Differentially Private Optimal Power Flow
for Distribution Grids
Vladimir Dvorkin Jr., Student member, IEEE,
Ferdinando Fioretto,
Pascal Van Hentenryck, Member, IEEE,
Jalal Kazempour, Senior Member, IEEE,
and Pierre Pinson, Fellow, IEEE

V. Dvorkin Jr., J. Kazempour, and P. Pinson are with the Technical University of Denmark,  Kgs. Lyngby, Denmark. F. Fioretto is with the Syracuse University, Syracuse, NY, USA. P. Van Hentenryck is with the Georgia Institute of Technology, Atlanta, GA, USA.

Although distribution grid customers are obliged to share their consumption data with distribution system operators (DSOs), possible leakage of this private data is often disregarded in operational routines of DSOs. This paper introduces a privacy-aware optimal power flow (OPF) model for distribution grids that secures customer privacy from unauthorised access to OPF solutions, e.g., current and voltage measurements. The model is based on the framework of differential privacy that allows to control the privacy risks for sensitive data sets by applying a carefully calibrated noise to the underlying computations. Unlike existing differentially private mechanisms, the model does not apply the noise to optimization parameters or results. Instead, it optimizes OPF variables as affine functions of the random noise that encodes privacy parameters. To ensure the feasibility of the randomized solution, the model makes use of chance constraints enforced on the grid limits. The model is further extended to control the optimality loss, induced by the random noise, as well as the variance of the OPF variables. The paper offers rigorous guarantees that the measurements of differentially private OPF solution do not leak customer data up to customer-specified privacy requirements.

Chance constraints, CVaR, differential privacy, distribution grid, optimal power flow, privacy, variance

§ INTRODUCTION

The increasing observability of distribution grids enables advanced operational practices for distribution system operators (DSOs). In particular, high-resolution voltage and current measurements available to DSOs allow for continuously steering the system operation towards an optimal power flow (OPF) solution <cit.>. However, when collected, these measurements expose distribution grid customers to privacy breaches. Several studies have shown that the measurements of OPF variables can be used by an adversary to identify the type of appliances and load patterns of grid customers <cit.>. The public response to these privacy risks has been demonstrated by the Dutch Parliament's decision to thwart the deployment of smart metering devices until the privacy concerns are resolved <cit.>.

Although grid customers tend to entrust DSOs with their data in exchange for a reliable energy supply, their privacy rights are often disregarded in operational routines of DSOs. To remedy this situation, this paper aims at augmenting OPF computations with the preservation of customer privacy in the following sense.

The right of grid customers to be secured from an unauthorized disclosure of sensitive information that can be inferred from the OPF solution.

To ensure this right, privacy needs to be rigorously quantified and guaranteed. Differential privacy (DP) <cit.> is a strong privacy notion that quantifies and bounds privacy risks in computations involving sensitive data sets. By augmenting the computations with a carefully calibrated random noise,
a DP algorithm guarantees that the noisy results do not disclose the attributes of individual items in a data set. Chaudhuri et al. <cit.> and Hsu et al. <cit.> introduced several mechanisms to solve optimization models while preventing the recovery of input data from optimization results. These mechanisms apply noise to either the parameters or the results of an optimization. The applied noise, however, fundamentally alters the optimization of interest. Therefore,  the direct application of these mechanisms to OPF problems has been limited: First, they may fail to provide a feasible solution for constrained optimization problems. To restore feasibility, they require an additional level of complexity such as the post-processing steps proposed in <cit.>. Second, although these mechanisms provide bounds on the worst-case performance, they do not consider the optimality loss as a control variable. As a result, they cannot trade off the expected and worst-case mechanism performance. Finally, the proposed mechanisms may overlook the impacts of noise on the variance of optimization results. Hence, their direct application to power systems may lead to undesired overloads of system components <cit.>.

Contributions: To contrast these limitations this paper proposes a novel differentially private OPF mechanism that does not inject noise to the optimization parameters or results. Instead, it attains DP by defining and optimizing OPF variables as affine functions of the noise. This approach allows for providing DP, while bypassing the above-mentioned theoretical drawbacks. More precisely, the paper makes the following technical contributions:

* The proposed mechanism produces a random OPF solution that follows a Normal distribution and ensure $(\varepsilon,\delta)-$differential privacy <cit.>. The parameters $\varepsilon$ and $\delta$, respectively, bound the multiplicative and additive differences between the probability distributions of OPF solutions obtained on adjacent data sets (e.g., different in load values).
The mechanism is particularly suitable for protecting grid loads from unauthorized access to OPF solutions, as a proper tuning of $(\varepsilon,\delta)$ makes randomized OPF solutions similar irrespective of the load data used in the computation. The proposed mechanism is further extended to accommodate customer privacy preferences, expressed by a parameter $\beta$. It provides a rigorous guarantee that the disclosure of either power injection or voltage related to node $i$ does not allow to distinguish the active load magnitude $d_{i}^{p}$ at this node from any other load magnitude in the range $[d_{i}^{p}-\beta_{i},d_{i}^{p}+\beta_{i}].$

Many concepts anticipated here. Also, should we give this mechanism a name?

* The mechanism enforces chance constraints on random OPF variables to ensure solution feasibility for a given constraint satisfaction probability. This way, it does not require a post-processing step to restore OPF feasibility, as for instance in <cit.> and <cit.>. Since the OPF variables are affine in the Gaussian noise, the chance constraints are reformulated into computationally efficient second-order cone constraints.

* The mechanism is also extended to control random OPF outcomes without losing the DP guarantees. Using results from stochastic programming <cit.>, the optimality loss induced by the noise is controlled using variance and conditional Value-at-Risk measures. Furthermore, following a variance-aware control from <cit.>, the mechanism achieves DP with a significantly smaller variance.

Related Work: Thanks to its strong privacy guarantees, DP has been recently applied to private OPF computations. In particular, Zhou et al. <cit.> build a mechanism that releases aggregated OPF statistics, e.g., aggregated load, while ensuring the privacy for individual loads, even if all but one loads are compromised.
The proposal by Fioretto et al. <cit.> and Mak et al. <cit.> provide a differentially private mechanism to release high-fidelity OPF datasets (e.g., load and network parameters) from real power systems while minimizing the risks of disclosing the actual system parameters. The mechanisms, however, are meant for the private release of aggregate statistics and input datasets and do not provide the OPF solution itself.

Private OPF computations have also been studied in a decentralized setting. Dvorkin et al. <cit.> designed a distributed OPF algorithm with a differentially private exchange of coordination signals, hence preventing the leakage of sensitive information in the algorithm subproblems. Han et al. <cit.> proposed a privacy-aware distributed coordination scheme for electrical vehicle charging. The privacy frameworks in <cit.> and <cit.>, however, are not suitable for centralized computations. In distribution systems, Zhang et al. <cit.>, among other proposals reviewed in <cit.>, designed a privacy-aware optimization of behind-the-meter energy storage systems to prevent the leakage of consumption data from the smart meter readings. However, they disregard OPF feasibility of distribution systems that has to be preserved in all circumstances.

Paper Organization: Following the preliminaries in Section <ref>, Section <ref> formalizes privacy goals and provides an overview of the proposed solution. Section <ref> provides the formulation of the proposed privacy-preserving OPF mechanism, whereas Section <ref> presents its theoretical properties and extensions. Finally, Section  <ref> provides numerical experiments.

§ PRELIMINARIES

§.§ Optimal Power Flow Problem

The paper considers a low-voltage radial distribution grid with
controllable distributed energy resources (DERs). A DSO is responsible for
controlling the DERs and supplying power from the high-voltage grid while meeting the technical limits of the grid. The grid is
modeled as a graph  $\Gamma\braceit{\set{N},\set{L}}$, where
$\set{N}=\{0, 1, \ldots, n\}$ is the set of nodes and $\set{L}=\set{N}\setminus \{0\}$ is the set of lines connecting those nodes.
The root node, indexed by $0$, is a substation with a large capacity
and fixed voltage magnitude $v_0 = 1$. The radial topology, depicted in
Fig. <ref>, associates each node $i \in \set{N}$ with the sets $\set{U}_{i}$ of upstream and $\set{D}_{i}$ of downstream nodes, as well as with the set $\set{R}_{i}$ of nodes on the path to the root node.

Each node $i$ is characterized by its fixed active $d_{i}^{p}$ and reactive $d_{i}^{q}$ power load and by its voltage magnitude $v_{i}\in[\underline{v}_{i},\overline{v}_{i}]$.
For modeling convenience, the voltage variables are substituted as $u_{i}=v_{i}^{2},\;\forall i\in\set{N}$.
A controllable DER sited at node $i$ outputs an amount of active
$g_{i}^{p}\in[\underline{g}_{i}^{p},\overline{g}_{i}^{p}]$ and
reactive $g_{i}^{q}\in[\underline{g}_{i}^{q},\overline{g}_{i}^{q}]$
power. Its active power cost is described by a quadratic function with positive linear and
quadratic coefficients $c_{1i}$ and $c_{2i}$ respectively.
To model the relation between active and reactive DER output, the paper assumes a constant power factor $\tan{\phi_{i}}$. Similarly, the constant power factor is assumed to hold for loads, such that each load $i$ can be described solely by its active component $d_{i}^{p}$.
The active and reactive power flows, $f_{\ell}^{p}$ and $f_{\ell}^{q}$, $\forall \ell \in \set{L}$ respectively, are constrained by the apparent power limit $\overline{f}_{\ell}$, and each line is characterized by its resistance $r_{\ell}$ and reactance $x_{\ell}$.

The distribution OPF model can be formulated as:

\begin{align}
\minimize{g^{\dag},f^{\dag},u}\;&\mysum{i\in\set{N}}(c_{1i}g_{i}^{p}+c_{2i}(g_{i}^{p})^{2})\label{label123}\\
\text{s.t.}
\;\;&g_{0}^{\dag}=\mysum{i\in\set{D}_{0}}(d_{i}^{\dag}-g_{i}^{\dag}),\;u_{0}=1,\label{det_OPF_root}\\
&f_{\ell}^{\dag}=d_{\ell}^{\dag}-g_{\ell}^{\dag} + \mysum{i\in\set{D}_{\ell}}(d_{i}^{\dag}-g_{i}^{\dag}), \;\forall \ell\in\set{L},\label{det_OPF_flow}\\
&u_{i} = u_{0} - 2\mysum{\ell \in\set{R}_{i}}(f_{\ell}^{p}r_{\ell}+f_{\ell}^{q}x_{\ell}),\;\forall i\in\set{N} \setminus \{0\},\label{det_OPF_voltage}\\
&(f_{\ell}^{p})^2 + (f_{\ell}^{q})^2 \leqslant\overline{f}_{\ell}^2,\;\forall \ell\in\set{L},\label{det_OPF_flow_limit}\\
&\underline{g}_{i}^{\dag} \leqslant g_{i}^{\dag}\leqslant \overline{g}_{i}^{\dag},\;\forall i\in\set{N},\label{det_OPF_gen_limit}\\
&\underline{v}_{i}^2 \leqslant u_{i} \leqslant \overline{v}_{i}^2,\;\forall i\in\set{N}\setminus \{0\},\label{det_OPF_voltage_limit}
\end{align}

where superscript $\dag=\{p,q\}$ indexes active and reactive power. The objective is to minimize the total operational cost subject to the OPF equations (<ref>)–(<ref>) and grid limits (<ref>)–(<ref>). The OPF equations balance the grid based on the LinDistFlow AC power flow equations <cit.>.

