once per day. The range of AMI
installations varies from basic hourly interval meters to real-time meters
with built-in two-way communication capabilities that allow for instantaneous
data recording and transmission.

Considering recent advancements, we propose an incentive-based program
accompanied by a novel ML-driven approach to address the existing challenges
and help to balance supply and demand in times of crisis. In particular, we
have the following contributions:

  * •

We propose an innovative incentive-based strategy designed to manage demand
and response effectively during critical scenarios. This approach, termed the
Incentive-Driven Load Balancer (ILB), strategically identifies households with
the flexibility to adjust their electricity consumption and then operates on a
meticulously designed incentive system, motivating these families to moderate
their energy usage in emergency situations.

  * •

We develop an ML-driven two-step framework for the efficient selection of
participants in the program. The framework involves two GNNs: a pattern
recognition GNN and a household selection GNN. The pattern recognition GNN is
a time-series forecasting dynamic graph with the goal of revealing
similarities among users based on attention mechanism, taking into account
socio-economic factors influencing the elasticity of demand. Meanwhile, the
household selection GNN models communities and aids in selecting suitable
households for the program based on the pattern recognition GNN’s output. Our
framework is thoughtfully developed to factor in geo-spatial neighborhoods.

  * •

We have devised and publicly shared a dataset based on the factors that
influence the elasticity of electricity demand for data mining objectives.
This dataset is greatly empowered by a sample from the synthetically generated
household-level data in [5], which offers a digital twin of residential energy
consumption. We further enrich this dataset by integrating education and
awareness factors from high schools [6], college education data from the
Census [7], median household income and unemployment statistics from the US
Department of Agriculture [8], and climate data [9].

  * •

We perform an extensive number of experiments on the household-level
electricity consumption of people in California, Michigan, and Texas. The
results demonstrate the significant effectiveness of the ILB program in
assisting communities during emergency situations.

##  II Related Work

###  II-A Demand Response Programs

A critical advantage of DR programs lies in their potential to substantially
mitigate peak demand - a pressing issue in electricity systems that
necessitates significant investment in underutilized infrastructure. Violette
et al. [10] have demonstrated the potential of DR programs to curtail peak
demand by up to $15\%$. Moreover, DR programs have been effective in enhancing
the reliability of the electricity system, with comprehensive reviews like the
one by Albadi and El-Saadany [11] elucidating a reduction in the frequency and
duration of blackouts.

Existing DR programs fall into two broad categories: price-based programs
(PBP) and incentive-based programs (IBP). PBP programs typically revolve
around dynamic pricing rates, with the objective of reducing consumption
during peak times. Notable approaches in this category include Time of Use
[1], where the rate changes based on the time of day, and Real-Time Pricing
[12], where the price fluctuates according to the market price of electricity.
The second category, i.e. IBP programs, often reward participants with
financial benefits either for their performance or for their participation.
Direct Load Control (DLC) is one such program that allows the utility to
control certain appliances within a consumer’s home, such as air conditioning,
in exchange for a financial incentive [13]. Similarly, the Demand
Bidding/Buyback program provides consumers the opportunity to bid for payments
in exchange for their desired load reductions [14].

###  II-B Time Series Forecasting

Time-series methods can be grouped into two broad categories: univariate time
series techniques and multivariate time series techniques. Univariate time
series methods focus on analyzing individual observations sequentially without
considering the correlations between different time series. ARIMA methods
[15], for instance, assume linearity, where the prediction is a weighted
linear sum of past observations. On the other hand, multivariate time series
techniques consider and model the interactions and co-movements among a group
of time series variables [16, 17]. Wan et al. [18] propose an encoder-encoder
model based on attention mechanism to capture correlations. The proposed end
to end approach include bi-directional long short-term memory networks (Bi-
LSTM) layers as the encoder network to adaptively learning long-term
dependency and hidden correlation features of multivariate temporal data. The
authors in [19] use filters to extract temporal patterns that remain
consistent over time. Then, apply attention mechanism to select pertinent time
series and utilizes their frequency domain information for multivariate
prediction.

Table I: Summary of Notations. Symbol |  Description
---|---
$n,\,m$ |  Number of households and neighborhoods
$\mathcal{U}=\\{u_{1},...,u_{n}\\}$ |  Set of users
$X_{d}^{\text{supply}},\,X_{d}^{\text{demand}}$ |  Supply and demand power on day $d$
$X_{d}^{u}$ |  Power consumption of user $u$ on day $d$
$k$ |  Number of features
$x^{u}$ |  User $u$ power consumption time series
$\mathcal{D},\mathcal{D}^{\prime}$ |  Set of all days and emergency days
$r_{\text{baseline}}^{u},\,r_{\text{emergency}}^{u}$ |  Baseline & emergency price rate on ILB participants
$I^{u}$ |  Incentive for household $u$
$e^{u}$ |  PE of household $u$
$A_{real},\,A_{est}$ |  Real and estimated similarity matrices

##  III Preliminaries

###  III-A Notation

Consider a map that contains $n$ households
$\mathcal{U}=\\{u_{1},...,u_{n}\\}$, which are distributed among $m$ non-
overlapping neighborhoods $\mathcal{N}=\\{N_{1},...,N_{m}\\}$. Each
neighborhood $N_{i}$ contains a population of size $|N_{i}|$. We represent the
hourly electricity consumption of household $u\in\mathcal{U}$ using a time
series $x^{u}\in\mathbb{R}^{1\times T}$, and we denote the set of all time
series by $X=(x^{1},...x^{n})\in\mathbb{R}^{n\times T}$. Additionally, we
model the total power consumption of user $u$ on day $i$ as $X_{i}^{u}$, and
the set of all days in a billing cycle by $\mathcal{D}$. Table I summarizes
the important notations used throughout the manuscript.

###  III-B Message Passing Framework

Most GNNs use message-passing and aggregation to learn improved node
representations, or so-called embeddings, of the graph. At propagation step
$i$, the embeddings of node $v$ is derived by:

$H_{v}^{i}=f(\text{aggr}(H_{u}^{i-1}|u\in N_{1}(v))).$ (1)

In the above formulation, $H_{v}^{i}$ represents the $i$-th set of features
for the nodes, $f(.)$ is the function used to transform the embeddings between
propagation steps, $N_{1}(.)$ retrieves the 1-hop embeddings of a node, and
aggr combines the embeddings of its 1-hop neighbors. For instance, in GCN,
node aggregation and message-passing are expressed as:

$H_{v}^{i}=\sigma(W\sum_{u\in
N_{1}(v)}\dfrac{1}{\sqrt{\hat{d}_{u}\hat{d}_{v}}}H_{u}^{i-1}),$ (2)

where $\sigma$ is the activation function, $W$ is a matrix of learnable
weights, $\hat{d}_{v}$ is the degree of node $v$. The propagation length of
the message-passing framework is commonly limited to avoid over-smoothing.

##  IV Proposed Scheme

In this section, we present our proposed emergency DR program.

###  IV-A Program Overview

With the rising trend of electricity consumption, combined with the limited
capacity of supply, electricity disruptions in household networks are becoming
inevitable. Suppose the utility provides $X_{i}^{\text{supply}}$ kWh of power
on a particular day $i$, but the electricity demand is $X_{i}^{\text{demand}}$
kWh, where $X_{i}^{\text{supply}}<X_{i}^{\text{demand}}$. This necessitates a
strategy to balance the supply and demand. Currently, there are two extreme
strategies in place: (I) implementing the same level of power outage for all
users by reducing each household’s electricity by
$(X_{i}^{\text{demand}}-X_{i}^{\text{supply}})/n$, and (II) focusing on only
$l$ households and cutting their power for a longer duration, while ensuring
uninterrupted power supply for the others. The second strategy results in an
outage of $(X_{i}^{\text{demand}}-X_{i}^{\text{supply}})/l$ in this group of
households.

The sudden and unexpected power cuts can disrupt the daily life of households,
regardless of their efforts to manage the power demand and response. In order
to tackle this issue, we propose a price-driven demand response program named
the Incentive-driven Load Balancer (ILB), which prioritizes the user’s
preferences. The ILB program enables households to voluntarily participate in
a program that offers them financial incentives upfront, in exchange for
paying higher rates during a few unanticipated days. The households are
informed of such occurrences a day before and encouraged to reduce their
electricity consumption during those times. For those who do not opt in to the
program, higher rates will be charged throughout the period to cover the
program’s expenses. The formal definition of the ILB program is provided in
Definition 1.

