# Steady state correlations and induced trapping of an inertial AOUP particle

N Arsha  K P Jepsin  M Sahoo<EMAIL_ADDRESS>Department of Physics,
University of Kerala, Kariavattom, Thiruvananthapuram-$695581$, India

###### Abstract

We study the dynamics of an inertial active Ornstein-Uhnlenbeck particle self-
propelling in a confined harmonic well. The transport behaviour of the
particle is investigated by analyzing the particle trajectories, steady state
correlations, and mean square displacement (MSD). The steady state correlation
functions for the position as well as velocity are exactly calculated using
different methods. We explore how the inertia affects the dynamical behaviour,
when the particle is confined in a harmonic trap as well as when it is set
free. From the exact calculation of MSD, it is observed that the initial time
regimes are ballistic for both harmonically confined particle and free
particle, whereas the long time regimes are diffusive for a free particle and
non diffusive for a harmonically confined particle. One of our interesting
observation is that the harmonically confined particle gets more and more
confined with increase in the self propulsion time or activity time of the
dynamics and finally it gets trapped for very large self propulsion time. For
a free particle, the velocity correlation decays by the complex interplay
between the inertial time scale and the self propulsion time scale of the
dynamics. Moreover, decorrelation in velocity happens only when these two time
scales are of equal order.

##  I Introduction

In recent years, there is an immense interest in the study of systems which
are driven far away from equilibrium. Active matter is one among these
systems, which is inherently driven from equilibrium. The term ’active’ refers
to the development of directed motion generated by the constituents of the
system by consuming energy from the local environment[1, 2]. Examples of such
active systems extend from motile cellular microorganisms like bacteria or
unicellular protozoa[3], artificially synthesized microswimmers like Janus
particle [4], micro robots [5], hexbugs [6], the flocking of birds[7], to
school of fishes[8]. One of the simplest and most extensively used standard
model, namely active Brownian particle (ABP) model has been developed to treat
the dynamical behaviour of particles in such active matter systems in both
single particle level as well as collective level [9, 10]. This term was first
introduced in Ref. [11], where both the rotational as well as translational
motion of the particles are taken into account. However, recently another
model namely Active Ornstein-Uhlenbeck particle (AOUP) model [12, 13, 14] is
also used for studying the dynamical behaviour and steady state correlations
of particles in such systems. The degree of irreversibility in the dynamical
behaviour of an AOUP particle is also discussed in Ref. [15]. This model was
initially proposed by Uhlenbeck and Ornstein to study the velocity
distribution of passive particles[16]. The advantage of AOUP model is that it
helps in the exact computation of analytical results [17, 18, 19, 20, 21, 22].
Recently, the use of AOUP model has resulted many of the important features of
active matters such as accumulation near boundary [23], motility induced phase
separation (MIPS) [24, 25, 26, 27] etc. However, in most of the studies,
inertia is not taken into account. Systems which are macroscopic in nature,
while self-propelling in a dilute/gaseous or lower viscous medium, the inertia
becomes prominent and plays an important role in the dynamics. The inertial
influence in such systems challenge the theoretical modelling of the dynamical
behaviour. Macroscopic swimmers [28, 29, 30, 31], flying insects [32] etc. are
some examples of this category, where inertia plays an important role. Hence
the introduction of inertia to the standard models is necessary for treating
the dynamics of such systems. Recently, inclusion of inertia in the ABP model
results some modification in the fundamental properties of active systems,
such as inertial delay [33], motility induced phase transitions [34] and so
on. So far the existing results on active matter systems are mostly on the
over damped regime of the particle dynamics. The inertial effects of active
matter systems have received only limited attention to the steady state
correlation studies. Without the knowledge of inertia and inertial effects,
the exact transport properties of a system will remain unexplored. Hence in
this work, we are interested the dynamics of an inertial active Ornstein-
Uhlenbeck particle moving in a viscous medium. We are mainly interested in
steady state correlation functions and the mean square displacement of the
dynamics. The exact computation of steady state correlation functions can
provide the complete information about the dynamical behaviour or transport
properties of the system. The analytical behaviour of steady state correlation
functions of an overdamped active Brownian particle as well as AOUP particle
are discussed in [35, 18]. In this paper, we consider the inertia in the AOUP
model and exactly compute the steady state correlation functions using various
methods which is discussed in the following sections.

##  II Model

We consider the motion of an inertial active Ornstein- Uhlenbeck particle
confined in an one dimensional harmonic trap, $U(x,t)$. The dynamics of the
particle can be described by the Langevin equation of motion [36, 37, 38, 22,
39], which is given by

$m\frac{dv}{dt}=-\gamma v-\frac{\partial U\left(x,t\right)}{\partial
x}+\sqrt{D}\eta\left(t\right).$ (1)

Here, $m$ and $v(t)$ represent the mass and instantaneous velocity of the
particle. $\gamma$ is the viscous coefficient of the medium. $\eta(t)$ is the
random noise which follows the Ornstein- Uhlenbeck process and satisfies the
properties $<\eta(t)>=0$ and
$<\eta(t)\eta(t^{\prime})>=\dfrac{1}{2t_{c}}e^{-\frac{|t-t^{{}^{\prime}}|}{t_{c}}}$,
with $t_{c}$ being the noise correlation time or activity time in the
dynamics. It is the time up to which there exists a finite correlation or
activity in the dynamics. As a result the particle persists to self propel
upto time $t_{c}$. A finite $t_{c}$ especially quantifies the activity or
correlation in the medium, which decays exponentially with $t_{c}$. The
angular brackets $\left\langle...\right\rangle$ represent the ensemble average
over the noise. $D$ is the parameter which remains constant through out the
dynamics and represents strength of the Ornstein-Uhnelbeck noise [40]. For a
finite $t_{c}$, the system is in non-equilibrium and can’t be mapped to the
thermal equilibrium limit of the dynamics. However, in the white noise limit,
i.e., $t_{c}\rightarrow 0$ limit, we consider $D=2\gamma k_{B}T$, with $T$ as
the temperature of the system, to have the thermal equilibrium limit of the
dynamics. The potential $U(x,t)(=\frac{1}{2}kx^{2})$ represents the harmonic
potential, with $k$ as the harmonic stiffness constant. Substituting
$k=m\omega_{0}^{2}$ or $U(x,t)=\frac{1}{2}m\omega_{0}^{2}x^{2}$ with
$\omega_{0}$ as the harmonic frequency, the dynamics [Eq. (1)] can take the
form as

$\begin{split}m\frac{dv}{dt}=-m\omega_{0}^{2}x-\gamma
v+\sqrt{D}\eta\left(t\right).\end{split}$ (2)

##  III RESULTS AND DISCUSSION

Figure 1:  Simulated two-dimensional(2D) trajectories of a single particle
moving in a harmonic confinement. (a), (b) and (c) are for $t_{c}=0.01$ and
(d),(e) and (f) are for $t_{c}=30$. The other common parameters are
$m=D=\omega_{0}=1$

By using the Fourier transform method formalism, the solution of the
dynamics[Eq. (2)] in Fourier space
$x(\omega)\left[=\int_{-\infty}^{\infty}e^{i\omega t}x(t)dt\right]$ and
$v(\omega)\left[=\int_{-\infty}^{\infty}e^{i\omega t}v(t)dt\right]$ can be
obtained as

$\displaystyle
x\left(\omega\right)=\dfrac{\sqrt{D}\eta\left(\omega\right)}{-m\omega^{2}+i\gamma\omega+\dfrac{\omega_{0}^{2}}{m}}$

and

$\displaystyle
v\left(\omega\right)=\dfrac{\omega\sqrt{D}\eta\left(\omega\right)}{im\omega^{2}+\gamma\omega-i\omega_{0}^{2}m},$

respectively with $\eta(\omega)\left[=\int_{-\infty}^{\infty}e^{i\omega
t}\eta(t)dt\right]$. The time evolution of the dynamics is mainly controlled
by the complex interplay of inertial force, self propulsion force and the
force exerted by the harmonic confinement. In Fig. 1 we show the simulated
trajectories of the motion [Eq. (2)] of a single particle in xy-plane for
different values of $\gamma$ and for different $t_{c}$ values. When $t_{c}$ is
very small, the particle is mostly influenced by the inertial force and the
harmonic confinement. Hence, for small $t_{c}$, when $\gamma$ is large, the
inertial influence is negligibly small and the particle behaves like an over
damped passive particle showing random trajectories. However, with decrease in
$\gamma$ value, the inertial influence becomes prominent and the particle
shows almost circular trajectories in the harmonic confinement. It changes
it’s direction slowly as compared to the higher $\gamma$ regimes resulting the
trajectories comparatively smoother. In the inertial regime (i.e., for low
$\gamma$ value), with increase in $t_{c}$ value, the particle gets more and
more confined. As a result for larger $t_{c}$, the circular trajectories get
squeezed. Next we are interested in exploring the steady state position
correlation and velocity correlation. The spectral density corresponding to
position is given by the relation $S_{x}(\omega)[=\left\langle
x(\omega)x^{*}(\omega)\right\rangle]$, which is calculated as

