# Power Allocation Algorithms for
Massive MIMO Systems with Multi-Antenna Users

Evgeny Bobrova,b, Boris Chinyaeva, Viktor Kuznetsovb,
Dmitrii Minenkova,c, Daniil Yudakova Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>a M.V. Lomonosov Moscow State
University;
b Huawei Technologies, Russian Research Institute, Moscow Research Center
c A. Ishlinsky Institute for Problems in Mechanics RAS

###### Abstract

Modern 5G wireless cellular networks use massive multiple-input multiple-
output (MIMO) technology. This concept entails using an antenna array at a
base station to concurrently service many mobile devices that have several
antennas on their side. In this field, a significant role is played by the
precoding (beamforming) problem. During downlink, an important part of
precoding is the power allocation problem that distributes power between
transmitted symbols. In this paper, we consider the power allocation problem
for a class of precodings that asymptotically work as regularized zero-
forcing. Under some realistic assumptions, we simplify the sum spectral
efficiency functional and obtain tractable expressions for it. We prove that
equal power allocation provides optimum for the simplified functional with
total power constraint (TPC). Also, low-complexity algorithms that improve
equal power allocation in the case of per-antenna power constraints (PAPC) are
proposed. On simulations using Quadriga, the proposed algorithms show a
significant gain in sum spectral efficiency while using a similar computing
time as the reference solutions.

###### keywords:

5G, MIMO, Multi-antenna UE, Precoding, Regularized Zero-Forcing, Power
Allocation, MMSE-IRC Detection, Constrained Optimization, Karush–Kuhn–Tucker
conditions, Asymptotics

##  1 Introduction

The massive multiple-input multiple-output (MIMO) systems have attracted a lot
of attention in both academia and industry since their first appearance [2,
25]. The main characteristic of the massive MIMO system is the large-scale
antenna arrays at the cellular base station (BS). Using a large number of
antennas, the massive MIMO system can exceed the achievable speed of a
conventional MIMO [16] system and simultaneously serves (with low power
consumption) several users.

A critical issue for improving the performance of wireless networks is the
efficient management of available radio resources [22]. Numerous works are
dedicated to optimal allocation of the radio resources, for example, power and
bandwidth to improve the performance of wireless networks [30]. An important
part of signal processing in downlink is precoding since with this procedure
we can focus transmission signal energy on smaller regions, which allows
achieving greater spectral efficiency with lower transmitted power [27, 2].
Various linear precodings allow directing the maximum amount of energy to the
user like Maximum Ratio Transmission (MRT) or completely get rid of inter-user
interference like Zero-Forcing (ZF) [29, 45]. The precoding problem is well-
studied (see e.g., overview [15, 47, 14] and textbooks [4, 35, 2] and
bibliography within), nonetheless there are open questions. For example, most
of the works consider the total power constraint (TPC) (see e.g., [11]), the
more realistic per-antenna power constraints (PAPC) are much less studied (see
e.g., [43, 5]).

An important component of the precoding procedure is the power allocation (PA)
problem that is widely discussed in the literature. In [13], by using either
the signal-to-noise ratio (SNR) or the outage probability as the performance
criteria, different power allocation (PA) strategies are developed to exploit
the knowledge of channel means. In [18] bounds on the channel capacity are
derived for a similar model with Rayleigh fading and channel state information
(CSI). The power allocation problem in a three-node Gaussian orthogonal relay
system is investigated in [23] to maximize a lower bound on the capacity. Two
power allocation schemes based on minimization of the outage probability are
presented in [46] for the case when the information of the wireless channel
responses or statistics is available at the transmitter. In [28] studies
optimal power allocation schemes in a multi-relay cooperating network
employing amplify-and-forward protocol with multiple source-destination pairs.
The work [32] advocates the use of deep learning to perform max-min and max-
prod power allocation in the downlink of Massive MIMO networks. In [37] the
total downlink power consumption at the access points is minimized,
considering both to transmit powers and hardware dissipation.

The most relevant works to the current paper are of E. Björnson et al. In [4,
sec. 7.1] the case of single-antenna user equipment (UE) is studied in detail,
targeting UE spectral efficiency and using multi-criteria optimization
approach and Pareto front analysis. In [5, p. 328] multi-antenna UEs are
considered, but they are supposed to get only one data channel (or stream).
The difficulty of the multi-antenna UE case is that the channels between
different antennas of one UE are often spatial correlated [6]. Therefore, the
matrix of the user channel is ill-conditioned (or even has incomplete rank)
thus one can not efficiently transmit data using the maximum number of
streams. To solve this problem, instead of the full matrix of the user
channel, vectors from its singular value decomposition (SVD) with the largest
singular values are used for precoding [34]. When the number of streams (UE
rank) is greater than one, it is necessary to consider the phenomenon of
effective SINR [17].

In this paper, we study the problem of power allocation (PA) of MIMO wireless
systems with users with multiple antennas and generalize the results of E.
Björnson et al. for the case of multi-antenna UEs with rank greater than one.
We present the novel solutions to the PA problem that maximize network
throughput in terms of spectral efficiency (SE) subject to either total or
per-antenna power constraints. The original problem is not convex, but we
managed to simplify it to a convex one with additional assumptions on the
system model, e.g., applying a specific class of detection. Under some natural
assumptions, we simplify the sum spectral efficiency functional and prove that
the uniform power allocation provides its optimum subject to TPC. For the case
of PAPC, we equivalently reformulate the optimization problem as the Lagrange
system of equations and write down the Karush–Kuhn–Tucker conditions. Here,
algorithmic solutions of PA problem are proposed assuming realistic PAPC. The
simulation results based on Quadriga channel simulator [19] show the
effectiveness of the proposed algorithmic approach in comparison with the
reference PA schemes. To the authors’ best knowledge, these mathematical
results are new.

The rest of this paper is organized as follows. After this Introduction,
Section 2 is devoted to the channel and system model where we introduce the
downlink MIMO channel model, reference precoding methods, various detection
schemes, and quality measures. In Section 3 we show a simplification of the PA
problem, where we describe asymptotic diagonalization property of precoding
matrices are used, proof of similarity of Conjugate and MMSE-IRC matrices, and
Effective SINR models. In Section 4 we consider the problem of the PA
algorithm under TPC and PAPC assumptions, where we describe equal power
allocation under the TPC, and the solution under the PAPC assumptions. We also
consider problem-solving taking into account Modulation and Coding Scheme
(MCS) (4.2). The numerical algorithm description is presented in Section
(4.5). Numerical experiments to compare considered algorithms are provided in
Section 5, and Section 6 contains a conclusion.

Algebraic notations and reference values are shown in Tab. 1.

Table 1: Algebraic notations together with the reference values . Symbols | Notations
---|---
$\bm{H}\in\mathbb{C}^{R\times T},\bm{W}\in\mathbb{C}^{T\times L},\bm{G}\in\mathbb{C}^{L\times R}$ | Channel, precoding and detection matrices
$\bm{w}_{n}\in\mathbb{C}^{T}$ |  $n$-th column of matrix $\bm{W}$
$\bm{h}_{k}\in\mathbb{C}^{T},\bm{w}^{k}\in\mathbb{C}^{L}$ |  $k$-th row of matrices $\bm{H},\bm{W}$
$h_{nm}\in\mathbb{C},w_{nm}\in\mathbb{C}$ |  $n,m$-th element of matrices $\bm{H},\bm{W}$
$\bm{S}={\rm diag}(s_{1},\dots,s_{L})\in\mathbb{C}^{L\times L}$ | Diagonal matrix of singular values
$K\;(=4)$ | the number of users
$T\;(=64)$ | the number of transmit antennas
$R\;(=16)$ | the total number of receive antennas
$R_{k}\;(=4)$ | the number of receive antennas for each user
$L\;(=8)$ | the total number of layers in the system
$L_{k}\;(=2)$ | the number of layers for each user
${\bf()}^{\mathrm{H}}$ | Complex conjugate operator
Figure 1: Multi-User precoding allows transmitting of different information to
different users simultaneously. Using the matrix $\bm{W}$ we can configure the
amplitude and phase of the beams presented on the picture. The problem is to
find the optimal precoding matrix $\bm{W}$ of the system given the target SE
function (11).

##  2 Channel and System Model

According to [35, 4, 1, 44] we consider a MIMO broadcast channel. Symbol
$\bm{r}\in\mathbb{C}^{L}$ is a received vector, and $\bm{x}\in\mathbb{C}^{L}$
is a sent vector, and $\bm{H}\in\mathbb{C}^{R\times T}$ is a channel matrix,
and $\bm{W}\in\mathbb{C}^{T\times L}$ is a Precoding matrix, and
$\bm{G}\in\mathbb{C}^{L\times R}$ is a block-diagonal detection matrix, and
$\bm{n}\sim\mathcal{CN}(0,I_{L})$ is a noise-vector. Note that the linear
precoding and detection are implemented by simple matrix multiplications. The
constant $T$ is the number of transmit antennas, $R$ is the total number of
receive antennas, and $L$ is the total number of transmitted symbols in the
system. Usually, they are related as $L\leq R\leq T$. Each of the matrices
$\bm{G},\bm{H},\bm{W}$ decomposes by $K$ users, so please see the scheme in
Fig. 2. The Multi-User MIMO model is described using the following linear
system:

$\bm{r}=\bm{G}(\bm{H}\bm{W}\bm{x}+\bm{n}).$ (1)

Throughout the paper, we assume that i) the channels of all users are subject
to uncorrelated Rayleigh fading and, for simplicity, all users are homogeneous
and experience statistically independent fading, ii) the transmitter has
perfect CSI of all downlink channels, this assumption is reasonable in time
division duplex (TDD) systems, which allows the transmitter to employ
reciprocity to estimate the downlink channels, and iii) each user only has
access to their own CSI, but not the CSI of the downlink channels of the other
users.

Figure 2: Linear system model.

###  2.1 Singular Value Decomposition of the Channel

The channel matrix for user $k$, $\bm{H}_{k}\in\mathbb{C}^{R_{k}\times T}$
contains channel vectors $\bm{h}_{i}\in\mathbb{C}^{T}$ by rows. The path loss
diagonal matrix $\bm{S}_{k}\in\mathbb{R}^{R_{k}\times R_{k}}$ contains $R_{k}$
singular values $\sigma_{kn}$ in decreasing order along its main diagonal. It
is convenient [34] to represent $\bm{H}_{k}$ via its Singular Value
Decomposition (SVD):

$\bm{H}_{k}=\bm{U}^{\mathrm{H}}_{k}\bm{S}_{k}\bm{V}_{k}.$

######  Lemma 1 (Main Decomposition).

[7] Denote $\bm{H}=[\bm{H}_{1},\dots,\bm{H}_{K}]\in\mathbb{C}^{R\times T}$ the
concatenation of individual channel rows $\bm{H}_{k}$. Similarly, $\bm{U}={\rm
bdiag}\\{\bm{U}_{1},\dots,\bm{U}_{K}\\}$, $\bm{S}={\rm
diag}\\{\bm{S}_{1},\dots,\bm{S}_{K}\\}$,
$\bm{V}=[\bm{V}_{1},\dots,\bm{V}_{K}]$. Then, the decomposition exists (see
Fig. 3):

$\bm{H}=\bm{U}^{\mathrm{H}}\bm{S}\bm{V},$

where the $\bm{H}\in\mathbb{C}^{R\times T}$, and $\bm{S}={\rm
diag}(\bm{S}_{k})\in\mathbb{C}^{R\times R}$, and $\bm{U}={\rm
bdiag}(\bm{U}_{k})\in\mathbb{C}^{R\times R}$ is block-diagonal unitary matrix,
$\bm{V}=[\bm{V}_{1},\dots,\bm{V}_{K}]\in\mathbb{C}^{R\times T}$ is the
concatenation of corresponding UE singular vectors and
$\bm{C}=\bm{V}\bm{V}^{\mathrm{H}}-\bm{I}\neq\bm{O}$.

Lemma 1 means that by collecting all users together, we can write a specific
channel matrix decomposition [7]. Note, that such decomposition is not a
convenient SVD of the channel matrix $\bm{H}$, and the matrix $\bm{V}$ is not
unitary. But it consists of the $K$ SVDs of the size $R_{k}\times T$ and has
block-diagonal unitary left matrix $\bm{U}$. We use this form in the
construction of the optimal detection matrix $\bm{G}$ [34].

Figure 3: A graphical illustration of the Main Decomposition Lemma 1.

Usually, the transmitter sends to UE several layers and the number of layers
(rank) is less than the number of UE antennas ($L_{k}\leq R_{k}$). In this
case, it is natural to choose for transmission the first $L_{k}$ vectors from
$\widetilde{\bm{V}}_{k}$ that correspond to the $L_{k}$ largest singular
values from $\widetilde{\bm{S}}_{k}$. Denote by
$\widetilde{\bm{S}}_{k}\in\mathbb{C}^{L_{k}\times L_{k}}$ the first $L_{k}$
largest singular values from $\bm{S}_{k}$, and by
$\widetilde{\bm{U}}_{k}^{\mathrm{H}}\in\mathbb{C}^{R_{k}\times
L_{k}},\;\widetilde{\bm{V}}_{k}\in\mathbb{C}^{L_{k}\times T}$ the first
$L_{k}$ left and right singular vectors that correspond to
$\widetilde{\bm{S}}_{k}$:

$\widetilde{\bm{S}}_{k}={\rm
diag}\\{s_{k,1},\dots,s_{k,L_{k}}\\},\quad\widetilde{\bm{U}}_{k}^{\mathrm{H}}=(\bm{u}_{k,1}^{\mathrm{H}},\dots,\bm{u}_{k,L_{k}}^{\mathrm{H}}),\quad\widetilde{\bm{V}}_{k}=[\bm{v}_{k,1};\dots;\bm{v}_{k,L_{k}}],$
(2)

i.e. ${\rm rank}\widetilde{\bm{V}}_{k}=L_{k}\leq R_{k}={\rm rank}\bm{V}_{k}$.
Numbers $L_{k}$ (and particular selection of $\widetilde{\bm{V}}_{k}$) are
defined during the Rank Adaptation problem that, along with Scheduler, is
solved before precoding. For the Rank adaptation problem, we refer for example
to [24] and in what follows we consider $L_{k}$, $\widetilde{\bm{V}}_{k}$
already chosen.