######  Definition 1.

(_Incentive-Driven Load Balancer_). Under this strategy, households are
allowed to voluntarily participate in a program, where they agree to pay
higher rates for a number of unplanned days during the upcoming billing cycle,
which they will be notified of only one day in advance. In return for their
participation, they receive a monetary incentive at the start of the program.
For the remaining duration of the billing cycle, the standard rates will be
applied. To cover the cost of the incentives, the electricity rate for all
other households is increased.

The aim of ILB to incentivize flexible users who opt-in to reduce their power
usage during emergency days, rather than enforcing power outages to balance
demand and supply, or force higher prices on all users. This is achieved by
charging the opt-in users higher rates, encouraging them to adjust their
consumption behavior. The incentives provided to the users needs to be
carefully determined — if it is too low, then not enough customers will accept
the offer resulting in an insufficient reduction of the demand during the
emergency days; if too high, it may potentially result in too high a number of
households signing up for the incentive making the program too expensive,
raising the burden on non-participating households beyond acceptable levels.

To quantify the utility of the proposed program, we propose the following two
indicators. The first utility function aims to reveal if offers are successful
in attracting customers.

######  Definition 2.

(_Utility: Acceptance Rate_). The acceptance rate indicates the percentage of
households who agree to participate in the program after receiving an offer.

$\text{Acceptance rate}=\frac{\\#\,\text{accepted offers}}{\\#\,\text{offers
made}}\times 100.$ (3)

The second utility focuses on the amount of reduction made in consumption
given the incentives formulated in Definition 3.

######  Definition 3.

(_Utility: Responsiveness Cost_). Let $\mathcal{U}^{\prime}$ denote the set of
users who voluntarily participate in the program, and
$\mathcal{D}^{\prime}={d_{i},...,d_{j}}$ denote the set of emergency days that
occurred during the billing cycle. The responsiveness cost of ILB is defined
as

$\text{Responsiveness Cost}=(\sum_{u\in\mathcal{U}^{\prime}}I^{u})/(\sum_{d\in
D^{\prime}}\sum_{u\in\mathcal{U}^{\prime}}\Delta X_{d}^{u}).$ (4)

In the above formulation, $\Delta X_{d}^{u}=\bar{X}_{i}^{u}-X_{i}^{u}$
represents the variation in the consumption of user $u$ on day $d$. Here, the
consumption of user $u$ is represented by $X_{d}^{u}$, and the adjusted
consumption due to their participation is represented by $\bar{X}_{d}^{u}$.
The symbol $I^{u}$ refers to the incentive provided to user $u$ at the
beginning.

The rationale behind the responsiveness utility function is to determine the
amount of power consumption reduction that can be achieved for a given amount
of incentives to participants. The utility company uses the following
optimization formulation to balance supply and demand while minimizing
incentive expenditure. The responsiveness cost is a metric that measures how
well this objective is met in practice.

Minimize $\displaystyle\sum_{u\in\mathcal{U}^{\prime}}I^{u}$ (5) subject to
$\displaystyle
X_{d}^{\text{demand}}-X_{d}^{\text{supply}}\leq\sum_{u\in\mathcal{U}^{\prime}}\Delta
X_{d}^{u},\,\,\,\,\,\,\forall\,d\in\mathcal{D}^{\prime}$

###  IV-B Pricing

As utility functions outlined in the previous section reveal, ILB requires a
diligent selection of participants and careful pricing of incentives. In this
subsection, we address the latter and explain how incentives are calculated,
and in Section V, the selection process of applicants is illustrated. Recall
that $\mathcal{D}^{\prime}\subset\mathcal{D}$ are the emergency days. For any
$u\in\mathcal{U}^{\prime}$, let us denote the price rate for emergency days
($d\in\mathcal{D}^{\prime}$) by $r_{\text{emergency}}^{u}$ and their regular
rate for $d\in D\textbackslash D^{\prime}$ by $r_{\text{baseline}}^{u}$. Thus,
by participating in the program the cost of user $u$ will be modified from
their baseline cost,

$Cost_{\text{baseline}}(u)=\sum_{d\in D}X_{d}^{u}\times
r_{\text{baseline}^{u}},$ (6)

to the modified cost of,

$Cost_{\text{ilb}}(u)=\sum_{d\in D\textbackslash D^{\prime}}X_{d}^{u}\times
r_{\text{baseline}}^{u}+\sum_{d\in D^{\prime}}\bar{X}_{d}^{u}\times
r_{\text{emergency}}^{u}-I^{u}.$ (7)

Once it comes to consideration, the household should understandably expect

$Cost_{\text{baseline}}(u)\geq Cost_{\text{ilb}}(u).$ (8)

Otherwise, the offer will not be beneficial for the user. The households who
are not part of the program will be charged by an extra charge of
$r_{\text{extra}}$ and pay the modified rate of,

$r_{others}=r_{\text{baseline}}+r_{\text{extra}}.$ (9)

The rate hike for users not included in the program is derived based on the
incentives provided in ILB, calculated as,

$r_{\text{extra}}=(\sum_{u\in\mathcal{U}^{\prime}}I^{u})/(\sum_{u\in\mathcal{U}\textbackslash\mathcal{U}^{\prime}}\sum_{d\in
D}X_{d}^{u}).$ (10)

###  IV-C Rate Hikes

A critical factor in the above formulation is the rate hikes on emergency
days. This factor directly influences the amount of demand response on such
days. We derive this rate based on the price elasticity of demand for
electricity.

######  Definition 4.

(_Elasticity of Demand_). The elasticity of demand refers to the degree to
which demand responds to a change in an economic factor.

The appropriate value of $r_{\text{emergency}}$ must be chosen such that the
increase in rates results in a change in consumers’ demand, which can be
measured by the price elasticity of demand. The price elasticity of demand is
derived by,

$PE=\dfrac{\%\,\text{Change in Quantity}}{\%\,\text{Change in price}}.$ (11)

The equation indicates the percentage of change in demanded quantity for a
given percentage of change in price. For example, if the price of electricity
increases by $10\%$, and the quantity of electricity demanded decreases by
$5\%$, the price elasticity of demand for electricity can be calculated as:

$PE=(-5\%/10\%)=-0.5$ (12)

The PE for electricity has been measured and estimated in many countries in
the world including the US. The factor is commonly considered for the short
term and long term. The average PE for electricity on the state level is
estimated to be $-0.1$ in the short-run and $-1.0$ in the long-run [20, 21].

It is crucial to note that the numbers mentioned earlier represent average
statistics for households. However, on an individual household level, the PE
for electricity can vary greatly, which can impact the necessary amount of
incentive required for them to accept an offer. Let us denote the PE for
household $u$ by $e^{u}$. Hence, given the goal of ILB that every household in
the program reduces its consumption by $i\%$, the percentage of change in
price for the household is calculated by

$\%\,\text{Change in price}=i/e^{u}.$ (13)

Once the percentage of change in price is calculated, the rates on emergency
days are calculated by

$r_{\text{emergency}}^{u}=(1+\%\,\text{Change in price})\times
r_{\text{baseline}}^{u}.$ (14)

Armed with this knowledge, applying inequality in the Equation 8, the minimum
incentive values can be derived as,

$\displaystyle Cost_{\text{baseline}}(u)\geq Cost_{\text{ilb}}(u)\rightarrow$
(15) $\displaystyle\begin{split}&\sum_{d\in D}X_{d}^{u}\times
r_{\text{baseline}}^{u}\geq\quad\sum_{d\in D\setminus
D^{\prime}}X_{d}^{u}\times r_{\text{baseline}}^{u}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{d\in
D^{\prime}}\bar{X}_{d}^{u}\times
r_{\text{emergency}}^{u}-I^{u}\rightarrow\end{split}$ (16) $\displaystyle
I^{u}\geq\sum_{d\in D^{\prime}}\bar{X}_{d}^{u}\times
r_{\text{emergency}}^{u}-\sum_{d\in D^{\prime}}X_{d}^{u}\times
r_{\text{baseline}}^{u}.$ (17)

Figure 1: Pattern Recognition Graph.