$S_{x}(\omega)=\frac{D\left\langle\eta(\omega)\eta^{*}(\omega)\right\rangle}{m^{2}\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+\omega^{2}\gamma^{2}},$
(3)

where

$\left\langle\eta(\omega)\eta^{*}(\omega)\right\rangle=\int_{-\infty}^{\infty}e^{i\omega
t}\left\langle\eta(t)\eta(0)\right\rangle dt.$

Substituting the integral value as
$\left\langle\eta(\omega)\eta^{*}(\omega)\right\rangle=\frac{1}{1+\omega^{2}t_{c}^{2}}$
and simplifying Eq. (3), $S_{x}(\omega)$ can be expressed as

$S_{x}\left(\omega\right)=\dfrac{D}{(1+\omega^{2}t_{c}^{2})\left\\{(m\omega_{0}^{2}-m\omega^{2})^{2}+\omega^{2}\gamma^{2}\right\\}}.$
(4)

Figure 2:  $S_{x}(\omega)$ [Eq. (4)] as a function of $\omega$ for different
values of $t_{c}$. The other fixed common parameters are $D=1$, $m=1$, and
$\gamma=1$.

In Fig. 2, we have plotted the spectrum corresponding to the position
$S_{x}(\omega)$ as a function of $\omega$ for various values $t_{c}$. For a
particular value of $t_{c}$, the spectrum shows a non monotonic behaviour with
$\omega$. For lower $t_{c}$ values, $S_{x}(\omega)$ shows a double peaking
behaviour. Subsequently with increase in $t_{c}$ value, the double peaking
behaviour gets suppressed and the spectrum becomes single peaked structure
across $\omega=0$. With further increase in $t_{c}$, the peak gets sharper
across $\omega=0$ as expected. This might be due to the reason that the
particle gets more and more confined in the well with increase in $t_{c}$
value. The same conclusion is drawn from the particle trajectory plot in Fig.
1 for larger $t_{c}$ value. When the correlation time approaches zero, the
active fluctuations become thermal and the particle behaves like a passive
particle. In this limit, the analytical expression for $S_{x}(\omega)$ is
consistence with that reported in Ref.[37] in the absence of any external
force and the spectrum shows the same behavior as reported in Ref. [37]. The
Optimum value of $\omega$ ($\omega_{m}$) at which the spectrum shows peaks can
be obtained by taking the time derivative of $S_{x}(\omega)$ with respect to
$\omega$ and equating it to zero, i.e.,
$\frac{dS_{x}(\omega)}{d\omega}|_{\omega=\omega_{m}}=0$. By simplifying this,
it is confirmed that the optimum $\omega$ depends on the parameters $t_{c}$,
$\omega_{0}$ and $\gamma$ values. With increase in $t_{c}$, $\omega_{m}$
decreases as a result the peak shifts towards $\omega=0$ and becomes single
peaked at $\omega=0$. Similarly the spectral density corresponds to velocity
$S_{v}(\omega)\left[=\left\langle v(\omega)v^{*}(\omega)\right\rangle\right]$
can be obtained as

$S_{v}\left(\omega\right)=\dfrac{D\omega^{2}}{(1+\omega^{2}t_{c}^{2})\left\\{(m\omega_{0}^{2}-m\omega^{2})^{2}+\omega^{2}\gamma^{2}\right\\}}$
(5)

Figure 3:  $S_{v}(\omega)$ [Eq. (5)] as a function of $\omega$ for different
values of $t_{c}$. The other fixed parameters are $D=1,m=1\text{ and
}\gamma=1$

In Fig. 3, we have shown the plot for $S_{v}(\omega)$ as a function of
$\omega$ for different values of $t_{c}$. For a given $t_{c}$ value, the
spectrum is symmetric across $\omega=0$ and shows a double peaking behaviour
with exponential tails. With further increase in $t_{c}$, the peaks get
suppressed as expected. At the same time, the peaks get narrow and shifts
towards lower values of $\omega$. For very large $t_{c}$ value, the influence
of the harmonic confinement becomes stronger. As a result the exponential
tails also get stiffer. This reflects the stronger effect of the barriers with
longer persistence of activity in the dynamics. Therefore, the particle gets
more and more confined with $t_{c}$. The optimum value of $\omega$
($\omega_{n}$)at which the spectrum shows maximum can be obtained by
simplifying $\dfrac{dS_{v}(\omega)}{d\omega}|_{\omega=\omega_{n}}=0$. It is
confirmed that the optimum $\omega$ ($\omega_{n}$) at which the spectrum shows
maximum decreases with $t_{c}$. Other than $t_{c}$, it also depend on the
parameters $\omega_{0}$ and $\gamma$.

The position correlation $C_{x}(t)$ and velocity correlation $C_{v}(t)$ can be
obtained by performing the inverse Fourier transform of $S_{x}({\omega})$ [Eq.
(4)] and $S_{v}(\omega)$ [Eq. (5)] as $\int_{-\infty}^{\infty}e^{-i\omega
t}S_{x}(\omega)d\omega$ and$\int_{-\infty}^{\infty}e^{-i\omega
t}S_{v}(\omega)d\omega$, respectively. Simplifying the integral
$\int_{-\infty}^{\infty}e^{-i\omega t}S_{x}(\omega)d\omega$, $C_{x}(t)$ can be
calculated as

$C_{x}(t)=\dfrac{t_{c}^{2}f_{1}}{2}e^{\dfrac{-t}{t_{c}}}+\dfrac{N}{4\omega_{0}^{2}}e^{\dfrac{-t}{2t_{m}}}\left(f_{2}\cos(\omega_{1}t)+f_{3}\sin(\omega_{1}t)\right),$
(6)

where

$\begin{split}f_{1}=\dfrac{Dt_{c}}{(m+mt_{c}^{2}\omega_{0}^{2})^{2}-\gamma^{2}t_{c}^{2})}\text{,
}N=\dfrac{D}{m\gamma\omega_{1}\left[1+t_{c}^{2}\left(2\omega_{0}^{2}-\dfrac{1}{t_{m}^{2}}+t_{c}^{2}\omega_{0}^{4}\right)\right]}\text{,
}\\\
f_{2}=\left[\omega_{1}\left(2+2\omega_{1}^{2}t_{c}^{2}-\dfrac{3t_{c}^{2}}{2t_{m}^{2}}\right)\right]\text{
and
}f_{3}=\left[\dfrac{1}{t_{m}}+t_{c}^{2}\left(\dfrac{3\omega_{1}^{2}}{t_{m}}-\dfrac{\gamma}{4t_{m}^{2}}\right)\right].\end{split}$

Similarly simplifying the integral $\int_{-\infty}^{\infty}e^{-i\omega
t}S_{v}(\omega)d\omega$, the velocity correlation $C_{v}(t)$ can also be
obtained as

$C_{v}(t)=\dfrac{-f_{1}}{2m^{2}}e^{\frac{-t}{t_{c}}}+\frac{N}{4}e^{\frac{-t}{2t_{m}}}\left(f_{4}\cos(\omega_{1}t)+f_{5}\sin(\omega_{1}t)\right),$
(7)

with

$f_{4}=\left[\omega_{1}\left(2+2t_{c}^{2}\omega_{1}^{2}+\dfrac{t_{c}^{2}}{2t_{m}^{2}}\right)\right]\text{
and
}f_{5}=\left[-\dfrac{1}{t_{m}}+t_{c}^{2}\left(\dfrac{\omega_{1}^{2}}{t_{m}}+\dfrac{1}{4t_{m}^{3}}\right)\right].$

In both Eqs. (6) and (7),
$\omega_{1}=\sqrt{\omega_{0}^{2}-\frac{1}{4t_{m}^{2}}}$ and
$t_{m}=\frac{m}{\gamma}$. Using the correlation matrix formalism method, we
have also obtained the same results for both $C_{x}(t)$ and $C_{v}(t)$, which
is shown in appendix-A. The results for both $C_{x}(t)$ and $C_{v}(t)$ are
well agreement with the athermal part of the correlations as reported in Ref.
[14]

Figure 4: Normalized $C_{x}(t)[\text{Eq}.~{}\eqref{cxtharmonic}]$ as a
function of $t$ in (a) and (c) and the normalized
$C_{v}(t)[\text{Eq}.~{}\eqref{cvtharmonic}]$ as a function of $t$ in (b) and
(d) are shown for a free particle and for a confined harmonic particle,
respectively for different $t_{c}$ values. The common parameters are
$D=\omega_{0}=m=1$.