###  2.2 Precoding Matrices

The precoding matrix $\bm{W}$ is responsible for the beamforming from the base
station to the users [10]. The linear methods for precoding do the following.
Firstly, the linear solutions obtain singular value decomposition for each
user
$\bm{H}_{k}=\bm{U}_{k}^{\mathrm{H}}\bm{S}_{k}\bm{V}_{k}\in\mathbb{C}^{R_{k}\times
T}$ (Lemma 1) and take the first $L_{k}$ singular vectors
$\widetilde{\bm{V}}_{k}\in\mathbb{C}^{L_{k}\times T}$ which attend to the
first $L_{k}$ greatest singular values [34]. All these matrices are
concatenated to the one matrix $\widetilde{\bm{V}}\in\mathbb{C}^{L\times T}$,
which is used as the main building block of these precoding constructions.
Finally, the precoding matrix is constructed from the obtained singular
vectors. We describe linear methods for constructing a precoding matrix.

We are considering precoding matrices in the following form:

$\bm{W}=\bm{W}^{\prime}\bm{P},\qquad\bm{W}^{\prime}=\bm{W}^{\prime}(\widetilde{\bm{V}}),$
(3)

where $\widetilde{\bm{V}}$ is taken from the specific SVD decomposition from
Lemma 1 and $\bm{P}$ is a diagonal matrix of power allocation.

Let us repeat some known algorithms of precoding that are usually considered
as baseline or reference. A Maximum Ratio Transmission (MRT) [29], Zero-
Forcing (ZF) [42], Regularized Zero-Forcing (RZF), [40] and Adaptive
Regularized Zero-Forcing [7] precoding algorithms are defined as follows :

$\bm{W}_{MRT}=\widetilde{\bm{V}}^{\mathrm{H}}\bm{P}$
$\bm{W}_{ZF}=\widetilde{\bm{V}}^{\dagger}\bm{P},\quad\widetilde{\bm{V}}^{\dagger}:=\widetilde{\bm{V}}^{\mathrm{H}}(\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}})^{-1}$
(4)
$\bm{W}_{RZF}=\widetilde{\bm{V}}^{\mathrm{H}}(\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}+\lambda\bm{I})^{-1}\bm{P}$
(5)
$\bm{W}_{ARZF}=\widetilde{\bm{V}}^{\mathrm{H}}(\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}+\lambda\bm{S}^{-2})^{-1}\bm{P}.$
(6)

###  2.3 Detection Matrices

After precoding and transmission, on the side of UE $k$, we have to choose a
detection matrix $\bm{G}_{k}\in\mathbb{C}^{L_{k}\times R_{k}}$, which takes
into account the rank of UE $L_{k}$. The way the UE performs detection
strongly affects overall performance, and different detection algorithms
require different optimal precoding matrices (see [20], where precoding is
chosen as a function of the detection matrix). The best way would be to
consistently choose precoding and detection, but this is hardly possible due
to the distributed nature of wireless communication. However, there are ideas
on how to set up a precoding matrix, assuming a specific detection method on
the UE side in the transmitter [33]. We do not consider such an approach in
our work, although it can be used to further improve our main proposal.

We assume the effective channel matrix $\bm{A}_{k}=\bm{H}_{k}\bm{W}_{k}$ to be
calculated on the UE side. The Minimum Mean Square Error (MMSE) detection for
the user $k$, where $\lambda\geq 0$ is the regularization value [26, 41],
performs as follows:

$\bm{G}_{k}^{\textit{MMSE}}(\lambda)=(\bm{A}_{k}^{\mathrm{H}}\bm{A}_{k}+\lambda\bm{I})^{-1}\bm{A}_{k}^{\mathrm{H}}$

In this paper, priority is given to the MMSE-Interference-Rejection-Combiner
(MMSE-IRC) detection [31]:

$\bm{G}_{k}^{\textit{IRC}}(\lambda)=\bm{A}_{k}^{\mathrm{H}}(\bm{A}_{k}^{\mathrm{H}}\bm{A}_{k}+\bm{R}_{uu}^{k}+\lambda\bm{I})^{-1}.$
(7)

And covariance matrix $\bm{R}_{uu}^{k}$ of total intra-user interference:

$\bm{R}_{uu}^{k}=\bm{H}_{k}(\bm{W}\bm{W}^{\mathrm{H}}-\bm{W}_{k}\bm{W}_{k}^{\mathrm{H}})\bm{H}_{k}^{\mathrm{H}}.$

To conduct analytical calculations, we assume virtual Conjugate Detection (CD)
in the following form [7]:

$\bm{G}^{C}_{k}={\bm{P}_{k}}^{-1}\widetilde{\bm{S}}_{k}^{-1}\widetilde{\bm{U}}_{k}=\bm{P}_{k}^{-1}\widehat{\bm{G}}^{C}_{k}\in\mathbb{C}^{L_{k}\times
R_{k}},$ (8)

where $\bm{P}_{k}$ is a corresponding to $k$-th user sub-matrix of matrix
$\bm{P}$ in equation (3).

###  2.4 Quality Measures

We measure the quality of precoding using well-known functions such as Signal-
to-Interference-and-Noise-Ratio (SINR) [39] and Spectral Efficiency (SE) [38].
These functions are based not on the actual sending symbols
$\bm{x}\in\mathbb{C}^{L\times 1}$, but some distribution of them [3]. Thus, we
get the common function for all assumed symbols, which can be sent using the
specified precoding matrix. We denote $\mathcal{L}_{k}$ as the set of symbols
for $k$-th user. The SINR function is defined as:

$\textit{SINR}_{l}(\bm{W},\bm{H}_{k},\bm{g}_{l},\sigma^{2}):=\dfrac{|\bm{g}_{l}\bm{H}_{k}\bm{w}_{l}|^{2}}{\sum_{i\neq
l}^{L}|\bm{g}_{l}\bm{H}_{k}\bm{w}_{i}|^{2}+\sigma^{2}\|\bm{g}_{l}\|^{2}},\quad\forall
l\in\mathcal{L}_{k}.$ (9)

According to the paper [17], the effective SINR for user $k$ is calculated
through the SINR at each layer of each Resource Block (RB) and as follows.
Functions $\beta=\beta(\textit{MCS})$ and
$\textit{MCS}=\textit{MCS}(\textit{SINR}^{eff}_{\beta})$ are table-defined
(see, e.g., Table 3 for $\beta(MCS)$). This model is called Exponential
effective SNR mapping (EESM) [21]. Assuming only one RB, we can define
$\textit{SINR}^{eff}_{\beta}$ as a self-consistent solution of the following
system:

$\textit{SINR}_{\beta,k}^{eff}(\bm{W},\bm{H}_{k},\bm{G}_{k},\sigma^{2})=-\beta\ln\left(\frac{1}{L_{k}}\sum\limits_{l\in\mathcal{L}_{k}}\exp\Big{\\{}-\frac{\textit{SINR}_{l}(\bm{W},\bm{H}_{k},\bm{g}_{l},\sigma^{2})}{\beta}\Big{\\}}\right)$
(10)

To get the SE function, we apply Shannon’s formula over all effective user
SINRs (20):

$\textit{SE}(\bm{W},\bm{H},\bm{G},\sigma^{2})=\sum_{k=1}^{K}L_{k}\log_{2}(1+\textit{SINR}_{\beta,k}^{eff}(\bm{W},\bm{H}_{k},\bm{G}_{k},\sigma^{2}))\rightarrow\max\limits_{\bm{W}}.$
(11)

###  2.5 Problem Statement

We consider the channel model in the form (1) that particularly means exact
measurements of the channel. To further simplify the problem we suppose
detection policy $\bm{G}=\bm{G}(\bm{H},\bm{W})$ to be a known function,
moreover we assume Conjugate Detection (8) that simplifies the channel model
to (16). Based on this channel model, we calculate SINR of transmitted symbols
by (9) and effective SINR of UE, which can be approximately calculated by (10)
and (20). We denote the total power of the system as, $P$ and assume that the
sent vector must be a unit norm:
$\mathbb{E}[\bm{x}\bm{x}^{\mathrm{H}}]=\bm{I}_{L}$.

The total power constraint and the more realistic per-antenna power
constraints (see [4]) impose the following conditions on the precoding matrix.
Since case $\bm{W}=\bm{W}^{\prime}(\bm{V})\bm{P}$ is considered in this paper,
conditions read:

${\rm(a)}\;\;\|\bm{W}^{\prime}\bm{P}\|^{2}\leq
P,\qquad\text{or}\qquad{\rm(b)}\;\;\|{\bm{w}^{\prime}}_{t}{\bm{p}}\|^{2}\leq
P/T,\quad t=1,\dots,T,$ (12)

where $\bm{P}={\rm diag}(\bm{p})={\rm
diag}\left(\sqrt{p_{1}}\dots\sqrt{p_{L}}\right)={\rm
diag}\left(\frac{\sqrt{\rho_{1}}}{\|{\bm{w}^{\prime}}_{1}\|}\dots\frac{\sqrt{\rho_{L}}}{\|{\bm{w}^{\prime}}_{L}\|}\right)$
is power allocation matrix and $P$ is total power of base station. The goal is
to find a precoding matrix that maximizes SE (11) given the power constraints:

$\textit{SE}(\bm{P})=\textit{SE}(\bm{W}^{\prime}\bm{P},\bm{H},\bm{G}(\bm{H},\bm{W}^{\prime}\bm{P}),\sigma^{2})\rightarrow\max\limits_{\bm{P}},\quad\text{subject
to ~{}\eqref{power_antenna} (a) or (b).}$ (13)

##  3 Simplifications of the Problem

###  3.1 Asymptotic Diagonalization Property of Precoding

######  Definition 2.

Let us denote $\lambda=\frac{\sigma^{2}}{P}\to 0$, and $\bm{P}>0$ is some
diagonal matrix. Define the property of asymptotic diagonalization of
$\widetilde{\bm{V}}$ as $\lambda\to 0$ of precoding matrix as follows:

$\widetilde{\bm{V}}\bm{W}=\left(\begin{array}[]{c}\widetilde{\bm{V}}_{1}\\\
\widetilde{\bm{V}}_{2}\\\ \ldots\\\ \widetilde{\bm{V}}_{K}\\\
\end{array}\right)\cdot\left(\bm{W}_{1},\bm{W}_{2}\ldots\bm{W}_{K}\right)=\bm{P}+\mathcal{O}(\lambda),\
\text{i.e.}\ \widetilde{\bm{V}}\bm{W}\sim\bm{P},\text{ as }\lambda\rightarrow
0$ (14)

Precoding algorithms: ZF (4), RZF (5), and ARZF (6) satisfy the property (14).
This can be easily shown with the following Lemma (it is similar to [7, Lemma
2]).

######  Lemma 3.

Consider square invertible complex matrices $\bm{M}$ and $\bm{N}$ of the same
size and rank. For any $0<\lambda\ll 1$ and $\det\bm{M}\neq 0$ the following
matrix identity is true:
$(\bm{M}+\lambda\bm{N})^{-1}=\bm{M}^{-1}-\lambda\bm{M}^{-1}\bm{N}\bm{M}^{-1}+\mathcal{O}(\lambda^{2})=\bm{M}^{-1}+\mathcal{O}(\lambda)$.

###### Proof.

$\bm{F}(\lambda)=(\bm{M}+\lambda\bm{N})^{-1},\text{ and
}\bm{F}^{\prime}(\lambda)=-(\bm{M}+\lambda\bm{N})^{-1}\bm{N}(\bm{M}+\lambda\bm{N})^{-1}$
$\bm{F}(\lambda)=\bm{F}(0)+\bm{F}^{\prime}(0)\lambda+\mathcal{O}(\lambda^{2}),\text{
where }\bm{F}(0)=\bm{M}^{-1},\text{ and
}\bm{F}^{\prime}(0)=-\bm{M}^{-1}\bm{N}\bm{M}^{-1}$
$(\bm{M}+\lambda\bm{N})^{-1}=\bm{M}^{-1}-\lambda\bm{M}^{-1}\bm{N}\bm{M}^{-1}+\mathcal{O}(\lambda^{2})=\bm{M}^{-1}+\mathcal{O}(\lambda)$

∎

For channel singular values $\widetilde{\bm{V}}$ such that the matrix
$\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}$ has a full rank, using
Lemma 3 for the algorithms ZF (4), RZF (5) and ARZF (6) we obtain:

$\widetilde{\bm{V}}{\bm{W}}_{ZF}=\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}(\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}})^{-1}\bm{P}={\bm{P}}$
$\widetilde{\bm{V}}{\bm{W}^{\prime}}_{RZF}=\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}(\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}+\lambda\bm{I})^{-1}\bm{P}={\bm{P}}+\mathcal{O}\left(\lambda\right)$
$\widetilde{\bm{V}}{\bm{W}^{\prime}}_{ARZF}=\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}(\widetilde{\bm{V}}\widetilde{\bm{V}}^{\mathrm{H}}+\lambda\bm{S})^{-1}\bm{P}={\bm{P}}+\mathcal{O}\left(\lambda\right)$

Thus, precodings ZF (4), RZF (5), and ARZF (6) satisfy property (14).

######  Remark 1.

In this case, matrix $\bm{P}$ of definition (14) coincides with matrix
$\bm{P}$ of Conjugate Detection (8).

###  3.2 The Similarity of Conjugate Detection and MMSE-IRC

In this section, we prove the similarity of MMSE-IRC (7) [31] and Conjugate
Detection (CD) (8) [7]. Detection CD does not depend on precoding and allows
to significantly simplify the considered problem (13). First, we prove some
useful properties about CD (compare with [7, Theorem 1]).

######  Lemma 4.

The detection matrix $\bm{G}$ is $\bm{G}^{C}$ (Conjugate Detection) if and
only if it satisfies the following property:

$\bm{G}=\bm{G}^{C}\Leftrightarrow\bm{G}\bm{H}={\bm{P}}^{-1}\widetilde{\bm{V}}\Leftrightarrow\forall
k:\bm{G}_{k}\bm{H}_{k}=\bm{P}_{k}^{-1}\widetilde{\bm{V}}_{k},$ (15)

where $\bm{P}$ is uniquely defined in (14), and the system model equation (1)
takes the form

$\bm{r}=\widetilde{\bm{V}}\bm{W}\bm{x}+\tilde{\bm{n}},\quad\tilde{\bm{n}}:={\bm{P}}^{-1}\widetilde{\bm{S}}^{-1}\widetilde{\bm{U}}\bm{n}.$
(16)

###### Proof.

Necessity. Using Lemma (1) we can write

$\bm{G}^{C}\bm{H}={\bm{P}}^{-1}\widetilde{\bm{S}}^{-1}\widetilde{\bm{U}}\bm{U}^{\mathrm{H}}\bm{S}\bm{V}={\bm{P}}^{-1}\widetilde{\bm{S}}^{-1}\big{[}\begin{array}[]{c|c}\bm{I}&\bm{O}\end{array}\big{]}\bm{S}\bm{V}={\bm{P}}^{-1}\widetilde{\bm{S}}^{-1}\widetilde{\bm{S}}\widetilde{\bm{V}}={\bm{P}}^{-1}\widetilde{\bm{V}},$

which immediately leads to (16).

