# Using conservative voltage reduction and dynamic thermal rating for
congestion management of power network

Ramin Nourollahi, Rasoul Esmaeilzadeh, Member, IEEE R. Nourollahi, is with the
Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz,
Iran. R. Esmaeilzadeh is with the Azarbaijan regional electric company.

###### Abstract

Increasing the amount of electric power that is used on the demand side has
brought more attention to the peak-load management of the distribution network
(DN). The creation of infrastructures for smart grids, the efficient
utilisation of the distributed network’s components, and the appropriate
administration of the distributed network would result in a valuable solution
for the operators of the distributed network. As a result, a framework for
peak-load management is given in this research. Within this framework, the
real-time rating of the components and the voltage-dependent characteristics
of the electric loads work together to assist the DN operator in effectively
navigating peak periods. The combination of the conservation voltage reduction
(CVR) and the dynamic thermal rating (DTR) of the components that make up the
DN produces outcomes that are more helpful than any of these factors alone
could provide. This is true even though each of these factors contributes to
the efficient functioning of the DN. According to the findings, as compared to
the individual implementation of CVR, the simultaneous utilisation of DTR and
CVR results in a cost-savings rise at peak events that is 58.75 percentage
points more than the individual implementation. In addition, a discussion is
offered concerning the current difficulties that are being experienced by the
feeders that are providing the voltage-dependent constant-power loads during
the utilisation of the CVR, which are handled by the dynamic rating of the
components that make up the DN.

###### Index Terms:

Conservation voltage reduction, dynamic thermal rating, distribution network,
demand response programs

##  I Introduction

The significant expansion of electric cars, distributed generation, and urban
development necessitates the establishment of communication and control
infrastructures to effectively manage the distribution networks (DNs) [1].
Conversely, the yearly rise in load demands and the phenomenon of global
warming pose a significant risk to the distribution networks (DNs) and their
associated equipment, particularly during hot summer days (6678820). Three
primary challenges to improving equipment capacity, whether via replacement or
addition, are the privatisation of distribution businesses, financial
constraints, and excessive prices. The present challenge has compelled the
owners of DNs to contemplate the optimisation of their current
infrastructures, perhaps necessitating the use of peak management strategies
by the DN operator during periods of heavy loading. The establishment of
communication and control infrastructures presents an advantageous prospect
for operators to use the sophisticated DN management system in order to
efficiently oversee the network during periods of heavy usage. The
comprehensive peak management system has a range of components, with
particular emphasis placed on demand response programmes (DRPs) and voltage
control programmes, which are deemed significant. In addition, the transformer
and line facilities of the DN are often constrained by their temperature
limitations in order to mitigate the risk of overheating incidents. Alongside
the constraint of line capacity, the thermal constraints might impose some
restrictions on the voltage control capabilities of distribution networks
(DNs). Additionally, the implementation of real-time monitoring of the
system’s state may significantly contribute to the effective management of the
DN’s operation, particularly during critical periods [2]. Therefore, the use
of dynamic thermal rating (DTR) as opposed to traditional rating systems has
potential for demand response (DN) in optimising energy management at elevated
load levels, particularly during peak periods.

###  I-A literature review

Voltage regulation in distribution networks (DN) refers to a control approach
used to ensure that the voltage inside the DN remains within a certain range.
The function of voltage regulation in the distribution network (DN) varies
depending on the temporary and long-term circumstances of the network. This
regulation involves adjusting the voltage levels, either by raising or
lowering them, to fulfil certain objectives within the DN [3]. Voltage
regulation may be carried out in order to regulate reactive power or to avoid
voltage drop in distribution network (DN) buses during periods of heavy
loading. Regarding load management, voltage control may be implemented by
conservative voltage reduction (CVR) in order to mitigate peak load levels and
achieve energy savings [4]. Experimental investigations have examined the
significant impact of Conservation Voltage Reduction (CVR) on the reduction of
energy consumption and demand levels in Distribution Networks (DN) [5]. The
assessment of the economic and engineering advantages of the CVR in the DN’s
projects has been conducted in a study by [6] alongside its technical merits.
Furthermore, the paper by [7] presents a cost-benefit analysis of CVR,
specifically focusing on the investment return derived from implementing CVR
on distribution feeders. Furthermore, the advantages of Capacitor Voltage
Regulation (CVR) have been substantiated in the context of unfavorable power
system issues, such as the mitigation of power loss as examined in the study
conducted by [8]. The aforementioned advantages of CVR mostly pertain to the
decrease in system demand and energy consumption resulting from voltage
lowering. Another potential use of the CVR technique is in the decrease of
peak loads in systems that are running in close proximity to their thermal
limits [9]. Various approaches were presented by [4] to measure the impacts of
the CVR (Conservation Voltage Reduction) at peak periods in the distribution
network. All of the approaches used in the study demonstrated the substantial
demand-saving effects of CVR at peak times, resulting in significant economic
gains.

Transitioning from the traditional static rating approach to the real-time
dynamic rating technique would result in an augmentation of the nominal
current capacity of distribution networks, particularly in relation to
transformers and other components. Multiple scholarly articles have presented
thermal models that may be used for the purpose of real-time monitoring of DTR
[10]. The Large Electric Systems Working Group (CIGRE) was first proposed by
the International Council as a means to enhance the capacity of power
transmission lines in a cost-effective way, particularly in comparison to
other expensive approaches such as network expansion planning [11]. The
findings of a concrete study on the DTR performed by the Electric Power
Research Institute are elucidated in [12]. According to the findings presented
in the study conducted by Douglass et al. (1996), the use of DTR techniques
has been seen to enhance the capacity of transmission lines by around 1-5%.
The DTR is influenced by several weather-related factors, such as wind speed
and direction, ambient temperature, and sun irradiation. In the study by [13],
the uncertainties associated with these parameters are addressed via the use
of a fuzzy-based technique. Furthermore, [14] employs three variants of
polynomial regressions in their study to develop a time-series-based approach
for forecasting the DTR values of electricity lines. The DTR significantly
impacts the reliability analysis of power lines.

In addition, the DRP incorporates the necessary programmes for managing peak
loads in the distribution network [15]. DN’s resilience, profitability, and
dependability are all increased as a result of the DRP programmes’ capacity to
reduce the amount of power used at crucial periods [16]. According to
reference [17], the network operator has a safe and efficient resource at
their disposal in the form of the load of the customers who are already
connected to the network in order to control the peak load. Grid operator
involvement in the smart grid residential customer’s load to critical peak
load management utilising the customer engagement plans is explored in the
paper cited above [18]. According to quote 24Alami2010, the incentive-based
interruptible/curtailable DRPs are regarded to peak-load management, in which
the practise of penalising consumers is considered in the event that they do
not react to load reduction. This is done so that peak load management may be
performed effectively.

Moreover, limited research has coordinated the CVR, DTR, and DRP methods in
the peak load management. In [19], DRP and CVR are coordinated into an
advanced DN management system to increase DN efficiency by minimizing the cost
of power consumption in the day-ahead market. Furthermore, most of the
reviewed DTR literature is related to the transmission network level, while
very limited research has studied the DTR in DN. In [20], a methodology has
been proposed for analyzing the potential advantages of the real-time-
monitoring-based DTR in the DN from the reliability point of view.