§.§ Differential Privacy

The paper uses the framework of differential privacy <cit.> to quantify and control the privacy risks of customer loads.

It considers data sets $D \in \mathbb{R}^n$ as $n$-dimensional vectors describing the active load values, denoted by $d_i$ for each node $i$. To protect the participation of the load in the $i^{\text{th}}$ entry of the data set, the following adjacency relation is introduced:

\begin{equation*}
\label{eq:adj_rel}
D \sim_\beta D' \Leftrightarrow \exists i
\textrm{~s.t.~} | d_i - {d}_{i}' | \leqslant \beta_i \;\land\;
d_j = {d}_{j}', \forall j \neq i,
\end{equation*}

where $D$ and $D'$ are two adjacent data sets, $\beta \in \RR^n$ is a vector of positive real values, and the values $d_i$ and $d_i'$ are the load values corresponding to customer $i$ in $D$ and $D'$, respectively.  The adjacency relation relates two load vectors that differ in at most one item, at position $i$, by a value not greater than $\beta_i$. This relation is used to capture differentially private mechanisms that protect an individual load irrespective of the information an adversary acquires about all other loads. This intuition is formalized in the following definition.

Given a value $\beta \in \RR^n_{+}$, a randomized mechanism ${\cal M} \!:\! \cD \!\to\! \cR$ with domain $\cD$ and range $\cR$ is $(\varepsilon, \delta)$-differential private if, for any output response $s \subseteq \cR$ and any two adjacent inputs $D \sim_\beta D' \in \RR^n$

\begin{equation*}
\mathbb{P}[{\cal M}(D) \in s] \leqslant e^\varepsilon \mathbb{P}[{\cal M}(D') \in s] + \delta,
\end{equation*}

where $\mathbb{P}$ denotes the probability over runs of ${\cal M}$.

The level of privacy is controlled by DP parameters $\varepsilon,\delta\in(0,1)$. The former corresponds to the maximal multiplicative difference in distributions obtained by the mechanism on adjacent data sets, whereas the latter defines the maximal additive difference. Consequently, smaller values of $\varepsilon$ and $\delta$ provide stronger privacy protection. Definition <ref> extends the metric-based differential privacy introduced by Chatzikokolakis et al. <cit.> to control of individual privacy risks.

The differentially private design of any mechanism is obtained by means of randomization using, among others, Laplace, Gaussian, or exponential noise <cit.>. The DP requirements for an optimization problem are achieved by introducing a calibrated noise to the input data <cit.> or to the output or objective function of the mechanism itself <cit.>.
Regardless of the strategy adopted to attain DP, the amount of noise to inject depends on the mechanism sensitivity.
In particular, the $L_2-$sensitivity of a mechanism ${\cal M}$ on $\beta$-adjacent data sets, denoted by $\Delta^\beta$, is defined as:

\[
\Delta^\beta = \max_{D \sim_\beta D'}
\left\|  {\cal M}(D) - {\cal M}(D')\right\|_2.
\]

This work makes use of the Gaussian mechanism, which provides $(\varepsilon,\delta)-$differential privacy.

Let $f$ be a function of interest that maps data sets to $\RR^n$. For  $\varepsilon\in(0,1)$ and $\gamma^{2}>2\ln(\frac{1.25}{\delta})$, the Gaussian mechanism that outputs $f + \xi$, with $\xi$ noise  drawn from the Normal distribution with $0$ mean and standard deviation $\sigma \geq \gamma\frac{\Delta^{\beta}}{\varepsilon}$
is $(\varepsilon,\delta)$-differentially private.

When the privacy mechanism produces solutions to an optimization problem, it is also important to quantify the optimality loss, i.e., the distance $\Theta:=\norm{\tilde{o}-o^*}$ between the value $\tilde{o}$ obtained by the mechanism and the optimal value $o^*$ obtained by the optimizer over the original data.

§ PROBLEM STATEMENT

In the context of the underlying dispatch problem, the DSO collects a
data set $D=\{d_i^{p}\}_{i \in \set{N}}$ of customer sensitive loads and dispatches the DER according to the
solution of the OPF model (<ref>).  The OPF model can be viewed as
a mapping $\OPF:D\mapsto s$ from the data set $D$ into an optimal OPF
solution $s^{\star}$. The solution is a tuple comprising generator set
points $\{g_i^p,g_i^q\}_{i \in\set{N}}$, power flows $\{f_\ell^p,
f_\ell^q\}_{\ell \in \set{L}}$, and voltages $\{u_i\}_{i \in \set{N}}$
which is depicted on the left plane in Fig. <ref>.
However, the release of $s^{\star}$ poses an important privacy threat:
an adversary with access to the items in $s^{\star}$ could
decode the customers activities
<cit.>. For instance, the voltage sags at a
node of interest discloses the activity of residential costumers
(e.g., charging an electrical vehicle). Voltages and flows (currents)
also encode information about the technology, production patterns, and
other commercial data of industrial customers <cit.>.

To minimize privacy risks, this work proposes a mechanism
$\tilde{{\cal M}}$ for the DSO that returns instead a sub-optimal, yet
feasible, solution $\tilde{s}$ at the expense of an optimality loss as
shown in Fig. <ref>.  A non-trivial benefit of
choosing $\tilde{s}$ over $s^{\star}$ is that the former includes a
particular perturbation of the optimal solution and thus carries less
information about the real load data $D$. For instance, the sub-optimal
solution can feature a more intensive deployment of the DERs to
compensate for the voltage sags instead of purchasing power from the
high-voltage grid. To ensure that $\tilde{{\cal M}}$ returns a
differentially private solution, $\tilde{s}$ has to follow a carefully
calibrated distribution as depicted on the right plane in
Fig. <ref>. In other words, the mechanism must satisfy
Definition <ref>, i.e., on adjacent load data sets, the
mechanism output should be a distribution (describing the generator outputs, flows, and voltages) that differ by at most
$\varepsilon$ and $\delta$ in multiplicative and additive terms
respectively. However, with smaller $\varepsilon$ and $\delta$, the
variance of the OPF solutions increases and hence, denoting with $\set{S}$ the feasible OPF space, the probability
$\mathbb{P}[\tilde{s}\notin\set{S}]$ of producing an infeasible
solution also increases. Therefore,
$\mathbb{P}[\tilde{s}\notin\set{S}]$ should remain small and, ideally,
controlled by the DSO. Finally, the last desired property is the
ability of the mechanism to control the induced optimality loss
$\Theta$ to ensure cost-effective grid operations. The following section provide a mathematical description of
mechanism $\tilde{{\cal M}}$ and details its application, while
Section <ref> reviews its desirable properties and

§ DIFFERENTIALLY PRIVATE OPF MECHANISM

§.§ Random Perturbation of OPF Solutions

To guarantee differential privacy, the proposed mechanism focuses on perturbing the OPF variables $\{g_{i}^{\dag}\}_{i\in\set{N}}$ and
$\{f_{i}^{\dag},u_{i}\}_{i \in \set{L}}$.
Notice however that the OPF equations
(<ref>)–(<ref>) couple these variables, thus, it suffices to perturb only one set of those variables to induce a change
in the overall solution. For instance, by perturbing the active power
flow $f_{i}^{p}$, the reactive power flow $f_{i}^{q}$, voltage
magnitude $u_{i}$, and DER output $g_{i}^{\dag}$ associated with node
$i$, need to adjust accordingly. The perturbation of power flows is also
convenient as the sensitivity of power flows to loads can be directly
upper-bounded by the load magnitudes, as described in Lemma <ref>.

As a result, the proposed mechanism perturbs the active power flows by
a random variable $\xi\in \RR^{|\set{L}|}$ that obeys a Normal
distribution $\mathcal{N}(0,\Sigma)$ with covariance matrix

We will refer to $\Sigma$, $\sigma^{2}$, and $\sigma$ interchangeably
to discuss the parameters of the perturbation. Moreover, the mechanism
imposes an affine policy to react to this perturbation, i.e., the
DSO dispatches the network DERs according to the following policy:

\begin{align}\label{affine_policy} \tilde{g}_{i}^{p} = g_{i}^{p} +
\mysum{\ell\in\set{D}_{i}}\alpha_{i\ell}\xi_{\ell} -
\mysum{\ell\in\set{U}_{i}}\alpha_{i\ell}\xi_{\ell},
\;\forall i\in\set{N},
\end{align}

where $\tilde{g}_{i}^{p}$ and $g_{i}^{p}$ are, respectively, the random and nominal active power outputs, and
$\alpha_{i\ell}$ is the portion of random perturbation $\xi_{\ell}$ provided by generator at node $i$, modeled as a free variable.
The sets $D_i$ and $U_i$ denote, respectively, the downstream and upstream nodes w.r.t. node $i$ in the network.
To make policy (<ref>) viable, the following balancing conditions are enforced:

\begin{align}\label{policy_balance}
\mysum{i\in\set{U}_{\ell}}\alpha_{i\ell}=1,
\;\mysum{i\in\set{D}_{\ell}}\alpha_{i\ell}=1,
\;\forall \ell\in\set{L},
\end{align}

such that the perturbation of flow $f_\ell^p$ causes the upstream DERs to adjust their aggregated output by $\xi_{\ell}$ and the downstream DERs to counterbalance this perturbation by $\xi_{\ell}$.

To provide a succinct representation of the randomized OPF variables, we introduce a topology matrix
$T \in \mathbb{R}^{|\set{N}|\times|\set{L}|}$ whose elements are such that:

\begin{equation*}
T_{i\ell} = \left\{
\begin{array}{rl}
1, & \text{if line $\ell$ is downstream w.r.t.~node $i$},\\
-1, & \text{if line $\ell$ is upstream w.r.t.~node $i$} \\
0, & \text{otherwise}.
\end{array} \right.
\end{equation*}

We further consider an auxiliary row vector
that returns a Schur product of $i^{\text{th}}$ row of $T$ and $i^{\text{th}}$ row of $\alpha$, and set $\rho_{i}^{q} = \rho_{i}^{p}\tan\phi_{i}$.
Using this notation, the perturbed OPF solution can be modeled in terms of the following random variables:

= g_i^†+ρ_i^†ξ,
∀i ∈N,

= f_ℓ^† - [ρ_ℓ^†+j∈D_ℓρ_j^†]ξ,

= u_i + 2j∈R_i
[r_j(ρ_j^p + k∈D_jρ_k^p) +

x_j(ρ_j^q +k∈D_jρ_k^q)]ξ,     ∀i ∈L,

where the randomized power flows $\tilde{f}_{i}^{\dag}$ are obtained
by substituting generator policy (<ref>) into
(<ref>), and randomized voltage magnitudes
$\tilde{u}_{i}$ are expressed by substituting $\tilde{f}_{i}^{\dag}$
into (<ref>).  Each variable is thus represented by
its nominal component and its random component whose realization
depends on $\xi$. By properly calibrating parameters $\xi$ (See
Section <ref> (Theorem <ref>)), the
randomized OPF solution in (<ref>) will provide
the required differential privacy guarantees for grid
customers. However, there is no guarantee that, after adding this
calibrated noise to an optimal dispatch, the randomized OPF solution
in (<ref>) will be a feasible dispatch. The next
section addresses this issue through a chance-constrained optimization
problem formulation.

§.§ The Chance-Constrained Optimization Problem

To obtain a feasible dispatch the proposed mechanism uses a chance-constrained optimization $\tilde{\OPF}$ whose optimal solution, when perturbed, will be feasible w.h.p.
The formulation of the chance-constrained optimization is given in
Fig. <ref>: It finds a solution to constraints
(<ref>) that also satisfies the grid limits up to
a DSO-specified probability. The formulation of the chance-constrained
optimization inserts equations (<ref>) into the
base OPF model (<ref>). It minimizes the expected cost, and
enforces chance constraints for the grid limits. These steps are
detailed below.