Sufficiency. Assume that (15) holds, then
$\widetilde{\bm{V}}=\bm{P}\bm{G}\bm{H}$, since the matrix $\bm{P}>O$. Then,
$\forall\bm{v}\in\widetilde{\bm{V}}$ expansion of vector $\bm{v}$ in basis
$\bm{H}$ is unique. The elements of the matrix $\bm{P}\bm{G}$ are the
coefficients of this expansion. Therefore, a matrix ${\bm{G}}$ with the
property (15) is unique.

The last equivalence in (15) is true due to the block diagonality of the
matrix $\bm{G}$. ∎

######  Theorem 5.

In assumption that $\bm{H}_{k}$ has the full rank and precoding $\bm{W}$ has
property (14), detection $\bm{G}^{\textit{IRC}}(\lambda)$ (7) asymptotically
equals to $\bm{G}^{C}$ (8), in other words
$\bm{G}^{\textit{IRC}}(\lambda)\sim\bm{G}^{C}\text{ as }\lambda\rightarrow 0$.

###### Proof.

We need the following consequence of the (14) property:

$\bm{W}\bm{W}^{\mathrm{H}}\widetilde{\bm{V}}_{k}^{\mathrm{H}}=\left(\sum_{v=1}^{K}\bm{W}_{v}\bm{W}_{v}^{\mathrm{H}}\right)\widetilde{\bm{V}}_{k}^{\mathrm{H}}\sim\bm{W}_{k}\bm{W}_{k}^{\mathrm{H}}\widetilde{\bm{V}}_{k}^{\mathrm{H}}\sim\bm{W}_{k}\bm{P}_{k}$
(17)

Taking into account the form of $\bm{R}_{uu}^{k}$ we can rewrite [9]:

$\bm{G}^{\textit{IRC}}_{k}(\lambda)=\left(\bm{H}_{k}\bm{W}_{k}\right)^{\mathrm{H}}(\bm{H}_{k}\bm{W}_{k}(\bm{H}_{k}\bm{W}_{k})^{\mathrm{H}}+\bm{R}_{uu}^{k}+\lambda\bm{I})^{-1}=\\\
=\left(\bm{H}_{k}\bm{W}_{k}\right)^{\mathrm{H}}(\bm{H}_{k}\bm{W}_{k}(\bm{H}_{k}\bm{W}_{k})^{\mathrm{H}}+\bm{H}_{k}(\bm{W}\bm{W}^{\mathrm{H}}-\bm{W}_{k}\bm{W}_{k}^{\mathrm{H}})\bm{H}_{k}^{\mathrm{H}}+\lambda\bm{I})^{-1}=\\\
=\left(\bm{H}_{k}\bm{W}_{k}\right)^{\mathrm{H}}(\bm{H}_{k}\bm{W}(\bm{H}_{k}\bm{W})^{\mathrm{H}}+\lambda\bm{I})^{-1}.$

Using (15), (17), Lemma 4 in the case $\lambda\rightarrow 0$ we obtain:

$\bm{G}_{k}^{C}=\bm{I}\bm{G}_{k}^{C}\bm{I}=\bm{P}_{k}^{-1}\bm{P}_{k}\bm{G}_{k}^{C}({\bm{H}}_{k}\bm{W}(\bm{H}_{k}\bm{W})^{\mathrm{H}})(\bm{H}_{k}\bm{W}(\bm{H}_{k}\bm{W})^{\mathrm{H}})^{-1}=\\{Eq.~{}\ref{IRC
4}\\}=\\\
=\bm{P}_{k}^{-1}\widetilde{\bm{V}}_{k}\bm{W}\bm{W}^{\mathrm{H}}\bm{H}_{k}^{\mathrm{H}}(\bm{H}_{k}\bm{W}(\bm{H}_{k}\bm{W})^{\mathrm{H}})^{-1}\sim\\{Eq.~{}\ref{IRC
2}\\}\sim\\\
\sim\bm{W}_{k}^{\mathrm{H}}\bm{H}_{k}^{\mathrm{H}}(\bm{H}_{k}\bm{W}(\bm{H}_{k}\bm{W})^{\mathrm{H}})^{-1}=\left(\bm{H}_{k}\bm{W}_{k}\right)^{\mathrm{H}}(\bm{H}_{k}\bm{W}(\bm{H}_{k}\bm{W})^{\mathrm{H}})^{-1}\sim\bm{G}^{\textit{IRC}}_{k}.$

∎

######  Remark 2.

The introduced CD detection is speculative: it hardly can be implemented in
practice. UE measures $\bm{H}_{k}\bm{W}_{k}$ via pilot signals instead of
$\bm{H}_{k}$. Nonetheless, it is very useful for theoretical research.
Moreover, the asymptotic behavior of MMSE and MMSE-IRC detection is similar to
that of CD (Sec. 3.2). Particularly, if precoding $\bm{W}$ is Zero-Forcing (4)
and the noise power is zero ($\sigma^{2}=0$), then
$\bm{G}^{\textit{IRC}}(\lambda)=\bm{G}^{C}$; if, additionally, precoding has
the full rank, then $\bm{G}^{\textit{MMSE}}(\lambda)=\bm{G}^{C}$.

######  Remark 3.

Lemma 4 shows that the assumption that UEs use CD on their side sufficiently
simplifies the initial problem, decreases its dimensions, and allows notation
to be uniform. Namely, we can work with user layers of shapes $L_{k}$ and $L$
instead of considering user antennas space. Note also that for precoding it is
sufficient to only perform Partial SVD of the channel
$\bm{H}_{k}\in\mathbb{C}^{R_{k}\times T}$, keeping just the first $L_{k}$
singular values and vectors for each user $k$:

$\bm{H}_{k}\approx\widetilde{\bm{U}}^{\mathrm{H}}_{k}\widetilde{\bm{S}}_{k}\widetilde{\bm{V}}_{k}.$

Based on this, in what follows we can omit the tilde and write
$\bm{U}_{k},\bm{S}_{k},\bm{V}_{k}$ instead of
$\widetilde{\bm{U}}_{k},\widetilde{\bm{S}}_{k},\widetilde{\bm{V}}_{k}$
correspondingly.

###  3.3 Low Correlated Users

We define an interference-correlation matrix as
$\bm{C}=\bm{V}\bm{V}^{\mathrm{H}}-\bm{I}$. In real networks, the set of UEs is
chosen by Scheduler and the number of layers of each UE is chosen to be fixed
by the Rank Selection algorithm. Both Scheduler and Rank Selection methods
provide $\|\bm{C}\|=\mathcal{O}(\lambda)$, where
$\lambda=\frac{\sigma^{2}}{P}$ is the noise-power ratio. Thus, we assume user
correlation to be low compared to noise power, which means
$\|\bm{C}\|=\mathcal{O}(\lambda)$.

######  Lemma 6.

For precoding $\bm{W}=\bm{W}^{\prime}\bm{P}$ satisfying the property (14) and
inference-correlation matrix $\bm{C}=\bm{V}\bm{V}^{\mathrm{H}}-\bm{I}$
satisfying $\|\bm{C}\|=\mathcal{O}(\lambda)$, is the noise-power ratio, it is
asymptotically true that
$\bm{G}^{C}\bm{H}\bm{W}=(1-\lambda)\bm{I}+\mathcal{O}(\lambda^{2})$.

###### Proof.

$\bm{V}\bm{W}^{\prime}=\bm{V}\bm{V}^{\mathrm{H}}(\bm{V}\bm{V}^{\mathrm{H}}+\lambda\bm{I})^{-1}=\\{Lemma~{}\ref{lemm
inv}\\}=\bm{V}\bm{V}^{\mathrm{H}}((\bm{V}\bm{V}^{\mathrm{H}})^{-1}-\lambda(\bm{V}\bm{V}^{\mathrm{H}})^{-2}+\mathcal{O}(\lambda^{2}))=\\\
=\bm{I}-\lambda(\bm{V}\bm{V}^{\mathrm{H}})^{-1}+\mathcal{O}(\lambda^{2})=\bm{I}-\lambda(\bm{C}+\bm{I})^{-1}+\mathcal{O}(\lambda^{2})=\bm{I}-\lambda(\bm{I}+\mathcal{O}(\|\bm{C}\|))+\mathcal{O}(\lambda^{2})=\\\
=(1-\lambda)\bm{I}+\lambda\mathcal{O}(\|\bm{C}\|)+\mathcal{O}(\lambda^{2})=\\{(\|\bm{C}\|)=\mathcal{O}(\lambda)\\}=(1-\lambda)\bm{I}+\mathcal{O}(\lambda^{2})$
(18)
$\bm{G}^{C}\bm{H}\bm{W}=\\{Lemma~{}\ref{lemma_conj_det}\\}=\bm{P}^{-1}\bm{V}\bm{W}=\bm{P}^{-1}\bm{V}\bm{W}^{\prime}\bm{P}=\\{Eq.~{}\ref{eq:singular_channel}\\}=\\\
=\bm{P}^{-1}(1-\lambda)\bm{I}\bm{P}+\bm{P}^{-1}\mathcal{O}(\lambda^{2})\bm{P}=(1-\lambda)\bm{I}+\mathcal{O}(\lambda^{2})$

∎

Using Lemma 6 we immediately get the following

######  Theorem 7.

For precoding $\bm{W}$ satisfying the property (14) and inference-correlation
matrix $\bm{C}=\bm{V}\bm{V}^{\mathrm{H}}-\bm{I}$ satisfying
$\|\bm{C}\|=\mathcal{O}(\lambda)$, where $\lambda=\frac{\sigma^{2}}{P}$ is the
noise-power ratio, formula for SINR (9) in the case of $\bm{G}^{C}$ (8)
detection will take the asymptotic form:

$\textit{SINR}_{l}(\bm{W},\bm{H}_{k},\bm{g}_{l}^{C},\sigma^{2})\sim\frac{p_{l}s_{l}^{2}}{\sigma^{2}}$
(19)

###### Proof.

$\textit{SINR}_{l}(\bm{W},\bm{H}_{k},\bm{g}_{l}^{C},\sigma^{2}):=\dfrac{|\bm{g}^{C}_{l}\bm{H}_{k}\bm{w}_{l}|^{2}}{\sum_{i=1,\neq
l}^{L}|\bm{g}^{C}_{l}\bm{H}_{k}\bm{w}_{i}|^{2}+\sigma^{2}\|\bm{g}^{C}_{l}\|^{2}}=\\{Lemma~{}\ref{lemma:singular_channel}\\}=\\\
=\dfrac{1-\lambda+\mathcal{O}(\lambda^{2})}{\mathcal{O}(\lambda^{2})+\dfrac{\sigma^{2}}{p_{l}s_{l}^{2}}}=\dfrac{1-\dfrac{\sigma^{2}}{P}+\mathcal{O}\Big{(}\dfrac{\sigma^{4}}{P^{2}}\Big{)}}{\mathcal{O}\Big{(}\dfrac{\sigma^{4}}{P^{2}}\Big{)}+\dfrac{\sigma^{2}}{p_{l}s_{l}^{2}}}\sim\dfrac{p_{l}s_{l}^{2}}{\sigma^{2}}$

∎

###  3.4 Effective SINR Models

In this subsection, we compare two models of Effective SINR from [21, 17, 7].
In theoretical calculations, model (10) is extremely inconvenient. To simplify
the formula of effective SINR (10), we average his $L_{k}$ per-symbol SINRs
(9) by the geometric mean, where $\mathcal{L}_{k}$ denotes the set of symbols
for $k$-th user:

$\textit{SINR}_{k}^{eff}(\bm{W},\bm{H}_{k},\bm{G}_{k},\sigma^{2})=\Big{(}{\prod\nolimits_{l\in\mathcal{L}_{k}}\textit{SINR}_{l}(\bm{W},\bm{H}_{k},\bm{g}_{l},\sigma^{2})}\Big{)}^{\frac{1}{L_{k}}},\quad\forall
l\in\mathcal{L}_{k}.$ (20)

To justify the close relationship of the various SINR averaging (10) and (20),
Fig. 4 shows the dependencies of $\textit{SINR}^{eff}(dB)$ for a user with
four antennas. The $x$ axis is the sum
$\textit{SINR}_{1}(dB)+\textit{SINR}_{2}(dB)+\textit{SINR}_{3}(dB)+\textit{SINR}_{4}(dB)$.
Fig. 4 shows the comparison of effective SINR in the form of the geometric
mean and the form of different MCS-$\beta$ values. Differences between various
effective SINRs can take values greater than five decibels. On the other hand,
points SINR with a large difference in the maximum and minimum values are
unusual in practice. The dependence of the effective SINR deviation on the
difference between the minimum and maximum SINR in Fig. 5 proves this fact.

Figure 4: Approximation of exponential model of $\textit{SINR}^{eff}$ (10)
realized with MCS Tables 1 and 2 using geometrical averaging (20). Figure 5:
Deviation between the minimum and maximum of $\textit{SINR}^{eff}$.

For precoding $\bm{W}$ satisfying the property (14) and from the formula for
SINR (19) with a CD using the geometric mean effective SINR model (20), we can
write the SINR for the $k$-th user as follows:

$\textit{SINR}^{eff}_{k}(\widetilde{\bm{S}}_{k},\bm{P}_{k},\sigma^{2})=\frac{1}{\sigma^{2}}\sqrt[L_{k}]{\prod\limits_{l=1}^{L_{k}}(s_{l}^{2}p_{l})}.$
(21)

The formula (21) reflects the channel quality for the specified user without
considering other users. The value of
$\textit{SINR}^{eff}_{k}(\widetilde{\bm{S}}_{k},\bm{P}_{k},\sigma^{2})$
depends on the singular values
$\widetilde{\bm{S}}_{k}\in\mathbb{R}^{L_{k}\times L_{k}}$ (related to matrices
$\bm{H}_{k}\in\mathbb{C}^{R_{k}\times T}$), the transmitted power $\bm{P}_{k}$
and noise $\sigma^{2}$. This function will be used in theoretical calculations
due to its simplicity.

###  3.5 Sum Spectral Efficiency Simplification

In this section we simplify optimization problem of Spectral Efficiency
maximization in case of Zero-Forcing algorithms with asymptotic
diagonalization property (14), Conjugate detection matrix (8), geometrical
averaging of effective SINR model (21) and low correlated users (see Lemma 6).