From the documents that were looked through, it is possible to draw the
conclusion that the CVR application in DN’s peak load management solutions has
not been explored in an adequate manner. In addition, DTR in DN has not been
given the proper attention it deserves, particularly in the peak load
management of DN, which has been preserved in its original state. The goal of
the writers of this research is to discuss the CVR-based peak management of
distribution networks while they are operating in overloaded situations. It is
conceivable for older distribution networks to experience overloading of their
substation transformers on hot summer days when there is a significant
increase in the amount of power that is being used by the network. In these
kinds of predicaments, the distribution network operator is required to
implement the emergency DRPs, which results in an increase in the overall
expense incurred by the distribution firm. In addition to that, this increase
in consumption results in an increase in the cost of acquiring electricity
from the network that is farther upstream. The CVR is built in such a manner
so as to achieve the lowest possible cost while not violating any of the
system indices in order to circumvent this issue and cut down on the expenses
associated with both operational and DRP charges.

Thus, the contributions of this paper can be summarized as below:

  * •

This paper addresses the peak management issue through voltage regulation and
possible thermal capacity of the network components that,to the best of our
knowledge , have not been considered for peak management in the literature.

  * •

CVR method implementation provides the opportunity for peak management at a
lower cost than NO-CVR condition.

  * •

Considering the thermal rating of distribution network components and their
dynamic characteristics, the CVR method with DTR standpoint releases the
components’ capacity and dramatically reduces the costs of peak management.

  * •

The relevant power flow algorithm based on backward-forward sweep is
introduced considering voltage reduction limits and dynamic line rating.

The remaining parts of the article are structured as described below. In
Section 2, we go into more detail about the issue at hand and provide the
mathematical formulation. In Section 3, we covered both the optimisation
approach and the power flow that was applied in order to address the issue.
Section 4 provides a validation of the suggested methodology, and Section 5
provides a representation of the eventual result.

##  II Problem formulation

Within the scope of this article, a peak-load management framework for DN is
put up for consideration. The suggested framework is comprised of three
different methods: DRT, DRP, and CVR, all of which are capable of being
implemented in the majority of DNs. At these crucial peak moments, the
emergent DRP option of mandatory load reduction is being investigated as a
possible means of reducing the overall stress on the network. The mathematical
formulation of the aforementioned approaches will be presented in this portion
of the article.

###  II-A Conservative voltage reduction

It is reasonable to say that the electrical loads of all feeders in DN are
dependent on the voltage level of the connecting bus, and this is true in both
directions. In addition, the electrical loads on the DN have a unique reaction
to the fluctuations in the network voltage. To represent the voltage
dependence of DN’active and reactive loads, many functions have been presented
in the investigations. These functions include the exponential model and the
ZIP model. In this research, the ZIP model of voltage dependence is examined
to model the active and reactive load changes against the voltage regulation.
The goal of this modelling is to investigate the relationship between the two.
You may see a representation of the mathematical formulation of the ZIP model
down below [21].

$\displaystyle{{P}_{n,h}}=P_{n,h}^{0}\left[{{C}_{{{z}_{p}}}}{{\left(\frac{{{V}_{n,h}}}{V_{n,h}^{0}}\right)}^{2}}+{{C}_{{{i}_{p}}}}\left(\frac{{{V}_{n,h}}}{V_{n,h}^{0}}\right)+{{C}_{{{p}_{p}}}}\right]$
(1)

$\displaystyle{{Q}_{n,h}}=Q_{n,h}^{0}\left[{{C}_{{{z}_{q}}}}{{\left(\frac{{{V}_{n,h}}}{V_{n,h}^{0}}\right)}^{2}}+{{C}_{{{i}_{q}}}}\left(\frac{{{V}_{n,h}}}{V_{n,h}^{0}}\right)+{{C}_{{{p}_{q}}}}\right]$
(2)

In (1) and (2), $P_{n,h}^{0}$, $Q_{n,h}^{0}$, and $V_{n,h}^{0}$ represent the
hourly active reactive and voltage of DN buses before implementation of the
CVR. In contrast, $P_{n,h}$, $Q_{n,h}$, and $V_{n,h}$ respectively refer to
the hourly active, reactive and voltage of the bus n by voltage reduction. The
impacts of voltage reduction on the active power of the DN can be determined
through the parameters ${{C}_{{{z}_{p}}}}$, ${{C}_{{{i}_{p}}}}$,
${{C}_{{{p}_{p}}}}$ that are the active power constants of ZIP loads.
Furthermore, the voltage-dependent behavior of the reactive loads determine
through the constants ${{C}_{{{z}_{q}}}}$, ${{C}_{{{i}_{q}}}}$ and
${{C}_{{{p}_{q}}}}$. The voltage dependency of the DN’s loads leads to
comprehensive changes in the distribution network, from power consumption to
current changes, that will be evaluated in the next sections through power
flow analysis.

###  II-B Dynamic thermal rating

The traditional static thermal rating of the DN is what is used to establish
the rating of the equipment in a way that is cautious and is based on the
parameters that are considered to be the worst possible weather, such as
greater temperatures and the lowest wind speed for overhead lines. In most
cases, the static rating is established independently for each season. The
operators of some DNs have the ability to manually adjust the equipment rating
depending on their own personal expertise as well as the current weather
conditions in order to make the most efficient use of the available network
capacity. Utilising DTR in order to unlock and put to good use the DN capacity
is one of the fundamental approaches that may be used when constructing the
infrastructures of a smart grid. In other words, the DN operator is able to
permit more power to be transported via the network components by making use
of the DTR, which may result in less load curtailment occurring during the on-
peak hours. The IEEE standards [22], [23], and [24] are used in this study to
simulate the DTR of the various components that make up the DN. In order for
the DTR to function properly, the time constants of the components are
necessary. The time constants of the component are determined by the
capacitors, and the transient responses of the component are corresponding to
the thermal resistance of the component. The time-constant definition is equal
to the needed time interval for changing the temperature from the beginning
temperature (temperature before step change in the load) to 63.2 percent of
the final temperature level (final temperature a while after step change in
the load). This time interval is equal to the final temperature a while after
the step change in the load. There is a disparity in the time constants
between the three elements that make up the DN: transformers, overhead lines,
and subterranean cables. Transformers, overhead lines, and subterranean lines
each have a thermal time constant that is about equal to 4 hours and 15
minutes, followed by 8 hours. This is true for each of the components that
were described. While a shorter time constant makes it easier for components
to cool down, a longer time constant makes them less sensitive to variations
in the load that occur over short periods of time. Figure 1 illustrates a
conventional thermal model for an underground cable based on the circumstances
that have been provided. The thermal equivalent circuit of the DN components
may be modelled in a manner similar to that of the subterranean cables seen in
Fig. 1. After calculating the time constant of the components, the hourly
rating of the components may be estimated by utilising the following formulae
in such a way that the hottest-spot temperature of each component should be
lower than its thermal limit (for example, ninety degrees Celsius). This
ensures that the components will continue to function properly.