##  V Implementation Framework

The success of the program relies heavily on the process of selecting
candidates. Utilizing both a sequential and machine learning approach can aid
in identifying when enough households have been efficiently selected, offered,
and accepted the incentive. This is essential in ensuring that a satisfactory
number of high-demand flexibility users have accepted the incentives and that
the expected reduction in demand meets the shortfall on emergency days.
Additionally, this method prevents over-recruitment, which could result in
unnecessary program expenses.

Our proposed approach for choosing potential participants for the ILB program
is based on two graph models: the pattern recognition GNN and the household
selection GNN. The dynamic pattern recognition model is designed to generate a
similarity matrix among users, which indicates the probability that a
household would respond positively to an offer based on the response of its
neighbors and other households. The household selection model, on the other
hand, is a node classification model that aims to identify the candidates who
are most likely to accept the offer.

###  V-A Pattern Recognition GNN

The main objective of the pattern recognition model is to create an accurate
similarity matrix that reflects the degree of similarity between two
individuals based on their socio-economic status and electricity consumption
pattern. This advance model helps us understand the likelihood of a household
accepting an offer, given the responses of those already queried. The
underlying assumption is that households with a high similarity score are
likely to respond similarly to the offer. Although there may be some margin of
error, a well-designed model can mitigate this issue to a reasonable extent.
Our proposed model considers three factors: (I) socio-economic factors
affecting demand response, (II) intra-series temporal correlations in time
series, and (III) inter-series correlations captured by an attention mechanism
that reveals pattern similarity. While the model is primarily designed to
predict future demand, the inter-series attention matrix enables enhanced
modeling of user similarity.

We use $A_{real}\in\mathbb{R}^{n\times n}$ and $A_{est}\in\mathbb{R}^{n\times
n}$ to denote the actual and predicted probability matrices of accepting the
offer, respectively. The element ($a_{ij}$) located in the $i$-th row and
$j$-th column of these matrices represents the likelihood of household $u_{i}$
accepting the offer, given that household $u_{j}$ has already accepted the
offer. An overview of the model is provided in Figure 1 and the components of
the model are elaborated layer by layer in the following.

####  V-A1 Intra-series Temporal Correlation Layer

The first layer of the model aims to capture the temporal correlation in each
time series. For a given window $s$, the training data
$X\in\mathbb{R}^{n\times s}$ would be input to the model. Each time series is
processed by two essential components within this layer: a Gated Recurrent
Unit (GRU) unit and a self-attention module.

The purpose of GRU units is to handle sequential data by temporal
dependencies. GRU has been shown to utilize less memory and perform faster
compared to Long Short Term Memory (LSTM). The self-attention components have
been included after GRU units to enhance their performance further. The input
data $X$ is fed into GRU units and self-attention components generating
embeddings $C=(c_{1},...,c_{s})\in\mathbb{R}^{s\times M}$ where $M$ is the
embedding size.

####  V-A2 Inter-series Correlation Layer

Next, the generated embeddings are fed to a multi-head attention layer [22].
The multi-head attention layer consists of multiple attention heads, each of
which learns to attend to different parts of the input sequence. This is the
critical step where the correlation and similarity between households are
captured. The ultimate output of this unit is the matrix $A_{\text{est}}$, or
the so-called attention matrix, representing the pairwise correlations of
households.

The input of the attention layer, as formulated in Equation 18 consists of
queries ($Q$), keys ($K$), and values ($V$), each of which is a sequence of
vectors. To understand the similarity between time series, all inputs are set
to the embedding matrix $C$ generated in the previous layer. The layer then
applies multiple attention heads, each of which computes a weighted sum of the
values using a query and key pair. The outputs from each attention head are
concatenated and linearly transformed to produce the final output of the
layer.

$A_{est}=MultiHead(Q,K,V)=(head_{1}\mathbin{\|}…\mathbin{\|}head_{j})W^{O},$
(18)

where each head consists of,

$head_{i}=Attention(QW_{i}^{Q},KW_{i}^{K},VW_{i}^{V}),$ (19)
$Attention\left(Q,\ K,\
V\right)=Softmax\left(\frac{QK^{T}}{\sqrt{M}}\right)V.$ (20)

In the above formulation, $W^{O}$, $W_{i}^{Q}$, $W_{i}^{K}$, and $W_{i}^{V}$
are weight matrices learned during the training.

####  V-A3 Socio-economic Factors

Next, the embedding produced in the preceding layers will be combined with the
socio-economic factors to serve as node features in the dynamic graph. It is
important to note that our focus is on factors that have been proven to have a
considerable effect on demand elasticity, not just on user consumption. For
instance, even though the number of individuals in a household can impact
consumption, it has not been identified as a significant coefficient of demand
elasticity in some studies, as cited in [23]. We have summarized the key
determinants in the following. (I) Income: Studies, such as the one by Brounen
et al.[24], have demonstrated that the level of income has a notable influence
on the elasticity of demand for electricity. (II) Demographic characteristics:
Factors such as race and age significantly impact household electricity
consumption[25, 23]. (III) Dwelling physical factors: Aspects including the
type of building, its size, and its thermal and quality characteristics are
linked to the amount of energy consumed by a household [26]. (IV) Climate:
Temperature has been demonstrated to impact electricity consumption
significantly [23]. (V) Living Area: The geographical location of households,
including regional variables such as urban and rural areas, cities and
counties, critically affect their elasticity of demand [23]. (VI) Awareness
and Education: The level of awareness about policies and the education level
of individuals can influence their electricity demand and compliance with ILB,
as shown by Du et al. [23], who found the coefficient representing the extent
to which users understand the policy to be a significant factor in demand
response.

Let us denote the socio-economic features for $i$-th household by
$\hat{c}_{i}\in\mathbb{R}^{\bar{M}}$. The concatenation of embeddings from
time series and socio-economic features leads to node features of the dynamic
graph, mathematically represented by,

$c_{i}^{\prime}=c_{i}||\hat{c}_{i}\in\mathbb{R}^{(M+\bar{M})},$ (21)

resulting in the final node feature matrix
$C^{\prime}=(c_{1}^{\prime},...,c_{n}^{\prime})\in\mathbb{R}^{n\times(M+\bar{M})}$.

####  V-A4 Dynamic Graph

In the final stage of the pattern recognition graph, the dynamic GNN is
formulated as $G(V,E)$, where $V$ represents the node set and $E$ represents
the edge set. To create the graph, each household is assigned a node, and
their node features are generated using the concatenated features derived from
Equation 21. The attention matrix $A_{est}$ obtained from Equation 18 is
utilized as the edge weights of the graph. Subsequently, the GNN block is
applied to the graph. The result of this process is the embeddings for the
nodes, represented by

$C^{\prime\prime}=F_{\text{GNN}}(G(V,E))\in\mathbb{R}^{n\times M"}.$ (22)

where $F_{\text{GNN}}$ denotes a custome GNN model. The graph uses the mean
square error (MSE) as its objective loss function to predict demand. After the
training process, the pattern recognition graph outputs the attention matrix
$A_{\text{est}}$, which is utilized to represent the similarity between
households.

Figure 2: Household Selection Graph.

###  V-B Household Selection GNN

The purpose of the household selection graph is to efficiently select the
users who have the highest likelihood of participating in the program.
Therefore, the graph is a node classification GNN conducting the semi-
supervised task. Let $G_{\text{HSG}}=(V_{\text{HSG}},E_{\text{HSG}})$ denote
the household selection graph. The similarity matrix generated based on the
pattern recognition graph represents the weighted edges of the graph, i.e.,
$E_{\text{HSG}}=A_{\text{est}}$. At this stage all socio-economic and power
consumption information have been incorporated in the similarity matrix. The
set of nodes in the graph consist of a single node for each household, i.e.,
$V_{\text{HSG}}=\mathcal{U}$. The one-hot encoding method is used as node
features of the graph. For example, if $4$ households exist in the network,
their feature set would be $0001,\,0010,\,0100,\,1000$.

The goal of this graph is to efficiently label nodes indicating acceptance or
rejection of an offer. Two steps are taken to conduct the labeling. First,
Spectral Graph Analysis is conducted on the graph to cluster nodes based on
the similarity. Such an approach allows for naive understanding of the node
labels. Based on the spectral graph analysis small portion of households in
each neighborhood are queried to understand the true labels. Then, those
labels are used in the graph to conduct the semi-supervised graph labeling.