In Fig. 4, both normalized $C_{x}(t)$ and $C_{v}(t)$ are plotted as a function
of $t$ for lower values of $\gamma$ in (a) and (b) and for larger values of
$\gamma$ in (c) and (d), respectively. For low viscous medium ($\gamma=0.1$),
where the inertial influence is prominent, both $C_{x}(t)$ and $C_{v}(t)$
persists with a finite value for initial time regimes and show an oscillatory
behaviour in the long time regime of the dynamics before approaching to zero.
For $C_{x}(t)$, the amplitude of the oscillation decreases with increase in
$t_{c}$ values and the curves shift in such a way that for very large $t_{c}$,
the correlation never approaches negative values (see Fig. 4(a)). However, for
$C_{v}(t)$, the amplitude of the oscillation decreases with time. At the same
time, the minimum value shifts towards right and approaches zero in the long
time limit, which can be reflected from Fig. 4(b). For the case of a highly
viscous medium (say for $\gamma=3.0$), where the inertial influence is
negligible, the $C_{x}(t)$ never becomes negative and show a finite constant
value before decaying to zero. With increase in $t_{c}$, the correlation
persists for a longer time before decaying to zero (as in Fig. 4(c)). On the
other hand, for very low $t_{c}$ value, $C_{v}(t)$ shows a damped oscillatory
behaviour. It decays faster, becomes negative, attains a minimum value and
then approaches zero. With further increase in $t_{c}$, the correlation
persists for longer time and the decay becomes slower. At the same time, the
minimum value shifts towards right and finally approaches zero for large
$t_{c}$ (see Fig. 4(d)). In $t_{c}\rightarrow{0}$ limit, the result for
$C_{v}(t)$ is consistent with that reported in Ref. 37 in the absence of any
external force.
When the particle is not confined by the harmonic potential, it can freely
self propel in the environment. The dynamics of the particle [Eq. (2)] in such
a case can be reduced as,

$m\frac{dv}{dt}=-\gamma v+\sqrt{D}\eta\left(t\right),\\\ ,$ (8)
$t_{c}\dot{\eta}(t)=-\eta(t)+\zeta(t).$ (9)

Figure 5: 2D single particle trajectories of a free particle. $t_{c}=0.1$ is
considered for (a),(b) and (c) and $t_{c}=10$ is for (d),(e) and (f),
respectively. The other common parameters are $m=D=1$

The dynamics [Eq. (8)] and [Eq. (9)] can be written in the form [14],

$\dot{w}=-A\cdot w+\sqrt{2}\sigma\cdot\eta,$ (10)

where,

$w=\left(\begin{array}[]{c}v\\\
\eta\end{array}\right),A=\left(\begin{array}[]{cc}\frac{\gamma}{m}&\frac{-\sqrt{D}}{m}\\\
0&\frac{1}{t_{c}}\end{array}\right),\text{and
}\sigma=\left(\begin{array}[]{cc}0&0\\\
0&\frac{1}{\sqrt{2}t_{c}}\end{array}\right)$

With the help of correlation matrix method formalism[14], by introducing the
correlation matrix $C$, Eq. (10) can take the form as

$A\cdot C+C\cdot A^{T}=2M,$ (11)

where $M=\sigma\cdot\sigma^{T}$ is the diffusion matrix and

$C=\begin{pmatrix}C_{vv}&C_{v\eta}\\\ C_{\eta v}&C_{\eta\eta}\end{pmatrix}.$

Solving Eq. (11), the elements of the correlation matrix can be obtained as
$C_{vv}=\frac{D}{2\gamma^{2}(t_{m}+t_{c})}$, $C_{v\eta}=C_{\eta
v}=\frac{\sqrt{D}}{2\gamma(t_{m}+t_{c})}$, and $C_{\eta\eta}=\frac{1}{2t_{c}}$
respectively. The time dependent auto correlation function for velocity can be
obtained from Eq. (8) by multiplying initial velocity $v_{0}$ to both the
sides of Eq. (8)and taking the time derivative of average of each terms as,

$\frac{d}{dt}\langle v(t)v(0)\rangle+\frac{\gamma}{m}\langle
v(t)v(0)\rangle=\frac{\sqrt{D}}{m}\langle\eta(t)v(0)\rangle.$ (12)

Similarly, from Eq. (9) the time derivative of $\langle\eta(t)v(0)\rangle$ can
be written as

$\frac{d}{dt}\langle\eta(t)v(0)\rangle=-\frac{1}{t_{c}}\langle\eta(t)v(0)\rangle.$
(13)

By solving the above equation, we get

$\displaystyle\langle\eta(t)v(0)\rangle=C_{\eta v}\exp{\frac{-t}{t_{c}}}.$
(14)

Now substituting the value for $C_{\eta v}$ in Eq. (13) and using the
expression of $\langle\eta(t)v(0)\rangle$ in Eq. (12), one can solve Eq.
(12)for the velocity correlation $C_{v}(t)$($\langle v(t)v(0)\rangle$) as

$C_{v}(t)=\dfrac{D}{2\gamma^{2}\left(t_{m}^{2}-t_{c}^{2}\right)}\left[t_{m}e^{-\dfrac{t}{t_{m}}}-t_{c}e^{-\dfrac{t}{t_{c}}}\right].$
(15)

Figure 6:  Normalized $C_{v}(t)$[Eq. (15)] as a function of $t$ for a free
particle for different $\gamma$ values in (a) and for different values of
$t_{c}$ in (b), respectively. The common parameters are $m=D=1$

The single particle trajectory of a freely moving self propelled particle in
xy-plane is shown in Fig. 5 for different values of $\gamma$ as well as for
different values of $t_{c}$. For a free particle, the motion is mainly
governed by the inertial force and the self propulsion force and hence the
time evolution of the dynamics is characterized by the complex interplay of
the inertial time scale ($t_{m}=\frac{m}{\gamma}$) and the self propulsion
time scale ($t_{c}$) of the dynamics. It is observed that for small $t_{c}$,
when $\gamma$ is large, the particle shows a random trajectory. With either
decrease in $\gamma$ or increase in $t_{c}$, the trajectory becomes smoother.
Further, in Fig. 6, we have plotted the normalized $C_{v}(t)$ as a function of
$t$ for different values of $\gamma$ in (a) and for different values of
$t_{c}$ in (b), respectively. Since the dynamics is governed by the inertial
time and self propulsion time scale, $C_{v}(t)$ decays by the complex
interplay of these two time scale. For any fixed $t_{c}$ or $\gamma$, it
maintains with the same value in the lower time regimes before decaying to
zero in the long time regimes. With increase in $\gamma$, $C_{v}(t)$ decays
faster (see Fig. 6(a)) for a fixed $t_{c}$ and it’s persistence with time
decreases whereas with increase in $t_{c}$ it persists for longer time before
decaying to zero as reflected from Fig. 6(b). It is always found to be
positive for any chosen value of $\gamma$ or $t_{c}$. For very lower value of
$t_{c}$, $C_{v}(t)$ is mainly controlled by the inertial time scale $t_{m}$.
Hence in $t_{c}\rightarrow 0$ limit, $C_{v}(t)$ displays the same behaviour as
in case of an inertial passive particle.