For any $x\gg 1$ it is true that: $\log(1+x)=\log x+O(x^{-1})$, and so

$SE(\bm{W},\bm{V},\sigma^{2})=\sum_{k=1}^{K}L_{k}\log_{2}(1+\textit{SINR}_{k}^{eff}(\bm{W},\bm{V}_{k},\bm{S}_{k},\sigma^{2}))=\\\
=\sum_{k=1}^{K}L_{k}\log_{2}(\textit{SINR}_{k}^{eff}(\bm{W},\bm{V}_{k},\bm{S}_{k},\sigma^{2}))+\sum_{k=1}^{K}\mathcal{O}(\textit{SINR}_{k}^{eff(-1)}(\bm{W},\bm{V}_{k},\bm{S}_{k},\sigma^{2}))$
(22)

We simplify the initial optimization problem by maximization of its leading
term:

$\sum_{k=1}^{K}L_{k}\log_{2}(\textit{SINR}_{k}^{eff}(\bm{W},\bm{V}_{k},\bm{S}_{k},\sigma^{2}))=\sum_{k=1}^{K}L_{k}\log_{2}\Big{(}{\prod\nolimits_{l\in\mathcal{L}_{k}}\textit{SINR}_{l}(\bm{W},\bm{H}_{k},\bm{g}_{l},\sigma^{2},P)}\Big{)}^{\frac{1}{L_{k}}}=\\\
=\sum_{k=1}^{K}\log_{2}{\prod_{l\in\mathcal{L}_{k}}\textit{SINR}_{l}(\bm{W},\bm{H}_{k},\bm{g}_{l},\sigma^{2},P)}\rightarrow\max\limits_{\bm{P}}$
(23)

These problems are not equivalent, although their solutions are close to each
other.

If we calculate $W$ by ZF algorithm (that gives zero interference) SINR are as
follows

$\textit{SINR}_{l}(\bm{W},\bm{v}_{l},s_{l},\sigma^{2})=\text{\\{Zero-Forcing
Algorithm\\}}=\frac{s_{l}^{2}}{\sigma^{2}}p_{l}$ (24)

and maximization of the leading term gives

$\sum_{k=1}^{K}\log_{2}\prod_{l\in\mathcal{L}_{k}}\textit{SINR}_{l}(\bm{W},\bm{v}_{l},s_{l},\sigma^{2})=\sum_{k=1}^{K}\log_{2}\prod_{l\in\mathcal{L}_{k}}\frac{s_{l}^{2}}{\sigma^{2}}p_{l}=\sum_{k=1}^{K}\log_{2}\prod_{l\in\mathcal{L}_{k}}\frac{s_{l}^{2}}{\sigma^{2}}\prod_{l\in\mathcal{L}_{k}}p_{l}=\\\
\sum_{j=1}^{K}\log_{2}\prod_{l\in\mathcal{L}_{k}}s_{l}^{2}-\sum_{j=1}^{K}\log_{2}\prod_{l\in\mathcal{L}_{k}}\sigma^{2}+\sum_{j=1}^{K}\log_{2}\prod_{l\in\mathcal{L}_{k}}p_{l}\rightarrow\max\limits_{\bm{P}}$

Finally, we can reduce tasks (13) (a) and (b) to the following problems:

$\sum_{k=1}^{K}\log_{2}\prod_{l\in\mathcal{L}_{k}}p_{l}=\log_{2}\prod_{l=1}^{L}p_{l}\rightarrow\max\limits_{\bm{P}},\quad\text{s.t.
}||\bm{W}||^{2}\leq P.$

##  4 Solutions of the Problem

According to [4, sec. 7] we consider equal transmit power strategy for all $K$
users. Such power allocation gives the maximum for a reasonable lower bound on
the $SE$ (11) under some feasible assumptions. Although this Power allocation
is not optimal, these heuristics provide a good suboptimal solution.

###  4.1 Simplified with Total Power Constraints

######  Theorem 8.

If $\bm{W}$ satisfies to the property (14) and $\bm{G}=\bm{G}^{C}$, assuming
model (10) of effective SINR, the equal PA (all $\|\bm{w}_{l}\|$ is equal,
namely, $p_{l}=P/L$) asymptotically provides maximum to the first optimization
problem:

$U=\sum_{n}SE_{n}\rightarrow\max,\quad||\bm{W}||^{2}\leq P.$

###### Proof.

Using asymptotic $\ln(1+\textit{SINR})=\ln(\textit{SINR})(1+O({\varepsilon}))$
for large SINR, conjugate detection, SINR estimation (24) for ZF algorithm and
considering coordinates $\rho$ we get first optimization problem (13):

$\prod_{l=1}^{L}\rho_{l}\rightarrow\max\limits_{\rho_{1}\dots\rho_{L}},\quad\text{s.t.
}\sum\limits_{l=1}^{L}||{\bm{w}^{\prime}}_{l}||^{2}\frac{\rho_{l}}{\|{\bm{w}^{\prime}}_{l}\|^{2}}=\sum\limits_{l=1}^{L}\rho_{l}\leq
P.$

It is an optimization problem of the maximal volume of the box with predefined
lengths of edges which solution is

$\forall l:\rho_{l}=P/L,\text{ and
}p_{l}=\frac{P/L}{||{\bm{w}^{\prime}}_{l}||^{2}}$ (25)

∎

######  Remark 4.

The original function (11), (20) asymptotically reaches its maximum at the
solution of the simplified problem (25).

###  4.2 EESM Model and Total Power Constraints

By analogy with the formulas (22 and 23) we can calculate Spectral efficiency
using physical MCS-$\beta$ model (10), where the parameter $\beta_{k}$ for
each $k=1\dotsc K$ depends on given $MCS$ and therefore depends on the
precoding matrix, in particular on power allocation variables $p_{l}$ for all
$l=1\dots L$:

$SE(\bm{W},\bm{H},\sigma^{2})=\sum_{k=1}^{K}L_{k}\ln(1+SINR_{k}^{eff})=\\\
=-\sum_{k=1}^{K}L_{k}\ln\left(1-\beta_{k}\log\left(\frac{1}{L_{k}}\sum\limits_{j=1}^{L_{k}}\exp\left(-\frac{SINR_{kj}}{\beta_{k}}\right)\right)\right)$
(26)

The function (26) is discontinuous. Nevertheless, if we fix $\beta_{k}$ for
all $k=1\dots K$, this function becomes smooth from $p_{k}$. For example, we
can take $\beta_{k}$ from point $\bm{P}=\bm{P}_{1}$ as
$p_{l}=\frac{P/L}{\|\bm{w^{\prime}}_{l}\|^{2}}$. Next, write SINR similar to
the formula (19) without interference as
$\textit{SINR}_{kl}=\frac{{p}_{l}}{\|\bm{g}_{l}\|^{2}\sigma^{2}}$
($\bm{g}_{l}$ does not depend on ${p}_{l}$). We can write Lagrangian for the
problem 13 (a):

$\mathcal{L}=-\sum_{k=1}^{K}L_{k}\ln\left(1-\beta_{l}\log\left(\frac{1}{L_{k}}\sum\limits_{j=1}^{L_{k}}\exp\left(-\dfrac{{p}_{j}}{\sigma^{2}\beta_{l}\|\bm{g}_{j}\|^{2}}\right)\right)\right)+\lambda_{i}\left(\sum\limits_{l=1}^{L}(\|{{\bm{w}^{\prime}}}_{l}\|^{2}{p}_{l})-P\right)$

And its partial derivatives concerning ${p}_{l}$.

$\mathcal{L}^{{}^{\prime}}_{{p}_{l}}=-\frac{\frac{1}{\sigma^{2}\|\bm{g}_{l}\|^{2}}x_{l}}{\left(1-\beta_{l}\ln\left(X_{k}\right)\right)X_{k}}+\sum\limits_{t=1}^{T}(\lambda_{t}|{w^{\prime}}_{tl}|^{2}),$
(27)

where
$x_{l}=\exp\left(-\frac{{p}_{l}}{\beta_{l}\sigma^{2}\|\bm{g}_{l}\|^{2}}\right)$
and
$X_{k}=\frac{1}{L_{k}}\sum\limits_{i=1}^{L_{k}}\exp\left(-\frac{{p}_{i}}{\beta_{l}\sigma^{2}\|\bm{g}_{i}\|}\right)$.

We can write Karush-–Kuhn-–Tucker conditions:

$\begin{cases}\mathcal{L}^{{}^{\prime}}_{{p}_{l}}=0,\quad l=1\dots L\\\
\lambda_{i}\left(\sum\limits_{l=1}^{L}(\|{{\bm{w}^{\prime}}}_{l}\|^{2}{p}_{l})-P\right)=0,\\\
\lambda_{i}\geq 0\end{cases}$

And its solution is (see proof in Appendix):

${p}_{l}=-\ln(x_{l})\beta_{k}\sigma^{2}\|\bm{g}_{l}\|^{2}=\sigma^{2}\|\bm{g}_{l}\|^{2}\left(\frac{\frac{P}{\sigma^{2}L}+\frac{1}{L}\sum\limits_{v=1}^{L}\|\bm{g}_{v}\|^{2}\|{{\bm{w}^{\prime}}}_{v}\|^{2}f_{v}}{\frac{1}{L_{k}}\sum\limits_{v\in\mathcal{L}_{k}}\left(\|\bm{g}_{v}\|^{2}\|{{\bm{w}^{\prime}}}_{v}\|^{2}\right)}-f_{l}\right),$
(28)

where:

$f_{l}=\beta_{k}\ln\left(\frac{\|\bm{g}_{l}\|^{2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}}{\frac{1}{L_{k}}\sum\limits_{v\in\mathcal{L}_{k}}\|\bm{g}_{v}\|^{2}\|{{\bm{w}^{\prime}}}_{v}\|^{2}}\right)+1$

###  4.3 Simplified with Per-Antenna Power Constraints

######  Theorem 9.

If $\bm{W}$ satisfies to the property (14) and $G=G_{C}$, assuming model (10)
of effective SINR, we can find a strict asymptotic solution of the second
optimization problem

$U=\sum_{n}SE_{n}\rightarrow\max,\quad||\bm{w}_{t}||^{2}\leq\frac{P}{T},\;t=1\dots
T.$

by solving the system of equations.

###### Proof.

The problem (13) (b) can be reduced to a task

$\sum_{l=1}^{L}\log({p}_{l})\rightarrow\max\limits_{\bm{P}},\quad\text{subject
to
}\sum\limits_{l=1}^{L}(|{{w^{\prime}}}_{tl}|^{2}{p}_{l})\leq\frac{P}{T}\;\forall
t=1\dots T$

To solve it, we can use the Karush-–Kuhn-–Tucker conditions. Lagrangian has
the form

$\mathcal{L}=-\sum_{l=1}^{L}\log({p}_{l})+\sum\limits_{t=1}^{T}\left(\lambda_{t}\left(\sum\limits_{l=1}^{L}(|{{w^{\prime}}}_{tl}|^{2}{p}_{l})-\frac{P}{T}\right)\right).$

If ${p}_{l}$ and $\lambda_{t}$ are the optimum of the optimization problem,
then they satisfy the following conditions

$\begin{cases}{p}_{l}\sum\limits_{t=1}^{T}|{w^{\prime}}_{tl}|^{2}\lambda_{t}=1,\quad
l=1\dots L\\\
\lambda_{t}\left(\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})-\frac{P}{T}\right)=0,\quad
t=1\dots T\\\ \lambda_{t}\geq 0,\quad t=1\dots
T\end{cases}\Leftrightarrow\begin{cases}\bm{A}^{T}\bm{\lambda}=1./{\bm{p}}\\\
\bm{\lambda}.*\left(\bm{A}{\bm{p}}-\bm{1}\frac{P}{T}\right)=0\\\
\bm{\lambda}\geq 0\end{cases}$ (29)

We take $\bm{A}=\\{a_{ij}=|{w^{\prime}}_{ij}|^{2}\\}$.

For geometric reasons, the original optimization problem has a solution;
therefore, there is at least one solution to the system (29).

The resulting system can be solved by brute force on the set of zeroed
lambdas. Let’s say we have non-zero $m$ lambdas. Consider the cases.

  * 1.

$m>L$, in this case the linear system
$\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})=\frac{P}{T},t=1\dots m$
will be inconsistent since the number of equations is greater than the number
of unknowns ($m>L$) and the system itself (29) will not have a solution.

  * 2.

$m=L$, in this case the linear system
$\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})=\frac{P}{T},t=1\dots m$
has exactly one solution, and the system itself (29) has at most one solution.

  * 3.

$1<m<L$. This case reduces to the system of quadratic equations. If
$\bm{A}^{\prime}$ is a matrix consisting of rows of matrix $\bm{A}$
corresponding to nonzero lambdas, then

$\begin{cases}{\bm{A}^{\prime}}^{T}\bm{\lambda}=1./{\bm{p}}\\\
\left({\bm{A}^{\prime}}{\bm{p}}-\bm{1}\frac{P}{T}\right)=0\\\
\end{cases}\Rightarrow\begin{cases}(\bm{A}^{\prime})^{\perp}(1./{\bm{p}})=0\\\
{\bm{A}^{\prime}}{\bm{p}}=\bm{1}\frac{P}{T}\\\ \end{cases}$

Here $\bm{A}^{\prime}\in\mathbb{C}^{m\times L}$ and
$(\bm{A}^{\prime})^{\perp}\in\mathbb{C}^{(L-m)\times L}$ is the orthogonal
complement to $\bm{A}^{\prime}$.

  * 4.

$m=1$, in this case, there are one nonzero lambda. Let $\lambda_{i}\neq 0$
therefore

${p}_{l}=\frac{1}{\lambda_{i}|{w^{\prime}}_{il}|^{2}}=\frac{P}{TL|{w^{\prime}}_{il}|^{2}}$
(30)

∎

###  4.4 EESM Model and Per-Antenna Power Constraints

In this section, we combine two ideas of previous sections. We calculate
Spectral efficiency (26) using exponential model (10) for the fixed $\beta$
value. Using this, we can write Lagrangian for the problem 13 (b) and its
partial derivatives concerning $p_{l}$:

$\mathcal{L}=-\sum_{k=1}^{K}L_{k}\ln\left(1-\beta_{l}\log\left(\frac{1}{L_{k}}\sum\limits_{j=1}^{L_{k}}\exp\left(-\dfrac{{p}_{j}}{\sigma^{2}\beta_{l}\|\bm{g}_{j}\|^{2}}\right)\right)\right)+\sum\limits_{t=1}^{T}\lambda_{t}\left(\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})-\frac{P}{T}\right)$

$\mathcal{L}^{{}^{\prime}}_{{p}_{l}}=-\frac{\frac{1}{\sigma^{2}\|\bm{g}_{l}\|^{2}}x_{l}}{\left(1-\beta_{l}\ln\left(X_{k}\right)\right)X_{k}}+\sum\limits_{t=1}^{T}(\lambda_{t}|{w^{\prime}}_{tl}|^{2})$

The Karush-–Kuhn-–Tucker conditions:

$\begin{cases}\mathcal{L}^{{}^{\prime}}_{{p}_{l}}=0,\quad l=1\dots L\\\
\lambda_{t}\left(\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})-\frac{P}{T}\right)=0,\quad
t=1\dots T\\\ \lambda_{t}\geq 0,\quad t=1\dots T\end{cases}$ (31)

The resulting system can be solved by brute force on the set of zeroed
lambdas. Let’s say we have non-zero $m$ lambdas. Consider the cases.