####  V-B1 Spectral Graph Analaysis

Spectral clustering methods rely on the spectrum (eigenvalues) of the data’s
similarity matrix to reduce its dimensions before clustering it in a lower
number of dimensions. The similarity matrix $A_{\text{est}}$ is given as input
to the clustering.

First, the method computes the normalized Laplacian of the graph as defined by
the following equation:

$L=I^{\prime}-D^{\prime\prime-1/2}A_{\text{est}}D^{\prime\prime-1/2}.$ (23)

In this equation, $I^{\prime}$ represents the identity matrix, while
$D^{\prime\prime}$ is defined as $diag(d)$, where $d(i)$ denotes the degree of
node $i$.

Secondly, it computes the first $k$ eigenvectors that correspond to the $k$
smallest eigenvalues of the Laplacian matrix. Once eigenvectors are derived, a
new matrix with the $k$ eigenvectors is generated, where each row is treated
as a set of features of the corresponding node in the graph. Finally, the
nodes are clustered based on these features using $k$-means algorithms into
$2$ clusters representing people who are likely to accept or reject the offer.

####  V-B2 Semi-Supervised Node classification

Up to this point, there have been no offers made to determine real-world
responses to offers. However, using spectral graph analysis, households have
been grouped into two clusters and assigned a label. It’s important to note
that the labels based on these clusters don’t necessarily indicate which group
is more likely to accept the offers. The ultimate goal at this stage is to use
the perceptions received from clustering to survey a small portion of
individuals from each of the $m$ neighborhoods, and use this data alongside
the adjacency matrix in a semi-supervised approach to identify individuals who
are likely to accept the offers.

To ensure that all parts of the graph are properly discovered and fairly
treated, it’s crucial to diversify the initial queries in all neighborhoods.
For this reason, in each neighborhood, $5\%$ of users from each cluster
generated by spectral graph analysis are selected to be offered incentives.
Querying this sample of users leads to real labels determined for $10\%$ of
the population. With this information in hand, a node classification GNN model
is applied on top of $G_{\text{HSG}}=(V_{\text{HSG}},E_{\text{HSG}})$ to
classify whether other users will accept the offer or not. The resulting
labels are used to make offers.

##  VI Experimental Evaluation

###  VI-A Datasets

We conduct our experiments by combining multiple datasets from various
sources, taking into consideration significant aspects such as geographical
diversity, the time series of household electricity consumption, and
socioeconomic factors.

Household-level Electricity Time-Series. The privacy concerns regarding
household-level electricity consumption have limited the availability of
publicly accessible datasets, with current datasets containing no more than
$50$ households. A comprehensive overview of these datasets can be found in
[27]. Nonetheless, the work in [5] has tackled this issue by creating a
synthetic dataset that covers households throughout the United States. This
dataset functions as a digital twin of a residential energy-use dataset within
the residential sector. For our experiments, we use the time-series data at
the household level from this dataset.

Education and Awareness. To indicate the level of awareness, the average ACT
scores are used, which are obtained from the EdGap dataset [6]. This dataset
contains socioeconomic features of high school students across the United
States. The geospatial coordinates of schools are derived by linking their
identification number to data provided by the National Center for Education
Statistics [7], and then the average ACT scores are calculated. Additionally,
the percentage of people who have completed some college education provided by
[8] is used to further expatiate on the level of awareness.

Median Household Income and Unemployment Percentage. The county-level
information for the median household income as well as the unemployment
percentage in 2014 is extracted from the US Department of Agriculture website
[8] and used as static features for counties.

Climate. Two key attributes provided by the National Centers for Environmental
Information [9] are used as indicators of climate in counties: average
temperature and precipitation amount.

Geographic Diversity. Three states of California (CA), Texas (TX), and
Michigan (MI) are selected for the purpose of experiments, including the first
five counties based on codes published by the U.S. Census Bureau [28]. The
geospatial location of counties used in experiments and their corresponding
statistics is provided in Figure 3.

(a) California

(b) Michigan

(c)  Texas

(d) California

(e) Michigan

(f)  Texas

Figure 3: Counties used for the purpose of experiments and their corresponding
average hourly consumption. The standard deviation of consumption is shown as
error bars.

###  VI-B Experimental Setup

Overview. The experiments are divided into two parts, each looking at
performance before and after applying the framework. Subsection VI-C
specifically looks at how well the ILB performs before the selection framework
is applied, and investigates how the program affects responsiveness cost and
how much it can reduce the total consumption given a certain amount of
incentive. Subsection VI-D focuses on the performance after the framework has
been applied in which factors such as responsiveness cost, the effects of rate
increases on those not participating in the program, and noise analysis in the
selection process are thoroughly evaluated.

Hyper-parameters Setting. The household electricity time series is used on an
hourly basis between September and December of $2014$. The number of
households participating in the program in each county is $50$, adding up to
$250$ households in each state. The dataset was separated into three parts:
training, evaluation, and testing sets, with a ratio of 7:2:1 respectively.
Z-normalization was applied to normalize the input data, and training was
conducted using the RMSProp optimizer with a learning rate of 3e-4. Training
took place over $100$ epochs, with a batch size of $32$. The number of
emergency days per month is set to three days. The number of participants in
the experiment is set to be one-quarter of the population being considered and
the default incentive provided to participants is $100$ dollars unless stated
otherwise. The electricity PE for households is randomly selected based on a
Gaussian distribution with a mean of $-0.25$ and a standard deviation of
$0.1$.

Hardware and Software Setup. Our experiments were performed on a cluster node
equipped with an $18$-core Intel i9-9980XE CPU, $125$ GB of memory, and two
$11$ GB NVIDIA GeForce RTX 2080 Ti GPUs. Furthermore, all neural network
models are implemented based on PyTorch version 1.13.0 with CUDA 11.7 using
Python version 3.10.8.

###  VI-C Performance Analysis of ILB

In this subsection, we focus on the ILB program’s performance independent of
the framework. Initially, we evaluate the ILB’s efficiency in terms of
responsiveness cost, and subsequently, the program’s effectiveness in total
demand reduction of electricity.

(a) California

(b) Michigan

(c) Texas

Figure 4: Evaluation of total responsiveness cost.

####  VI-C1 Responsiveness Cost

Figure 4 illustrates the results for the ILB program’s responsiveness cost.
Recall that responsiveness reflects the total incentive amount offered in
relation to the observed reduction, thus a lower cost indicates better
performance. The figure showcases the performance across three states:
California, Michigan, and Texas. In each subfigure, given a specified
incentive (highlighted with a bar) offered to the entire community, each
individual assesses the benefit of participating in the program. This
assessment is conducted using Equation 17, which allows us to derive the
community’s acceptance rate and represent it on the $x$-axis, while the
corresponding responsiveness cost is plotted on the $y$-axis. The figure
reveals that an increase in the incentive amount expectedly raises the
acceptance rate, but it also results in a higher responsiveness cost. The rate
of increase for responsiveness cost tends to be lower for smaller incentive
amounts and grows for larger values.

####  VI-C2 Total Demand Reduction

Figure 5 shows the assessment of the total percentage reduction in consumption
on emergency days. The structure of subfigures is analogous to Figure 4, but
they illustrate the impact on the total consumption reduction instead of
responsiveness cost on the $y$-axis. An increasing trend is observed across
all three states indicating that a rise in the incentive amount and acceptance
rate leads to a higher reduction in consumption on emergency days, enhancing
the agility and performance of the electricity network. This pattern hints at
a crucial trade-off between responsiveness cost and overall demand reduction,
which becomes increasingly apparent as the incentives and participant count
increase.

(a) California

(b) Michigan

(c) Texas

Figure 5: Evaluation of total demand reduction.

###  VI-D Evaluating the Integrated Framework and ILB

Unlike the previous subsection, where offers were made to every household,
this subsection evaluates performance based on a proposed framework that
incorporates socio-economic factors and other determinants affecting the
elasticity of demand for electricity when selecting participants.

(a) California

(b) Michigan

(c) Texas

Figure 6: Responsiveness performance of ILB.

(a) California

(b) Michigan

(c) Texas

Figure 7: Rate hike on nonparticipants in ILB.