§.§.§ Objective Function Reformulation

The chance-constrained formulation minimizes the expected cost

\[
\mathbb{E}_{\xi}\Big[\mysum{i\in\set{N}}(c_{1i}\tilde{g}_{i}^{p} + c_{2i}(\tilde{g}_{i}^{p})^{2})\Big].
\]

Due to the linearity of expectations, it can be separated into linear and quadratic terms.
The linear term rewrites as

\mathbb{E}_{\xi}[c_{1i}\tilde{g}_{i}^{p}] = c_{1i}g_{i}^{p},\;\forall i\in\set{N},

due to the zero-mean distribution of $\xi$. The quadratic term is the expectation of the squared random variable, which expresses analytically through the second moment as:

\begin{align*}
\mathbb{E}_{\xi}[c_{2i}(\tilde{g}_{i}^{p})^2]=&\mathbb{E}_{\xi}[c_{2i}(g_{i}^{p}+\rho_{i}^{p}\xi)^2] \nonumber\\
=&c_{2i}(g_{i}^{p})^2 + (c_{2i}\rho_{i}^{p})\Sigma(c_{2i}\rho_{i}^{p})^{\top},\;\forall i\in\set{N},
\end{align*}

where second term follows from the fact that $\mathbb{E}[\xi^2]=\sigma^2.$

§.§.§ Approximation of the Quadratic Power Flow Constraints

The left-hand side of apparent power flow limit

(\tilde{f}^p_\ell)^2 + (\tilde{f}^p_\ell)^2 \leq \bar{f}^2_\ell,

exhibits a quadratic dependency on random variable $\xi$, for which no tractable reformulation is known. To resolve this issue, the above quadratic constraint is replaced by the inner polygon <cit.>, which writes as

\begin{align*}
\gamma_{c}^{p}\tilde{f}_{i}^{p} + \gamma_{c}^{q}\tilde{f}_{i}^{q} + \gamma_{c}^{s}\overline{f}_{i}\leqslant0,\;\forall i\in\set{L},\forall c\in\set{C},
\end{align*}

where $\gamma_{c}^{p}, \gamma_{c}^{q}$, and $\gamma_{c}^{s}$ are the coefficients for each side $c$ of the polygon. The details of this approximation are given in <cit.>. The cardinality $|\set{C}|$ is arbitrary, but a higher cardinality brings a better accuracy. The experimental results in this paper choose $|\set{C}|=12.$

§.§.§ Conic Reformulation of Linear Chance Constraints

For the normally distributed random variable $\xi$, the chance constraint of the form $\mathbb{P}_{\xi}[\xi^{\top}x\leqslant b]\geqslant1-\eta$ is translated into a second-order cone constraint as <cit.>:

$$ z_{\eta}\norm{\text{Std}(\xi^{\top}x)}_{2}\leqslant b - \mathbb{E}_{\xi}[\xi^{\top}x],$$

where $z_{\eta}=\Phi^{-1}(1-\eta)$ is the inverse cumulative distribution function of the standard Gaussian distribution at the $(1-\eta)$ quantile.
To enable the DSO to control the probability of constraint violation, the individual chance constraints on the generation, voltage, and power flow variables are formulated in a conic form in Equations (<ref>)–(<ref>), introduced next.

\begin{align}
&\tilde{\OPF}(D)\coloneqq \bigg\{\minimize{\set{V}=\{g^{\dag},f^{\dag},u,\rho^{\dag}\}}\;\mysum{i\in\set{N}}\Big(c_{1i}g_{i}^{p} + c_{2i}(g_{i}^{p})^2 \nonumber\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+ (c_{2i}\rho_{i}^{p})\Sigma(c_{2i}\rho_{i}^{p})^{\top}\Big)\label{cc_OPF_ref_obj}\\
& z_{\eta^{g}}\normtwo{\big}{\rho_{i}^{\dag}\sigma}_{2}\leqslant\overline{g}_{i}^{\dag}-g_{i}^{\dag},\;\forall i\in\set{N},\label{cc_OPF_ref_gen_max}\\
& z_{\eta^{g}}\normtwo{\big}{\rho_{i}^{\dag}\sigma}_{2}\leqslant g_{i}^{\dag} - \underline{g}_{i}^{\dag},\;\forall i\in\set{N},\label{cc_OPF_ref_gen_min}\\
&z_{\eta^{u}}\Big\lVert\Big[\mysum{j\in\set{R}_{i}}\big[r_{j}\big(\rho_{j}^{p} +\mysum{k\in\set{D}_{j}}\rho_{k}^{p}) + x_{j}\big(\rho_{j}^{q} +\mysum{k\in\set{D}_{j}}\rho_{k}^{q}\big)\big]\Big]\sigma\Big\rVert_{2}\nonumber\\
&\quad\leqslant \textstyle\frac{1}{2}\left(\overline{u}_{i}-u_{i}\right), \;\forall i\in\set{L},\label{cc_OPF_ref_volt_max}\\
&z_{\eta^{u}}\Big\lVert\Big[\mysum{j\in\set{R}_{i}}\big[r_{j}\big(\rho_{j}^{p} +\mysum{k\in\set{D}_{j}}\rho_{k}^{p}) + x_{j}\big(\rho_{j}^{q} +\mysum{k\in\set{D}_{j}}\rho_{k}^{q}\big)\big]\Big]\sigma\Big\rVert_{2}\nonumber\\
&\quad\leqslant \textstyle\frac{1}{2}\left(u_{i} - \underline{u}_{i}\right), \;\forall i\in\set{L},\label{cc_OPF_ref_volt_min}\\
&\quad-\gamma_{c}^{p}f_{\ell}^{p} - \gamma_{c}^{q}f_{\ell}^{q} - \gamma_{c}^{s}\overline{f}_{\ell},\;\forall \ell\in\set{L},\forall c\in\set{C}\label{cc_OPF_ref_flow}\\
&\bigg\}\mapsto \;\optimal{\set{V}}=\{\optimal{g}^{\dag},\optimal{f}^{\dag},\optimal{u},\optimal{\rho}^{\dag}\}.\nonumber
\end{align}

The Chance-Constrained Optimization at the Core of the Privacy-Preserving Mechanism.

§.§ The Privacy-Preserving Mechanism

The privacy-preserving mechanism $\tilde{{\cal M}}$ is depicted in
Fig. <ref>. The first step defines the
covariance matrix $\Sigma$ that encodes the DP parameters
$\varepsilon, \delta$, and adjacency parameter $\beta$. The mechanism
then solves the optimization (<ref>) to obtain an
optimal solution $\optimal{\set{V}}$ which will be feasible for any
realization of $\xi\sim\mathcal{N}(0,\Sigma)$ up to given constraint
violation probabilities. In the last step, the mechanism samples $\xi$
to obtain the final OPF solution using Equations

Clearing = [rectangle, rounded corners = 5, minimum width=10, minimum height=10,text centered, draw=black, fill=white!30,line width=0.3mm]

[align=center] (step_1) [Clearing]
Define parameters

of perturbation $\Sigma$
[align=center] (step_2) [Clearing, xshift = 3cm]
Compute optimal

solution to (<ref>)

[align=center] (step_3) [Clearing, xshift = 6.25cm]
Sample $\xi\sim \mathcal{N}(0,\Sigma).$
Obtain OPF solution
using equations (<ref>)
[transform canvas=xshift=0cm,->,>=stealth,line width=0.2mm] (step_1) – node[midway, above, sloped] $\Sigma$  (step_2) ;
[transform canvas=xshift=0cm,->,>=stealth,line width=0.2mm] (step_2) – node[midway, above, sloped] $\optimal{\set{V}}$  (step_3) ;
[draw,->,>=stealth, line width=0.2mm] (step_1.south) –++ (0,-0.5) -| (step_3.south) node[midway, right, yshift=7] $\Sigma$  ;

The Privacy-Preserving Mechanism $\tilde{{\cal M}}$.

§ MECHANISM PROPERTIES

This section first discusses the feasibility and privacy guarantees
provided by mechanism $\tilde{{\cal M}}$. It then explains how to
extend the mechanism in order to control of the OPF variance and the
optimality loss. For reference, the summary of all mechanisms and
their properties is given in Table <ref>.

Notation     ${\cal M}$      $\tilde{{\cal M}}$     $\tilde{{\cal M}}_{\text{a}}$     $\tilde{{\cal M}}_{\text{b}}$     $\tilde{{\cal M}}_{\text{c}}$     $\tilde{{\cal M}}_{\text{d}}$

Formulation     (<ref>)     (<ref>)     (<ref>)     (<ref>)     (<ref>)     (<ref>)

OPF feasibility     $+$     $+$     $+$     $+$     $+$     $+$

DP guarantees     –     $+$     $+$     $\pm$     $+$     $+$

Variance control     –     –     $+$     $+$     –     –

Opt. loss control     –     –     –     –     $+$     $+$

Summary of the Properties for the Mechanisms.

This table is not clear. The mechanisms are never discussed. Also what does a $\pm$ means as for DP guarantees?

§.§ Feasibility and Privacy Guarantees

After solving the chance-constrained optimization, the mechanism can be viewed as specifying a probability density function for the OPF variables in  (<ref>). The DSO then obtains a realization of the OPF variables by sampling random variable $\xi$. By design, the sampled solution is guaranteed to satisfy grid limits and customer loads up to the DSO-specified violation probabilities $\eta^{g},\eta^{u}$ and $,\eta^{f}$, of the generator, voltage, and flow constraints.

The privacy guarantees, in turn, depend on the specification of DP parameters $(\varepsilon,\delta)$ and the vector of adjacency coefficients $\beta.$ For simplicity, the DP parameters are assumed to be uniform for all customers and specified by the DSO, whereas customer privacy preferences are expressed in the submitted adjacency coefficients. In this setting, the load of every customer $i$ is guaranteed to be indistinguishable from any other load in the range $[d_{i}^{p}-\beta_{i},d_{i}^{p}+\beta_{i}]$ in the release of OPF solution related to node $i$ up to DP parameters $(\varepsilon,\delta)$. This guarantee is formalized by the following result.

This statement is wrong as it is. Cannot use equalities
Let $\varepsilon,\delta\in(0,1)$ and $\sigma_{i} \geqslant
\beta_{i}\sqrt{2\text{ln}(1.25/\delta)}/\varepsilon, \;\forall
i\in\set{L}.$ Then, if mechanism $\tilde{{\cal M}}$ returns an optimal
solution, it is $(\varepsilon, \delta)$-differentially private for
$\beta$-adjacent load datasets. That is, the probabilities of
returning the same power flow solution $\tilde{f}^{p}$ on two
$\beta$-adjacent datasets $D$ and $D'$ are such that

\begin{equation*}
\mathbb{P}[\tilde{{\cal M}}(D) \in \tilde{f}^{p}] \leqslant e^\varepsilon \mathbb{P}[\tilde{{\cal M}}(D') \in \tilde{f}^{p}] + \delta,
\end{equation*}

where $\mathbb{P}$ denotes the probability over runs of $\tilde{{\cal M}}$.