Similarly the steady state correlation $C_{x}(t)$ corresponding to position of
a free particle can be calculated as

$C_{x}(t)=x_{0}^{2}+2x_{0}t_{m}v_{0}+\left(t_{m}v_{0}\right)^{2}+C_{x}^{{}^{\prime}}(t),$
(16)

where
$C_{x}^{{}^{\prime}}(t)=\dfrac{D}{\gamma^{2}}\left[t-t_{m}-t_{c}^{2}+\dfrac{t_{m}^{3}}{t_{c}^{2}-t_{m}^{2}}e^{\dfrac{-t}{t_{m}}}-\dfrac{t_{c}}{2}e^{\dfrac{-t}{t_{c}}}\left\\{1+t_{m}^{\prime}\right\\}\right]$
with $t_{m}^{\prime}=\dfrac{t_{m}}{t_{c}+t_{m}}$. It is worth to note that
$C_{x}(t)$ depends on the initial velocity of the particle. Although
$C_{x}^{{}^{\prime}}$ is part of the correlation which depends on both $t_{m}$
and $t_{c}$ values, it is mainly controlled by the strength of the AOUP noise,
$D$. Further using Kubo’s theorem, we have calculated the diffusion
coefficient as $D_{f}=\int_{0}^{\infty}\left\langle v(t)v(0)\right\rangle$
=$\dfrac{D}{2\gamma^{2}}$. The diffusion coefficient depends only on the
strength of noise as well as on the viscous drag of the medium. However, it is
independent of the mass of the particle as reported in Ref. [40].

Next we are interested in the exact computation of MSD ($\langle\Delta
x^{2}(t)\rangle$) for both harmonically confined particle and free particle.
The MSD for a harmonically confined particle ($\Delta x^{2}(t)\rangle_{h}$) is
calculated as

$\langle\Delta x^{2}(t)\rangle_{h}=\langle(x(t)-x_{0})^{2}\rangle$ (17)
$\begin{split}\langle\Delta
x^{2}(t)\rangle_{h}&=\frac{e^{\frac{-t}{t_{m}}}}{4\omega_{1}^{2}t_{m}^{2}}\left[-2t_{m}x_{0}\omega_{1}e^{\frac{t}{2t_{m}}}+2t_{m}x_{0}\omega_{1}\cos(\omega_{1}t)+(2t_{m}v_{0}+x_{0})\sin(\omega_{1}t)\right]^{2}-\frac{D}{\gamma^{2}t_{m}^{2}\omega_{1}^{2}}\left[\frac{\left(2-\frac{t_{c}}{t_{m}}\right)e^{\frac{-t}{t_{m}}}}{\left(\frac{t_{c}}{t_{m}}-2\right)^{2}+4t_{c}^{2}\omega_{1}^{2}}\right]\\\
&+\frac{D}{\gamma^{2}t_{m}^{2}\omega_{1}^{2}}\left\\{\frac{8\omega_{1}^{2}\left(1+\frac{t_{c}}{t_{m}}\right)}{\left[4\omega_{1}^{2}+\frac{1}{t_{m}^{2}}\right]\left[\left(2+\frac{t_{c}}{t_{m}}\right)^{2}+4t_{c}^{2}\omega_{1}^{2}\right]}+\frac{8t_{c}\omega_{1}\left[2t_{c}\omega_{1}\cos(\omega_{1}t)+\left(2+\frac{t_{c}}{t_{m}}\right)sin(\omega_{1}t)\right]e^{-t\left(\frac{1}{t_{c}}+\frac{1}{tm}\right)}}{\left[\left(-2+\frac{t_{c}}{t_{m}}\right)^{2}+4t_{c}^{2}\omega_{1}^{2}\right]\left[\left(2+\frac{t_{c}}{t_{m}}\right)^{2}+4t_{c}^{2}\omega_{1}^{2}\right]}\right\\}\\\
&+\frac{D}{2\gamma^{2}\omega_{1}^{2}t_{m}^{2}}\frac{e^{\frac{-t}{t_{m}}}\left[\left(\frac{-4}{t_{m}}+\frac{2t_{c}}{t_{m}^{2}}-8t_{c}\omega_{1}^{2}\right)\cos(2\omega_{1}t)+8\omega_{1}\left(1-\frac{t_{c}}{t_{m}}\right)sin(2\omega_{1}t)\right]}{\left[4\omega_{1}^{2}+\frac{1}{t_{m}^{2}}\right]\left[\left(-2+\frac{t_{c}}{t_{m}}\right)^{2}+4t_{c}^{2}\omega_{1}^{2}\right]}\end{split}$
(18)

By expanding equation. (18) in the powers of $t$, it can be expressed as,

$\langle\Delta
x^{2}(t)\rangle_{h}=v_{0}^{2}t^{2}-v_{0}\left(\frac{v_{0}}{t_{m}}+x_{0}\omega_{0}^{2}\right)t^{3}+\frac{1}{12}\left[\frac{3D}{2t_{m}^{2}t_{c}\gamma^{2}}+\frac{7v_{0}^{2}}{t_{m}^{2}}-4v_{0}^{2}\omega_{0}^{2}+\frac{10v_{0}x_{0}\omega_{0}^{2}}{t_{m}}+3x_{0}^{2}\omega_{0}^{4}\right]t^{4}+O\left(t^{5}\right)$
(19)

In the long time limit (i.e., $t\rightarrow\infty$ limit), $\langle\Delta
x^{2}(t)\rangle_{h}$ is found to be

$\langle\Delta
x^{2}(t)\rangle_{h}=x_{0}^{2}+\frac{D\left(1+\frac{t_{c}}{t_{m}}\right)}{4\gamma^{2}t_{m}^{2}\omega_{0}^{2}\left[\frac{1}{t_{m}}+\frac{t_{c}}{t_{m}}\left(t_{c}\omega_{0}^{2}+\frac{1}{t_{m}}\right)\right]}.$
(20)

Similarly, the MSD for a free particle is obtained as

$\begin{split}\langle\Delta
x^{2}(t)\rangle_{f}=\frac{D}{\gamma^{2}}\left[t-2t_{m}-t_{c}+t_{c}e^{\frac{-t}{t_{c}}}\right]+v_{0}^{2}t_{m}^{2}(e^{\frac{-t}{t_{m}}}-1)\\\
+\frac{Dt_{c}t_{m}}{2\gamma^{2}(\text{tc}^{2}-\text{tm}^{2})}\left[t_{m}e^{\frac{-t}{tc}}+3t_{c}e^{\frac{-t}{t_{m}}}+tc-
t_{c}e^{-t(\frac{1}{t_{c}}+\frac{1}{t_{m}})}\right]-\frac{Dt_{m}^{2}}{4\gamma^{2}(t_{c}^{2}-t_{m}^{2})}\left[t_{m}+t_{c}e^{\frac{-t}{t_{m}}}+3t_{m}e^{\frac{-t}{t_{m}}}\right]\end{split}$
(21)

Figure 7: MSD as a function of $t$ in (a) for a free particle [Eq. (21)] and
in (b) for a confined harmonic particle [Eq. (18)], respectively for differnt
values of $t_{c}$. The common parameters are $m=D=\gamma=1$

.