####  VI-D1 Responsiveness Cost

The performance evaluation in terms of responsiveness cost is depicted in
Figure 6. The $x$-axis represents the reduction in consumption by ILB program
participants during emergency days, while the $y$-axis corresponds to the
responsiveness cost. A quarter of households with the highest scores based on
our framework are selected to participate in the program. As expected, once
the amount of reduction by participants increases, a considerable decrease in
responsiveness is observed in all three states. The horizontal red lines in
the figure indicate the total reduction in demand for the whole community. The
intersection of the curve and the horizontal line shows when the reduction in
demand by ILB users accounts for a certain amount of total reduction in demand
specified by the red line. The figures demonstrate that even small reductions
of $10$ to $20$ percent by participants can lead to a significant reduction of
approximately $5$ to $10$ percent in total demand. When the reduction in ILB
participants is $50\%$, the total reduction accounts for $20$ to $25$ percent.
Therefore, the proposed approach is a viable approach to effectively manage
supply and demand during emergencies and prevent severe outages across states
and counties.

####  VI-D2 Rate Hikes on Non-participants

Figure 7 shows the impact of the ILB program on non-participants. It displays
the rate hikes in $c/kWh$ for non-participants at different percentages of ILB
participation and corresponding incentives paid. Consistent with previous
experiments, the PE for electricity is considered, and participants with the
highest scores based on our framework are selected for the program. As it can
be seen in the figure, increasing the percentage of participants and the
amount of incentives paid leads to a higher cost for non-participants.
However, the figure illustrates that in all three states, with a billing
period of one month, by keeping the incentive amount within the range of $100$
to $200$ dollars, it is possible to only sacrifice a minimal cost on hourly
consumption of non-participants.

####  VI-D3 Candidate Selection and Noise Analysis

In Table II, the proposed framework’s effectiveness for selecting candidates
to participate in ILB is presented alongside noise analysis on the similarity
matrix. To introduce inaccuracies in the attention matrix, a uniform random
noise is added to each entry of the matrix, following a distribution of
$Uniform(0,b)$ where $b$ is set to three values: $25\%,50\%$ and $75\%$ of the
average value of the attention matrix. The results demonstrate that when the
attention matrix is accurate, the model’s accuracy is approximately $90\%$.
However, as inaccuracies increase, the model’s performance consistently
declines across all three states. However, it is notable that even when the
amount of inaccuracy is large, it does not severely impact the performance of
the final model and the accuracy remains in an acceptable range.

Table II: Accuracy of ILB candidate selection framework. Percentage of Noise | Accuracy
---|---
| California | Texas | Michigan
0 % | 88 | 90 | 90
25 % | 84 | 86 | 87
50 % | 81 | 79 | 86
75 % | 73 | 78 | 84

##  VII conclusion

In conclusion, this paper introduced ILB, a novel program specifically
designed for efficient demand management and response during crisis events.
The ILB program encourages flexible households with the potential to lower
their energy demand through incentives, thus facilitating effective demand
reduction and preparing them for unforeseen circumstances. We also developed a
two-step machine learning framework for the selection of participants. This
framework, which incorporates a graph-based method to pinpoint households
capable of readily modifying their energy usage, leverages two GNNs - one for
recognizing patterns and another for household selection. Our comprehensive
experiments on household-level electricity consumption across CA, MI, and TX
provided compelling evidence of the ILB program’s significant capacity to aid
communities during emergency situations.

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# Are James Webb Space Telescope observations consistent with warm dark
matter?

Bruce Hoeneisen

( Universidad San Francisco de Quito, Quito, Ecuador
Email<EMAIL_ADDRESS>
16 January 2024 )

###### Abstract

We compare observed with predicted distributions of galaxy stellar masses
$M_{*}$ and galaxy rest-frame ultra-violet luminosities per unit bandwidth
$L_{UV}$, in the redshift range $z=2$ to 13. The comparison is presented as a
function of the comoving warm dark matter free-streaming cut-off wavenumber
$k_{fs}$. For this comparison the theory is a minimal extension of the Press-
Schechter formalism with only two parameters: the star formation efficiency,
and a proportionality factor between the star formation rate per galaxy and
$L_{UV}$. These two parameters are fixed to their values obtained prior to the
James Webb Space Telescope (JWST) data. The purpose of this comparison is to
identify if, and where, detailed astrophysical evolution is needed to account
for the new JWST observations.

Keywords
James Webb Space Telescope, JWST, Warm Dark Matter.

##  1 Introduction

Early James Webb Space Telescope (JWST) observations were surprising: they
reveal galaxies at high redshifts ($z\gtrsim 10$) with greater number
densities than predicted by the “$\Lambda$CDM cosmology”. A list of surprises
form the first observations of JWST, and of several proposed modifications of
the theory to account for this data, is summarized in [1]. The “$\Lambda$CDM
cosmology” that is actually compared with the new JWST data is an extension of
the 6-parameter $\Lambda$CDM cosmology with a dozen, or so, astrophysical
parameters [2].

The purpose of the present study is two-fold: a) To extend the comparison
between observations and predictions to include warm dark matter, and b) to
make the comparison with a “first-order” prediction, that has as few,
physically motivated, adjustable parameters as possible, with these parameters
determined numerically from observations prior to JWST data, leaving zero new
degrees of freedom for the comparison. The idea is to clearly identify regions
of parameter space that require a more detailed cosmological and/or
astrophysical description to understand the evolution of the observations.

We focus our attention on distributions of the galaxy stellar masses $M_{*}$
and of the galaxy rest-frame ultra-violet (UV) luminosities per unit bandwidth
$L_{UV}$, that can be predicted with the Press-Schechter formalism given three
parameters: the warm dark matter power spectrum cut-off wavenumber $k_{fs}$,
the stellar formation efficiency $f_{*}$, and a proportionality factor between
the star formation rate per galaxy (SFR) and $L_{UV}$.

The outline of the article is as follows. In Section 2 we compare observed and
predicted distributions of $M_{*}$ and $L_{UV}$ for redshifts $z$ in the range
2 to 13. The “first order” theory turns out to be in agreement with most of
the data. There are, however, three parameter regions with tensions. In
Section 3 we describe the data. In Section 4 we present the details of how we
include warm dark matter in the Press-Schechter formalism. With these
preparations we are able to discuss the observed tensions in Section 5. We
close with conclusions.

##  2 Comparison of observed distributions of $M_{*}$ and $L_{UV}$ with
predictions

In Figure 1 we compare observed distributions of galaxy stellar masses $M_{*}$
(top panel) and galaxy rest-frame ultra-violet luminosities per unit bandwidth
$L_{UV}$ (bottom panel) with “first-order” predictions at redshift $z=6$. The
data is obtained from Hubble Space Telescope (HST) observations (black
squares) [3] [4], from the continuity equation [5] (red triangles), and from
the James Webb Space Telescope (JWST) observations (green triangles) [6]. A
description of this data is postponed to Section 3.

Figure 1: Comparison of predicted and observed distributions of
$M/M_{\odot}=10^{1.5}M_{*}/M_{\odot}$ (top panel) and
$L_{UV}/L_{\odot}=10^{9.6}\textrm{SFR}/(M_{\odot}/\textrm{yr})$ (bottom panel)
for redshift $z=6$. Data are from the Hubble Space Telescope ($M_{*}$ from [3]
and $L_{UV}$ from [4]) (black squares), from the continuity equation [5] (red
triangles), and from the James Webb Space Telescope (green triangles) [6].

Experimenters obtain the galaxy absolute magnitude $M_{1600,AB}$ in the AB
system [7] at the rest frame wavelength $\approx 1600$ Å in the ultra-violet
(UV). This absolute magnitude is related to the galaxy radiated power per unit
bandwidth $L_{1600}$ as follows:

$M_{1600,AB}=+51.6-2.5\log_{10}{L_{UV}},$ (1)

where $L_{UV}\equiv L_{1600}/(\textrm{erg s}^{-1}\textrm{ Hz}^{-1})$ (from the
definition of absolute $M_{AB}$ in [7]). In comparison, the absolute magnitude
of the Sun at wavelength $3900$ Å is [8]

$M_{3900,\odot,AB}=5.9=+51.6-2.5\log_{10}{L_{\odot}},$ (2)

where $L_{\odot}\equiv L_{3900,\odot}/(\textrm{erg s}^{-1}\textrm{ Hz}^{-1})$.
Note that the solar flux power per unit wavelength is a maximum near $3900$ Å.
We plot distributions of $M_{*}/M_{\odot}$ and $L_{UV}/L_{\odot}$.
$L_{UV}/L_{\odot}$ is obtained from $M_{1600,AB}$ as follows:

$M_{1600,AB}\equiv 5.9-2.5\log_{10}{\left(\frac{L_{UV}}{L_{\odot}}\right)}.$
(3)

This definition of $L_{UV}/L_{\odot}$, that is somewhat arbitrary, is chosen
because it gives a physical sense of luminosity in solar units, and also
because $M/M_{\odot}\approx L_{UV}/L_{\odot}$ at $z=6$, see Figure 1.