$\displaystyle{{\theta}_{j}}({{t}_{I}})={{\theta}_{d,j}}+{{\theta}_{amb}}({{t}_{I}})+\sum\limits_{k}{{{\theta}_{j,k}}({{t}_{I}})}$
(3)

$\displaystyle{{\theta}_{j,k}}({{t}_{I}})={{\theta}_{j,k}}({{t}_{I-1}})-$
$\displaystyle{{T}_{j,k}}{{W}_{c}}({{\theta}_{c}}({{t}_{I-1}})){{e}^{-\frac{{{t}_{I}}-{{t}_{I-1}}}{{{\tau}_{k}}}}}$
(4) $\displaystyle+{{T}_{j,k}}{{W}_{c}}({{\theta}_{c}}({{t}_{I-1}}))$

In (3), ${{\theta}_{j}}({{t}_{I}})$ refers to the temperature of the node $j$
between all nodes indicated in Fig. 1 at time ${{t}_{I}}$ . Moreover,
${{\theta}_{j,k}}$ represents the raised temperature arising from the
dielectric losses in the node $j$ . Furthermore, ${{\theta}_{amb}}$ and
${{\theta}_{j,k}}$ respectively indicate the ambient temperature and
temperature difference among nodes $j$ and $k$ . The nodal time constant is
also indicated with ${{\tau}_{k}}$ in the above formulation. Besides, the
control parameter ${{T}_{j,k}}$ is used in (4) to control the exponential
equations, which subscripts $j$ and $k$ represent the considered node and the
ladder circuit’s thermal loop, respectively. Finally,
${{W}_{c}}({{\theta}_{c}}({{t}_{I-1}}))$ refers to the conductor loss in
conductor temperature ${{\theta}_{c}}({{t}_{I-1}})$ at time ${{t}_{I-1}}$ .

The computation of the hourly rating of DN’s components is negatively affected
by the inclusion of real-time DTR. This computation involves accessing the
previous hour’s data as well as the initial state thermal conditions of the
components and the environmental states such as ambient temperature, wind
speed, and sun radiation. Hence, in accordance with the dynamic alterations in
the loading and environmental conditions of the component, the thermal
resistances and capacitance of the components are modified using (1) and (2).
The computation of the updated ratings for overhead lines and transformers in
the next hour is facilitated by the utilisation of IEEE standards 738 [22] and
IEEE standard C57.91-2011 [23]. These standards take into account various
factors such as the initial and previous hour thermal status, as well as
environmental conditions including ambient temperature, solar irradiation, and
wind.

###  II-C Objective function

The distribution network operator seeks to manage the peak load in a way that
minimum cost is incurred by applying the curtailment and the least acceptable
system indices are met. Suppose a day that the network is in overloading
condition for the peak hour period of
${{\Omega}_{h}}=\\{{{h}_{1}},...,{{h}_{n}}\\}$ known as event hours. The total
cost function $\Gamma$ over this period and its relevant constraints are given
in (5)-(10).

$\displaystyle\Gamma=\sum\limits_{h\in{{\Omega}_{h}}}{\rho_{h}^{G}P_{h}^{G}+}\sum\limits_{h\in{{\Omega}_{h}}}{\sum\limits_{i\in{{\Omega}_{c}}}{\rho_{i,k}^{cur}\,{{\chi}_{i,h}}P_{i,h}^{0}}}$
(5)

Subject to

$\displaystyle
P_{h}^{G}=\sum\limits_{n\in{{\Omega}_{L}}}{(1-{{\chi}_{n,h}}){{P}_{n,h}}}+{{S}_{base}}\sum\limits_{S\in{{\Omega}_{S}}}{{{r}_{s}}I_{s,h}^{2}}$
(6)

$\displaystyle\left[{{\mathbf{V}}_{\mathbf{h}}}\mathbf{,}{{\mathbf{I}}_{\mathbf{h}}}\right]\mathbf{=f(\chi,P}_{\mathbf{h}}^{\mathbf{0}}\mathbf{,Q}_{\mathbf{h}}^{\mathbf{0}}\mathbf{,Z,}V_{h}^{Sub}\mathbf{)}$
(7)

$\displaystyle{{V}^{\min}}\leq{{V}_{n,h}}\leq{{V}^{\max}}$ (8)

$\displaystyle{{I}_{s,h}}\times{{I}_{base}}<I_{s}^{R}{{u}_{s}}(\theta,w,\varphi,h)I_{h}^{Tr}<I_{s}^{Tr,R}{{u}_{Tr}}(\theta,w,\varphi,h)$
(9)

$\displaystyle{{\chi}_{i,h}}\leq MCL$ (10)

In (5), $\rho_{h}^{G}$is the hourly market energy price and $P_{h}^{G}$ is the
hourly purchased power from the upstream grid. Set of ${{\Omega}_{h}}$ denotes
the set of load points that are participated in the curtailment program,
$\rho_{i,k}^{cur}$ is the penalty price for every kW of load curtailment,
considering the type of customers e.g. residential or industrial. Variable
${{\chi}_{i,h}}$ determines the curtailment amount of each load and
$P_{i,h}^{0}$is the hourly baseline load. Equation (6) states the hourly
purchased power that is sum of the CVR-effected loads in the presence of
curtailment and total network loss. ${{\Omega}_{L}}$and ${{\Omega}_{S}}$ are
respectively the set of load points and sections, ${{S}_{base}}$is base power,
${{r}_{s}}$is the Per-unit section resistance and ${{I}_{s,h}}$ is the Per-
unit current of sections. Function $f(.)$, gives the bus voltages and section
currents through the power flow in substation voltage of $V_{h}^{sub}$.
$\mathbf{Z}$ is the impedance vector of the distribution network. Constraints
(8)-(9) state the limitations on bus voltage, section current and HV/MV
transformer current $I_{h}^{Tr}$. Parameters ${{V}^{\min}}$, ${{V}^{\max}}$and
${{I}_{base}}$ respectively denote the voltage bounds and base current.
$I_{s}^{R}$and $I_{s}^{Tr,R}$ respectively stand for the rating current limits
of feeders and HV/MV transformers. Both ${{u}_{s}}(\theta,w,\varphi,h)$ and
${{u}_{Tr}}(\theta,w,\varphi,h)$ are the functions of temperature, wind speed,
solar irradiation and hour. Inequality (10) states that the curtailment amount
is restricted within a maximum curtailment level (MCL) must not exceed this
value.

##  III solving algorithm

The user’s text is already academic in nature. As demonstrated in the
preceding section, the aforementioned peak management issue exhibits
nonlinearity and is characterised by intricate power flow limitations. In
order to achieve this objective, the present study suggests the utilisation of
the particle swarm optimisation algorithm (PSO) as a computational meta-
heuristic approach, particularly suitable for issues involving continuous
variables.The fundamental principle underlying Particle Swarm Optimisation
(PSO) involves the establishment of an initial population of solutions
referred to as particles, which undergo repetitive movements aimed at
converging towards the optimal solution. The process of convergence is
achieved by the utilisation of updating equations for the velocity and
position of particles, as seen in equations (11) and (12).