The proof relies on two intermediate results summarized in Lemmas
<ref> and <ref> in the
appendix.  The first lemma shows that the standard deviation of power
flow related to customer $i$ is at least as much as
$\sigma_{i}$. Therefore, by specifying $\sigma_{i}$, the DSO attains
the desired degree of randomization. The second lemma shows that
$\beta_{i}\geqslant\Delta_{i}^{\beta}$, i.e., if $\sigma_{i}$ is
parameterized by $\beta_{i}$, then $\sigma_{i}$ is also parameterized
by sensitivity $\Delta_{i}^{\beta}$, required by the Gaussian
mechanism in Theorem <ref>. The full proof, is available
in an online Appendix <cit.>.

§.§ OPF Variance Control

Due to the radial topology of distribution grids, the flow perturbations in lines along the same radial branch may accumulate larger flow variance than that intended by covariance matrix $\Sigma$. This section extends the privacy preserving mechanism $\tilde{\OPF}$ to reduce the overall flow variance while still preserving privacy guarantees. Two strategies are analyzed to achieve this goal.

§.§.§ Total Variance Minimization

To reduce the variation of OPF variables, this paper exploits a result from variance-aware control of networks from
<cit.>. The variance of flows can be
reduced by the following optimization:

\begin{align}
\tilde{\OPF}_{\text{a}}\coloneqq\bigg\{\minimize{\set{V}\cup \{t\}}\quad&
\text{Obj. function}\;\eqref{cc_OPF_ref_obj} + \mysum{\ell\in\set{L}}\psi_{\ell}t_{\ell}\\
\text{s.t.}\quad&\normtwo{\Big}{\big[\rho_{\ell}^{p}+\mysum{i\in\set{D}_{\ell}}\rho_{i}^{p}\big]\sigma}_{2}\leqslant t_{\ell},\;\forall\ell\in\set{L},\label{total_variance_control_flow_std}\\
\end{align}

where the decision variable $t_{\ell}$ represents the standard deviation
of the active power flow in line $\ell$, which is penalized in the
objective function by a non-negative parameter $\psi_{\ell}$. By
choosing $\psi_{\ell},\forall \ell\in\set{L}$, the DSO trades off the
total variance and the expected operational cost.  Note that, by
Lemmma <ref>, optimization
(<ref>) does not violate the privacy guarantees.

§.§.§ Pursuing Target Variance

This strategy solely perturbs the flow in the selected line of the radial branch (e.g., adjacent to the customer with the strongest privacy requirement) and constrains DERs to maintain the flow variance in each line as required by the original covariance matrix $\Sigma$. It specifies a new matrix $\hat{\Sigma}=\text{diag}([\hat{\sigma}_{1}^2,\dots,\hat{\sigma}_{|\set{L}|}^2])$, $\mathbbm{1}^{\top}\hat{\sigma}^2\leqslant \mathbbm{1}^{\top}\sigma^2,$ and solves

\begin{align}
\tilde{\OPF}_{\text{b}}\coloneqq\bigg\{\minimize{\set{V}\cup \{t,\tau\}}\quad&
\text{Obj. function}\;\eqref{cc_OPF_ref_obj} + \mysum{\ell\in\set{L}}\psi_{\ell}\tau_{\ell}\\
\text{s.t.}\quad&\normtwo{\Big}{\big[\rho_{\ell}^{p}+\mysum{i\in\set{D}_{\ell}}\rho_{i}^{p}\big]\hat{\sigma}}_{2}\leqslant t_{\ell},\;\forall\ell\in\set{L},\label{target_variance_control_flow_std}\\
&\normtwo{\big}{t_{\ell}-\sigma_{\ell}}_{2}\leqslant \tau_{\ell},\;\forall \ell\in\set{L},\label{target_variance_control_var_distance}\\
&\text{Equations}\;\eqref{cc_OPF_ref_det_con}-\eqref{cc_OPF_ref_flow} \;\text{with}\; \hat{\sigma}\bigg\}.
\end{align}

Here, $t_{\ell}$ returns the resulting flow standard deviation, while constraint (<ref>) yields the distance $\tau_{\ell}$ between the resulting standard deviation and original one $\sigma_{\ell}=\Sigma_{\ell,\ell}^{1/2}$ required to provide customer at node $\ell$ with differential privacy. By penalizing this distance in the objective function, the DSO attains privacy at a smaller amount of random perturbations.

As optimization (<ref>) acts on covariance matrix $\hat{\Sigma}$ instead of $\Sigma$, the privacy guarantees need to be verified post factum, i.e., by verifying that $t_{\ell}\geqslant\sigma_{\ell},\;\forall\ell\in\set{L}.$

I suggest to remove this remark. It is strange to talk about privacy to be computed a-posteriori and we also never really use it.

§.§ Optimality Loss Control

The application of mechanism $\tilde{\cal{M}}$ necessarily leads to an
optimality loss compared to the solution of privacy-agnostic mechanism
$\OPF$. In this section, we abuse notation and use $\OPF(D)$ and
$\tilde{{\cal M}}(D)$ to denote, respectively, the costs of the OPF optimal solutions and that of mechanism $\tilde{{\cal M}}$ on a data set $D$. The optimality loss is measured in expectation as

as $\OPF(D)$ always provides a deterministic
solution. However, the worst-case realization of operational cost in
$\tilde{{\cal M}}(D)$ may significantly exceed the expected value and
lead to a larger optimality loss. To this end, this section introduces
two control strategies for the optimality loss by optimizing variance
and conditional Value-at-Risk (CVaR) measures
<cit.>. The strategies are streamlined by
observing that

$$\text{Var}[\Theta]=\text{Var}[\OPF(D)-\tilde{{\cal M}}(D)]=\text{Var}[\tilde{{\cal M}}(D)],$$

such that the optimality loss control boils down to the integration of the two measures into DP mechanism $\tilde{{\cal M}}$. Variance and CVaR optimization for quadratic cost functions, however, involves non-convexities. Therefore, to preserve the convex properties of the OPF mechanism, the strategies are solely applied to the linear DER cost functions.

§.§.§ Controlling the Variance of Cost

For the total operational cost, the expected value finds as

\begin{align*}
\mu_{c}:=\mathbb{E}[c_{1}^{\top}\tilde{g}^{p}]=\mathbb{E}[c_{1}^{\top}(g^{p}+\rho\xi)]=c_{1}^{\top}g^{p}
\end{align*}

for zero-mean $\xi$, and the standard deviation finds as

\begin{align*}
\sigma_{c}:=\text{Std}[c_{1}^{\top}(g^{p}+\rho\xi)]=\text{Std}[(c_{1}^{\top}\circ\rho)\xi]=\norm{(c_{1}^{\top}\circ\rho)\sigma}_{2}.
\end{align*}

For some parameter $\theta\in[0,1]$, the DSO trades off the mean and
standard deviation of the loss by solving

\begin{align}
\tilde{\OPF}_{\text{c}}\coloneqq\bigg\{\minimize{\set{V}\cup \{\mu_{c},\sigma_{c}\}}\quad&(1-\theta)\mu_{c}+\theta\sigma_{c}\label{cost_std_optimization_obj}\\
\text{s.t.}\quad&\norm{(c_{1}^{\top}\circ\rho)\sigma}_{2}\leqslant\sigma_{c},\\
\end{align}

§.§.§ Controlling the Conditional Value at Risk

An alternative strategy is to control the optimality loss in
$(1-\varrho)\%$ of the worst-case outcomes, which can be achieved by
using conditional VaR $(\text{CVaR}_{1-\varrho})$ risk measure. For a
normally distributed random variable $\xi$, $\text{CVaR}_{1-\varrho}$
admits the following convex formulation <cit.>:

\begin{align*}
\text{CVaR}_{1-\varrho} := \mu_{c} + \sigma_{c}\phi\left(\Phi^{-1}(\varrho)\right)/(1-\varrho),
\end{align*}

where $\mu_{c}$ and $\sigma_{c}$ respectively represents the expected value and standard deviation of operational cost, while  $\phi$ and $\Phi^{-1}(\varrho)$ respectively denote the probability density function and the inverse cumulative distribution function at the $\varrho$ quantile of the standard Normal distribution. For some parameter $\theta\in[0,1]$, the DSO trades off the mean and $\text{CVaR}_{1-\varrho}$ of optimality loss by solving

\begin{align}
\tilde{\OPF}_{\text{d}}\coloneqq\bigg\{\minimize{\set{V}\cup \{\mu_{c},\sigma_{c}\}}&(1-\theta)\mu_{c}+\theta\left[\mu_{c} + \textstyle\sigma_{c}\frac{\phi\left(\Phi^{-1}(\varrho)\right)}{(1-\varrho)}\right]\\
\text{s.t.}\quad&\norm{(c_{1}^{\top}\circ\rho)\sigma}_{2}\leqslant\sigma_{c},\\
\end{align}

The optimality loss control by means of (<ref>)
and (<ref>) does not violate the privacy
guarantees as per Lemma <ref>.

§ NUMERICAL EXPERIMENTS

Section <ref> first illustrates the
application of mechanism $\tilde{{\cal M}}$ to obfuscate the load
pattern of grid customers in voltage measurements. Section
<ref> illustrates the privacy and
feasibility guarantees of the mechanism. Section
<ref> explains its variance control
properties. Finally, Section <ref> presents
the optimality loss trade-offs that the mechanism offers to the DSO.

§.§ Experiment Setting

The experiments consider a radial feeder introduced in <cit.>
with adjusted parameters taken from repository <cit.>.
Each node is equipped with a DER with quadratic and linear cost
coefficients drawn from Normal distributions $c_{2i}\sim
\mathcal{N}(0.05,0.01)$ 0.85$/kWh$^2$ and $c_{1i}\sim
\mathcal{N}(0.01,0.02)$ 0.85$/kWh respectively. By
default, the DP parameters are set to $\varepsilon\rightarrow1,
\delta\rightarrow0.75,$ whereas the adjacency parameter $\beta$ varies
depending on the experiment. The constraint violation probabilities
are $\eta^{g}=2\%, \eta^{u}=2\%$ and $\eta^{f}=5\%$.

I think that $\delta$ is too large. It should be a value in $O(1/n)$ with $n$ being the number of customers.

§.§ Illustrative Example

Consider a customer at node 1 with a maximum load of 80
0.85kW. Its load pattern was obtained by multiplying the
maximum load by $k(t)$ multiplier with three periodic components:

\begin{align*}
k(t)=&\textstyle\frac{9}{10}\text{max}\left\{|\sin{\frac{3}{10^2}t}|,\frac{95}{100}\right\} +\frac{1}{10^2}\sin{\frac{5}{10^2}t}+\frac{1}{10^3}\sin{\frac{1}{2}t}
\end{align*}

where $t$ is a time step. The information about this pattern is
leaking through the voltage measurements $v_{1}$ as depicted on the
left plot in Fig. <ref>. The figure shows how the voltage readings are
obfuscated for different assignment of $\beta_{1}$. By setting
$\beta_{1}\rightarrow0.5$ 0.85kW, the presence of the
smallest periodic component in the load dataset is hidden through
randomization, whereas the presence of two remaining components can be
clearly distinguished. By increasing $\beta_{1}$, the medium and
larger periodic components are gradually obfuscated as well.

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$\pm$ 3 Std;

Voltage magnitude at node 1 as a function of adjacency $\beta_{1}$: all periodic components are gradually obfuscated with an increasing adjacency.