The predictions are an extension of the Press-Schechter formalism to include
warm dark matter (as described in Section 4). The “warmness” of the dark
matter is defined by the comoving power spectrum cut-off wavenumber $k_{fs}$
due to dark matter free-streaming in and out of density minimums and maximums.
We present predictions corresponding to $k_{fs}=1,2,4$ and 200 Mpc-1. The
latter large value of $k_{fs}$, corresponding to negligible free-streaming, is
identified with the cold dark matter $\Lambda$CDM cosmology. We present Press-
Schechter predictions [9], and two Sheth-Mo-Tormen ellipsoidal collapse
extensions [10] [11]. Presenting three predictions for each $k_{fs}$
illustrates the uncertainties of these predictions. The predictions obtain the
distributions of the linear perturbation total (dark matter plus baryon)
masses $M$ as defined by the Press-Schechter formalism (see Section 4). The
predicted number of galaxies per unit volume and per decade of $M$,
$(1/V)\cdot dn/d\log_{10}(M/M_{\odot})$, with units
$[\textrm{dex}^{-1}\textrm{Mpc}^{-3}]$, is a function of $M$, $z$ and
$k_{fs}$. Therefore it is still necessary to find relations between $M_{*}$
and $L_{UV}$ with $M$. We consider the simplest relations, i.e. $M_{*}$
proportional to $M$, and $L_{UV}$ proportional to the star formation rate per
galaxy (SFR), with units $\left[M_{\odot}/\textrm{yr}\right]$. The
proportionality of $L_{UV}$ with SFR is justified because $L_{UV}$ is
dominated by large mass stars with lifetimes $\tau$ short compared to the age
of the universe $t(z)$ at redshift $z$ [12]. Then

$\displaystyle M_{*}$ $\displaystyle\equiv$ $\displaystyle 10^{-a}M\equiv
f_{*}\frac{\Omega_{b}}{\Omega_{c}+\Omega_{b}}M,$ (4)
$\displaystyle\frac{L_{UV}}{L_{\odot}}$ $\displaystyle\equiv$ $\displaystyle
10^{b}\frac{\textrm{SFR}}{M_{\odot}/\textrm{yr}},$ (5)

which define the parameters $a$, $f_{*}$ and $b$. $\Omega_{b}$ and
$\Omega_{c}$ are the mean densities of baryons and dark matter in units of the
critical density (throughout we use the notation and parameter values of [7]).
$f_{*}$ is defined to be the “star formation efficiency”. Given $(M,z,k_{fs})$
we predict

$\textrm{SFR}\equiv
10^{-a}\frac{M}{\tau}\frac{dn/d\log_{10}(M/M_{\odot})-dn^{\prime}/d\log_{10}(M/M_{\odot})}{dn/d\log_{10}(M/M_{\odot})},$
(6)

where $dn/d\log_{10}(M/M_{\odot})$ is calculated at $(M,z,k_{fs})$, and
$dn^{\prime}/d\log_{10}(M/M_{\odot})$ is calculated at
$(M,z^{\prime},k_{fs})$, where the age of the universe at redshift $z$ is
$t(z)$, and the age of the universe at redshift $z^{\prime}$ is
$t(z^{\prime})=t(z)-\tau$, in the limit of small $\tau$, i.e. $\tau\ll t(z)$.

In summary, the predictions for each $k_{fs}$ depend on only the two
parameters $a$ and $b$. These parameters are in principle functions of $M$,
$z$ and $k_{fs}$. However, for the purpose of comparisons with the data, we
assume that $a$ and $b$ are constants. Note that varying $a$ and $b$ shifts
the predictions in Figure 1 to the right or to the left. For the comparisons
we choose values of $a$ and $b$ obtained prior to JWST data [13] [14]:

$a=1.5\qquad\textrm{and}\qquad b=9.6.$ (7)

These values of $a$ and $b$ define our “first-order” predictions. $a=1.5$
corresponds to a star formation efficiency $f_{*}=0.20$ (taken to be
independent of $z$ in our “first-order” predictions!). For a Salpeter initial
mass function (IMF), and $L_{UV}$ measured at a rest frame wavelength $\approx
1500\AA$, the following $L_{UV}$ is obtained in [12]:

$L_{UV}=8\times 10^{27}\frac{\textrm{SFR}}{M_{\odot}/\textrm{yr}}.$ (8)

Then, from (2), (5) and (8), we obtain $b=9.6$. The value of $k_{fs}$ measured
prior to JWST data is [14]

$k_{fs}=2.0^{+0.8}_{-0.5}\textrm{ Mpc}^{-1}.$ (9)

Comparisons of observed distributions of $M_{*}/M_{\odot}$ and
$L_{UV}/L_{\odot}$ with first-order predictions, for redshift $z$ in the range
2 to 13, are presented in Figures 1 to 5. Sometimes we omit the “Press-
Schechter” and “Ellipsoidal Collapse, $\nu$” predictions for clarity. For
future reference we also present predictions for $z=15$ in Figure 5. The
predictions are presented for $M\gtrsim M_{vd}$, where $M_{vd}$ is the
velocity dispersion limit of validity of the predictions [14]: for $M\lesssim
M_{vd}$ the density fluctuation does not collapse gravitationally. We find
agreement of the measurements with the first-order predictions in most of the
parameter space, so these predictions are a good starting point to develop a
more detailed theory.

Figure 2: Comparison of predicted and observed distributions of
$M/M_{\odot}=10^{1.5}M_{*}/M_{\odot}$ (left panels) and
$L_{UV}/L_{\odot}=10^{9.6}\textrm{SFR}/(M_{\odot}/\textrm{yr})$ (right panels)
for redshift $z=2$ (top row), 3 and 4 (bottom row). Data are from the Hubble
Space Telescope (black squares) ($M_{*}$ from [3] and $L_{UV}$ from [4]), from
the continuity equation [5] (red triangles), and from the James Webb Space
Telescope (green triangles) [6].

Figure 3: Comparison of predicted and observed distributions of
$M/M_{\odot}=10^{1.5}M_{*}/M_{\odot}$ (left panels) and
$L_{UV}/L_{\odot}=10^{9.6}\textrm{SFR}/(M_{\odot}/\textrm{yr})$ (right panels)
for redshift $z=5$ (top row), 7 and 8 (bottom row). Data are from the Hubble
Space Telescope (black squares) ($M_{*}$ from [3] and $L_{UV}$ from [4]), from
the continuity equation [5] (red triangles), and from the James Webb Space
Telescope (green triangles) ($M_{*}$ from [6] and $L_{UV}$ from [15], [16],
[17]).

Figure 4: Comparison of predicted and observed distributions of
$M/M_{\odot}=10^{1.5}M_{*}/M_{\odot}$ (left panels with no data) and
$L_{UV}/L_{\odot}=10^{9.6}\textrm{SFR}/(M_{\odot}/\textrm{yr})$ (right panels)
for redshift $z=9$ (top row), 10 and 11 (bottom row). Data are from the Hubble
Space Telescope (black squares) ($L_{UV}$ from [4]), and from the James Webb
Space Telescope (green triangles) ($L_{UV}$ from [15], [16], [17], [18], [19],
[20], [21]).

Figure 5: Comparison of predicted and observed distributions of
$M/M_{\odot}=10^{1.5}M_{*}/M_{\odot}$ (left panels with no data) and
$L_{UV}/L_{\odot}=10^{9.6}\textrm{SFR}/(M_{\odot}/\textrm{yr})$ (right panels)
for redshift $z=12$ (top row), 13 and 15 (bottom row). Data are from the James
Webb Space Telescope (green triangles) ($L_{UV}$ from [15], [16], [17], [18],
[19], [21], [22], [23]).