  * 1.

$m>L$, in this case the linear system
$\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})=\frac{P}{T},t=1\dots m$
will be inconsistent since the number of equations is greater than the number
of unknowns ($m>L$) and the system itself (31) will not have a solution.

  * 2.

$m=L$, in this case the linear system
$\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})=\frac{P}{T},t=1\dots m$
has exactly one solution, and the system itself (31) has at most one solution.

  * 3.

$1<m<L$. If $T^{\prime}$ is the set of indexes of nonzero lambda, then this
case reduces to the system of following equations:

$\begin{cases}\mathcal{L}^{{}^{\prime}}_{{p}_{l}}=0\\\
\sum\limits_{l=1}^{L}(|{w^{\prime}}_{tl}|^{2}{p}_{l})=\frac{P}{T},\quad t\in
T^{\prime}\\\ \lambda_{t}\geq 0,\quad t\in T^{\prime}\end{cases}$

  * 4.

$m=1$, in this case there are one nonzero lambda. Let $\lambda_{i}\neq 0$
therefore

${p}_{l}=-\ln(x_{l})\beta_{k}\sigma^{2}\|\bm{g}_{l}\|^{2}=\sigma^{2}\|\bm{g}_{l}\|^{2}\left(\frac{\frac{P}{\sigma^{2}TL}+\frac{1}{L}\sum\limits_{v=1}^{L}\|\bm{g}_{v}\|^{2}|{{w^{\prime}}}_{iv}|^{2}f_{v}}{\frac{1}{L_{k}}\sum\limits_{v\in\mathcal{L}_{k}}\left(\|\bm{g}_{v}\|^{2}|{{w^{\prime}}}_{iv}|^{2}\right)}-f_{l}\right),$
(32)

where:

$f_{l}=\beta_{k}\ln\left(\frac{\|\bm{g}_{l}\|^{2}|{{w^{\prime}}}_{il}|^{2}}{\frac{1}{L_{k}}\sum\limits_{v\in\mathcal{L}_{k}}\|\bm{g}_{v}\|^{2}|{{w^{\prime}}}_{iv}|^{2}}\right)+1$

The proof is similar to the proof of formula (28) which can be found in the
Appendix.

###  4.5 Heuristic Algorithms, based on KKT-analysis

Input: Channel $\bm{H}$ and its decomposition
$\bm{H}=\bm{U}^{\mathrm{H}}\bm{S}\bm{V}$ by Lemma 1, precoding matrix
$\bm{W}(\bm{V})$, station power $P$, number of base station antennas $T$ and
noise $\sigma^{2}$;

Calculate starting point
$\bm{P}_{1}:(\bm{p}_{1})_{l}=\frac{P}{T\|\bm{w}_{l}\|^{2}}$

Calculate the hyperplane on which the square of the starting point lies. Index
of this hyperplane $i(\bm{P}_{1})=\arg\max(row\\_norm(\bm{W}\bm{P}_{1}))$;

Calculate optimal point on this hyperplane
$\bm{P}_{2}:({\bm{p}}_{2})_{l}=\frac{P}{TL|w_{il}|^{2}}$

if  _$\bm{P}_{2}$ satisfies to Power Constraints_ then

return _$\bm{W}_{opt}=\bm{W}\bm{P}_{2}$_

else

Calculate direction vector $\bm{d}=\bm{P}_{2}^{2}-\bm{P}_{1}^{2}$

Calculate first intersection $\bm{P}_{opt}^{2}$ on a beam
{$\bm{P}_{1}^{2}+\alpha\bm{d}|\alpha>0$} with other hyperlanes.

return _$\bm{W}_{opt}=\bm{W}\bm{P}_{opt}$_

end if

Algorithm 1 Heuristic Intersection Method of Power Allocation using Conjugate
Detection and effective SINR as the geometrical mean

Input: Channel $\bm{H}$ and its decomposition
$\bm{H}=\bm{U}^{\mathrm{H}}\bm{S}\bm{V}$ by Lemma 1, station power $P$, noise
$\sigma^{2}$;

Define smooth precoding function $\bm{W}(\bm{V})$;

Define smooth detection function $\bm{G}(\bm{H},\bm{W})$ using MMSE-IRC (7) or
CD (8);

Define smooth target function $J(\bm{P})$. For example,
$J^{\textit{SE}}(\bm{P})=\textit{SE}(\bm{W}^{\prime}\bm{P},\bm{H},\bm{G},\sigma^{2})$
using (9), (11) and (10);

Calculate starting point
$\bm{P}_{1}:(\bm{p}_{1})_{l}=\frac{P/T}{\|\bm{w}_{l}\|^{2}}$ (30)

Calculate the hyperplane on which the square of the starting point lies. Index
of this hyperplane $i(\bm{P}_{1})=\arg\max(row\\_norm(W\bm{P}_{1}))$;

Calculate the optimal point on this hyperplane
$\bm{P}_{2}=\arg\max(J^{\textit{SE}}(\bm{P}))$ (32)

if  _$\|\bm{P}_{1}\|_{\infty} <0$_ then

return _$\bm{W}_{opt}=\bm{W}\bm{P}_{1}$_

end if

if  _$\bm{P}_{2}$ satisfies to Power Constraints_ then

return _$\bm{W}_{opt}=\bm{W}\bm{P}_{2}$_

else

Calculate direction vector $\bm{d}=\bm{P}_{2}^{2}-\bm{P}_{1}^{2}$

Calculate first intersection $\bm{P}_{opt}^{2}$ on a beam
{$\bm{P}_{1}^{2}+\alpha\bm{d}|\alpha>0$} with other hyper-lines.

return _$\bm{W}_{opt}=\bm{W}\bm{P}_{opt}$_

end if

Algorithm 2 Heuristic Intersection Method of Power Allocation using MMSE IRC
Detection and exponential effective SINR (10) with MCS-$\beta$ Tab. 3 Figure
6: Example of the random generation of users for Urban setup. There are two
buildings, and the users are assigned to either a cluster in a building or to
the ground near the building.

The algorithms take equalizing powers as the first approximation of the vector
$\bm{p}$ (See Point $1$ on Fig. 7). Then it finds the hyperplane on which the
given point lies and searches on this hyperplane for the optimal (Point $2$).
To find the optimal point, we use the formula (30) for the first algorithm and
by formula (32) for the second algorithm.

If the obtained point is satisfied with the Power Constrains, then this is the
result of the algorithm. This point may not be satisfied with the Power
Constraints. In this case, we construct a beam from the starting point to the
optimal point. The first intersection with other hyperplanes (Point $3$) is a
result of the algorithms. The formula for Point $2$ can be negative or zero.
In this rare case, the result of the algorithm is Point $1$.

An example of the power distribution obtained using this method is shown in
Fig. 8. This figure shows the ratio of power gains using Alg. 1 (ZF IM CD)
compared to the equal power allocation method (ZF EQ). The SINR values in dB
of each layer are also given for comparison. It is shown that the SINR values
increase for those layers for which the power increases. And vice versa, the
SINR decreases for those layers for which the power decreases. The total
precoding power and also increases with the use of Alg. 1.

##  5 Simulation Results

We compare different power allocations applied to the different reference
algorithms (ZF, RZF, ARZF). Primarily, the comparison involves precoding with
base powers (native methods without PA) and the power equalization algorithm
(25). Also, we consider some algorithms based on Karush–Kuhn–Tucker conditions
(29).

We describe the generation process in detail in our work [7]. To generate
channel coefficients, we use Quadriga [19], open-source software for
generating realistic radio channel impulse responses. We consider the Urban
Non-Line-of-Sight [12] scenarios. For each seed, we generate the random sets
of user positions (see example in Fig 6 [7]) and compute channel matrices for
the obtained user configurations. The parameters of the experiments are listed
in Table 1.

In Figs. 9, 10 11, we present numerical simulations of the proposed
Intersection Method (1) ”ZF IM CD” and ”ZF IM IRC” with its modifications to
MCS-$\beta$ model (2) ”ZF IM MCS CD”, ”ZF IM MCS IRC” and reference Naive ”ZF”
and Equal Power ”ZF EQ” methods. All pictures show the average spectral
efficiency gain over the Equal Power method. Fig. 9 shows SE gain assuming
Geometric Mean Effective SINR (20), while Figs 10 and 11 assume the
Exponential Averaging model (10). Both ”ZF IM” and ”ZF IM IRC” algorithms
provide better power allocation (PA) under per-antenna power constraint
(PAPC), which means better value of Spectral Efficiency (11) of the obtained
precoding in comparison to the Naive ZF and Equal Power ZF EQ methods.

These experiments claim that the proposed method ”ZF IM CD” outperforms the
reference ”ZF EQ” up to 5% at the low SUSINR region ($<5$ dB) and up to 2% at
high ($>20$ dB). The modification of the algorithm ”ZF IM MCS IRC” provides
better results up to 6% at the low SUSINR region.

Finally, it is experimentally proved out that the modification ”ZF IM MCS CD”
in case of table 2 MCS-$\beta$ values (see Tab. 3) provides better results
than ”ZF IM CD”. This can be explained by the fact that final quality in this
case is also measured using MCS Exponential Average Effective SINR (10), which
is explicitly used in ”ZF IM MCS CD”. For both cases table 1 and table 2
MCS-$\beta$ values, the method ”ZF IM MCS CD” outperforms ”ZF IM CD” since ”ZF
IM MCS CD” takes into account MCS-$\beta$ values inside its formulas.

Figure 7: Geometrical illustration of the PA Intersection Method (Alg. 1) with
a small layer space ($L=3$). Point $1$ is the first approximation for solution
of the algorithm and lies on the green hyperplane. Point $2$ is the optimal
Point on the green hyperplane, which we obtain as a solution to the Lagrange
problem (30). The red hyperplane contains the closest intersection of the beam
from Point $1$ to Point $2$ concerning other hyperplanes. Point $3$ is the
intersection of the beam from Point $1$ to Point $2$ on the red hyperplane. If
Point $3$ lies between Points $1$ and $2$, it is a solution of the algorithms,
otherwise, a solution is Point $2$. Figure 8: Ratio of power gains using Alg.
1 ”ZF IM CD” compared to equal power allocation ”ZF EQ”. Figure 9: Average SE
(11) gains using Geometric Mean Effective SINR (20) and the different Power
Allocation algorithms. Figure 10: Average SE (11) gains using Exponential
Averaging SINR (10) using table 1 MCS-$\beta$ values (see Tab. 3) and the
different Power Allocation algorithms. Figure 11: Average SE (11) gains using
Exponential Averaging SINR (10) using table 2 MCS-$\beta$ values (see Tab. 3)
and the different Power Allocation algorithms.

##  6 Conclusion

In this work, we studied the power allocation (PA) problem of wireless MIMO
systems with multi-antenna users. We presented solutions to the PA problem,
which maximize the network throughput in terms of spectral efficiency (SE).
Since the initial problem is non-convex, it was equivalently reformulated as a
Lagrange problem by simplifying the MMSE-IRC detection and the SE function. We
obtained (i) an equal power exact solution of PA under total power constraint
(TPC) and (ii) an algorithmic solution of PA for per-antenna power constraints
(PAPC) based on a system of Lagrange equations and geometric view of the
problem. Simulation results show the effectiveness of the proposed algorithmic
approach over the reference PA schemes. The proposed PA method makes it
possible to improve the quality of MIMO systems in the future.