§.§ Privacy and Feasibility Guarantees

To illustrate the privacy guarantees, consider a grid customer at the
end of distribution feeder (node 7), where voltage magnitudes are
often close to the boundaries of feasible region. For
$\beta_{7}\rightarrow \frac{1}{2}d_{7}^{p}$, there exist two adjacent
load datasets $D'$ and $D''$ containing $d_{7}^{\prime
p}=\frac{1}{2}d_{7}^{p}$ and $d_{7}^{\prime\prime
p}=\frac{3}{2}d_{7}^{p}$ respectively. The privacy-agnostic
mechanism returns the following voltage magnitudes on the three

\begin{align*}
\OPF(D') = 1.0964,\;
\OPF(D) =& 1.0938,\;
\OPF(D'')= 1.0912,
\end{align*}

clearly distinguishing the differences in datasets through voltage
readings.  The differentially private mechanism $\tilde{{\cal M}}$,
obfuscates the load value used in computation as displayed in
Fig. <ref>. It displays the DP property
according to Definition <ref>: the randomization of voltage
outcomes enables to make the load indistinguishable from any other
load in the range $[\frac{1}{2}d_{7}^{p},\frac{3}{2}d_{7}^{p}]$ up to
privacy parameters $(\varepsilon,\delta)$. With
$\delta\rightarrow0.75$, the distributions slightly stand out from one
another, whereas $\delta\rightarrow0.05$ yields a larger noise
magnitude that overlaps the support of the three
distributions. Notice, the OPF solution to be implemented is a sample
drawn from these distributions. By observing a single sample, an
adversary cannot infer from which distribution (on what dataset) it
was obtained. Finally, mechanism $\tilde{{\cal M}}$ ensures the
probability of a feasible voltage solution, i.e.,
$\tilde{v}_{7}\in[0.9,1.1]$, remains at least $(1-\eta^{u})=98\%$.

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The overlay of voltage probability densities yielded by differential private mechanism $\tilde{{\cal M}}$ on the three $\beta-$adjacent load datasets.

§.§ OPF Variance Control

This section extends the analysis to variance-aware OPF computations
by means of mechanisms $\tilde{{\cal M}}_{\text{a}}$ and $\tilde{{\cal
M}}_{\text{b}}$.  what are the rows of that
table. Please specify.Table <ref> contrasts the
solutions provided by $\OPF$ and the privacy-aware mechanisms with and
without variance control.  I do not understand this
sentence. What does encode mean? How does sigma encode anything? In
the third column, the standard deviation of perturbations $\sigma$
encodes DP parameters $(\varepsilon,\delta)$ and customer-specific
adjacency set to $\beta\rightarrow15\%$ of the nominal load.

Mechanism $\tilde{{\cal M}}$ yields an expected optimality loss of
$\mathbb{E}[\Theta]=31.0\%$. However, the flow variance exceeds the
intended quantities, putting more stress on power grid. To
minimize the negative impact of the noise, the DSO may apply instead
the variance-aware mechanisms $\tilde{{\cal M}}_{\text{a}}$ or
$\tilde{{\cal M}}_{\text{b}}$. The former trades off the cost of
operation and sum of standard deviations of power flows. By setting
$\psi_{i}\rightarrow10^5$ in (<ref>), the DSO
reduces the standard deviation of power flows by 41% at a small
increase in expected optimality loss. The OPF variance can be
further reduced by perturbing flows only in selected edges according
to mechanism $\tilde{{\cal M}}_{\text{b}}$. Here, the DSO solely
perturbs edges $1$, $9$, $5$, $12$ and constrains DERs to maintain the
target standard deviation of power flows $\sigma$ across the grid. The
results show that with almost no difference in cost, the solution
provides significantly smaller variance for OPF variables. Moreover, as
per Remark <ref>, the privacy guarantees hold as
resulting standard deviations are at least as much as target values

Solutions for different OPF mechanisms. The mean values are followed by standard deviations in parentheses

3*$i$     3*$d_{i}^{p}$     3*$\sigma_{i}$     3c|Privacy-agnostic     3c|Private – $\tilde{{\cal M}}$     3c|Private – $\tilde{{\cal M}}_{\text{a}}$     3cPrivate – $\tilde{{\cal M}}_{\text{b}}$

3c|$\Theta=0\%$     3c|$\mathbb{E}[\Theta]=31.0\%,\sum_{i}\text{Std}(f_{i}^{p})=\textbf{65.5}$     3c|$\mathbb{E}[\Theta]=33.9\%,\sum_{i}\text{Std}(f_{i}^{p})=\textbf{38.9}$     3c$\mathbb{E}[\Theta]=34.5\%,\sum_{i}\text{Std}(f_{i}^{p})=\textbf{26.0}$

$g_{i}^{p}$     $f_{i}^{p}$     $v_{i}$     $g_{i}^{p}$     $f_{i}^{p}$     $v_{i}$     $g_{i}^{p}$     $f_{i}^{p}$     $v_{i}$     $g_{i}^{p}$     $f_{i}^{p}$     $v_{i}$

0                         -                         -                          85.7          -            1.00         117 (15.2)     -          1.00 (0)             117 (15.1)          -          1.00 (0)            117 (15.1)       -          1.00 (0)

1                         79.4                      12                       0.0           55.9         1.00         5.1 (2.5)      71.9 (12)      0.99 (0.007)           18.7 (9)              72.2 (12)      0.99 (0.007)     23.8 (11.5)        71.9 (12)      0.99 (0.008)

2                         0.0                       0.0                        0.0           27.9         1.03         0.0 (0)          -2.3 (9.5)     0.99 (0.022)         0.0 (0)               11.5 (3)       0.97 (0.012)     0.0 (0)            16.5 (0.7)     0.96 (0.008)

3                         2.0                       0.3                        2.7           27.9         1.09         0.0 (0)          -2.2 (9.5)     1.00 (0.045)         1.0 (0.5)             11.4 (3)       0.95 (0.02)      0.0 (0)            16.5 (0.7)     0.93 (0.01)

4                         1.7                       0.3                        28.8          25.3         1.10         14.2 (6.9)     -9.0 (7.3)     1.01 (0.047)           0.0 (0)               4.1 (1.3)      0.95 (0.02)      0.0 (0)            5.8 (0.5)      0.93 (0.01)

5                         2.9                       0.4                        0.0           5.3          1.10         0.1 (0.1)      3.5 (0.7)      1.01 (0.047)           0.1 (0)               2.4 (1.3)      0.95 (0.021)     1.0 (0.5)          4.1 (0.5)      0.93 (0.01)

6                         2.2                       0.3                        0.0           2.3          1.10         1.4 (0.7)      0.8 (0.7)      1.01 (0.047)           2.6 (1.3)             -0.4 (1.3)     0.95 (0.022)     0.0 (0)            2.2 (0.3)      0.93 (0.01)

7                         1.3                       0.2                        5.9           5.3          1.10         1.7 (0.8)      -0.4 (0.8)     1.00 (0.047)           1.1 (0.5)             0.3 (0.5)      0.95 (0.021)     0.0 (0)            1.3 (0.2)      0.92 (0.01)

8                         2.4                       0.4                        0.0           2.1          1.09         0.0 (0)        4.6 (2.3)      1.00 (0.046)           0.0 (0)               6.3 (1.4)      0.95 (0.021)     0.0 (0)            8.6 (0.4)      0.92 (0.01)

9                         2.3                       0.3                        0.0           1.3          1.09         0.0 (0)        2.6 (1.5)      1.00 (0.046)           0.0 (0)               3.7 (1)        0.95 (0.021)     0.1 (0.1)          5 (0.4)        0.92 (0.01)

10                        2.2                       0.3                        4.4           1.6          1.09         2.2 (1.1)      0.4 (1.5)      1.00 (0.048)           0.7 (0.3)             1.4 (1)        0.95 (0.022)     0.7 (0.3)          2.8 (0.3)      0.92 (0.01)

11                        1.3                       0.2                        0.0           1.4          1.09         0.9 (0.4)      0.4 (0.4)      1.00 (0.047)           1.4 (0.7)             -0.1 (0.7)     0.95 (0.022)     0.0 (0)            1.3 (0.2)      0.92 (0.01)

12                        62.2                      9.4                        14.7          30.1         1.00         8.7 (4.3)      45.1 (9.4)     0.99 (0.006)           16.6 (8)              45.0 (9.5)     0.99 (0.006)     19.4 (9.4)         45.3 (9.4)     0.99 (0.006)

13                        0.1                       0.0                        5.2           19.8         1.04         1.7 (0.8)      -8.3 (5.1)     1.01 (0.017)           0.0 (0)               -0.6 (1.4)     1.00 (0.009)     0.0 (0)            2.3 (0)        0.99 (0.006)

14                        2.2                       0.3                        14.8          14.1         1.06         9.0 (4.3)        -6.7 (4.3)     1.02 (0.022)         3.0 (1.5)             -0.7 (1.4)     1.00 (0.011)     0.0 (0)            2.2 (0.3)      0.99 (0.006)

The table should be better described in the text, if we have space. I suggest sacrificing some detail, or content, perhaps in Section V to better explain the results.

§.§ Optimality Loss Control

This section concludes by showing how to control the optimality loss
using mechanisms $\tilde{{\cal M}}_{\text{c}}$ and $\tilde{{\cal
M}}_{\text{d}}$. Table <ref> collects the optimally
loss relative to the privacy-agnostic solution when setting the
adjacency to $\beta_{i}\rightarrow d_{i}^{p},\forall
$\beta_{1},\beta_{12}\rightarrow0$[Customers 1 and 12 are
excluded as either $\beta_{1}\rightarrow d_{1}^{p}$ or $\beta_{12}
\rightarrow d_{12}^{p}$ is a too strong requirement to be
accommodated, i.e., the models are infeasible.]. Table
<ref> presents the optimality loss for the linear DER
cost functions ($c_{2}\rightarrow0$), which is significantly smaller
than that for quadratic DER cost functions in Table
<ref>. By optimizing the cost variance, the
standard deviation of loss reduces in $\theta$ at the expense of
increasing expected value. By optimizing CVaR, the DSO trades off
the expected loss with that across 5% or 2% of the worst-case
scenarios. With increasing $\theta,$ it increases the expected loss
but more significantly improves the outcomes in the worst

Optimality loss control, $\Theta$ [%]

2*$\theta$     2c$\tilde{{\cal M}}_{\text{c}}$     2c$\tilde{{\cal M}}_{\text{d}}$, $\varrho=95\%$     2c$\tilde{{\cal M}}_{\text{d}}$, $\varrho=98\%$

exp.     std     exp.     $\text{CVaR}_{5\%}$     exp.     $\text{CVaR}_{2\%}$

0.0     3.88     0.97     3.88     5.89     3.88     6.23

0.2     3.89     0.92     3.92     5.62     3.94     5.83

0.4     4.00     0.67     4.08     5.24     4.17     5.29

0.6     4.57     0.13     4.38     4.94     4.56     4.89

0.8     4.84     0.00     4.60     4.83     4.66     4.85

1.0     4.84     0.00     4.68     4.82     4.84     4.84

The proof of Theorem <ref> relies on Lemmas <ref> and <ref>. The first lemma shows that the standard deviation of power flow related to customer $i$ is at least as much as $\sigma_{i}$. Therefore, by specifying $\sigma_{i}$, the DSO attains the desired degree of randomization.

If OPF mechanism (<ref>) returns optimal solution, then $\sigma_{\ell}$ is the lower bound on $\text{Std}[\tilde{f}_{\ell}^{p}]$.