The power series expansion of (21) in the powers of $t$, can be written as

$\langle\Delta
x^{2}(t)\rangle_{f}=t^{2}v_{0}^{2}-\frac{t^{3}v_{0}^{2}}{t_{m}}+t^{4}\left(\frac{D}{8m^{2}t_{c}}+\frac{7v_{0}^{2}}{12t_{m}^{2}}\right)+O\left(t^{5}\right)$
(22)

In Fig. 7, we have shown the $\langle\Delta x^{2}(t)\rangle_{f}$ as a function
of $t$ both for a free particle(Fig. 7(a)) and a confined harmonic particle
(Fig. 7(b)) for different values of $t_{c}$. The calculation results for
$\langle\Delta x^{2}(t)\rangle$ for both free and harmonically confined
particle confirms that in the $t\rightarrow 0$ limit, $\langle\Delta
x^{2}(t)\rangle$ is proportional to $t^{2}$ and hence the initial time regimes
are ballistic in nature. The expression of MSD for a free particle in the long
time limit (i.e., $t\rightarrow\infty$ limit) is obtained as $\langle\Delta
x^{2}(t)\rangle_{f}=\frac{Dt}{\gamma^{2}}$, i.e., proportional to $t$.
However, it approaches a constant value for a harmonically confined particle.
Hence, the long time regimes for a free particle are diffusive where as for a
confined harmonic particle, it is non diffusive in nature. Moreover, for a
free particle the initial time regime as well as the long time regimes are not
dependent on $t_{c}$, but there is a dependence of $t_{c}$ in the intermediate
time regimes. The MSD for a harmonically confined particle in the time
asymptotic limit depends on $t_{c}$ and gets suppressed with increase in
$t_{c}$ value and finally approaches zero in $t_{c}\rightarrow\infty$ limit.
This confirms that the particle gets more and more confined around the mean
position of the well for longer persistence of activity in the medium.

##  IV Conclusion

In this work, we have studied the dynamics of a harmonically confined particle
following the AOUP process. With the help of both numerical simulation and
exact analytical calculations, we have explored the behaviour of particle
trajectories, the steady state correlation functions, and mean square
displacement both for a free particle and a harmonically confined particle.
From the simulated trajectories, it is confirmed that the dynamics of a
harmonic particle is mainly characterized by the inertial time, self
propulsion time, and harmonic time scale. Similarly, for a free particle it is
governed by the complex interplay of the inertial time and the self propulsion
time. One of our important observations is that for a harmonic particle, in
lower viscous medium, the particle makes circular trajectories due to strong
influence of inertia. Further with increase in the self propulsion time of the
dynamics, the circular trajectories get squeezed. This confirms that the
particle gets a stronger confinement with the longer persistence of activity
in the medium. The same conclusion is drawn from the behaviour of mean square
displacement, which gets suppressed with increase in the self propulsion time.
The steady state correlation functions shows oscillatory behaviour in the
inertial regime and this oscillatory behaviour disappears either with
viscosity of the medium or with self propulsion time of the dynamics. However,
the velocity correlation shows a damped oscillatory behaviour in the over
damped regime. In contrast to that for a free particle, the steady state
velocity correlation is always found to be finite and positive before decaying
to zero in the long time limit. When the inertial time and the self-propulsion
time are of equal order, velocity of a free particle at two different times
get decorrelated. Further, it would be interesting to explore the relaxation
behaviour of the system in various limits of the dynamics.

##  V Acknowledgment

MS acknowledges the INSPIRE Faculty award (IFA 13 PH-66) by the Department of
Science and Technology and the UGC Faculty recharge research grant for the
financial support. We dedicate this work to our mentor, teacher and friend
Arun Jayannavar, who passed away recently.

## Appendix A

For the confined harmonic particle, the dynamics [Eq. (2)] can be expressed in
the form of a vector equation as[14],

$\dot{n}=-A^{\prime}\cdot n+\sqrt{2}\sigma^{\prime}\cdot\eta^{\prime}$ (23)

where,

$n=\left(\begin{array}[]{c}x\\\ v\\\
\eta\end{array}\right),A^{\prime}=\left(\begin{array}[]{ccc}0&-1&0\\\
\frac{k}{m}&\frac{\gamma}{m}&-\frac{\sqrt{D}}{m}\\\
0&0&\frac{1}{t_{c}}\end{array}\right)\text{and
}\sigma^{\prime}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&\frac{1}{\sqrt{2}t_{c}}\end{array}\right)$

Using the correlation matrix method formalism, by introducing a correlation
matrix $(C^{\prime})$, Eq. (23)can take a new from as,

$A^{\prime}\cdot C^{\prime}+C^{\prime}\cdot A^{\prime T}=2M^{\prime},$ (24)

where $M^{{}^{\prime}}=\sigma^{{}^{\prime}}\cdot\sigma^{{}^{\prime}T}$ is the
diffusion matrix for confined particle and

$C^{{}^{\prime}}=\begin{pmatrix}C^{{}^{\prime}}_{xx}&C^{{}^{\prime}}_{xv}&C^{{}^{\prime}}_{x\eta}\\\
C^{{}^{\prime}}_{vx}&C^{{}^{\prime}}_{vv}&C^{{}^{\prime}}_{v\eta}\\\
C^{{}^{\prime}}_{\eta x}&C^{{}^{\prime}}_{\eta
v}&C^{{}^{\prime}}_{\eta\eta}\end{pmatrix}.$

Solving Eq. (24), the elements of correlation matrix can be obtained as,
$C^{\prime}_{xx}=0$, $C^{\prime}_{xv}=\frac{(m+\gamma
t_{c})}{2\gamma^{2}(m+\gamma t_{c}+kt_{c}^{2})}=-C^{\prime}_{vx}$,
$C^{\prime}_{x\eta}=C^{\prime}_{\eta x}=\frac{\sqrt{D}t_{c}}{2(m+\gamma
t_{c}+kt_{c}^{2})}$, $C^{\prime}_{vv}=\frac{D}{2\gamma(m+\gamma
t_{c}+kt_{c}^{2})}$, $C^{\prime}_{v\eta}=C^{\prime}_{\eta
v}=\frac{\sqrt{D}}{2(m+\gamma t_{c}+kt_{c}^{2})}$ and
$C^{\prime}_{\eta\eta}=\frac{1}{2t_{c}}$. The time dependent equation for
velocity auto correlation function $\langle v(t)v(0)\rangle$ can be obtained
from the Eq. (2) as,

$\begin{split}\frac{d^{2}}{dt^{2}}\langle
v(t)v(0)\rangle+\frac{\gamma}{m}\frac{d}{dt}\langle
v(t)v(0)\rangle+\frac{k}{m}\langle v(t)v(0)\rangle=\\\
\frac{\sqrt{D}}{m}\langle\eta(t)v(0)\rangle.\end{split}$ (25)

Solving the time dependent equation for $\langle\eta(t)v(0)\rangle$, one can
obtain

$\langle\eta(t)v(0)\rangle=C^{\prime}_{\eta v}\exp{-\frac{t}{t_{c}}}.$

Now substituting the value of $C^{\prime}_{\eta v}$ in the above equation, one
can obtain

$\langle\eta(t)v(0)\rangle=\frac{\sqrt{D}}{2(m+\gamma
t_{c}+kt_{c}^{2})}\exp{-\frac{t}{t_{c}}}$ (26)

Further substituting $\langle\eta(t)v(0)\rangle$ and solving Eq. (25), the
correlation corresponds to velocity $C_{v}(t)=\langle v(t)v(0)\rangle$ can be
obtained as

$C_{v}(t)=\dfrac{-f_{1}}{2m^{2}}e^{\dfrac{-t}{t_{c}}}+\frac{N}{4}e^{\dfrac{-t}{2t_{m}}}\left(f_{4}\cos(\omega_{1}t)+f_{5}\sin(\omega_{1}t)\right),$
(27)

which is same as what we get using the Fourier transform method.

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# A Two-Layer Framework with Battery Temperature Optimal Control and Network
Optimal Power Flow

Anshuman Singh, Wang Peng, and Hung D. Nguyen⋆ ⋆ Corresponding authorAuthors
are with School of EEE, NTU, Singapore<EMAIL_ADDRESS>epwang,
<EMAIL_ADDRESS>
###### Abstract

Battery energy storage is an essential component of a microgrid. The working
temperature of the battery is an important factor as a high-temperature
condition generally increases losses, reduces useful life, and can even lead
to fire hazards. Hence, it is indispensable to regulate the temperature
profile of the battery modules/packs properly in the battery energy storage
during the operation. In view of this, a two-layer optimal control and
operation scheme is proposed for a microgrid with energy storage. In the first
layer, an optimal control model is formed to derive the optimal control policy
that minimizes the control efforts, consisting of the fan speed and battery
current magnitude, in order to achieve a temperature distribution reference
over the battery modules. In the second layer, the system operator of the
microgrid performs an optimal power flow to search for the optimal temperature
distribution reference used in the first stage and the corresponding operating
current of the battery that minimize the operation cost of the entire
microgrid system. This two-layer scheme offers a great computational benefit
that allows for large-scale integration of batteries. A case study is
performed on the proposed two-layer model to illustrate its performance.