There are however three discrepant regions:

  1. 1.

Discrepancies of $L_{UV}$ distributions at $L_{UV}/L_{\odot}\gtrsim 10^{10.7}$
and $3\lesssim z\lesssim 9$.

  2. 2.

Discrepancies of $M_{*}$ distributions at $z\lesssim 5$.

  3. 3.

Discrepancies with preliminary JWST observations of $L_{UV}$ distributions at
$z\gtrsim 11$.

The first and second discrepancies are common to cold and warm dark matter.

We will discuss these discrepancies in Section 5. However, before doing so, we
need to understand the data sources, and the warm dark matter extension of the
theory.

##  3 Data

For $z=2$ to 10 we obtain the distributions of magnitude $M_{1600,AB}$ from
Table 4 of [4] (for $z=10$ the results are from [24]). These measurements have
black square markers in Figures 1 to 4. The data in these references are
obtained from Hubble Space Telescope (HST) observations in approximately 11
filter bands (to obtain the photometric $z$ with stellar population synthesis
(SPS) models), i.e.
$UV_{275},B_{435},V_{606},z_{850},V_{606},UV_{336},I_{814},J_{125}$,
$Y_{105},H_{160},JH_{140}$, and from Spitzer Space Telescope (SST)
observations.

For $z=4$ to 8 we obtain the distributions of galaxy stellar mass
$M_{*}/M_{\odot}$ from Table 2 or Figure 9 of [3] (identified by black square
markers in distributions of $M_{*}/M_{\odot}$). The analysis in [3] obtains
relations between $M_{*}/M_{\odot}$ and magnitude $M_{UV}$, for each
photometric $z$, fitting HST images taken with 10 filters, using an SPS model.
Then the distributions of $M_{*}/M_{\odot}$ are obtained with the
distributions of magnitude $M_{UV}$ in [25]. As a cross-check we mention that
the thick gray lines in Figure 9 of [3], corresponding to $\Lambda$CDM
predictions, are in agreement with our $\Lambda$CDM predictions.

We obtain distributions of galaxy stellar mass $M_{*}/M_{\odot}$ for $z=2$ to
8 from Figure 4 of [5], and distributions of star formation rates (SFR) from
Figure 1 of [5] (red triangles). These distributions are obtained using the
continuity equation for the stellar masses of galaxies, with inputs and
comparisons to a large number of sources [5].

For $z=4$ to 8 we obtain the distributions of galaxy stellar mass
$M_{*}/M_{\odot}$ from Figure 5 of [6] (green triangles). These measurements
are based on approximately 3300 galaxy images taken with the James Webb Space
Telescope Near Infrared Camera (NIRCam). This data is complemented by the HST
Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey (CANDELS). The
label “$z=6$” in, e.g. Figure 1, means $z=6$ for the predictions, and
$5.5<z<6.5$ for the measurements. We have neglected the corresponding bias.

Measurements of $L_{UV}$ for $z\geq 8$ from JWST data are obtained from [15],
[16], [17], [18], [19], [20], [21], [22], [23] (red triangles). See [26] for
an analysis in the $\Lambda$CDM scenario. The measurements for $z\gtrsim 11$
need to be considered as “preliminary”, as stated by their authors, for
several reasons:

  * •

The measurements at lower $z$ are counting experiments: the galaxy candidates
are assigned to bins of magnitude and redshift $(M_{UV},z)$ and are then
counted. At the high-$z$ frontier, the bins $(M_{UV},z)$ are increased in size
so the mean number of counts does not drop below $\approx 1$. The analysis
ceases to be a counting experiment, and becomes an extrapolation of a Press-
Schechter-like fit to lower $z$ data, keeping all parameters fixed except the
normalization. The tension arises at the transition between these two methods.

  * •

To illustrate the low numbers of events, let us mention that the measurements
in [15] are based on 33, 22, 16 and 3 galaxy candidates, after correcting for
completeness, in the bins $7.5<z<8.5$, $8.5<z<9.5$, $9.5<z<11.5$ and
$11.5<z<13.5$, respectively. Therefore, where the apparent tension between
observations and “first-order” predictions of $L_{UV}$ arise, i.e. $z\gtrsim
11$, there are only about 3 galaxy candidates with photometric redshift. On
the other hand, a single galaxy correctly assigned to a bin, i.e. with
spectroscopically confirmed redshift $z$, and a luminosity distribution
consistent with stellar expectations (limiting the possible contribution from
active galactic nuclei (AGN)), could rule out a theory if the theory predicts
one galaxy with a probability less than, say, 0.6%.

  * •

To illustrate the difficulties with the photometric classification of galaxies
let us mention that Table 4 of [16] presents an assessment of the “purity” and
“completeness” of the selected samples by several authors. A good feeling of
the uncertainties is quoted from [16]: “Using all of these samples we then
derive UV LF and luminosity density results at $z\geq 8$, finding substantial
differences. For example, including the full set of “solid” and “possible”
$z\geq 12$ candidates from the literature, we find UV LF and luminosity
densities which are $\approx 7\times$ and $\approx 20\times$ higher than
relying on the “robust” candidates alone. These results indicate the evolution
of the UV LF and luminosity densities at $z\geq 8$ is still extremely
uncertain, emphasizing the need for spectroscopy and deeper NIRCam$+$optical
imaging to obtain reliable results.” Note that the measurements of $L_{UV}$ in
Figures 4 and 5 are based on photometric redshift measurements.

  * •

If AGN contribute to the observed UV luminosity, then the measurements can
drop by 0.4 dex on average, or up to 4 dex for individual galaxies [27].

  * •

Corrections for dust attenuation are very uncertain unless data is available
in a wide range of wavelengths [5].

  * •

The measurements of $L_{UV}$ presented in Figures 3 to 5 for $z\geq 8$ are not
independent as they use overlapping data sets. At $z\geq 11$ there are more
measurements than galaxy candidates.

  * •

Cosmic variance becomes important at $z\gtrsim 14$ [28].

The experimental determinations of the galaxy stellar masses $M_{*}$ depend on
stellar population synthesis (SPS) models and spectral energy distributions
(SED) models that use images with several filters, and therefore do not
include dead star remnants (ejected baryons into inter-stellar space, white
dwarfs, neutron stars, and black holes). So measured and predicted $M_{*}$ do
not include dead star remnants.

##  4 Predictions

This section describes the extension of the Press-Schechter formalism that we
use to include warm dark matter (see [14] and [29] for more details). The
Press-Schechter prediction [9] is

$\frac{1}{V}\frac{dn}{d\ln{M}}=\frac{\bar{\rho}_{m}}{M}\frac{d\ln(\sigma^{-1})}{d\ln
M}f_{\textrm{PS}}(\nu),$ (10)

where

$f_{\textrm{PS}}(\nu)=\sqrt{\frac{2}{\pi}}\nu\exp{\left(-\frac{\nu^{2}}{2}\right)},$
(11)

and

$\nu\equiv\frac{1.686}{\sigma(M,z,k_{fs})}.$ (12)

The factor 1.686 is obtained analytically for spherical collapse with cold
dark matter, and becomes valid for warm dark matter when $M\gtrsim M_{vd}$,
where $M_{vd}$ is the velocity dispersion cut-off mass. Predictions are
presented only for $M>M_{vd}$, see Table 1 of [14]. In the spirit of the
present study, we define our “first-order” prediction, for comparison
purposes, with the factor 1.686 unchanged, i.e. independent of $M$ and
$k_{fs}$. The Sheth-Mo-Tormen ellipsoidal collapse extensions [10] [11] are
obtained by replacing $f_{\textrm{PS}}(\nu)$ by $f_{\textrm{EC}}(\nu)$:

$f_{\textrm{EC}}(\nu)=0.322\left[1+\tilde{\nu}^{-0.6}\right]f_{\textrm{PS}}(\tilde{\nu}),$
(13)

with $\tilde{\nu}=\nu$. Good fits to simulations are obtained with
$\tilde{\nu}=0.84\nu$ [11]. The factor 0.84 depends on the algorithm used to
identify the collapsed halos.