$\displaystyle Ve{{l}_{n,t+1}}=\omega Ve{{l}_{n,t}}+$
$\displaystyle{{c}_{1}}{{r}_{1}}(pbes{{t}_{n,t}}-{{Y}_{n,t}})$ (11)
$\displaystyle+{{c}_{2}}{{r}_{2}}(gbes{{t}_{t}}-{{Y}_{n,t}})$

$\displaystyle{{Y}_{n,t+1}}={{Y}_{n,t}}+Ve{{l}_{n,t+1}}$ (12)

In the provided equations, the variables $Ve{{l}_{n,t}}$ and ${{Y}_{n,t}}$
denote the velocity and position of the n’th particle in the t’th iteration,
correspondingly. The variables $pbest$ and $gbest$ denote the optimal
positions discovered by individual particles and the overall best position,
respectively. The symbol $\omega$ represents the inertia weight, whereas
${{r}_{1}}$ and ${{r}_{2}}$ are random numbers that follow a uniform
distribution. The acceleration coefficients are denoted as $c_{1}$ and
$c_{2}$. To effectively tackle the matter of peak management, the particle
placements demonstrate variability contingent upon the operator’s chosen
method. The fundamental purpose of the no-CVR technique is to ascertain the
curtailment amounts of the generated solutions. On the other hand, the CVR and
DTR methods prioritise the identification of the most favourable per-unit
voltage at the substation. The configuration of particles during each hour of
the event can be observed in Figure 2.

Figure 1: Structure of particles for each event hour

The CVR problem necessitates the utilisation of power flow analysis in order
to determine the voltages at each bus and the currents flowing through each
branch, as well as to identify any instances of voltage or current violation
within the network. Due to the presence of curtailment in the loads, the use
of linear power flow is rendered ineffective, resulting in the emergence of a
complex nonlinear restriction. Therefore, it appears that mathematical
programming optimisation models are not well-suited for addressing this
particular challenge, however heuristic optimisation methods can be highly
advantageous in resolving similar issues. The backward-forward sweep method is
an iterative approach commonly employed for the numerical resolution of power
flow problems in radial distribution networks. The power flow solution
technique presented in this study is founded on the utilisation of the bus
injection branch current (BIBC) matrix $\Psi$, which has a primary dimension
of ${{N}_{S}}\times{{N}_{B}}$ [25]. The provided binary matrix represents the
currents of each section in the network in relation to the injected currents
of the load points located downstream of each section. The matrix is
structured as follows. It should be noted that when constructing the BIBC
matrix, it is necessary to exclude the first column in order to obtain a
square matrix.

$\displaystyle\Psi=\left(\begin{matrix}0&1&\cdots&1\\\ 0&1&\ddots&0\\\
0&0&\cdots&1\\\ \end{matrix}\right)$ (13)

In order to determine the BIBC matrix of a radial distribution network, a
straightforward approach is presented in the research paper by Teng et al.
(2003) [25]. However, it is imperative to have a consistent numbering system
for both nodes and sections, commencing with the substation and extending to
the nodes further downstream. In order to address this limitation, a
comprehensive algorithm is introduced in Algorithm 1, which facilitates the
generation of the BIBC matrix. Algorithm 1 utilises the adjacency matrix $A$
to represent the network, where the substation nodes are uniformly designated
with the value 1. Algorithm 1 is founded on the identification of leaf nodes
$\Xi$ and their corresponding parent nodes $s$ in the updated tree, achieved
by the iterative process illustrated in Figure 3.

Figure 2: Updated tree of BIBC matrix Algorithm 1 Algorithm of BIBC matrix
construction

0: Matrix A representing network topology

0: BIBC matrix Initialisation :

1: Construct the adjacency matrix $A$

2: Create an empty cell array $\overrightarrow{\varphi}$ in length of network
nodes

3: $n={{N}_{S}}$ LOOP Process

4: while $(\Xi\neq\\{\\})$ do

5: $\Xi=\arg find(\sum\nolimits_{j}{{{A}_{i,j}}=1})$

6: for $\,(i=1,i\leq\left|\Xi\right|)$ do

7: $r=\Xi\\{i\\}$

8: $\phi\\{r\\}=[\phi\\{r\\},r]$

9: $\Psi(n,\phi\\{r\\})=1$

10: $s=\arg find(A(:,r)=1)$

11: $A(s,r)=0,A=(r,s)=0$

12: $\phi\\{s\\}=[\phi\\{s\\},\phi\\{r\\}]$

13: $n=n-1$

14: end for

15: end while

16: return $BIBCMatrix$

Adjacent to the generation of the BIBC matrix, the backward-forward sweep can
be executed using the approach outlined in approach 2. as the present
iterative approach, all variables and parameters are expressed as per-unit.
The technique commences by constructing a diagonal matrix ${{Z}_{D}}$, where
each element corresponds to the impedance of the respective sections, as
indicated in equation (14). Next, we generate a square matrix $\Upsilon$ using
equation (15) for utilisation in the method. The calculation of bus voltages
is performed with respect to the per-unit voltage of the substation, denoted
as ${{V}^{sub}}$. The functions ${{g}^{P}}$ and ${{g}^{Q}}$ defined in
equations (1) and (2) are used to determine the precise values of active and
reactive power for the ZIP loads in every iteration. It is important to take
into account the reduced capabilities of load points when analysing the power
flow through the vector variable $\chi$. Next, the current injection vector
for each load is computed using the Hadamard product symbol $\odot$. After the
completion of each iteration, the bus voltages are changed using the equation
specified in line 8 of the method.

Algorithm 2 Algorithm of backward-forward sweep

0: BIBC matrix

0: Results of power flow analysis (nodal voltage, branch current, loss)
Initialisation:

1: Construct the matrix $\Upsilon$

2: Create the vector of${{V}_{0}}=\overrightarrow{1}\times{{V}^{sub}}$ with
length of number of load points and initiate the vector
${{\mathbf{V}}_{\mathbf{B}}}$ as the bus voltages with
${{\mathbf{V}}_{\mathbf{B}}}={{\mathbf{V}}_{0}}$

3: $iter=1$ LOOP Process

4: while $(iter\leq Maxiter)$ do

5:
$\mathbf{P}={{g}^{P}}({{\mathbf{V}}_{\mathbf{B}}}\mathbf{,}{{\mathbf{C}}_{{{\mathbf{z}}_{\mathbf{p}}}}}\mathbf{,}{{\mathbf{C}}_{{{\mathbf{i}}_{\mathbf{p}}}}}\mathbf{,}{{\mathbf{C}}_{{{\mathbf{p}}_{\mathbf{p}}}}})\odot(1-\chi)\odot{{\mathbf{P}}^{\mathbf{0}}}$

6:
$\mathbf{Q}={{g}^{Q}}({{\mathbf{V}}_{\mathbf{B}}}\mathbf{,}{{\mathbf{C}}_{{{\mathbf{z}}_{q}}}}\mathbf{,}{{\mathbf{C}}_{{{\mathbf{i}}_{q}}}}\mathbf{,}{{\mathbf{C}}_{{{\mathbf{p}}_{q}}}})\odot(1-\chi)\odot{{\mathbf{P}}^{\mathbf{0}}}$

7:
$S=P+jQ,I={{\left(s\odot{{\left[\frac{1}{V_{B}^{1}},\frac{1}{V_{B}^{2}},...,\frac{1}{V_{B}^{N}}\right]}^{T}}\right)}^{*}}$

8: ${{\mathbf{V}}_{B}}={{\mathbf{V}}_{0}}-\Upsilon\mathbf{I}$

9: end while

10: return $\mathbf{V},\,\mathbf{I}$

$\displaystyle{{Z}_{D}}=\left(\begin{matrix}{{z}_{1}}&\cdots&0\\\
\vdots&\ddots&\vdots\\\ 0&\cdots&{{z}_{N}}\\\ \end{matrix}\right)$ (14)

$\displaystyle\Upsilon={{\Psi}^{T}}.{{Z}_{D}}.\Psi$ (15)

Figure 4 illustrates the comprehensive flowchart for addressing the issue of
peak management within an overloaded distribution network. The ideal solution,
which includes the substation voltage and curtailment parameters, is obtained
for each event hour. In the event that the solutions produced by the Particle
Swarm Optimisation (PSO) algorithm fail to adhere to the prescribed voltage
and current constraints, a penalty is applied to the cost function.