## References

  * [1] A.C. Aitken, _IV.—On least squares and linear combination of observations_ , Proceedings of the Royal Society of Edinburgh 55 (1936), pp. 42–48.
  * [2] J.G. Andrews, S. Buzzi, W. Choi, S.V. Hanly, A. Lozano, A.C. Soong, and J.C. Zhang, _What will 5G be?_ , IEEE Journal on selected areas in communications 32 (2014), pp. 1065–1082.
  * [3] E. Björnson, M. Bengtsson, and B. Ottersten, _Optimal multiuser transmit beamforming: A difficult problem with a simple solution structure [lecture notes]_ , IEEE Signal Processing Magazine 31 (2014), pp. 142–148.
  * [4] E. Björnson, J. Hoydis, and L. Sanguinetti, _Massive MIMO networks: Spectral, energy, and hardware efficiency_ , Foundations and Trends in Signal Processing 11 (2017), pp. 154–655.
  * [5] E. Björnson and E. Jorswieck, _Optimal resource allocation in coordinated multi-cell systems_ , Now Publishers Inc, 2013.
  * [6] E. Björnson, E. Jorswieck, and B. Ottersten, _Impact of spatial correlation and precoding design in OSTBC MIMO systems_ , IEEE Transactions on Wireless Communications 9 (2010), pp. 3578–3589.
  * [7] E. Bobrov, B. Chinyaev, V. Kuznetsov, H. Lu, D. Minenkov, S. Troshin, D. Yudakov, and D. Zaev, _Adaptive regularized zero-forcing beamforming in Massive MIMO with multi-antenna users_ , arXiv preprint arXiv:2107.00853 (2021).
  * [8] E. Bobrov, D. Kropotov, and H. Lu, _Massive MIMO adaptive modulation and coding using online deep learning algorithm_ , IEEE Communications Letters (2021).
  * [9] E. Bobrov, D. Kropotov, S. Troshin, and D. Zaev, _L-BFGS precoding optimization algorithm for massive MIMO systems with multi-antenna users_ (2021).
  * [10] E. Bobrov, A. Markov, and D. Vetrov, _Variational autoencoders for studying the manifold of precoding matrices with high spectral efficiency_ , arXiv preprint arXiv:2111.15626 (2021).
  * [11] F. Boccardi and H. Huang, _Optimum power allocation for the MIMO-BC zero-forcing precoder with per-antenna power constraints_ , in _2006 40th Annual Conference on Information Sciences and Systems_. IEEE, 2006, pp. 504–504.
  * [12] F. Bohagen, P. Orten, and G. Oien, _Construction and capacity analysis of high-rank line-of-sight MIMO channels_ , in _IEEE Wireless Communications and Networking Conference, 2005_ , Vol. 1. IEEE, 2005, pp. 432–437.
  * [13] X. Deng and A.M. Haimovich, _Power allocation for cooperative relaying in wireless networks_ , IEEE Communications Letters 9 (2005), pp. 994–996.
  * [14] S. Dhakal, _High rate signal processing schemes for correlated channels in 5G networks._ (2019).
  * [15] N. Fatema, G. Hua, Y. Xiang, D. Peng, and I. Natgunanathan, _Massive MIMO linear precoding: A survey_ , IEEE systems journal 12 (2017), pp. 3920–3931.
  * [16] X. Ge, R. Zi, H. Wang, J. Zhang, and M. Jo, _Multi-user massive MIMO communication systems based on irregular antenna arrays_ , IEEE Transactions on Wireless Communications 15 (2016), pp. 5287–5301.
  * [17] Z. Hanzaz and H.D. Schotten, _Analysis of effective SINR mapping models for MIMO OFDM in LTE system_ (2013), pp. 1509–1515.
  * [18] A. Host-Madsen and J. Zhang, _Capacity bounds and power allocation for wireless relay channels_ , IEEE transactions on Information Theory 51 (2005), pp. 2020–2040.
  * [19] S. Jaeckel, L. Raschkowski, K. Börner, and L. Thiele, _QuaDRiGa: A 3-D multi-cell channel model with time evolution for enabling virtual field trials_ , IEEE Transactions on Antennas and Propagation 62 (2014), pp. 3242–3256.
  * [20] M. Joham, W. Utschick, and J.A. Nossek, _Linear transmit processing in MIMO communications systems_ , IEEE Transactions on signal Processing 53 (2005), pp. 2700–2712.
  * [21] S. Lagen, K. Wanuga, H. Elkotby, S. Goyal, N. Patriciello, and L. Giupponi, _New radio physical layer abstraction for system-level simulations of 5G networks_ , in _ICC 2020-2020 IEEE International Conference on Communications (ICC)_. IEEE, 2020, pp. 1–7.
  * [22] L. Le and E. Hossain, _Multihop cellular networks: Potential gains, research challenges, and a resource allocation framework_ , IEEE Communications Magazine 45 (2007), pp. 66–73.
  * [23] Y. Liang and V.V. Veeravalli, _Gaussian orthogonal relay channels: Optimal resource allocation and capacity_ , IEEE Transactions on Information Theory 51 (2005), pp. 3284–3289.
  * [24] N.H. Mahmood, G. Berardinelli, F.M. Tavares, M. Lauridsen, P. Mogensen, and K. Pajukoski, _An efficient rank adaptation algorithm for cellular MIMO systems with IRC receivers_ , in _2014 IEEE 79th Vehicular Technology Conference (VTC Spring)_. IEEE, 2014, pp. 1–5.
  * [25] T.L. Marzetta, _Noncooperative cellular wireless with unlimited numbers of base station antennas_ , IEEE transactions on wireless communications 9 (2010), pp. 3590–3600.
  * [26] A.H. Mehana and A. Nosratinia, _Diversity of MMSE MIMO receivers_ , IEEE Transactions on information theory 58 (2012), pp. 6788–6805.
  * [27] H.Q. Ngo, E.G. Larsson, and T.L. Marzetta, _Energy and spectral efficiency of very large multiuser MIMO systems_ , IEEE Transactions on Communications 61 (2013), pp. 1436–1449.
  * [28] D.H. Nguyen and H.H. Nguyen, _Power allocation in wireless multiuser multi-relay networks with distributed beamforming_ , IET communications 5 (2011), pp. 2040–2051.
  * [29] T. Parfait, Y. Kuang, and K. Jerry, _Performance analysis and comparison of ZF and MRT based downlink massive MIMO systems_ , in _2014 sixth international conference on ubiquitous and future networks (ICUFN)_. IEEE, 2014, pp. 383–388.
  * [30] K.T. Phan, T. Le-Ngoc, S.A. Vorobyov, and C. Tellambura, _Power allocation in wireless multi-user relay networks_ , IEEE Transactions on Wireless Communications 8 (2009), pp. 2535–2545.
  * [31] B. Ren, Y. Wang, S. Sun, Y. Zhang, X. Dai, and K. Niu, _Low-complexity MMSE-IRC algorithm for uplink massive MIMO systems_ , Electronics Letters 53 (2017), pp. 972–974.
  * [32] L. Sanguinetti, A. Zappone, and M. Debbah, _Deep learning power allocation in massive MIMO_ , in _2018 52nd Asilomar conference on signals, systems, and computers_. IEEE, 2018, pp. 1257–1261.
  * [33] S. Shi, M. Schubert, and H. Boche, _Downlink MMSE transceiver optimization for multiuser MIMO systems: Duality and sum-MSE minimization_ , IEEE Transactions on Signal Processing 55 (2007), pp. 5436–5446.
  * [34] L. Sun and M.R. McKay, _Eigen-based transceivers for the MIMO broadcast channel with semi-orthogonal user selection_ , IEEE Transactions on Signal Processing 58 (2010), pp. 5246–5261.
  * [35] D. Tse and P. Viswanath, _Fundamentals of wireless communication_ , Cambridge university press, 2005.
  * [36] TSG RAN; NR;, _Physical layer procedures for data (release 16) v16.0.0_ , 3GPP TS 38.214 (2019).
  * [37] T. Van Chien, E. Björnson, and E.G. Larsson, _Joint power allocation and load balancing optimization for energy-efficient cell-free massive MIMO networks_ , IEEE Transactions on Wireless Communications 19 (2020), pp. 6798–6812.
  * [38] S. Verdú, _Spectral efficiency in the wideband regime_ , IEEE Transactions on Information Theory 48 (2002), pp. 1319–1343.
  * [39] B. Wang, Y. Chang, and D. Yang, _On the SINR in massive MIMO networks with MMSE receivers_ , IEEE Communications Letters 18 (2014), pp. 1979–1982.
  * [40] Z. Wang and W. Chen, _Regularized zero-forcing for multiantenna broadcast channels with user selection_ , IEEE Wireless Communications Letters 1 (2012), pp. 129–132.
  * [41] D. Wubben, R. Bohnke, V. Kuhn, and K.D. Kammeyer, _Near-maximum-likelihood detection of MIMO systems using MMSE-based lattice-reduction_ , in _2004 IEEE International Conference on Communications (IEEE Cat. No. 04CH37577)_ , Vol. 2. IEEE, 2004, pp. 798–802.
  * [42] T. Yoo and A. Goldsmith, _On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming_ , IEEE Journal on selected areas in communications 24 (2006), pp. 528–541.
  * [43] W. Yu, _Uplink-downlink duality via minimax duality_ , IEEE Transactions on Information Theory 52 (2006), pp. 361–374.
  * [44] A. Zaidi, F. Athley, J. Medbo, U. Gustavsson, G. Durisi, and X. Chen, _5G Physical Layer: principles, models and technology components_ , Academic Press, 2018.
  * [45] J. Zhang, S. Chen, R.G. Maunder, R. Zhang, and L. Hanzo, _Regularized zero-forcing precoding-aided adaptive coding and modulation for large-scale antenna array-based air-to-air communications_ , IEEE Journal on Selected Areas in Communications 36 (2018), pp. 2087–2103.
  * [46] Y. Zhao, R. Adve, and T.J. Lim, _Improving amplify-and-forward relay networks: optimal power allocation versus selection_ , in _2006 ieee international symposium on information theory_. IEEE, 2006, pp. 1234–1238.
  * [47] K. Zheng, L. Zhao, J. Mei, B. Shao, W. Xiang, and L. Hanzo, _Survey of large-scale MIMO systems_ , IEEE Communications Surveys & Tutorials 17 (2015), pp. 1738–1760.

## Statements and Declarations

### Funding

Authors have received research support from Huawei Technologies.

### Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

## Appendix

###  6.1 Search of MCS-$\beta$ Effective SINR

The values of $\beta$ for Modulation and Coding Scheme (MCS) [8] are taken
from Tab. 3. There are different $\beta$ values for different MCSes [21]. The
Table 3 shows $\beta$ values, which corresponds to Tables 5.1.3.1-1 to
5.1.3.1-2 in [36]. The MCS value depends on the radio quality and therefore on
$\textit{SINR}^{eff}_{\beta}$.

Thus, $\textit{SINR}^{eff}_{\beta}$ can be found by simple iteration method on
the equation (10), initializing $\textit{SINR}^{eff}_{\beta}$ by geometrical
average using (20) and then taking $\beta=\beta(\textit{MCS})$ from Tab. 3 and
$\textit{MCS}=\textit{MCS}(\textit{SINR}^{eff}_{\beta})$ from Tab. 3.

Also note that low values of $\textit{SINR}^{eff}_{\beta}$ (up to -5dB)
indicate that the user is almost out of service, and high values of
$\textit{SINR}^{eff}_{\beta}$ (after 23dB) do not make much sense.

MCS |  $\beta$-table 1 |  $\beta$-table 2
---|---|---
0 | 1.6 | 1.6
1 | 1.61 | 1.63
2 | 1.63 | 1.67
3 | 1.65 | 1.73
4 | 1.67 | 1.79
5 | 1.7 | 4.27
6 | 1.73 | 4.71
7 | 1.76 | 5.16
8 | 1.79 | 5.66
9 | 1.82 | 6.16
10 | 3.97 | 6.5
11 | 4.27 | 10.97
12 | 4.71 | 12.92
13 | 5.16 | 14.96
14 | 5.66 | 17.06
15 | 6.16 | 19.33
16 | 6.5 | 21.85
17 | 9.95 | 24.51
18 | 10.97 | 27.14
19 | 12.92 | 29.94
20 | 14.96 | 56.48
21 | 17.06 | 65
22 | 19.33 | 78.58
23 | 21.85 | 92.48
24 | 24.51 | 106.27
25 | 27.14 | 118.74
26 | 29.94 | 126.36
27 | 32.05 | 132.54
Table 2: Optimal $\beta$ values for each MCS.

MCS | SE-table 1 | SE-table 2
---|---|---
0 | 0.2344 | 0.2344
1 | 0.3066 | 0.377
2 | 0.377 | 0.6016
3 | 0.4902 | 0.877
4 | 0.6016 | 1.1758
5 | 0.7402 | 1.4766
6 | 0.877 | 1.6953
7 | 1.0273 | 1.9141
8 | 1.1758 | 2.1602
9 | 1.3262 | 2.4063
10 | 1.3281 | 2.5703
11 | 1.4766 | 2.7305
12 | 1.6953 | 3.0293
13 | 1.9141 | 3.3223
14 | 2.1602 | 3.6094
15 | 2.4063 | 3.9023
16 | 2.5703 | 4.2129
17 | 2.7305 | 4.5234
18 | 3.0293 | 4.8164
19 | 3.3223 | 5.1152
20 | 3.6094 | 5.332
21 | 3.9023 | 5.5547
22 | 4.2129 | 5.8906
23 | 4.5234 | 6.2266
24 | 4.8164 | 6.5703
25 | 5.1152 | 6.9141
26 | 5.332 | 7.1602
27 | 5.5547 | 7.4063
Table 3: Optimal SE values for each MCS.

###  6.2 Derivation of the eq. (28)

From the identity (27) $\mathcal{L}^{{}^{\prime}}_{{p}_{l}}=0$:

$x_{l}=(1-\beta_{k}\ln(X_{k}))X_{k}\beta_{k}\sigma^{2}\|\bm{g}_{l}\|^{2}\lambda_{i}\|{{\bm{w}^{\prime}}}_{l}\|^{2}$
(33)

Taking average of (33):

$X_{k}=\frac{1}{L_{k}}\sum\limits_{l\in\mathcal{L}_{k}}x_{l}\Leftrightarrow
X_{k}=(1-\beta_{k}\ln(X_{k}))X_{k}\frac{1}{L_{k}}\sum\limits_{l\in\mathcal{L}_{k}}\left(\sigma^{2}s_{l}^{-2}\lambda_{i}\|{{\bm{w}^{\prime}}}_{l}\|^{2}\right)$
(34)

Dividing (33) by (34) we get:

$\frac{x_{l}}{X_{k}}=\frac{\sigma^{2}s_{l}^{-2}\lambda_{i}\|{{\bm{w}^{\prime}}}_{l}\|^{2}}{\frac{1}{L_{k}}\sum\limits_{v\in\mathcal{L}_{k}}\left(\sigma^{2}s_{v}^{-2}\lambda_{i}\|{{\bm{w}^{\prime}}}_{v}\|^{2}\right)}=\frac{s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}}{\frac{1}{L_{k}}\sum\limits_{v\in\mathcal{L}_{k}}\left(s_{v}^{-2}\|{{\bm{w}^{\prime}}}_{v}\|^{2}\right)}$
(35)

From (34):

$X_{k}=\exp\left(\frac{1}{\beta_{k}}-\frac{1}{\beta_{k}\frac{1}{L_{k}}\sum\limits_{l\in\mathcal{L}_{k}}\left(\sigma^{2}s_{l}^{-2}\lambda_{i}\|{{\bm{w}^{\prime}}}_{l}\|^{2}\right)}\right)$
(36)

Taking into account
$\sum\limits_{l=1}^{L}(\|{{\bm{w}^{\prime}}}_{l}\|^{2}{p}_{l})=P$ we obtain:

$\sum\limits_{l=1}^{L}(\|{{\bm{w}^{\prime}}}_{l}\|^{2}{p}_{l})=-\sum\limits_{k=1}^{K}\sum\limits_{l\in\mathcal{L}_{k}}\left[\sigma^{2}s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}\left(1-\frac{1}{\frac{1}{L_{k}}\sum\limits_{v=1}^{L_{k}}\left(\sigma^{2}s_{v}^{-2}\lambda_{i}\|{{\bm{w}^{\prime}}}_{v}\|^{2}\right)}\right)\right]-\\\
-\sum\limits_{l=1}^{L}\beta_{k}\sigma^{2}s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}\ln\left(\frac{s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}}{\frac{1}{L_{k}}\sum\limits_{v=1}^{L_{k}}s_{v}^{-2}\|{{\bm{w}^{\prime}}}_{v}\|^{2}}\right)=\\\
=\lambda_{i}^{-1}L-\sum\limits_{l=1}^{L}\sigma^{2}s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}-\sum\limits_{l=1}^{L}\beta_{k}\sigma^{2}s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}\ln\left(\frac{s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}}{\frac{1}{L_{k}}\sum\limits_{v=1}^{L_{k}}s_{v}^{-2}\|{{\bm{w}^{\prime}}}_{v}\|^{2}}\right)=P$

$\lambda_{i}^{-1}=\frac{P}{L}+\frac{1}{L}\sum\limits_{l=1}^{L}\sigma^{2}s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}\left(\beta_{k}\ln\left(\frac{s_{l}^{-2}\|{{\bm{w}^{\prime}}}_{l}\|^{2}}{\frac{1}{L_{k}}\sum\limits_{v\in\mathcal{L}_{k}}s_{v}^{-2}\|{{\bm{w}^{\prime}}}_{v}\|^{2}}\right)+1\right)$
(37)

Substituting (37) into (36) and (36) into (35) we get the required expressions
for $x_{l}$ and then for $p_{l}$.