###### Index Terms:

Battery energy storage system, Optimal control, Thermal modelling

##  I Introduction

The presence of battery energy storage in a microgrid increases the
flexibility of operation. With the global trend of higher integration of
renewable energy sources that are variable and intermittent, the role of the
energy storage system (ESS) has become more crucial [1]. The temperature of
the individual cells of the battery decides the operating efficiency and the
degradation rate. High operating temperatures lead to increased resistance of
cells that causes more heat generation inside the cell. Also, the degradation
rate of a cell is increased at higher operating temperatures. Hence, the ESS
should be operated in conditions where the temperature is in a safe range.

In general, the optimal power flow (OPF) models for a microgrid having energy
storage have only State of charge constraints in their mathematical model [2,
3, 4, 5]. The temperature constraints of battery are usually ignored due to
difficulty in the thermal modelling of the individual battery modules and
including them in the OPF model. Authors in [6] have tried to design a
controller for the energy storage system (ESS) of a residential consumer such
that the bill of energy consumption is reduced. Here, modelling of battery
temperature and aging of battery is also included. Authors in [7] have tried
to formulate an optimal control problem such that the time required to charge
the battery and the degradation in that time interval is minimized. Since both
the objectives are conflicting, hence weight in the objective decides the
control action. In both the above papers, only one battery is considered and
implementation at the grid level is difficult because of the large number of
parameters involved.

At the level of the battery, thermal modelling is done using simulation
software that uses computational fluid dynamics (CFD) modelling. This process
is time-consuming and can not be synchronised with the OPF which takes place
in shorter time intervals. The problem can be simplified by considering the
collection of all battery modules as a single battery and represent it with a
single temperature. However, it may lead to underestimation or overestimation
based on the model. In this work, a lumped thermal model is presented which
analyses temperature at discrete points instead of modelling temperature of
each cell inside a battery module [8].

The heat is generated in the battery due to resistive heating, over-potential
loss, and change in entropy [9]. Since the size of ESS is large at the scale
of the grid, only resistive heating is considered as the cause of heat
generation. In heat generation due to resistive heating, the amount of heat
generated by a battery is directly proportional to the square of the magnitude
of the current. Hence, the ESS controller is only concerned about the
magnitude of the current flowing through the battery. Unlike other control
models that decide when and how much the ESS should charge or discharge, the
proposed optimal control model in this paper only optimizes the magnitude of
the current flowing through the battery. With this simplification, the
explicit optimal control laws can be derived and conveniently used in the
network optimal power flow. Then the charging or discharging decision will be
made by the system operator at the level of the microgrid.

The contributions of this paper are summarised as follows.

  * •

The dependence of the battery module temperature profile on the battery
current magnitude and fan speed is established.

  * •

A two-level model having optimal control of batteries and optimal power flow
for a microgrid is modelled. This two-level optimization framework can be
solved effectively and thus paying the way to incorporate more batteries with
complicated and dense mesh for temperature computation in the network’s
optimal operation.

The remainder of this paper is organised as the followings. In Section II,
thermal modelling of battery modules and network modelling of microgrid is
formulated. In Sections III and IV, operation of microgrid using combined
formulation and bi-level formulation is presented. A case study is performed
on the proposed model and the results are discussed in Section V where the
impact of proposed model on the temperature of the battery and overall cost is
measured and compared with the combined formulation. Finally, Section VI
provides the summary and discusses the future scope of this work.

##  II Mathematical Formulation

###  II-A Thermal modelling of battery modules

Figure 1: Configuration of N battery modules in a row

Steady-state modelling of heat transfer is considered in this paper. Figure 1
presents a configuration of battery modules in a battery rack. Each battery
module is represented by a node located at the centre of the module and the
surface temperature of the battery. It is considered that $N$ battery modules
are kept in a straight line without any air gap between them. The energy
balance at each node can be written as

$\dot{Q}+\sum_{out}Q=0\\\ $ (1)

where $\dot{Q}$ is the heat generated inside the battery module. Each node
transfers heat to the neighbouring node through conduction from the front and
back face. Heat is also lost to air through convection from the side and top
faces. The magnitude of heat loss through radiation is small compared to
conduction and convection hence ignored to keep equations linear. At steady
state, heat transfer through conduction between node $i$ and node $i+1$ can be
written as shown in (2). Heat lost to air can be written as shown in (3).

$\displaystyle Q_{cond}$
$\displaystyle=\frac{k_{b}}{l}A_{cond}(T_{i}-T_{i+1})$ (2) $\displaystyle
Q_{conv}$ $\displaystyle=hA_{conv}(T_{i}-T_{a})$ (3)

where $T_{a}$ is the ambient temperature, $l$ is the length of battery, and A
is the area perpendicular to the direction of heat transfer. $k_{b}$ is the
conduction heat transfer coefficient of battery casing. $h$ is the convective
heat transfer coefficient and assumed to be varying linearly with fan speed
$u_{f}$ as

$\displaystyle h=h_{0}(1+\lambda(u_{f}-u_{f_{0}})).$ (4)

Heat generation due to resistive heating is only considered in this paper and
can be written as

$\displaystyle\dot{Q}=I_{b}^{2}R_{b}$ (5)

where $R_{b}$ is the resistance and $I_{b}$ is the current through the
battery. The top covering of the battery module is considered to be an
insulating surface and there is no heat transfer between the bottom surface
and the rack. The energy balance for the 1st battery can be written as for
$k_{b}^{\prime}={k_{b}A_{1}}/{l}$

$\displaystyle\dot{Q_{1}}=hA_{1}(T_{1}-T_{a})+2hA_{2}(T_{1}-T_{a})+k_{b}^{\prime}(T_{1}-T_{2})$
(6)

The energy balance for all batteries from 2 to $N-1$ can be written as

$\displaystyle\dot{Q_{i}}=k_{b}^{\prime}(T_{i}-T_{i-1})+k_{b}^{\prime}(T_{i}-T_{i+1})+2hA_{2}(T_{i}-T_{a})$
(7)

The energy balance for the last battery can be written as

$\displaystyle\dot{Q_{N}}=k_{b}^{\prime}(T_{N}-T_{N-1})+hA_{1}(T_{N}-T_{a})+2hA_{2}(T_{N}-T_{a})$
(8)

The equation for node $i$ can be generalised as a linear algebraic equation as
shown below

$\displaystyle
b_{i1}T_{1}+b_{i2}T_{2}+...+b_{in}T_{N}+b_{i(N+1)}T_{a}=I^{2}_{i}R_{b_{i}}$
(9)

Collectively, this can be represented by a system of equations

$\displaystyle B\,T=C.$ (10)

Here, the size of B matrix is $(N,\,1)$ column vector representing the heat
generation within N battery modules.

In the following, we further represent the two coefficients matrices $B$ and
$C$ in terms of the control inputs $u=[u_{f},u_{I}]$, which consists of the
fan speed $u_{f}$ and squared current $u_{I}=I_{b}^{2}$, for regulating the
temperature $T$ of the battery’s cells. The resulting representation will be
leveraged to derive the optimal control policy in Section IV-A that minimizes
the control efforts while achieving the desired temperature profile.

The sensitivity matrix B is a linear function of heat transfer coefficient $h$
which in turn is a linear function of fan speed $u_{f}$. Hence, matrix B can
be written as

$B=B_{0}+Ku_{f}.$ (11)

The constant matrix $B_{0}$ is used in the absence of a fan. The coefficient
$K$ reflects the effectiveness of the forced ventilation while varying the fan
speed $u_{f}$.

The matrix C depends on the squared battery current and resistance of battery.
The resistance of battery is an affine function of temperature, i.e.,
$R=R_{ref}\left(1+\alpha_{T}(T-T_{ref})\right)$ where the subscript ref
denotes the reference values, typically at $25^{\circ}$. $\alpha_{T}$ is the
temperature coefficient of resistance; for the battery material, we choose
$\alpha_{T}=0.004$. The resistance of battery can be reduced to
$R=R_{0}+\alpha T$ where $R_{0}=R_{ref}(1-\alpha_{T}T_{ref})$ and
$\alpha=\alpha_{T}R_{ref}$. Hence, matrix C can be written as

$\displaystyle C=u_{I}(R_{0}+\alpha T).$ (12)

As a result, the temperature profile relation (10) can be rewritten as:

$(B_{0}+Ku_{f})T=u_{I}(R_{0}+\alpha T).$ (13)

Rearranging (13) and introducing $M=B_{0}+Ku_{f}-\alpha u_{I}$ and
$\beta=u_{I}R_{0}$, yields:

$MT=\beta$ (14)

which will be used to design the optimal control policy in Section IV-A below.