Figure 3: Solving algorithm of the problem.

##  IV Simulation and Results

###  IV-A System under study

The proposed Finnish DN bus, known as the 144 bus, is being seen as a suitable
subject for examining the primary case studies discussed in this work under
conditions of overload. Figure 5 depicts a concise visual representation of
the system being investigated. The peak load of the test system under
consideration is 11 MW per year. Additional information regarding the system
being investigated, including as interruption charges, bus data, branch data,
line size, and current rating, can be found in the document referenced as
cite35Aaltodoc. As seen in Figure 5, the Finnish DN system for the 144 bus
configuration has a single primary distribution substation operating at 110/20
kV. This substation is responsible for providing power to 144 secondary
substations operating at 20/0.4 kV. The overall configuration of the system is
radial in nature. In addition, the data pertaining to weather conditions, such
as ambient temperature, sun irradiation, and wind speed, were sourced from the
publication by Finnish et al. (2018) [26]. The weather data pertaining to the
most extreme temperature recorded throughout the summer of 2019 is utilised in
the simulation procedure.

Figure 4: Single line representation of the under study test system.

###  IV-B Numerical results

The obtained results are represented in three case studies as below.

  * •

Case 1: Operation of DN in normal mode without CVR and DTR

  * •

Case 2: Operation of DN by implementation of CVR without considering DTR
during the peak times (hour 10- hour 21)

  * •

Case 3: Operation of DN by implementation of CVR considering DTR during the
peak times (hour 10 to hour 21)

Furthermore the problems related to increased lines current by implementation
of the CVR due to the constant power loads will be discussed in the following
of results.

####  IV-B1 Cost results

The primary benefits of the proposed CVR-DTR, integrated with the DRP
framework, are evident in the reduction of operational expenses for the
distribution network. The cost findings of the introduced case studies are
depicted in Figure 6-(a) (see Fig. 5-(a)). Based on the information shown in
Figure 6-(a), it is evident that the operational cost of the DN is $476,000,
$293,000, and $51,400 during the peak hours (from hour 10 to hour 21) for
examples 1 to 3, respectively. Based on the aforementioned findings, it can be
inferred that the deployment of the CVR alone leads to a decrease of 38.45% in
system operation costs during peak times. However, this cost reduction can be
further increased to 89.2% by concurrently implementing both DTR and CVR.
Furthermore, it is important to note that the decrease in cost reduction
observed in case 2 compared to case 3 can be attributed to the findings
depicted in Figure 6-(b), which play a crucial role in accurately interpreting
the results. Based on the findings shown in Figure 6-(b), it is evident that a
limitation exists which hinders further lowering of voltage in example 2. This
constraint is associated with certain buses that possess substantial constant-
power loads. The excessive drop in voltage at the substation might lead to
increased currents flowing through the substation transformer and its
associated branches, potentially exceeding their thermal current restrictions.
Therefore, the ability of the substation transformer and branches to adjust
their thermal rating dynamically enables a greater reduction in substation
voltage in scenario 3, as depicted in Figure 6-(b) (see Fig. 5-(b)). Given the
significance of this matter, there has been an increased focus on the ongoing
developments pertaining to the following.

Figure 5: Cost and voltage results of each case study.

####  IV-B2 Substation power and load curtailment

To effectively demonstrate the optimal utilisation of the substation and lines
capacity in various scenarios, the conveyed power originating from the 110/20
kV substation is visually depicted in Figure 7. As previously said, the
utilisation of Dynamic Voltage Regulation (DTR) during the deployment of
Constant Voltage Regulation (CVR) results in increased cost reduction as a
consequence of the enhanced flexibility in voltage reduction. The decrease in
cost can be attributed to the less load curtailment observed in example 3 in
comparison to case 2. To clarify, the adoption of Demand Turn Reduction (DTR)
will result in a decrease in load curtailment during the events. Additionally,
the enhanced thermal capacity of DN’s components will enable the provision of
electricity to a greater number of loads. This fact is depicted in Figure 7.
Based on the findings presented in Figure 7, it can be inferred that the
dynamic thermal rating of the distribution network’s components, specifically
the branches and substation transformer as discussed in this study, leads to a
substantial increase in the provided load through the substation in case 3 as
compared to case 2. In order to effectively showcase the effects of CVR
(Conservation Voltage Reduction) and DTR (Demand Time Response) in peak
duration management, we give the illustrative Figure 8, which depicts the load
curtailment seen in each of the case studies. Based on the information shown
in Figure 8, there is a modest decrease in curtailed loads observed in
instance 2 when compared to case 1. Moreover, a notable decrease is shown as a
result of the probable impacts of the DTR in scenario 3, as depicted in Figure
8.

Figure 6: Carried power from the 110/20 kV substation during the event. Figure
7: curtailed loads of buses in each case studies.

####  IV-B3 Voltages and currents analysis

The figure labelled as Figure 9 illustrates the voltage levels of the buses in
the various case studies conducted during two specific event hours, namely 10
and 14. These case studies differ in terms of temperature, wind speed, and
irradiation. Based on the information presented in Figure 9, it is apparent
that the application of CVR results in a noticeable decrease in voltage in
case 2 when compared to case 1. The voltage reduction observed at both 10:00
and 14:00 hours is nearly same, as depicted in Figure 9. This similarity is
evident in both scenarios 1 and 2. On the other hand, in scenario 3, the
varying local weather conditions (including wind, solar radiation, and
temperature) between hour 10 and hour 14 contribute to an increased potential
for voltage reduction in the majority of buses. Additionally, the current
flowing through the branch in case studies 2 and 3 is depicted in Figure 10.
Based on the information presented in Figure 10, it is evident that in Case 3,
there is an increased allowance for currents to flow through the branches,
particularly in the branches located in close proximity to the substation.
This is mostly attributed to the potential effects of Distributed Temperature
Regulation (DTR).

Figure 8: Buses voltages in each case study. Figure 9: Branches current in
each case studies.

####  IV-B4 Effects of DN’s demand factor on CVR potential

As previously stated, the potential of the CVR (Conservation Voltage
Reduction) in a Distribution Network (DN) is contingent upon the loading
condition of the entire network, as it is constrained by the thermal limit of
the DN’s components. Figure 11 illustrates the potential of Demand Response
(DR) in the context of Conservation Voltage Reduction (CVR) across various
demand parameters. This observation is of particular significance when
examining the efficacy of CVR in managing peak loads within the Distribution
Network (DN). The increase in the DN demand factor leads to an observable rise
in the minimum voltage of DN, as depicted in Figure 11. A considerable
increase in the minimum CVR voltage is observed with a demand factor of 95%.
Therefore, it can be inferred from Figure 11 that the use of CVR-based peak
load management in distribution networks (DN) can be substantially mitigated
by enhancing the loading condition of the DN.