# Divide-Conquer Transformer Learning for Predicting Electric Vehicle Charging
Events
Using Smart Meter Data ††thanks: This work was supported in part by the
Australian Research Council (ARC) Discovery Early Career Researcher Award
(DECRA) under Grant DE230100046.

Fucai Ke1, Hao Wang2,3,* *Corresponding author: Hao Wang
(hao.wang2@monash.edu). 1Department of Human Centred Computing, Faculty of
Information Technology, Monash University, Australia
2Department of Data Science and AI, Faculty of Information Technology, Monash
University, Australia
3Monash Energy Institute, Monash University, Australia

###### Abstract

Predicting electric vehicle (EV) charging events is crucial for load
scheduling and energy management, promoting seamless transportation
electrification and decarbonization. While prior studies have focused on EV
charging demand prediction, primarily for public charging stations using
historical charging data, home charging prediction is equally essential.
However, existing prediction methods may not be suitable due to the
unavailability of or limited access to home charging data. To address this
research gap, inspired by the concept of non-intrusive load monitoring (NILM),
we develop a home charging prediction method using historical smart meter
data. Different from NILM detecting EV charging that has already occurred, our
method provides predictive information of future EV charging occurrences, thus
enhancing its utility for charging management. Specifically, our method,
leverages a self-attention mechanism-based transformer model, employing a
“divide-conquer” strategy, to process historical meter data to effectively and
learn EV charging representation for charging occurrence prediction. Our
method enables prediction at one-minute interval hour-ahead. Experimental
results demonstrate the effectiveness of our method, achieving consistently
high accuracy of over 96.81% across different prediction time spans. Notably,
our method achieves high prediction performance solely using smart meter data,
making it a practical and suitable solution for grid operators.

###### Index Terms:

Electric vehicle (EV), EV charging forecasting, nonintrusive load monitoring
(NILM), smart meter, transformer

##  I Introduction

As the popularity of electric vehicles (EVs) continues to surge, EV charging
demand is increasing significantly. Uncoordinated EV charging poses
substantial pressure on the power grid, primarily due to the considerable
increase in peak load on distribution networks [1, 2]. The lifespan of network
assets can be significantly impacted by overloading, as a result of increasing
EV charging demand at homes [3, 4, 5]. The growing adoption of EVs presents
new challenges for distribution system operators to plan and operate their
systems. Therefore, it is essential to advance predictive models for EV
charging events or called activities. Such predictive EV charging information
is crucial for ensuring the stability and reliability of the network and holds
substantial value for system operators [6, 7].

In recent years, there has been a large body of literature on various methods
for predicting EV charging demand and enhancing EV charging management [8].
Such prediction studies often relied on a wide range of factors, including
traffic data, ride-hailing trip data, arrival time, and charging rates [9, 10,
11, 12, 13], and utilized various optimization algorithms or machine learning
models to predict different charging variables like charging rate [14],
charging duration [15] and charging power [16]. All these methods provide
useful insights into EV charging demand, primarily at charging stations [17].
However, a substantial number of charging events/activities occur at home
rather than at charging stations, making the above existing studies less
evident to system operators. More importantly, the above studies leveraged
historical EV charging data to make predictions. However, behind-the-meter at-
home charging is highly invisible to the operators and home charging data may
not be available, making home charging event prediction even more challenging.

Several recent non-intrusive methods have gained prominence in various
domains, for example, enhancing grid management through EV charging detection
and disaggregation [18, 19, 20, 21, 22]. These innovative studies have
demonstrated commendable performance in post-charging analysis or real-time
detection, but a critical aspect remains unexplored: predicting future EV
charging events, particularly in the context of at-home charging. In contrast
to post-charging behavioral analysis, predictive information of future EV
charging events plays a vital role in energy management. Therefore, _our work
aims to fill this void by developing an effective algorithm for home EV
charging event prediction using solely historical smart meter data._

This paper introduces a novel method, namely Divide-Conquer Transformer EV
charging event prediction (DCT-EV), for predicting EV charging events several
minutes ahead with high accuracy based on smart meter data. We evaluate our
proposed method using a real-world dataset and compare it with four
representative machine learning and deep learning models for different
prediction requirements (minutes ahead). The main contributions of this paper
are summarized as follow.

  * •

New practical task: Unlike existing studies on predicting EV charging at
charging stations based on historical charging data, our work focuses on a
more challenging and practical task of predicting home charging events using
historical residential meter data. Our work builds upon the NILM concept to
provide predictive information about future EV charging events, deviating from
post-charging analysis.

  * •

Non-intrusive, minute-interval, high-accuracy model: DCT-EV, a self-attention-
based model with a transformer backbone, divides long historical smart meter
time-series into multiple sub-sequences and processes them in parallel. This
approach enables effective handling of meter data to serve EV charging event
prediction.

  * •

Extensive experiments and insights into EV charging event prediction using
real-world datasets: Our method, DCT-EV, outperforms benchmark models,
establishing a new state-of-the-art in providing both 10-minute and 60-minute
ahead EV charging event prediction. Extensive experiments provide valuable
insights into 1) the impact of input history length on prediction performance,
2) the trade-off between model performance and the size of predictive time
span, and 3) probability threshold selection.

The remainder of the paper is organized as follows. Section II presents the
design of our DCT-EV model. Section III presents the experimental results,
including comparisons between DCT-EV and benchmark models on a real-world
dataset and the discussion of the evaluation results. Section IV concludes the
paper and outlines some future work directions.

##  II Methodology

The high-accuracy prediction of at-home EV charging events holds substantial
value for energy management and grid operations. In this section, we present
the structure of our proposed DCT-EV and its working principles in predicting
future charging events using smart meter data, more specifically household
load data. The model consists of three main components: the load embedding
module, the EV representation learning module, and charging event prediction
module, as illustrated in Figure 1.

Figure 1: DCT-EV structure and application.

In the electricity load embedding module, we first divide the lengthy smart
meter data sequence into a few overlapping short sequences and then project
the human-understandable and observable signals into high-dimension embedding
vectors using convolutional layers. This approach can substantially reduce the
computational resource and mitigate the inherent inductive bias for modeling
load patterns. In the EV representation learning module, we boost the model’s
capacity by incorporating a self-attention-based transformer encoder. The
encoder generates representations by taking into account the information from
all sub-sequences. The charging event prediction module then predicts the
near-future EV charging event and forecasts the probability of active EV
charging over upcoming time steps.

###  II-A Load Embedding Module

The objective of the load embedding module is to partition the long historical
load sequence into several overlapping short sequences, drawing inspiration
from [23, 24, 25]. This module can reduce computational resource consumption
and enable the subsequent module to process the entire sequence in parallel.

The load embedding module takes each time series of electricity load from
smart meter data as an input sequence, denoted as
$\bm{x}={x_{1},...,x_{t},...,x_{T}}\in\mathbb{R}^{1\times T}$, where $T$ is
the length of the input sequence. The long input sequence $\bm{x}$ is divided
into $N=\frac{T-L}{stride}+1$ number of overlapped sub-sequences, denoted as
$\bar{\bm{x}}=\\{\bm{x}^{1},...,\bm{x}^{n},...,\bm{x}^{N}\\}\in\mathbb{R}^{N\times
L}$, based on sub-sequence length $L$ and a stride size of $stride$, where
$\bm{x}^{n}=\\{x^{n}_{1},...,x^{n}_{l},...x^{n}_{L}\\}\in\mathbb{R}^{1\times
L}$.

Each sub-sequence $\bm{x}^{n}$ is then projected to a high-dimensional latent
space
$\bm{\chi}^{n}=\\{\chi^{n}_{1},...,\chi^{n}_{d},...,\chi^{n}_{D}\\}\in\mathbb{R}^{1\times
D}$ using a convolutional layer as follows

$\begin{split}\bm{\chi}^{n}=\bm{x}^{n}\mathbf{\kappa},\end{split}$ (1)

where $D$ represents the dimension of the latent space, and
$\mathbf{\kappa}\in\mathbb{R}^{L\times D}$ is the convolution layer kernal
matrix.

The utilization of sub-sequence partitioning reduces information redundancy by
treating each sub-sequence as a token instead of a single electricity load,
thereby reducing the count of input tokens from $T$ to $N$. The subsequent
component, the EV representation learning module and charging event prediction
module, constitute a self-attention backbone, demanding substantial memory
resources, depending upon the number of the input tokens. When calculating the
attention scores, the computational resources required for these two different
sequence sizes exhibit a notable difference, with approximately
$T^{2}>>N^{2}$, showing the advantage of our sub-sequence partitioning.

###  II-B EV Representation Learning Module

The backbone of EV representation learning module is the self-attention
mechanism, which is also a key element of the Transformer architecture [26].
We harness the attention mechanism to facilitate representation learning.
Subsequently, we employ a lightweight linear layer to interpret the essential
information and make predictions.

The $N$ embeddings of in-parallel sub-sequence are distinguished by their
temporal order of interactions. To capture the intrinsic temporal dependencies
within time series data, we implement positional encoding, a technique
employing fixed sinusoids of varying frequencies [26], to model the relative
ordering of time series data as

$\begin{split}\bm{\chi}_{\text{pos}}:=\bm{\chi}+\bm{b}^{\text{pos}}.\end{split}$
(2)

Multi-head attention layer serves as a representation learner for EV charging
behavior by computing the attention scores for all contextualized embeddings
of each sub-sequence. Each sub-sequence is mapped into the queries, keys, and
values in each head $h$. More specifically, the mapping can be expressed as

$\begin{split}&\bm{Q}_{h}=\bm{\chi}_{\text{pos}}\bm{W}_{h,Q},\leavevmode\nobreak\
\bm{K}_{h}=\bm{\chi}_{\text{pos}}\bm{W}_{h,K},\leavevmode\nobreak\
\bm{V}_{h}=\bm{\chi}_{\text{pos}}\bm{W}_{h,V},\end{split}$ (3)

where $\bm{W}_{Q},\bm{W}_{K},\bm{W}_{V}\in\mathbb{R}^{D\times D_{m}}$ are
learnable mapping matrices, with $D_{m}$ being the dimension after mapping,
and $\bm{K}^{{}^{\prime}}$ is the transpose of $\bm{K}$. Then, the scaled dot-
product attention scores $\boldsymbol{\alpha}_{h}\in\mathbb{R}^{N\times N}$
and the hidden representation of load record
$\boldsymbol{\psi}_{h}\in\mathbb{R}^{N\times D_{m}}$ are computed as

$\begin{split}&\boldsymbol{\alpha}_{h}=\text{Softmax}\bigg{(}\frac{\bm{Q}_{h}\bm{K}^{{}^{\prime}}_{h}}{\sqrt{d_{\bm{K}^{{}^{\prime}}_{h}}}}\bigg{)};\boldsymbol{\psi}_{h}=\boldsymbol{\alpha}_{h}\times\bm{V}_{h},\end{split}$
(4)

where $d_{\bm{K}^{{}^{\prime}}_{h}}=D_{m}$ is the dimension of the key
embedding matrix in head $h$. Thus the final attention is
$\boldsymbol{\psi}\in\mathbb{R}^{N\times D}$ after summarizing all
$\boldsymbol{\psi}_{h}$.

Feed-forward neural network (FFN) processes $\boldsymbol{\psi}$, which
represents the output after layer normalization across all heads, and passes
it through a two-layer multi-layer perceptron with ReLU activation to obtain
the final output of the interaction encoder as

$\begin{split}&\bm{Z}:=\text{ReLu}(\boldsymbol{\psi}\bm{\Phi}_{1}+\bm{b}_{1})\bm{\Phi}_{2}+\bm{b}_{2},\\\
\end{split}$ (5)

where $\bm{\Phi}_{1}\in\mathbb{R}^{D\times
D^{\prime}_{m}},\bm{\Phi}_{2}\in\mathbb{R}^{D^{\prime}_{m}\times D}$ and
$\bm{b}_{1}\in\mathbb{R}^{D^{\prime}_{m}},\bm{b}_{2}\in\mathbb{R}^{D}$ are
respectively the learnable weight matrices and bias vectors of the FFN, while
$\text{ReLu}(\cdot)$ denotes the ReLU activation function.

As a result, the representation of each sub-sequence is denoted by
$\bm{Z}\in\mathbb{R}^{N\times D}$, with each vector in the high-dimensional
representation encapsulating the information of a sub-sequence.

###  II-C Charging Event Prediction Module

Building upon the well-learned representation $\bm{Z}$, which is generated by
the previous two modules processing the input load sequence, the charging
event prediction module employs a trainable, lightweight fully connected
network to obtain the final predicted probability of EV charging state in the
near future.

The flattening technique is applied to reshape $\bm{Z}$, resulting in
$\bm{Z}\in\mathbb{R}^{(N*D)\times 1}$. For each future time step ($m$-th), the
probability of EV charging event being “active” is computed by

$\hat{y}_{m}=\sigma(W^{\prime}_{m}\bm{Z}),$ (6)

where $W^{\prime}_{m}\in\mathbb{R}^{(N*D)\times 1}$ is the weight matrix for
$m$-th minute prediction, and $\sigma(\cdot)$ is the sigmoid activation
function.

###  II-D Loss Function

In the training of DCT-EV, the objective is to minimize the binary cross-
entropy loss between the ground-truth labels and the corresponding prediction
as follows

$\mathcal{L}=-\sum_{i}{\sum_{m}(y_{i,m}\log(\hat{y}_{i,m}))+((1-y_{i,m})\log(1-\hat{y}_{i,m}))},$
(7)

where $y_{i,m}$ is the EV charging event state in the next $m$-th minute for
user $i$, and $\hat{y}_{i,m}$ is the corresponding prediction.

##  III Numerical Results and Analysis

In this section, we present the experimental settings, evaluation metrics, and
corresponding results and discussions. The simulation experiment employs DCT-
EV to predict future EV charging event in minute-interval based on historical
meter data. As we will show in this section, DCT-EV can accurately predict
near-future EV charging event.

###  III-A Experimental Settings and Evaluation

In our experiment, we preprocess minute-interval residential electricity meter
data with EV charging load records in 2018 from Pecan Street [27]. The dataset
provides smart meter data aligned with EV charging profiles, which will be
used to construct EV charging label for the experiment. After analyzing EV
charging profiles, we set the EV charging event label as “charging” when the
EV load is higher than 3 kW, and “not charging” otherwise. Note that our
supervised learning method replies on the EV charging label to learn an
effective prediction model.