###  II-B Network modelling of microgrid

The relationship between power and voltage is given by the linear branch flow
model for all $(i,j)\in U$, where $U$ is the set of all lines.

$\displaystyle P_{ij}+p^{g}_{j}-p^{d}_{j}=\sum_{k:(j,k)\in U}P_{jk},$ (15)
$\displaystyle Q_{ij}+q^{g}_{j}-q^{d}_{j}=\sum_{k:(j,k)\in U}Q_{jk},$ (16)
$\displaystyle v_{j}=v_{i}-2(r_{ij}P_{ij}+x_{ij}Q_{ij}),$ (17) $\displaystyle
P_{grid}=P_{12}.$ (18)

The notations are defined below.
$p^{g}_{j}/q^{g}_{j}:$ active/reactive power generated at bus j
$p^{d}_{j}/q^{d}_{j}:$ active/reactive power demand at bus j,
$r_{ij}/x_{ij}:$ resistance/reactance of line between bus i and j
$v_{i}:$ square of voltage magnitude at bus i
$P_{ij}/Q_{ij}:$ active/reactive power line flow from bus i to bus j The
constraints related to battery storage are as follows

$\displaystyle E_{t+1}=E_{t}+(P_{bc}\eta_{bc}-P_{bd}/\eta_{bd})\Delta t,$ (19)
$\displaystyle P_{bc}+P_{bd}=V_{ess}I_{ess},$ (20)
$\displaystyle\alpha_{bc}+\alpha_{bd}\leq 1.$ (21)

Here $E_{t+1}/E_{t}$ is the battery capacity at time interval $t+1/t$
respectively. $\eta_{bc}$ is the efficiency of battery charging and
$\eta_{bd}$ is the efficiency of battery discharging. $V_{ess}$ and $I_{ess}$
are overall voltage and current through ESS depending upon series-parallel
configuration of battery modules. $\alpha_{bc}/\alpha_{bd}$ are binary
variables associated with charging/discharging process.

##  III Mixed Optimal operation of microgrid

###  III-A Mixed Optimal Formulation

Optimal power flow is a fundamental problem for the operation that optimizes
the operation cost while satisfying network and operational constraints [10,
11]. The OPF framework proposed in this work is an extension of the
conventional OPF problem that accommodates the battery temperature
constraints. This new framework is called the mixed or combined optimal
operation formulation because this is a unified optimization formulation
crossing the thermal and electrical domains. This framework can be modelled as

$\displaystyle\min\,C_{grid}+C_{ess}.$ (22)

$C_{grid}$ is the energy import cost from the utility grid and $C_{ess}$ is
the cost of operation of ESS.

$\displaystyle C_{grid}=\sum_{T}P_{grid}f_{e}\Delta t$ (23) $\displaystyle
C_{ess}=\sum_{T}P_{ess}f_{ess}\Delta t$ (24)

where $P_{grid}$ is the net power exchange from the grid and $f_{e}$ is the
rate of power exchange ($/MW) from the grid. $P_{ess}$ is the net power
exchange by ESS and $f_{ess}$ is the corresponding rate of power exchange
($/MW). The time window $\Delta t$ is the duration that one shot of the mixed
OPF concerns. Typical time window can be 5 minutes. The constraints in this
mixed model are network constraints (15 \- 21), thermal modelling constraints
and operational limits.

###  III-B Limitations of the mixed formulation

The mixed formulation minimizes the cost of operation, mostly from the
electrical domain, while satisfying the constraints including constraints on
the temperature of each battery module in the thermal domain. This unified
optimization problem considers all possible information from both domains and
thus bringing potentially better optimal solutions if the optimization can be
solved properly. However, there are certain limitations to the implementation
of this formulation. Firstly, for a large number of batteries in the network,
the size of the optimization model will be huge. If a dense mesh is used for
the temperature profile computation, the number of considered temperature
points/variables is large. Solving such large-scale and complex optimization
problems is not numerically efficient while there is no guarantee that a high-
quality optimal solution can be found. Secondly, the system operator may not
be aware of the private information within the battery such as the
relationships among battery current, fan speed, and temperature distribution.
Therefore, this work proposes a two-layer optimization framework to handle the
operation in both thermal and electrical domains.

##  IV Two-layer Optimization

###  IV-A Optimal control of temperature in the thermal domain

Assume that the battery’s current temperature profile is $T$ and the desired
profile is $T^{\star}$ satisfying the thermal management requirement. To drive
the battery’s temperature profile to $T^{\star}$, one needs to control the fan
speed as well as the battery’s current. Let such required control efforts be
$\Delta u=[\Delta u_{f},\Delta u_{I}]$. We introduce the following relation
with its proof presented in Appendix VII

$KT\,\Delta u_{f}-(\alpha T+R_{0})\Delta u_{I}=-M(T^{\star}-T).$ (25)

Introducing $a=-M(T^{\star}-T)/(KT)$ and $b=(\alpha T+R_{0})/(KT)$, (25) leads
to

$\Delta u_{f}=a+b\,\Delta u_{I}.$ (26)

The optimal control is designed to optimize the control effort $\Delta
u=[\Delta u_{f},\Delta u_{I}]$ that sufficiently brings the battery’s current
temperature profile $T$ to the desired profile $T^{\star}$.

$\displaystyle\min_{\Delta u}$ $\displaystyle(\Delta u_{f})^{2}+c\,(\Delta
u_{I})^{2}$ (27) subject to $\displaystyle\Delta u_{f}=a+b\,\Delta u_{I}$

where $c>0$ is the weight of the corresponding objective component. This
weight can also be chosen to reflect how a change in the fan speed leads to a
change in the battery current in case the fan is powered by the battery.

####  IV-A1 Optimal temperature control policy

The optimal policy that optimizes (27) is as below.

$\displaystyle\Delta u_{f}$ $\displaystyle=a-ab^{2}/(b^{2}+c),$ (28)
$\displaystyle\Delta u_{I}$ $\displaystyle=-ab/(b^{2}+c).$

To prove this optimal policy, one can replace $\Delta u_{f}$ in the objective
function of (27) by $a+b\,\Delta u_{I}$ to make it a univariate function of
$\Delta u_{I}$. Taking the derivative of the resulting univariate function
with respect to $\Delta u_{I}$ and setting it to zero, one can arrive at the
optimal policy (28).

Recall that, for a given current temperature profile $T$, the term $a$ depends
affinely on the temperature setting $T^{\star}$. The optimal policy (27)
provides an explicit form in terms of the temperature setting $T^{\star}$,
i.e., $\Delta u=\Delta u(T^{\star})$. In other words, for a given temperature
setting $T^{\star}$, the optimal policy (28) will tell the optimal amount of
control effort needed to achieve such a desired temperature profile from the
current temperature.

###  IV-B Optimal power flow in the electrical domain

Once the optimal control policy is established in the lower layer, the optimal
control efforts and the temperature profile reference will be further
optimized in the upper optimization layer with OPF. The output of the upper
layer contains the charging/discharging decision and the temperature reference
settings. For the upper layer, the optimization objective is to minimize the
total cost of operation.

$\displaystyle\min$ $\displaystyle C_{grid}+C_{ess}$ (29) s.t.
$\displaystyle\text{OPF constraints:}\,(\ref{OPF1})-(\ref{OPF2})$
$\displaystyle\text{Opt. control policy}\,\eqref{eq:opt_sol}:\,\Delta u=\Delta
u(T^{\star})$

Here vector $\Delta u$ contains the fan speed and battery current magnitude
increments needed to provided at the current state to achieve the desired
temperature profile $T^{\star}$. The operational limit on all system variables
can be modelled as an inequality constraint given as

$\displaystyle\underline{x}_{p}\leq x_{p}\leq\overline{x}_{p}$ (30)

here, $x_{p}$ is the set of all system variables.