Figure 10: Effects of demand factor on CVR.

####  IV-B5 Current changes of DN under CVR considering the DTR

To effectively demonstrate the variations in currents of the feeders resulting
from the adoption of CVR, a visual representation in the form of Figure 11 is
utilised, employing a colour map for enhanced clarity. As anticipated, a
higher magnitude of current variation is typically observed in the feeders
closest to the substation. Figure 11 visually emphasises the initial feeders
using warm colours, representing the more pronounced fluctuations in current
inside these particular feeders. Based on the information presented in Figure
11, it can be observed that only a subset of the feeders have encountered
significant fluctuations in current, which has subsequently hindered the
successful implementation of CVR.

Figure 11: Current change in DN’s feeders.

##  V Conclusion

This article investigates three peak load control strategies of the
distribution network (DN), including CVR, DTR, and DRP. The CVR has been
extensively examined and introduced in past scholarly works, but there has
been comparatively less focus on the application of the DTR in managing peak
loads in Distribution Networks . In addition to examining the direct impacts
of the DTR on peak load management of the Distribution Network (DN), our
investigations extend beyond prior individual studies by exploring the
integration of the DTR with CVR, which offers further advantages in terms of
voltage reduction potential within the DN. According to the findings
presented, the utilisation of the individual CVR results in a cost reduction
of 38.45% during peak hours. In contrast, the implementation of the CVR
considering DTR leads to a significantly higher cost reduction of 89.2% within
the same peak duration. Moreover, the augmentation of the feeder’s current due
to the implementation of the CVR has been substantiated by the findings
presented in the given results. Consequently, the examination of the DTR has
been undertaken as a potential resolution to address this issue. The suggested
peak management framework offers the advantage of reducing load curtailment
during peak events. This reduction is achieved through the simultaneous
utilisation of the CVR and DTR techniques. Moreover, according to the
findings, the procurement of power from the energy market experiences an
increase during periods of high demand through the utilisation of the CVR and
DTR strategies. It is noteworthy to acknowledge that there was a significant
level of load curtailment before to the implementation of CVR and DTR.
However, the introduction of DTR and CVR has effectively decreased this
curtailment, resulting in the provision of power to a greater number of loads.

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In this section, we will show lower bounds for two learning problems. First,
we show $\mathsf{QPStat}$ query complexity lower bounds for average-case
learning brickwork random quantum circuits with depth at least logarithmic and
at most linear in the number of qubits. Next, we build upon the QSQ hardness
of shadow tomography from arunachalam2024role to show the hardness of a
related problem, quantum threshold search. This result has implications for
the hardness of quantum hypothesis selection, which is solved as a subroutine
in a learning algorithm of zhao2023learning, making it unlikely that such an
algorithm can be used in the quantum statistical query setting. Shallow
brickwork-random quantum circuits Before stating the main result of this
section, we state some important definitions. [Unitary $t$-Designs] We denote
the haar measure over $U(N)$ by $\mu_{H}$. The $t$-th moment superoperator
with respect to a distribution $\nu$ over $U(N)$ is defined as

$\mathcal{M}_{\nu}^{(t)}(A)=\underset{U\sim\nu}{\mathbf{E}}[U^{\otimes
t}A(U^{\dagger})^{\otimes t}]=\int U^{\otimes t}A(U^{\dagger})^{\otimes
t}d\nu(U)$ (1)

$\nu$ is said to be an exact unitary $t$-design if and only if

$\mathcal{M}_{\nu}^{(t)}(A)=\mathcal{M}_{\mu_{H}}^{(t)}(A)$ (2)

Similarly, $\nu$ is said to be an additive $\delta$-approximate unitary
$t$-design if and only if

$\norm{\mathcal{M}_{\nu}^{(t)}(A)-\mathcal{M}_{\mu_{H}}^{(t)}(A)}_{\diamond}\leq\delta$
(3)

We denote an exact unitary $t$-design by $\mu_{H}^{(t)}$ and an additive
$\delta$-approximate unitary $t$-design by $\mu_{H}^{(t,\delta)}$. [Brickwork
random quantum circuits] Denote by $\mathsf{RQC}(n,d)$ the measure over
brickwork random quantum circuits of $n$ qubits with depth $d$.
$\mathsf{RQC}(n,d)$ consists of unitaries of the form

$U=(I_{2}\otimes U_{2,3}^{(d)}\otimes
U_{4,5}^{(d)}\otimes\dots)(U_{1,2}^{(d-1)}\otimes
U_{3,4}^{(d-1)}\otimes\dots)\dots(U_{1,2}^{1}\otimes U_{3,4}^{1}\otimes\dots)$
(4)

where $U_{i,j}^{(l)}$ are 2-qubit unitaries distributed according to the Haar-
measure over $\mathcal{U}(4)$ and $I_{2}$ is the identity on a single qubit
(Figure 1). $\mathsf{RQC}(n,d)$ for any $d>0$ forms an exact unitary 1-design.
At infinite depth, the distribution over brickwork random quantum circuits
converges to the Haar measure.

[width = 0.4]img/brqc.png

Figure : Structure of brickwork random quantum circuits. Each box is a 2-qubit
Haar-random gate.

Now, we will state our result on the query-complexity lower bound for learning
shallow brickwork random quantum circuits. [Average-case lower bound for
shallow BRQCs] Let
$N=2^{n},0<\tau\leq\epsilon\leq\frac{1}{3}\left(1-\frac{1}{N}\right),\mathsf{RQC}(n,d)$
be an ensemble of $n$-qubit brickwork-random quantum circuits of depth $d$,
with

$\frac{\log(n)}{\log(5/4)}\leq d\leq\frac{n+\log(n)}{\log(5/4)}$ (5)

Assume there exists an algorithm that with probability $\beta$ over
$U\sim\mathsf{RQC}(n,d)$ and probability $\alpha$ over its internal randomness
produces a hypothesis $\hat{\mathcal{E}}$ such that
$d_{\diamond}(\hat{\mathcal{E}},U(.)U^{\dagger})\leq\epsilon$, using $q$
queries with tolerance $\tau$ to $\mathsf{QPStat}_{U}$. Then, it holds

$q+1\geq\Omega\left(\frac{(2\alpha-1)\beta}{n}\tau^{2}\left(\frac{5}{4}\right)^{d}\right)$
(6)

Before proving this theorem, we will state some useful lemmas. We start by
stating the general lower bound for learning processes up to diamond distance
using QPSQs from [wadhwa2024learning]. [General lower bound for QPSQ-learning
within $d_{\diamond}$, cf. [wadhwa2024learning]] Let $0<\tau\leq\epsilon$,
$\mathcal{C}\subseteq\mathcal{S}_{N}\rightarrow\mathcal{S}_{N}$ be a set of
quantum processes , and $\mu$ some measure over $\mathcal{C}$. Assume there
exists an algorithm that with probability $\beta$ over $\mathcal{E}\sim\mu$
and probability $\alpha$ over its internal randomness produces a hypothesis
$\hat{\mathcal{E}}$ such that
$d_{\diamond}(\hat{\mathcal{E}},\mathcal{E})\leq\epsilon$, using $q$ queries
with tolerance $\tau$ to $\mathsf{QPStat}_{\mathcal{E}}$. Then, for every
$\Phi:\mathcal{S}_{N}\rightarrow\mathcal{S}_{N}$, it holds