Consider a single flow perturbation with $\xi_{\ell}\sim\mathcal{N}(0,\sigma_{\ell}^2)$ and $\xi_{j}=0,\;\forall j\in\set{L}\backslash \ell$. The standards deviation of active power flow (<ref>) in optimum finds as

\begin{align}
&\text{Std}\Big[\opt{f_{\ell}^{p}} - \Big[\overset{\star}{\rho_{\ell}^{p}}+\mysum{j\in\set{D}_{\ell}}\overset{\star}{\rho_{j}^{p}}\Big]\xi\Big]=\text{Std}\Big[\Big[\overset{\star}{\rho_{\ell}^{p}}+\mysum{j\in\set{D}_{\ell}}\overset{\star}{\rho_{j}^{p}}\Big]\xi\Big]=\text{Std}\Big[\mysum{j\in\set{D}_{\ell}}\overset{\star}{\alpha}_{j\ell}\xi_{\ell}\Big]\overset{\eqref{policy_balance}}{=}
\text{Std}\Big[\xi_{\ell}\Big]=
\sigma_{\ell},
\end{align}

where the second to the last equality follows from balancing conditions (<ref>).
As for any pair $(\ell,j)\in\set{L}$ the covariance matrix returns $\Sigma_{\ell,j}=0$, $\sigma_{\ell}$ is a lower bound on $\text{Std}[\tilde{f}_{\ell}^{p}]$ in the optimum for any additional perturbation in the network.

The result of Lemma <ref> holds independently from the choice of objective function and is solely driven by the feasibility conditions.

The second lemma shows that $\beta_{i}\geqslant\Delta_{i}^{\beta}$, i.e., if $\sigma_{i}$ is parameterized by $\beta_{i}$, then $\sigma_{i}$ is also parameterized by sensitivity $\Delta_{i}^{\beta}$.

Let $D$ and $D'$ be two adjacent datasets differing in at most one load $d_i^p$ by at most $\beta_i > 0$. Then,

\Delta_i^\beta = \max_{\ell \in \set{L}} \norm{\OPF(D)_{|f_\ell^p} - \OPF(D')_{|f_\ell^p}}_{2} \leqslant \beta_i

where the notation $\OPF(\cdot)_{|f_\ell^p}$ denotes the value of the optimal active power flow on line $\ell$ returned by the computation $\OPF(\cdot)$.

Let $\opt{f_\ell^p}$ be the optimal solution for the active power flow in line $\ell$ obtained on input dataset $D = (d_1^p, \ldots, d_n^p)$.
From OPF equation (<ref>), it can be written as

\opt{f_\ell^p}=d_{\ell}^{p} - \opt{g_{\ell}^{p}} + \mysum{i\in\set{D}_{\ell}}(d_{i}^{p}-\opt{g_{i}^{p}}),

which expresses the flow as a function of the downstream loads and the optimal DER dispatch.
A change in the active load $d_\ell^p$ translates into a change of power flow as

\begin{align}
\frac{\partial \opt{f_\ell^p}}{\partial d_{\ell}^{p}} &=
\underbrace{
\frac{\partial d_{\ell}^{p}} {\partial d_{\ell}^{p}}}_{1}
\frac{\partial\opt{g_\ell^p}}{\partial d_{\ell}^{p}}
\mysum{i\in\set{D}_{\ell}}
\Big(
\underbrace{
\frac{\partial d_{i}^{p}} {\partial d_{\ell}^{p}}}_{0}
\frac{\partial\opt{g_i^p}} {\partial d_{\ell}^{p}}
\Big)
1 - \frac{\partial\opt{g_\ell^p}}{\partial d_{\ell}^{p}}
- \mysum{i\in\set{D}_{\ell}}
\frac{\partial\opt{g_i^p}} {\partial d_{\ell}^{p}},\label{eq_ap_2_summ}
\end{align}

where the last two terms are always non-negative due to convexity of model (<ref>).  The value of (<ref>) attains maximum when

\begin{equation}
\label{eq:app_2}
\opt{g_{k}^{p}} = \overline{g}_{k}^{p}
\mapsto
\frac{\partial \opt{g_{k}^{p}}} {\partial d_{\ell}^{p}}
= 0,
\;\; \forall k \in \{\ell\} \cup \set{D}_{\ell}.
\end{equation}

Therefore, by combining (<ref>) with (<ref>) we obtain the maximal change of power flows as

\frac{\partial \opt{f_{\ell}^{p}}} {\partial d_{\ell}^{p}}= 1.

Since the dataset adjacency relation considers loads $d_\ell^p$ that differ by at most $\beta_\ell$, it suffices to multiply the above by $\beta_\ell$ to attain the result. It finds similarly that for a $\beta_i$ change of any load $i\in\set{N}$, all network flows change by at most $\beta_i$.

Consider a customer at non-root node $i$. Mechanism $\tilde{{\cal M}}$
induces a perturbation on the active power flow $f_{i}^{p}$ by a random variable $\xi_{i}\sim\mathcal{N}(0,\sigma_{i}^{2})$. The randomized active power flow $f_{i}^{p}$ is then given as follows:

\begin{align*}
\tilde{f}_{i}^{p} &=
\opt{f_{i}^{p}} - \big[\opt{\rho_{i}^{p}}
+ \mysum{j\in\set{D}_{i}} \opt{\rho_{j}^{p}}\big]\xi
\end{align*}

where $\star$ denotes optimal solution for optimization variables. For privacy parameters $(\varepsilon,\delta)$, the mechanism specifies

\sigma_{i} \geqslant \beta_{i}\sqrt{2\text{ln}(1.25/\delta)}/\varepsilon, \;\forall i\in\set{L}.

As per Lemma <ref>, we know that $\sigma_{i}$ is the lower bound on the standard deviation of power flow $f_{\ell}^{p}$. From Lemma <ref> we also know that the sensitivity $\Delta_{i}^{\beta}$ of power flow in line $i$ to load $d_{i}^{p}$  is upper-bounded by $\beta_{i}$, so we have

\begin{align*}
\label{power_flow_min_std}
\text{Std}[\tilde{f}_{i}^{p}] &\geqslant\sigma_{i}\geqslant\Delta_{i}^{\beta}\sqrt{2\text{ln}(1.25/\delta)}/\varepsilon.
\end{align*}

Since the randomized power flow follow is now given by a Normal distribution with the standard deviation $ \text{Std}[\tilde{f}_{i}^{p}]$ as above, by Theorem <ref>, mechanism $\tilde{{\cal M}}$ satisfies $(\varepsilon,\delta)$-differential privacy for each grid customer up to adjacency parameter $\beta$.


# Energy asymptotics for the strongly damped Klein-Gordon equation

Haidar Mohamad<EMAIL_ADDRESS>Katholische Universität Eichstätt-
Ingolstadt
Auf der Schanz 49
85049 Ingolstadt
Germany

###### Abstract.

We consider the strongly damped Klein Gordon equation for defocusing
nonlinearity and we study the asymptotic behaviour of the energy for periodic
solutions. We prove first the exponential decay to zero for zero mean
solutions. Then, we characterize the limit of the energy, when the time tends
to infinity, for solutions with small enough initial data and we finally prove
that such limit is not necessary zero.

Keywords— Klein-Gordon equation, Strong and weak damping, Energy decay.

##  1\. Introduction

We consider the damped nonlinear Klein-Gordon equation

$\partial_{t}^{2}\psi+L\partial_{t}\psi-\Delta\psi+\psi+|\psi|^{p}\psi=0,\,\,\psi(0)=\psi_{0},\,\,\partial_{t}\psi(0)=\psi_{1},$
(1)

where $\psi:\mathcal{C}(\mathbb{R}^{+},X),$ $X$ is a Banach subspace in
$L^{2}(\Omega\subset\mathbb{R}^{d})$ and $L:D(L)\subset L^{2}(\Omega)\to
L^{2}(\Omega)$ is some non-negative operator. Damped semi-linear wave
equations particularly equation (1) have gained a lot of attention numerically
and analytically. The related literature is extensive. See for example [6, 14,
16, 4, 8, 9]. The associated energy $E\in\mathcal{C}(X,\mathbb{R}^{+})$ is
given by

$E(\psi)=\frac{1}{2}\int_{\Omega}|\partial_{t}\psi|^{2}+|\psi|^{2}+|\nabla\psi|^{2}+\frac{1}{p+2}\int_{\Omega}|\psi|^{p+2}.$
(2)

The linear part of the system is dissipative in the sense that the linear
semigroup action loses energy. Indeed, we have atleast formally

$\frac{{\rm d}}{{\rm
d}t}E(\psi(t))=-\int_{\Omega}|\sqrt{L}\,\partial_{t}\psi|^{2}\leq 0.$ (3)

In general, damping can be weak when the semigroup generated by the linear
part of the equation is merely continuous (example $L=\gamma(x)I$ with
$\gamma(x)\geq 0$ ), and strong when the semigroup is compact (example
$L=-\Delta+\gamma(x)I$ with $\gamma(x)\geq 0$ ). It has been proven that when
$L=\gamma(x)$ is a ”positive multiplicator”, namely,

$\forall x\in\omega\subset\Omega,\gamma(x)\geq\alpha>0$ (4)

for some open set $\omega$ and a positive $\alpha,$ the energy $E$ decays
exponentially to zero. We refer to [1, 3, 7] for related results. Another
related results of polynomially and exponentially decays a for slightly
different types of damping are proved by J. Royer [13] for the linear Klein-
Gordon equation and by [10] for the nonlinear equation.

The energy asymptotics for the strongly damped equation hasn’t gained enough
attention. R. Xu and W. Lian [15] have recently considered the following
equation

$\partial_{t}^{2}\psi-\omega\Delta\partial_{t}\psi+\mu\psi-\Delta\psi+\psi+f(\psi)=0,\omega\geq
0$

with logarithmic nonlinearity given by $f(\psi)=-\ln(|\psi|)\psi$ and they
proved, under the strong assumption $\mu>\omega\lambda_{1},$ that the related
energy decays exponentially to zero for three different initial energy levels
determined by the minima of the potential energy on the so-called Nehari
manifold [11, 12]. $\lambda_{1}$ is the first eigenvalue of the
operator$-\Delta$ under the homogeneous Dirichlet boundary conditions.
Considering a functional setting similar to that used in [15] , S.M.S.
Cordeiro et al. [2] proved the exponential decay to zero of the energy related
to the strongly damped ($L=-\Delta$) Klein-Gordon equation of Kirchhoff-
Carrier type

$\partial_{t}^{2}\psi-\Delta\partial_{t}\psi-M\left(\|\nabla\psi\|\right)\Delta\psi+M_{1}\left(\|\psi\|\right)\
\psi-\ln(|\psi|^{2})\psi=0,$

where $M,M_{1}$ are two continuous non-negative functions defined on
$[0,+\infty).$ However, depending on the initial data, the energy in the
strongly damped case ($L=-\Delta$) might decay to some conserved quantity
different from zero. For example, the initially non-zero-average periodic
solutions defined on the $d-$torus $\Omega=\mathbb{T}^{d}$ have such quantity.
This observation has not been studied so far and will be the focus of this
paper. To clarify our purpose, let’s consider first the linear equation
defined by the linear part of (1) and denote

$\theta(t):=\fint\psi(t):=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}\psi(t,x){\rm
d}x$

which corresponds to the time-dependent zero Fourier coefficient of
$\psi(t,\cdot).$ Thus, the zero average function $\phi:=\psi-\theta$ satisfies
the linear system

$\partial_{t}^{2}\phi-\Delta\partial_{t}\phi-\Delta\phi+\phi=0,$ (5)

which retrieves the exponential decay of $E(\phi)$ to zero. Moreover,
$E(\psi)$ decays to a conserved positive quantity; more specifically

$E(\psi)=E(\phi)+\frac{(2\pi)^{d}}{2}(|\theta|^{2}+|\theta^{\prime}|^{2})$

and

$|\theta(t)|^{2}+|\theta^{\prime}(t)|^{2}=|\theta(0)|^{2}+|\theta^{\prime}(0)|^{2}.$

Indeed, $\theta$ satisfies the differential equation

$\theta^{\prime\prime}+\theta=0.$

In the spectral level, the damping effect of $L=-\Delta$ is caused by its non-
zero eigenvalues, whereas $\theta$ oscillates independently, so that quantity
$|\theta(t)|^{2}+|\theta^{\prime}(t)|^{2}$ remains conserved. This separation
is however not clear in the case of the nonlinear equation. The rest of the
paper is organised as follows: We prove first that the Cauchy problem of (1)
($L=-\Delta$) is globally well posed on $H^{2}(\mathbb{T}^{d}).$ Then, we
prove that the energy of zero-average solutions decays exponentially to zero.
Finally, we study the energy asymptotics for small initial data solutions.