These predictions depend on the variance of the linear relative density
perturbation
$\delta(\textbf{x})\equiv(\rho(\textbf{x})-\bar{\rho})/\bar{\rho}$:

$\sigma^{2}(M,z,k_{\textrm{fs}})=\frac{f^{2}}{(2\pi)^{3}(1+z)^{2}}\int_{0}^{\infty}4\pi
k^{2}dkP(k)\tau^{2}(k)W^{2}(k).$ (14)

$\sigma^{2}(M,z,k_{\textrm{fs}})$ depends on the linear total (dark matter
plus baryon) mass scale $M$ at redshift $z$, and on the comoving cut-off
wavenumber $k_{fs}$ due to dark matter free-streaming. $P(k)$ is the comoving
power spectrum of linear density perturbations in the cold dark matter
$\Lambda$CDM cosmology [30]. $W(k)$ is a window function that defines the mass
scale $M$. $\tau^{2}(k)$ is the power spectrum cut-off factor due to dark
matter free-streaming. The factor $f$ is due to the cosmological constant:
$f=0.79,0.99,1.00$ for $z=0,2,\gg 2$ [30]. Three window functions are
considered: the top-hat in r-space, the sharp-k (or top-hat in k-space), and
the Gaussian window function:

$W(k)=\exp{\left(-\frac{k^{2}}{2k_{0}^{2}}\right)},\qquad
M=\frac{4}{3}\pi\left(\frac{1.555}{k_{0}}\right)^{3}\bar{\rho}_{m}.$ (15)

$\bar{\rho}_{m}$ is the total (dark matter plus baryon) mean density. We use
the following form for the free-streaming cut-off factor:

$\displaystyle\tau^{2}(k)$ $\displaystyle=$
$\displaystyle\exp{\left(-\frac{k^{2}}{k^{2}_{\textrm{fs}}(t_{\textrm{eq}})}\right)}\qquad\textrm{
if }k<k_{\textrm{fs}}(t_{\textrm{eq}}),$ (16) $\displaystyle=$
$\displaystyle\exp{\left(-\frac{k^{n}}{k^{n}_{\textrm{fs}}(t_{\textrm{eq}})}\right)}\qquad\textrm{
if }k\geq k_{\textrm{fs}}(t_{\textrm{eq}}).$

At the time $t_{eq}$ when the matter density begins to dominate, $\tau^{2}(k)$
has the approximate form (16) with $n=2$ [31]. Thereafter, $\tau^{2}(k)$
develops a non-linear regenerated “tail” with $n$ measured to be in the range
0.5 to 1.1 [14]. In the present study we take $n=1$, and use the Gaussian
window function (see studies in [14]). We have verified that the predictions
change negligibly for $0.5<n<1.1$, and also if the sharp-k window function is
used with $n=1$ [14]. The amplitude of $P(k)$ is adjusted for each $k_{fs}$ so
that the relative density root-mean-square fluctuation $\sigma$, calculated
with the top-hat window function with radius $r=8/h=8/0.674$ Mpc, is
$\sigma_{8}=0.811$ [7].

##  5 Discussion

Let us now discuss the discrepancies found in Figures 1 to 5.

1\. Discrepancies of $L_{UV}$ distributions at $L_{UV}/L_{\odot}\gtrsim
10^{10.7}$ and $3\lesssim z\lesssim 9$.
This discrepancy is observed in HST and JWST data. This discrepancy is studied
in [5], and is (apparently) due to the dust correction of the measured SFR.
When the dust correction is based only on the UV slope, the results are
inconsistent with other data sets, and the dust-corrected SFR falls short of
multi-wavelength determinations at high SFR (dominated by dusty star forming
progenitors of present-day quiescent galaxies). A reliable dust correction
needs, in addition to the UV images, also optical, radio, H$\alpha$, mid-IR 24
$\mu$m, and far-IR images. The results of the continuity equation [5], based
on multi-wavelength dust corrections, are indeed consistent with the
predictions as shown in Figures 1 to 3 (see the red triangles in $L_{UV}$
distributions).

2\. Discrepancies of $M_{*}$ distributions at $z\lesssim 5$. This discrepancy
can be understood qualitatively, at least in part, as follows (this is my
tentative understanding). In the warm dark matter scenario, the first galaxies
to form have $M\approx M_{vd}\approx 2\times 10^{8}M_{\odot}$ at $z=4$,
increasing to $\approx 2\times 10^{9}M_{\odot}$ at $z=10$ (see Table 1 of [14]
for $k_{fs}=2$ Mpc-1). Thereafter, the formation of galaxies proceeds
hierarchically as larger and larger perturbation masses $M$ become non-linear
and collapse due to gravity. Therefore, low mass halos become part of higher
and higher mass halos as time goes on, and so become under-counted. In other
words, the distribution of low $M_{*}$ galaxies becomes ill-defined at late
times.

3\. Discrepancies of preliminary JWST observations of $L_{UV}$ distributions
at $z\gtrsim 11$. For $6\lesssim z\lesssim 10$ the JWST data is in agreement
with predictions for $k_{fs}\approx 2$ Mpc-1 (see green triangles in Figures 1
to 4). Preliminary JWST observations at $z=11,12$ and 13 are in mild tension
with the predictions. However these observations need spectroscopic
confirmation of the redshifts, and higher statistics, before any conclusions
can be presented, see discussion in Section 3 and in [16].

##  6 Conclusions

We have presented comparisons of measured distributions of galaxy luminosities
per unit bandwidth $L_{UV}$ and galaxy stellar masses $M_{*}$ with “first-
order” predictions, as a function of $k_{fs}$, for $z$ in the range 2 to 13.
The only outstanding tension for JWST observations corresponds to $L_{UV}$
with $z\gtrsim 11$. However, these measurements are still preliminary until
spectroscopic confirmation of $z$ is obtained, and of limited significance (as
stated by the authors, e.g. [16])

We conclude that the “first-order” predictions with the measured
$k_{fs}=2.0^{+0.8}_{-0.5}$ Mpc-1 [14], and two parameters $a=1.5$ and $b=9.6$,
obtained prior to JWST data [13] [14], and assumed to be constants independent
of $(M,z,k_{fs})$, are in agreement with the current data within their
theoretical and observational uncertainties. This result is indeed surprising,
considering the large range of $z$ and $M$, with constant star formation
efficiency $f_{*}$ and constant $b$, and such a basic and simple “first order”
prediction model with no new degrees of freedom! The “first-order” predictions
are therefore a useful starting point to include more detailed astrophysical
models to account for more precise future observations, and to describe other
observables, and their evolution.

Can dark matter be cold? Consider, as an example, Figure 1. The distributions
of $M_{*}/M_{\odot}$ and of $L_{UV}/L_{\odot}$ are nicely consistent with
$k_{fs}\approx 2$ Mpc-1 and a constant star formation efficiency $f_{*}\approx
0.20$. If, however, dark matter is cold, i.e. if $k_{fs}$ is very large, then
both $M_{*}$ and $L_{UV}$ data tells us that $f_{*}$ at $M\approx
10^{9}M_{\odot}$ is approximately $1/10$th of $f_{*}$ at $M\gtrsim
10^{11}M_{\odot}$. Is this possible? Supernova and active galactic nuclei
(AGN) winds are invoked to explain this low $f_{*}$ at low $M_{*}$. However,
three additional and independent indications that dark matter may be warm,
with $k_{fs}\approx 2$ Mpc-1, are i) the observed distributions of
$M=10^{1.5}M_{*}$ have a cut-off at approximately $M_{vd}$ (less massive
galaxies would have to be “stripped-down” galaxies [29]), ii) the measurements
of spiral galaxy rotation curves [32] [33], and iii) the re-ionization optical
depth [14] [34]. Item i) is related to the “missing satellite” problem, and
ii) is related to the “core-cusp” problem.

If the preliminary tension of $L_{UV}$ with $z\gtrsim 11$ is confirmed by
future observations and analysis, and becomes significant, then we need to let
the parameters $a$ and $b$ become functions of $z$, with $b-a$ growing from
8.1 at $z\lesssim 10$ to approximately 9.1 at $z\gtrsim 11$ (for the case
$k_{fs}\approx 2$ Mpc-1). In other words, we would have to allow the star
formation efficiency $f_{*}$ to increase above 0.20 at $z\gtrsim 11$, and/or
allow first stars to be more massive and luminous than at lower $z$. Let us
work and see.

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