$q+1\geq(2\alpha-1)\frac{\beta-\mathbf{Pr}_{\mathcal{E}\sim\mu}(d_{\diamond}(\mathcal{E},\Phi)<2\epsilon+\tau)}{\underset{\rho,O}{\max\text{}}\mathbf{Pr}_{\mathcal{E}\sim\mu}(|Tr(O\mathcal{E}(\rho))-Tr(O\Phi(\rho))|>\tau)}$
(7)

We will refer to the term in the denominator,

$\underset{\rho,O}{\max\text{}}\mathbf{Pr}_{\mathcal{E}\sim\mu}(|Tr(O\mathcal{E}(\rho))-Tr(O\Phi(\rho))|>\tau)$

as the maximally distinguishable fraction. We refer to the term in the
numerator,

$\mathbf{Pr}_{\mathcal{E}\sim\mu}(d_{\diamond}(\mathcal{E},\Phi)<2\epsilon+\tau)$

as the trivial fraction. To prove theorem , we will bound both the maximally
distinguishable fraction and the trivial fraction with respect to the
maximally depolarizing channel as the reference channel. In order to bound the
maximally distinguishable fraction, we will first need results on the first
and second-order moments over the Haar measure. We state without proof the
necessary results from [wadhwa2024learning, mele2024introduction] in the
following lemma. [Moments over the Haar-measure, cf. [wadhwa2024learning,
mele2024introduction]] The first moment superoperator of the Haar measure
$\mu_{H}$ over $U(N)$ is the maximally depolarizing channel.

$\mathcal{M}_{\mu_{H}}^{(1)}=\Phi_{dep}$ (8)

Further, for all $O\in H_{N,N}$ and $\rho\in\mathcal{S}_{N}$,

$\underset{U\sim\mu_{H}}{\mathbf{E}}[Tr(OU\rho U^{\dagger})]=\frac{Tr(O)}{N}$
(9)

The second moment is given by

$\underset{U\sim\mu_{H}}{\mathbf{E}}[Tr(OU\rho
U^{\dagger})^{2}]=\left(\frac{N-Tr(\rho^{2})}{N(N^{2}-1)}\right)Tr(O)^{2}+\left(\frac{NTr(\rho^{2})-1}{N(N^{2}-1)}\right)Tr(O^{2})$
(10)

and the variance is bounded by

$\underset{U\sim\mu_{H}}{\mathbf{Var}}[Tr(OU\rho
U^{\dagger})]\leq\frac{1}{N+1}$ (11)

Now, we will prove a bound on the variance of expectation values over
$\mathsf{RQC}(n,d)$ [Bounded variance for $\mathsf{RQC}(n,d)$] For depth

$\frac{\log(n)}{\log(5/4)}\leq d\leq\frac{n+\log(n)}{\log(5/4)}$ (12)

the variance over $\mathsf{RQC}(n,d)$ is bounded by

$\underset{U\sim\mathsf{RQC(n,d)}}{\mathbf{Var}}[Tr(OU\rho
U^{\dagger})]=\mathcal{O}\left(n\left(\frac{4}{5}\right)^{d}\right)$ (13)

$\forall O\in H_{2^{n},2^{n}}$ with $\norm{O}_{\infty}\leq 1$ and
$\forall\rho\in\mathcal{S}_{2^{n}}$. [Proof sketch] Bounds on low-depth
moments for BRQCs have been shown in [nietner2023average, barak2020spoofing]
by counting partitions over a statistical mechanical model, a technique
developed originally in [hunter2019unitary]. We use the same technique to show
an analogous bound for our problem:

$\underset{U\sim\mathsf{RQC}(n,d)}{\mathbf{E}}[Tr(OU\rho
U^{\dagger})^{2}]\leq\left(1+\left(\frac{4}{5}\right)^{d}\right)^{n/2}\underset{U\sim\mu_{H}}{\mathbf{E}}[Tr(OU\rho
U^{\dagger})^{2}]$ (14)

Then, using the fact that brickwork random quantum circuits form exact
1-designs at any depth, and substituting in the moments from Lemma equationx2,
the desired result is obtained from direct computation. We show the detailed
proof in Appendix LABEL:app:shallow-var. In order to bound the trivial
fraction, we will use the following lemma from [wadhwa2024learning] [Unitaries
are far from the maximally depolarizing channel, cf. [wadhwa2024learning]] For
any unitary $U\in U(N)$, we have :

$d_{\diamond}(U(.)U^{\dagger},\Phi_{dep})\geq 1-\frac{1}{N}$ (15)

Finally, we are in a position to prove Theorem  [Proof of Theorem ] As stated
earlier, we will use the maximally depolarizing channel as our reference
channel $\Phi$ in Lemma 6. For
$0<\tau\leq\epsilon\leq\frac{1}{3}\left(1-\frac{1}{N}\right)$, Lemma 14 tells
us that $d_{\diamond}(U(.)U^{\dagger},\Phi_{dep})>2\epsilon+\tau$ for all
$U\sim\mathsf{RQC}(n,d)$. Thus, the trivial fraction is 0.

As $\mathsf{RQC}(n,d)$ is a 1-design, the expected channel over this ensemble
is the maximally depolarizing channel. Thus, we use Chebyshev’s inequality to
bound the maximally distinguishable fraction.

$\split\underset{U\sim\mathsf{RQC}(n,d)}{\mathbf{Pr}}(|Tr(OU(\rho)U^{{\dagger}})-Tr(O\Phi_{dep}(\rho))|>\tau)&\leq\frac{\mathbf{Var}_{U\sim\mathsf{RQC}(n,d)}[Tr(OU\rho
U^{\dagger})]}{\tau^{2}}\\\
=\mathcal{O}\left(\frac{n}{\tau^{2}}\left(\frac{4}{5}\right)^{d}\right)$ (16)

Now, using the bounds on the maximally distinguishable fraction and the
trivial fraction, we obtain the desired result

$q+1\geq\Omega\left(\frac{(2\alpha-1)\beta}{n}\tau^{2}\left(\frac{5}{4}\right)^{d}\right)$
(17)

##  Hardness of Threshold Search and implications for learnability of shallow
circuits

In this section, we will show the hardness of quantum threshold search and
discuss its implications for the learnability of shallow circuits.
Arunachalamet al. [arunachalam2024role] showed the hardness of shadow
tomography of mixed states $\rho$ from access to $\mathsf{QStat}_{\rho}$.
Given access to copies of $\rho$, an efficient algorithm for shadow tomography
is known, using a reduction to threshold search [buadescu2021improved,
aaronson2018shadow] through a mistake-bounded online learning algorithm
[aaronson2018online]. We recall the algorithm for shadow tomography and then
show that the reduction from shadow tomography to threshold search is
efficient in $\mathsf{QStat}_{\rho}$ queries. This implies that threshold
search itself must be a hard problem to solve with quantum statistical
queries. We start by recalling the problems of shadow tomography and threshold
search. We also recall the related problem of quantum hypothesis selection
which we will discuss later. [Shadow Tomography cf. [aaronson2018shadow]]
Given parameters $0<\epsilon,\delta<1$, access to an unknown $N$-dimensional
quantum state $\rho$, as well as observables $O_{1},\dots,O_{M}\in H_{N,N}$,
each of which has bounded operator norm $\|O_{i}\|_{\infty}\leq 1$ and
expectation value $Tr(O_{i}\rho)$, output numbers $b_{1},\dots,b_{M}\in[-1,1]$
such that $|b_{i}-Tr(O_{i}\rho)|\leq\epsilon$ for all $i$, with success
probability at least $1-\delta$. [Enhanced Quantum Threshold Search cf.
[buadescu2021improved]] Given parameters $0<\epsilon,\delta<1/2$, access to an
unknown $N$-dimensional quantum state $\rho$, observables
$O_{1},\dots,O_{M}\in H_{N,N}$, each of which has bounded operator norm
$\|O_{i}\|_{\infty}\leq 1$, and thresholds
$0\leq\theta_{1},\dots,\theta_{M}\leq 1$, output