##  V Numerical simulations

###  V-A Test system

A case study is performed to evaluate the effect of operating battery with and
without local temperature control. A 33 bus radial power distribution network
is considered [12]. Bus 1 is connected to the utility grid and a battery
energy storage of rating 66.304 kWh is considered to be located at bus 6.
There are 10 battery modules of rating 25.9 V/64 Ah each kept in a row
constituting one level of the battery rack as shown in Fig. 1. All the modules
in a row are connected in series. There are 4 levels or rows in the rack
having a similar configuration and are connected in parallel to each other.
Assuming symmetry in the operation of the 4 levels, only the temperature
distribution of one row is considered and rest of the rows follow same
pattern. Initial temperature distribution is shown in Fig. 2. The height of
each battery module is 230 mm. The other parameters of the test system are
given in Table I. The simulations are performed on MALAB R2019a using Yalmip
toolbox and Gurobi as the solver.

Figure 2: Initial temperature profile TABLE I: Data of test system Ambient temperature ($T_{a}$) | K | 308
---|---|---
Battery voltage rating ($V_{b}$) | V | 259
Buying/selling rate at grid ($P_{grid}$) | $/MW | 30/26
Buying/selling rate for ESS ($P_{ess}$) | $/MW | 26/32
Charging efficiency ($\eta_{c}$) | % | 0.95
Coefficient of conduction $(k_{b})$ | W/mk | 205
Coefficient of convection loss ($h_{0}$) | $W/m^{2}K$ | 5
Initial SOC ($E_{t}$) | kWh | 40
Discharging efficiency ($\eta_{d}$) | % | 0.95
ESS rated voltage | V | 259
Fan speed coefficient of h $(\lambda)$ | min/rotations | 0.01814
Initial fan speed $(u_{f})$ | rpm | 2000
Initial current through each battery ($u_{i}$) | A | 50
Maximum SOC ($\overline{E}$) | kWh | 66.304
Maximum charge power of ESS $(\overline{P_{bc}})$ | kW | 60
Maximum discharge power of ESS $(\overline{P_{bd}})$ | kW | 60
Maximum bus voltage ($\overline{V}$) | p.u. | 1.1
Minimum SOC ($\underline{E}$) | kWh | 5
Minimum bus voltage ($\underline{V}$) | p.u. | 0.9
Resistance of each module ($R_{ref}$) | ohm | 0.1
Temperature coefficient of resistance $(\alpha_{T})$ | $K^{-1}$ | 0.004

###  V-B Discussion

Fig. 3 shows the change in control variables for different values of the
weight of objective ($c$) and a given desired temperature of $0.95T$. The
decoupled formulation refers to the two-layer optimization framework. Fig. 4
shows the change in control variables for different values of desired
temperature for a given weight of 0.25. These figures shows the trade-off
between the two control efforts i.e. varying the fan speed versus varying the
current. The analysis of control effort is significant because there is a cost
associated with each control effort. Operating battery at higher temperatures
will provide more output and in turn more revenue but also increases the
degradation rate thereby reducing the remaining useful life.

Fig. 5 shows the total cost of operation of the microgrid for different values
of the weight of objective ($c$). It is observed that there is not a
significant variation in cost. Also, the cost is higher in case of less
desired changes in current magnitude. The effects will be more visible when
there are more than one ESS and each ESS operator try to locally optimize its
fan speed and ESS current.

Figure 3: Change in control variables with variation in weight ($T^{\star}$ =
0.95 T) Figure 4: Changes in control variables with variation in desired
temperature (for c = 0.25) Figure 5: Total cost of operation versus weight
(for $T^{\star}$ = 0.95 T)

##  VI Conclusion

In this paper, a simple two-layer mathematical formulation is discussed for
optimal operation in a microgrid having energy storage. In the lower layer or
layer 1, the decision on values of battery current magnitude and fan speed is
taken such that the battery modules’ temperature profile reaches the desired
profile that is compliant with the thermal management requirement. These
values are passed on to the microgrid operator who performs optimal power flow
in the upper level, level 2. A case study is performed on the proposed method
of operation and the results are discussed. The future scope of this paper
includes more detailed thermal modelling of individual battery cells and
dynamic thermal modelling of batteries. Also, the modelling of the remaining
useful life and the cost associated with the degradation of battery can be
included in the optimal control model.

##  VII Appendix

This appendix shows the proof of the relation (25). Perturbing (14), we have

$(M+\Delta M)(T+\Delta T)=\beta+\Delta\beta.$ (31)

Expanding (31) and assuming that $\Delta M\,\Delta T$ is small enough to be
ignored, one arrives at the following linearized form:

$M\,\Delta T+T\,\Delta M=\Delta\beta.$ (32)

Using $\Delta T=T^{\star}-T$, $\Delta M=K\Delta u_{f}-\alpha\Delta u_{I}$, and
$\Delta\beta=R_{0}\Delta u_{I}$, (32) becomes

$M\,(T^{\star}-T)+(K\Delta u_{f}-\alpha\Delta u_{I})\,T=R_{0}\Delta u_{I}$
(33)

which directly leads to (25), i.e.,

$KT\,\Delta u_{f}-(\alpha T+R_{0})\Delta u_{I}=-M(T^{\star}-T).$

## Acknowledgment

This work is supported by NTU SUG, MOE Tier-1 2019-T1-001-119, EMA & NRF EMA-
EP004-EKJGC-0003.

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# HCC Is All You Need: Alignment—The Sensible Kind Anyway—Is Just Human-
Centered Computing

Eric Gilbert
School of Information & CSE
University of Michigan
<EMAIL_ADDRESS>

###### Abstract

This article argues that AI Alignment is a type of Human-Centered Computing.

##  1 Argument

The argument of this very short paper is that the problem academic AI has
termed “alignment” is just a type of Human-Centered Computing (HCC). HCC is an
existing academic field.111HCC contains Human-Computer Interaction (HCI).

The term “alignment” rose to prominence among AGI and crypto
researchers/enthusiasts/grifters [3]—and has been linked with eugenics
traditions [18]. Nevertheless, it has jumped into academic discourse and
become an umbrella term for creating AI systems that respect “human intentions
and values” [24]. I address this latter, arguably more sensible kind of
alignment currently happening in academia and industry. Here, alignment often
quickly becomes specific technical problems, such as how to learn reward
functions that correspond to what users want (e.g., [29]), or how to construct
models that can explain themselves to people (e.g., [31]). However, the high-
level goal is broader: to bring a complex technology into concert with what
people want it to do.

This is literally what HCC is. There are whole journals and conferences (e.g.,
CHI, CSCW, UIST, FAccT). There is a whole program at the U.S. National Science
Foundation [13].

For over 40 years, HCC has struggled with, and made progress on, “aligning”
different technologies to people. Key issues include: Who, exactly, are we
talking about (e.g., [2, 5, 6, 12, 25, 39, 43])? How do we know what they want
(e.g., [9, 14, 17, 22, 27, 38, 42])? How stable is what they want to do with
technology (e.g., [40])? Can they help us design it (e.g., [20, 21, 26, 32,
35])? What are the different ways a technology can be designed (e.g., [7,
34])? How do we know if it’s good for people (e.g., [10, 33])? What are the
limits of design- and tech-centric approaches (e.g., [16, 19, 36, 44])? Is it
possible to avoid baking systemic oppression into technology (e.g., [4, 8, 11,
15, 23, 28, 30, 37, 41])?

Casting alignment as HCC invites it to draw upon the considerable theories,
methods, and findings of HCC, and the fields from which it borrows (e.g., STS,
Communication, Ethics, etc.)—instead of re-inventing them under new names.
Perhaps we don’t need the word “alignment” at all.

##  2 Argument as equation

In an effort to observe norms, I have also stated the argument above as the
following equation:

$\\{(AI,L):L\in\mathscr{L},AI\in\underset{AI^{\prime}}{\arg\min}\int_{\mathscr{H}}L(AI^{\prime},h)\,dh\\}\subset
HCC$ (1)

where $\mathscr{L}$ represents a family of human-AI loss functions, and
$\mathscr{H}$ is the space of what people want.

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