### Useful notations

  * •

The norm $\|\cdot\|$ refers to the $L^{2}$ norm.

  * •

The constant $C$ changes in the estimates from line to line unless otherwise
noted.

##  2\. The Cauchy problem

In this section we study the Cauchy problem for (1) on
$\mathcal{C}(\mathbb{R}^{+},H^{2}(\mathbb{T}^{d})).$

We have

######  Theorem 1.

Assume that $d\leq 3,$ $p\geq 0$ and $\psi_{0},\psi_{1}\in
H^{2}(\mathbb{T}^{d}).$ Equation (1) has a unique global solution
$\psi\in\mathcal{C}^{1}(\mathbb{R}^{+},H^{2}(\mathbb{T}^{d}))$ such that
$\psi(0)=\psi_{0}$ and $\partial_{t}\psi(0)=\psi_{1}.$ Moreover, there exists
a constant $C>0$ such that

$\|\psi(t)\|_{H^{2}}\leq
C\left(\|\psi_{0}\|_{H^{2}}+\|\psi_{0}\|^{\frac{p}{2}+1}_{p+2}+\|\psi_{1}\|\right)\,\,\forall
t\in\mathbb{R}^{+}.$ (6)

###### Proof.

Written as a first order system, equation (1) takes the abstract form

$\partial_{t}\Psi+\mathcal{A}\Psi+F(\Psi)=0$ (7)

with

$\Psi:=\left(\begin{matrix}\psi\\\
\partial_{t}\psi\end{matrix}\right),\,\,\mathcal{A}:=\left(\begin{matrix}0&-1\\\
-\Delta+1&-\Delta\end{matrix}\right),\,\,\text{and}\,\,F(\Psi):=\left(\begin{matrix}0\\\
|\psi|^{p}\psi\end{matrix}\right).$

We cast (7) in its mild formulation

$\Psi(t)={\rm e}^{-t\mathcal{A}}\Psi(0)-\int_{0}^{t}{\rm
e}^{(\tau-t)\mathcal{A}}F(\Psi(\tau)){\rm d}\tau,$ (8)

where

${\rm e}^{-t\mathcal{A}}={\rm
e}^{\frac{1}{2}t\Delta}\left(\begin{matrix}\cosh(tA)-\frac{1}{2}\Delta
A^{-1}\sinh(tA)&A^{-1}\sinh(tA)\\\
(\Delta-1)A^{-1}\sinh(tA)&\cosh(tA)+\frac{1}{2}\Delta
A^{-1}\sinh(tA)\end{matrix}\right)$

with

$A:=\frac{1}{2}\sqrt{\Delta^{2}+4\Delta-4}.$

Taking into account that $d\leq 3,$ it is classical that the map

$\Psi\mapsto F(\Psi)$

leaves $X_{2}:=H^{2}\times H^{2}$ invariant, and is Lipschitz continuous on
the bounded subsets of $X_{2}.$ Consequently, (8) has a maximal solution
$\Psi\in\mathcal{C}([0,T^{\ast}),X_{2}),$ which blows up as $t$ approaches
$T^{\ast}$ if $T^{\ast}<\infty.$ We prove now that $T^{\ast}=\infty.$ Denote

$J(\psi):=\frac{1}{2}\|\Delta\psi-\partial_{t}\psi\|^{2}+\frac{1}{2}\|\psi\|^{2}_{H^{1}}+\frac{1}{p+2}\|\psi\|_{p+2}^{p+2}.$

It is clear that

$J(\psi)=E(\psi)+\frac{1}{2}\|\Delta\psi\|^{2}-{\rm
Re}\int_{\mathbb{T}^{d}}\partial_{t}\psi\Delta\bar{\psi}.$

Differentiating $J$ with respect to time for $t<T^{\ast}$ and using (3), we
find that

$\displaystyle\frac{{\rm d}}{{\rm d}t}J(\psi)$ $\displaystyle=$
$\displaystyle\frac{{\rm d}}{{\rm d}t}E(\psi)+{\rm
Re}\int_{\mathbb{T}^{d}}\partial_{t}\Delta\psi\Delta\bar{\psi}-\partial^{2}_{t}\psi\Delta\bar{\psi}-\partial_{t}\psi\Delta\partial_{t}\bar{\psi}$
$\displaystyle=$ $\displaystyle{\rm
Re}\int_{\mathbb{T}^{d}}(-\Delta\psi+\psi+|\psi|^{p}\psi)\Delta\bar{\psi}$
$\displaystyle=$
$\displaystyle-\|\Delta\psi\|^{2}-\|\nabla\psi\|^{2}-\int_{\mathbb{T}^{d}}|\psi|^{p}|\nabla\psi|^{2}+p|\psi|^{p-2}|{\rm
Re}(\psi\nabla\bar{\psi})|^{2}$

which means that $t\mapsto J(\psi(t))$ is decreasing and we have

$J(\psi(t))\leq J(\psi(0))\,\,\forall t\in[0,T^{\ast}).$

Consequently, there exists a constant $C>0$ depending on $\psi_{0}$ and
$\psi_{1}$ which can be expressed in the form of the r.h. side of (6) such
that

$\|\psi(t)\|_{H^{2}}\leq C\,\,\forall t\in[0,T^{\ast}).$ (9)

Using (8) together with (9), we find that there exists a constant $C_{1}>0$
depending on $\psi_{0}$ and $\psi_{1}$ such that for any $0<T<T^{\ast},$ we
have

$\|\Psi(t)\|_{L^{\infty}([0,T],X_{2})}\leq\|\Psi(0)\|_{X_{2}}+C_{1}T,$

which implies that $\Psi$ can be extended as a global solution in
$\mathcal{C}(\mathbb{R}^{+},X_{2}).$ ∎

######  Remark 2.

Giving more restrictions on the nonlinearity, a global well posedness result
for the Cauchy problem of (1) could be established on a larger spacial domain.
If, for example, $\psi_{0},\psi_{1}\in H^{1}(\mathbb{T}^{d})$ and

$0<p\begin{cases}\leq\frac{4}{d-2}&\text{if $d\geq 3$,}\\\ <\infty&\text{if
$d=1,2,$}\end{cases}$

then (1) has a unique global solution

$\psi\in\mathcal{C}(\mathbb{R}^{+},H^{1}(\mathbb{T}^{d}))\cap\mathcal{C}^{1}(\mathbb{R}^{+},L^{2}(\mathbb{T}^{d}))\cap\mathcal{C}^{2}(\mathbb{R}^{+},H^{-1}(\mathbb{T}^{d}))$

with $\partial_{t}\psi\in L^{2}_{{\rm
loc}}(\mathbb{R}^{+},H^{1}(\mathbb{T}^{d})).$ The proof for such result
follows the same steps of Theorem 3.1 in [5] for the the existence of a unique
maximal solution. The fact that the energy is decreasing implies the boundness
of $t\mapsto\|\psi(t)\|_{H^{1}}$ which in terns imply that the solution is
global. However, the spacial regularity given by Theorem 1 is needed in the
main result presented in Theorem 4.

##  3\. Asymptotic behaviour of energy

We consider now the energy decay for zero mean solutions.

######  Theorem 3.

Let $\psi$ denote a solution of (1) in the same functional settings mentioned
in Remark 2. Assume further that

$\fint\psi(t)=0,\,\,\forall t\in\mathbb{R}^{+}.$ (10)

Then, there exists $C,\alpha>0$ such that

$E(\psi(t))\leq C{\rm e}^{-\alpha t},\,\,\forall t\in\mathbb{R}^{+}.$ (11)

###### Proof.

The proof uses the same technique used to prove Proposition 2.6 in [7]
together with the Poincaré inequality

$\|\partial_{t}\psi\|=\left\|\partial_{t}\left(\psi-\fint\psi\right)\right\|\leq
C\|\nabla\partial_{t}\psi\|$

for some constant $C>0.$ We introduce the modified energy

$E_{\varepsilon}(\psi):=E(\psi)+\varepsilon\int_{\mathbb{T}^{d}}{\rm
Re}(\bar{\psi}\partial_{t}\psi)$

with $\varepsilon>0.$ For $\varepsilon$ small enough, there exist two
constants $C_{1},C_{2}>0$ depending on $\varepsilon$ such that

$C_{1}(\|\psi\|^{2}+\|\partial_{t}\psi\|^{2})\leq\frac{1}{2}(\|\psi\|^{2}+\|\partial_{t}\psi\|^{2})+\varepsilon\int_{\mathbb{T}^{d}}{\rm
Re}(\bar{\psi}\partial_{t}\psi)\leq
C_{2}(\|\psi\|^{2}+\|\partial_{t}\psi\|^{2}),$

which means that $E_{\varepsilon}$ is equivalent to $E$ for small enough
$\varepsilon$ and it is sufficient to prove the exponential decay for
$E_{\varepsilon}.$ We have

$\displaystyle\frac{{\rm d}}{{\rm d}t}E_{\varepsilon}(\psi(t))$
$\displaystyle=$ $\displaystyle\frac{{\rm d}}{{\rm
d}t}E(\psi(t))+\varepsilon\|\partial_{t}\psi\|^{2}+\varepsilon\int_{\mathbb{T}^{d}}{\rm
Re}(\bar{\psi}\partial^{2}_{t}\psi)$ $\displaystyle=$
$\displaystyle-\|\nabla\partial_{t}\psi\|^{2}+\varepsilon\|\partial_{t}\psi\|^{2}-\varepsilon\|\psi\|^{2}-\varepsilon\|\nabla\psi\|^{2}$
$\displaystyle-\varepsilon\|\psi\|_{p+2}^{p+2}-\varepsilon\int_{\mathbb{T}^{d}}{\rm
Re}(\nabla\bar{\psi}\nabla\partial_{t}\psi)$ $\displaystyle\leq$
$\displaystyle-(C^{-1}-\varepsilon-\frac{\varepsilon}{2}C^{-1})\|\partial_{t}\psi\|^{2}-\varepsilon\|\psi\|^{2}-\frac{\varepsilon}{2}\|\nabla\psi\|^{2}-\varepsilon\|\psi\|_{p+2}^{p+2}$
$\displaystyle\leq$ $\displaystyle-\beta E_{\varepsilon}(\psi(t))$

with $\beta=\beta(\varepsilon)>0$ which implies the exponential decay of
$E_{\varepsilon}$ and thus of $E.$ ∎

We study now the energy decay for solutions with small initial data. Namely,

######  Theorem 4.

Let $\psi_{0},\psi_{1}\in H^{2}(\mathbb{T}^{d}).$ Denote $\psi$ the solution
of (1) given by Theorem 1 such that