  * •

“$|Tr(O_{j}\rho)-\theta_{j}|>\frac{3}{4}\epsilon$ and in fact
$|Tr(O_{j}\rho)-\mu_{j}^{\prime}|\leq\frac{1}{4}\epsilon$” for some particular
$j$ and value $\mu_{j}^{\prime}$; or else,

  * •

“$|Tr(O_{i}\rho)-\theta_{i}|\leq\epsilon$ for all $i$”

with probability of a false output at most $\delta$. [Quantum Hypothesis
Selection, cf. [buadescu2021improved]] Given parameters
$0<\epsilon,\delta<1/2$, access to an unknown $N$-dimensional quantum state
$\rho$, and $M$ fixed hypothesis states $\sigma_{1},\dots,\sigma_{M}$, with
$\eta=\min_{i}\left[d_{tr}(\rho,\sigma_{i})\right]$, output an index
$k\in[M]$, such that with probability at least $1-\delta$,

$d_{tr}(\rho,\sigma_{k})\leq C\eta+\epsilon$ (18)

for some small constant $C$. Now, we state the result of [arunachalam2024role]
on the hardness of shadow tomography from $\mathsf{QStat}_{\rho}$ queries.
[Hardness of Shadow Tomography from QSQs, cf. [arunachalam2024role]] Any
algorithm that use $\mathsf{QStat}_{\rho}$ queries of tolerance at least
$\tau$ and predicts $Tr[P\rho]$ up to error $\epsilon$ for all $4^{n}-1$ non-
identity Pauli strings $P\in\mathcal{P}_{n}\backslash\\{I\\}$ with high
probability requires $\Omega(\tau^{2}2^{2n}/\epsilon^{2})$ queries. The result
of [arunachalam2024role] implies that no algorithm can solve shadow tomography
significantly more efficiently than simply querying $\mathsf{QStat}_{\rho}$
with every single observable.

We briefly describe the reduction from Shadow Tomography to Quantum Threshold
Search as shown in [buadescu2021improved]. This algorithm for Shadow
Tomography involves multiple rounds of quantum threshold search interleaved
with the online learning algorithm. Each round consists of one “mistake”,
where the learner outputs an index where the expectation value deviates from
the threshold. In such a case, the learner must estimate an
$\epsilon/4$-estimate of the expectation value (which can be done with a
single query to $\mathsf{QStat_{\rho}}$). Such a mistake occurs only
$\mathcal{O}((\log N)/\epsilon^{2})$ many times. Thus, we obtain the following
relation between the QSQ complexity of shadow tomography and quantum threshold
search. [Quantum Threshold Search vs Shadow Tomography] Let $q^{shadow}$ and
$q^{thresh}$ be the number of $\mathsf{QStat}_{\rho}$ queries needed to solve
Problems 1 and 1 respectively for an unknown state $\rho\in\mathcal{S}_{N}$
and a common list of observables $O_{1},\dots,O_{M}\in H_{N,N}$. Then, it
holds that

$q^{thresh}+1\geq q^{shadow}\Omega\left(\epsilon^{2}/\log N\right)$ (19)

Now, using Lemmas 18 and 18 we state the hardness of Threshold Search.
[Hardness of Quantum Threshold Search] Any algorithm that use
$\mathsf{QStat}_{\rho}$ queries of tolerance at least $\tau$ and solves
Problem 1 up to error $\epsilon$ for all $4^{n}-1$ non-identity Pauli strings
$P\in\mathcal{P}_{n}\backslash\\{I\\}$ and all choices of
$\theta_{1},\dots,\theta_{4^{n}-1}$ with high probability requires
$\Omega(\tau^{2}2^{2n}/n)$ queries. We would like to point out an interesting
implication of this result in the context of learning shallow circuits. Zhao
et al. [zhao2023learning] showed an algorithm for learning unitaries
consisting of $G$ two-qubit gates up to average-distance (Definition
LABEL:def:avg-dist) using $\mathcal{O}(G\log(G))$ black-box queries. Given
that a circuit of constant depth can have at most $\mathcal{O}(n)$ gates, it
is a natural question whether the algorithm of [zhao2023learning] can also be
extended to the statistical query setting. A desirable property of this
algorithm is that it is nearly query-optimal in the black-box setting.
Unfortunately, using Theorem 19, we conjecture that such an extension to the
statistical query setting is unlikely.

The algorithm of [zhao2023learning] involves casting out an
$\epsilon$-covering net over the space of $G$-gate unitaries with respect to
$d_{avg}$, followed by quantum hypothesis selection on the output states of
the elements of the net. As the number of elements of the net $\mathcal{M}$
may be exponential in $G$, quantum hypothesis selection must be carried out
using $\mathcal{O}(\log(\mathcal{M}))$ queries. However, the authors in
[zhao2023learning] do not show an explicit net for this construction. Further,
in [zhao2023learning], quantum hypothesis selection is carried out using
classical shadow tomography. From the worst-case hardness of shadow tomography
shown in [arunachalam2024role], we see that without knowing the additional
properties of the net, one cannot use a shadow tomography algorithm for this
purpose. The original methods for quantum hypothesis selection in
[buadescu2021improved] used either shadow tomography or (a decision variant
of) quantum threshold search as subroutines. As a result, the worst-case
hardness of threshold search from Theorem 19 and the result of
[arunachalam2024role] prevent us from showing an efficient algorithm for
quantum hypothesis selection. Thus, without considering explicit constructions
for covering nets, one cannot extend existing techniques to perform quantum
hypothesis selection efficiently with statistical queries. Even with such
information, quantum hypothesis selection may prove to be hard. Thus, we
conjecture that no efficient QPSQ analogue for the algorithm of
[zhao2023learning] exists. As the robustness of our framework does not apply
here, it may be interesting to study the noise-tolerance of their algorithm
using other methods.

[width = 0.4]img/threshold-shadow.png

Figure : Relation between Problems 1, 1, 1. Given an algorithm to solve its
decision variant, Quantum Threshold Search can be solved with a single
additional query to $\mathsf{QStat}$. Shadow tomography requires running
quantum threshold search $\mathcal{O}((\log N)/\epsilon^{2})$ times,
increasing queries by that factor. Known algorithms for quantum hypothesis
selection use shadow tomography and the decision variant of threshold search,
which are both hard with QSQs.