To set up our experiment, we select homes with EVs and filter out homes
without a ‘charging’ label in their records, resulting in 22 households with
$8,063,175$ electricity usage records for the experiment. For the selected
households, we extract $402,296$ EV charging records in total. In the real-
world scenario, meter data is accessible to operators, but the EV charging
records are not. Therefore, we use electricity usage as the model input, and
EV charging records are merely used for validation. We divide the data into
training and testing sets with an 80:20 ratio. The first 80% of each home’s
records are for model training, the remaining 20% are reserved for testing.

The dataset exhibits an imbalance, primarily because EV charging events occur
for only a few hours per week, resulting in limited EV charging records. As a
classification problem, i.e., predicting future EV charging events, we utilize
a set of performance metrics, such as recall, precision, accuracy (ACC), area
under the ROC curve (AUC), average precision (AP), and F1 score (F1), along
with mean squared error (MSE), to assess the model’s overall performance. All
these metrics are classical indicators for assessing model performance in
machine learning classification problems [28].

In our experiment, the model outputs a probability value representing the
likelihood of an active charging event. To compute true positives, true
negatives, false positives, and false negatives– necessary for subsequent F1
score, ACC<EMAIL_ADDRESS>means threshold is 0.5. calculations, we adhere to a
standard evaluation setup with a threshold set at 0.5. Accordingly, all
predictions with a probability exceeding 0.5 are considered indicative of an
“active” event, while those with a probability less than or equal to 0.5 are
interpreted as “inactive” events.

###  III-B Results and Discussions

Figure 2: EV charging event prediction for home $\\#$4526. TABLE I:
Performance comparisons between our DCT-EV method and baselines, including
Random Forest, XGBoost, DNN, and LSTM. (Red: maximum and Blue: minimum.)

Predication Span | Model Name |<EMAIL_ADDRESS>| AUC | AP |<EMAIL_ADDRESS>| MSE
---|---|---|---|---|---|---
1-10 minutes | Random Forest | 22.14% | 56.50% | 10.76% | 98.55% | 0.0144
XGBoost | 83.43% | 88.34% | 71.16% | 98.63% | 0.0137
DNN | 83.32% | 97.55% | 85.89% | 98.45% | 0.0122
LSTM | 83.91% | 97.73% | 86.89% | 98.74% | 0.0103
DCT-EV | 87.35% | 97.90% | 88.55% | 98.80% | 0.0102
1-60 minutes | DNN | 49.74% | 88.24% | 55.20% | 96.39% | 0.0281
LSTM | 55.94% | 88.96% | 60.06% | 96.68% | 0.0260
DCT-EV | 63.78% | 90.07% | 61.59% | 96.81% | 0.0251

DCT-EV exhibits outstanding performance, as illustrated in Figure 2, achieving
high scores in AUC, AP, F1, ACC, surpassing all compared methods, and
establishing a new state-of-the-art benchmark. These results highlight the
robustness of the proposed method in predicting EV charging events based on
historical smart meter records.

Figure 3: The impact of input history length on the model performance for
predicting EV charging event in the next 60 minutes.

####  III-B1 Model Comparisons

To validate the effectiveness of our method, we conduct comprehensive
comparative studies with representative baseline models, including Random
Forest [29], XGBoost [30], Deep Neural Networks (DNN) [31], and Long Short-
Term Memory (LSTM) [32]. As depicted in Table I, it is evident that DCT-EV
outperforms the other models, exhibiting higher F1 scores, AUC, AP, and ACC,
as well as lower MSE.

####  III-B2 Impact of Input Length on Model Performance

The input length can significantly impact model performance, as it influences
the amount of information available to the model. As shown in Figure 3, the F1
score increases as the length of the size grows, but it plateaus when the size
exceeds 180 minutes. While more information contributes to better prediction
performance, it also increases computational resources needed. Hence, in this
paper, we choose a length size of 180 minutes to tradeoff the performance and
computational burden for our subsequent experiments.

TABLE II: The impact of the near-to-distant future prediction on model
performance. Threshold @.5. (Red: maximum and Blue: minimum.)

m-th | AUC | AP | F1 | Precision | Recall | ACC | MSE
---|---|---|---|---|---|---|---
1-min | 99.63% | 94.87% | 90.88% | 91.36% | 90.40% | 99.15% | 0.0068
2-min | 98.96% | 92.67% | 89.52% | 90.39% | 88.67% | 99.03% | 0.0079
3-min | 98.50% | 90.97% | 88.70% | 89.59% | 87.83% | 98.95% | 0.0086
4-min | 97.93% | 89.20% | 87.73% | 88.93% | 86.56% | 98.86% | 0.0094
5-min | 97.30% | 87.28% | 86.70% | 88.06% | 85.38% | 98.77% | 0.0103
6-min | 96.85% | 85.55% | 85.81% | 87.08% | 84.58% | 98.68% | 0.0110
7-min | 96.41% | 83.90% | 84.79% | 86.11% | 83.51% | 98.59% | 0.0118
8-min | 95.95% | 82.41% | 83.84% | 85.25% | 82.48% | 98.51% | 0.0125
9-min | 95.46% | 80.71% | 82.75% | 84.24% | 81.31% | 98.41% | 0.0133
10-min | 94.78% | 78.75% | 81.59% | 83.31% | 79.94% | 98.31% | 0.0142

####  III-B3 Trade-off Between Model Performance and the Size of Predictive
Time Span

Our experiments evaluate model performance over two different prediction time
spans: next 10 minutes and next 60 minutes. Obviously, all models exhibit
better performance in the next 1-10 minutes than 1-60 minutes as illustrated
in table II. Due to the relatively lower performance of the two machine
learning models–XGBoost and random forest in the 1-10 minutes task, we only
conduct experiments for the 1-60 minutes ahead prediction for DNN, LSTM, and
DCT-EV. All models’ performance drop in the 1-60 minutes task; therefore,
predicting minute-interval events far into the future is indeed a more
challenging task.

Furthermore, we conduct a series of experiments focused solely on predicting
the state in the next m-th minute ($1\leq m\leq 10$), as shown in Table II.
The results reveal the same phenomena as discussed above.

Figure 4: Impact of probability threshold on the performance (F1 score) of
DCT-EV, DNN, and LSTM for prediction tasks with 10/60 minutes ahead.

####  III-B4 Probability Threshold Selection

As discussed before, the model’s output is the probability of EV charging
event. The selection of the probability threshold is crucial to determine
whether the prediction is “active” or “inactive”. We calculate the models’
performance (e.g., F1 score) for two different prediction time spans. As
depicted in Figure 4, the optimal threshold for DCT-EV falls between 0.6 and
0.8, whereas for DNN and LSTM, it lies in the range of 0.2 to 0.4.
Furthermore, in comparison to DNN and LSTM, DCT-EV exhibits greater robustness
to threshold selection effects.

##  IV Conclusion

Accurate prediction of EV charging events is paramount for EV energy
management and grid operations. Existing EV charging prediction methods,
tailored to charging stations, overlooked the need for at-home EV charging
predictions, especially when historical EV charging profile is not available.
To address this problem, we introduced DCT-EV, a deep learning model
leveraging parallel processing and self-attention mechanisms, to predict
minute-by-minute future EV charging events with remarkable accuracy, exceeding
96%. Importantly, DCT-EV relies solely on historical smart meter data instead
of EV charging profile, making it a valuable tool for grid management in the
era of high EV adoption.

## References

  * [1] Zhuowei Luo, Yonghua Song, Zechun Hu, Zhiwei Xu, Xia Yang, and Kaiqiao Zhan.  Forecasting charging load of plug-in electric vehicles in china.  In 2011 IEEE Power and Energy Society General Meeting, pages 1–8. IEEE, 2011.
  * [2] Md Shariful Islam, N Mithulananthan, and Krischonme Bhumkittipich.  Feasibility of pv and battery energy storage based ev charging in different charging stations.  In 2016 13th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), pages 1–6. IEEE, 2016.
  * [3] Matteo Muratori.  Impact of uncoordinated plug-in electric vehicle charging on residential power demand.  Nature Energy, 3(3):193–201, 2018.
  * [4] Md Murshadul Hoque, Mohsen Khorasany, Reza Razzaghi, Mahdi Jalili, and Hao Wang.  Network-aware coordination of aggregated electric vehicles considering charge-discharge flexibility.  IEEE Transactions on Smart Grid, 2022.
  * [5] Jiayan Liu, Jan Peper, Gang Lin, Yang Zhou, Shrutarv Awasthi, Yong Li, and Christian Rehtanz.  A planning strategy considering multiple factors for electric vehicle charging stations along german motorways.  International Journal of Electrical Power & Energy Systems, 124:106379, 2021.
  * [6] Oliver Frendo, Nadine Gaertner, and Heiner Stuckenschmidt.  Improving smart charging prioritization by predicting electric vehicle departure time.  IEEE Transactions on Intelligent Transportation Systems, 22(10):6646–6653, 2020.
  * [7] Dingsong Cui, Zhenpo Wang, Peng Liu, Shuo Wang, Yiwen Zhao, and Weipeng Zhan.  Stacking regression technology with event profile for electric vehicle fast charging behavior prediction.  Applied Energy, 336:120798, 2023.
  * [8] Kejun Qian, Chengke Zhou, Malcolm Allan, and Yue Yuan.  Load model for prediction of electric vehicle charging demand.  In 2010 International Conference on Power System Technology, pages 1–6. IEEE, 2010.
  * [9] Qiang Xing, Zhong Chen, Ziqi Zhang, Xueliang Huang, Zhaoying Leng, Kehui Sun, Yanxi Chen, and Haiwei Wang.  Charging demand forecasting model for electric vehicles based on online ride-hailing trip data.  IEEE Access, 7:137390–137409, 2019.
  * [10] Mariz B Arias and Sungwoo Bae.  Electric vehicle charging demand forecasting model based on big data technologies.  Applied energy, 183:327–339, 2016.
  * [11] Gan Li and Xiao-Ping Zhang.  Modeling of plug-in hybrid electric vehicle charging demand in probabilistic power flow calculations.  IEEE Transactions on Smart Grid, 3(1):492–499, 2012.
  * [12] Ran Wang, Ping Wang, and Gaoxi Xiao.  Two-stage mechanism for massive electric vehicle charging involving renewable energy.  IEEE Transactions on Vehicular Technology, 65(6):4159–4171, 2016\.
  * [13] Lei Yu, Tianyang Zhao, Qifang Chen, and Jianhua Zhang.  Centralized bi-level spatial-temporal coordination charging strategy for area electric vehicles.  CSEE Journal of Power and Energy Systems, 1(4):74–83, 2015.
  * [14] Yongquan Nie, CY Chung, and NZ Xu.  System state estimation considering ev penetration with unknown behavior using quasi-newton method.  IEEE Transactions on Power Systems, 31(6):4605–4615, 2016.
  * [15] Olivier Beaude, Samson Lasaulce, Martin Hennebel, and Ibrahim Mohand-Kaci.  Reducing the impact of ev charging operations on the distribution network.  IEEE Transactions on Smart Grid, 7(6):2666–2679, 2016.
  * [16] Yuchen Tang, Jin Zhong, and Math Bollen.  Aggregated optimal charging and vehicle-to-grid control for electric vehicles under large electric vehicle population.  IET Generation, Transmission & Distribution, 10(8):2012–2018, 2016\.
  * [17] Pengcheng You, Zaiyue Yang, Mo-Yuen Chow, and Youxian Sun.  Optimal cooperative charging strategy for a smart charging station of electric vehicles.  IEEE Transactions on Power Systems, 31(4):2946–2956, 2015.
  * [18] Xudong Wang, Guoming Tang, Yi Wang, Srinivasan Keshav, and Yu Zhang.  Evsense: A robust and scalable approach to non-intrusive ev charging detection.  In Proceedings of the Thirteenth ACM International Conference on Future Energy Systems, pages 307–319, 2022.
  * [19] Shengyi Wang, Liang Du, Jin Ye, and Dongbo Zhao.  A deep generative model for non-intrusive identification of ev charging profiles.  IEEE Transactions on Smart Grid, 11(6):4916–4927, 2020.
  * [20] Zhilin Zhang, Jae Hyun Son, Ying Li, Mark Trayer, Zhouyue Pi, Dong Yoon Hwang, and Joong Ki Moon.  Training-free non-intrusive load monitoring of electric vehicle charging with low sampling rate.  IECON Proceedings (Industrial Electronics Conference), 1:5419–5425, 2014.
  * [21] Run Zhou, Yue Xiang, Yang Wang, Yuan Huang, and Shiwei Xia.  Non-intrusive extraction and forecasting of residential electric vehicle charging load.  In 2020 IEEE Sustainable Power and Energy Conference (iSPEC), pages 2141–2146. IEEE, 2020.
  * [22] Cameron Martin, Fucai Ke, and Hao Wang.  Non-intrusive load monitoring for feeder-level ev charging detection: Sliding window-based approaches to offline and online detection.  In 2023 IEEE 7th Conference on Energy Internet and Energy System Integration (EI2). IEEE, 2023.
  * [23] Haoyu He, Jianfei Cai, Zizheng Pan, Jing Liu, Jing Zhang, Dacheng Tao, and Bohan Zhuang.  Dynamic focus-aware positional queries for semantic segmentation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11299–11308, 2023.
  * [24] Haoyu He, Jing Liu, Zizheng Pan, Jianfei Cai, Jing Zhang, Dacheng Tao, and Bohan Zhuang.  Pruning self-attentions into convolutional layers in single path.  arXiv preprint arXiv:2111.11802, 2021.
  * [25] Yuqi Nie, Nam H Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam.  A time series is worth 64 words: Long-term forecasting with transformers.  arXiv preprint arXiv:2211.14730, 2022.
  * [26] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  Advances in neural information processing systems, 30, 2017.
  * [27] The complete data available at.  https://www.pecanstreet.org/dataport/.  Pecan Street: Austin, TX, USA, Accessed in Sep. 2019.
  * [28] Mohammad Hossin and Md Nasir Sulaiman.  A review on evaluation metrics for data classification evaluations.  International journal of data mining & knowledge management process, 5(2):1, 2015.
  * [29] Leo Breiman.  Random forests.  Machine learning, 45:5–32, 2001.
  * [30] Tianqi Chen and Carlos Guestrin.  Xgboost: A scalable tree boosting system.  In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pages 785–794, 2016.
  * [31] Fionn Murtagh.  Multilayer perceptrons for classification and regression.  Neurocomputing, 2(5-6):183–197, 1991.
  * [32] Sepp Hochreiter and Jürgen Schmidhuber.  Long short-term memory.  Neural computation, 9(8):1735–1780, 1997.