$\psi(0)=\psi_{0},\,\,\partial_{t}\psi(0)=\psi_{1}.$

Denote further

$\theta:=\fint\psi,\,\,\phi:=\psi-\theta\,\,\,\text{and}$
$Q(\theta):=(2\pi)^{d}\,\left(\frac{1}{2}|\theta|^{2}+\frac{1}{2}|\theta^{\prime}|^{2}+\frac{1}{p+2}|\theta|^{p+2}\right).$

Then, if $\|\psi_{0}\|_{H^{2}},\|\psi_{1}\|_{H^{2}}$ are small enough, there
exist $\tilde{C},\beta,\alpha>0$ depending on $\psi_{0}$ and $\psi_{1}$ and
there exists $C>0$ such that

$\|\phi\|_{H^{2}}+\|\partial_{t}\phi\|_{H^{2}}\leq
C(\|\phi(0)\|_{H^{2}}+\|\partial_{t}\phi(0)\|_{H^{2}}){\rm e}^{-\beta t}.$
(12)

Moreover, the limit $\lim_{t\to\infty}Q(\theta(t))$ exists and we have

$|E(\psi(t))-Q(\theta(t))|\leq\tilde{C}\,{\rm e}^{-\alpha\,t}.$ (13)

###### Proof.

Denote $\phi=\psi-\theta$ and $f(z)=|z|^{p}z.$ Thus, $(\phi,\theta)$ satisfies
the system

$\displaystyle\partial_{t}^{2}\phi-\Delta\partial_{t}\phi-\Delta\phi+\phi+f(\psi)-\fint
f(\psi)=0\,,$ (14a) $\displaystyle\theta^{\prime\prime}+\theta+\fint
f(\psi)=0\,.$ (14b)

Denote

$\Phi:=\left(\begin{matrix}\phi\\\
\partial_{t}\phi\end{matrix}\right)\,\,\text{and}\,\,G(\psi):=\left(\begin{matrix}0\\\
f(\psi)-\fint f(\psi)\end{matrix}\right).$

Thus, as in (8), $\Phi$ satisfies the mild equation

$\Phi(t)={\rm e}^{-t\mathcal{A}}\Phi(0)-\int_{0}^{t}{\rm
e}^{(\tau-t)\mathcal{A}}G(\psi(\tau)){\rm d}\tau.$ (15)

Using the Pioncaré inequality, there is a constant $C>0$ such that

$\|G(\psi)\|_{H^{2}}\leq C\|\nabla f(\psi)\|_{H^{1}}.$ (16)

Moreover, we have

$|\nabla f(\psi)|\leq(p+1)|\psi|^{p}|\nabla\psi|=(p+1)|\psi|^{p}|\nabla\phi|,$

and

$|\Delta
f(\psi)|\leq(p+1)(|\psi|^{p}|\Delta\psi|+p|\psi|^{p-1}|\nabla\psi|^{2})=(p+1)(|\psi|^{p}|\Delta\phi|+p|\psi|^{p-1}|\nabla\phi|^{2}).$

Thus, using the interpolation inequality

$\||\nabla\psi|^{2}\|\leq\sqrt{2}\|\nabla\psi\|\|\Delta\psi\|$

together with the Sobolev inequality

$\|\psi\|_{L^{\infty}}\leq C\|\psi\|_{H^{2}},$

there exists a constant $C>0$ such that

$\|\nabla f(\psi)\|_{H^{1}}\leq C\|\psi\|^{p}_{H^{2}}\|\phi\|_{H^{2}}.$ (17)

For any $\varphi\in H^{2}(\mathbb{T}^{d})$ with $\fint\varphi=0,$ we have

$\|{\rm e}^{\frac{1}{2}t\Delta}\sinh(tA)\varphi\|_{H^{2}}\leq{\rm
e}^{-\frac{1}{2}t}\|\varphi\|_{H^{2}},\,\,\,\|{\rm
e}^{\frac{1}{2}t\Delta}\cosh(tA)\varphi\|_{H^{2}}\leq{\rm
e}^{-\frac{1}{2}t}\|\varphi\|_{H^{2}},$

which implies that, for any $\Upsilon\in X_{2}$ with $\fint\Upsilon=0,$ we
have

$\left\|{\rm e}^{-t\mathcal{A}}\Upsilon\right\|_{X_{2}}\leq C{\rm
e}^{-\frac{1}{2}t}\|\Upsilon\|_{X_{2}}$ (18)

for some $C>0.$ Then, combining (15), (16), (17) and (18) together with
estimate (6), we get

$\displaystyle\|\Phi(t)\|_{X_{2}}$ $\displaystyle\leq$ $\displaystyle
C_{1}{\rm e}^{-\frac{1}{2}t}\|\Phi(0)\|_{X^{2}}+C_{2}\int_{0}^{t}{\rm
e}^{-\frac{1}{2}(t-\tau)}\|\psi(\tau)\|^{p}_{H^{2}}\|\Phi(\tau)\|_{X_{2}}{\rm
d}\tau$ $\displaystyle\leq$ $\displaystyle C_{1}{\rm
e}^{-\frac{1}{2}t}\|\Phi(0)\|_{X_{2}}+C_{2}C(\psi_{0},\psi_{1})\int_{0}^{t}{\rm
e}^{-\frac{1}{2}(t-\tau)}\|\Phi(\tau)\|_{X_{2}}{\rm d}\tau,$

for some constants $C_{1},C_{2}>0.$ Since $H^{2}(\mathbb{T}^{d})$ is embedded
continuously in $L^{p+2}(\mathbb{T}^{d})$ and using (6), we find that for
small enough $\|\psi_{0}\|_{H^{2}},\|\psi_{1}\|,$ we have

$\beta:=\frac{1}{2}-C_{2}C(\psi_{0},\psi_{1})>0,$

which implies, using Gronwall inequality,

$\|\Phi(t)\|_{X^{2}}\leq C_{1}\|\Phi(0)\|_{X^{2}}{\rm e}^{-\beta t}.$ (19)

Since

$\left||\phi+\theta|^{p+2}-|\theta|^{p+2}\right|\leq(p+2)2^{p}|\phi|(|\phi|^{p+1}+|\theta|^{p+1})$

and using the continuous embedding $H^{2}(\mathbb{T}^{d})\subset
L^{p+2}(\mathbb{T}^{d})$ together with (19), we get

$\displaystyle|E(\psi(t))-Q(t)|$ $\displaystyle\leq$ $\displaystyle
E(\phi(t))+\frac{1}{p+2}\left|\|\psi(t)\|_{p+2}^{p+2}-(2\pi)^{d}|\theta(t)|^{p+2}\right|$
(20) $\displaystyle\leq$ $\displaystyle C{\rm e}^{-\alpha t}$

for some constants $C,\alpha>0$ depending on $\|\psi_{0}\|_{H^{2}}$ and
$\|\psi_{1}\|.$ Since $t\mapsto E(\psi(t))$ is decreasing (see (3)), the limit
$\lim_{t\to\infty}Q(t)$ exists and (13) holds. ∎

We prove in the following theorem that $E(\psi)$ doesn’t decay necessarily to
zero.

######  Theorem 5.

Assume that $\frac{3}{2}\leq p\leq 4$. Keeping the other settings of Theorem
4, there exist $\psi_{0},\psi_{1}\in H^{2}(\mathbb{T}^{d})$ for which we have

$\lim_{t\to\infty}Q(t)>0.$

###### Proof.

Writing $\psi=\phi+\theta$ and using the Taylor series with integral
remainder, we find that

$\displaystyle f(\psi)$ $\displaystyle=$ $\displaystyle f(\theta)+\frac{{\rm
d}}{{\rm d}x}f(x\phi+\theta)|_{x=0}+\int_{0}^{1}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}f(x\phi+\theta)|(1-x){\rm d}x$ $\displaystyle=$
$\displaystyle|\theta|^{p}\theta+\frac{\varepsilon}{2}|\theta|^{p-2}((p+2)|\theta|^{2}\phi+p\,\theta^{2}\bar{\phi})$
$\displaystyle+$
$\displaystyle\frac{p(p+2)}{4}\int_{0}^{1}|x\phi+\theta|^{p-2}(2(x\phi+\theta)|\phi|^{2}+\overline{(x\phi+\theta)}\phi^{2})(1-x){\rm
d}\sigma$ $\displaystyle+$
$\displaystyle\frac{p(p-2)}{4}\int_{0}^{1}|x\phi+\theta|^{p-4}(x\phi+\theta)^{3}\bar{\phi}^{2}(1-x){\rm
d}\sigma.$

Since $\frac{3}{2}\leq p\leq 4,$ for any $x,y,z\in\mathbb{C},$ we have

$|z||y|^{p-1}\leq C(|y|+|z|^{2}+|y|^{p+2})$

and then

$\displaystyle|x|^{2}|x+y|^{p-1}|z|$ $\displaystyle\leq
C(|x|^{p+1}+|x|^{2})(|z||y|^{p-1}+|z|)$ $\displaystyle\leq
C(|x|^{p+1}+|x|^{2})(|z|+|y|+|z|^{2}+|y|^{p+2})$

for some constant $C=C(p)>0.$ Thus, we can write

$\displaystyle|Q^{\prime}|$
$\displaystyle=(2\pi)^{d}\left|{\rm{Re}}\left(\left(\fint
f(\psi)-|\theta|^{p}\theta\right)\bar{\theta^{\prime}}\right)\right|$
$\displaystyle\leq
C\fint\int_{0}^{1}|\phi|^{2}|x\phi+\theta|^{p-1}|\theta^{\prime}|(1-x){\rm
d}x$ $\displaystyle\leq
C(|\theta^{\prime}|+|\theta|+|\theta^{\prime}|^{2}+|\theta|^{p+2})\fint(|\phi|^{p+1}+|\phi|^{2})$
$\displaystyle\leq C(\|\phi\|_{p+1}^{p+1}+\|\phi\|^{2})(Q+\sqrt{Q}),$

where $C=C(p)>0.$ Denote now $S:=\frac{1}{2}Q$ and
$\eta(t):=\frac{1}{2}C\int_{0}^{t}(\|\phi(\tau)\|_{p+1}^{p+1}+\|\phi(\tau)\|^{2}){\rm
d}\tau.$ Thus, we have

$|S^{\prime}|\leq\frac{1}{2}C(\|\phi\|_{p+1}^{p+1}+\|\phi\|^{2})(S+1).$

A direct application of Gronwall lemma implies

$S(t)\geq{\rm
e}^{-\eta(t)}S(0)-\frac{1}{2}C\int_{0}^{t}(\|\phi(\tau)\|_{p+1}^{p+1}+\|\phi(\tau)\|^{2}){\rm
e}^{(\eta(\tau)-\eta(t))}{\rm d}\tau.$

Using (19) together with the Sobolev embedding $H^{2}(\mathbb{T}^{d})\subset
L^{p+1}(\mathbb{T}^{d}),$ we find that there exist
$\Gamma=\Gamma(\|\phi(0)\|_{H^{2}},\|\partial_{t}\phi(0)\|_{H^{2}})>0$ and
$\gamma=\gamma(\|\psi_{0}\|_{H^{2}},\|\psi_{1}\|)>0$ such that

$\eta(t)\leq\frac{\Gamma}{\gamma},\,\,\forall t\geq 0.$

Moreover, following the development of the constant $\beta$ in (19) and for
fixed $(\theta(0),\theta^{\prime}(0))\neq(0,0),$ we have

$\lim_{(\|\phi(0)\|_{H^{2}},\|\partial_{t}\phi(0)\|_{H^{2}})\to(0,0)}\frac{\Gamma}{\gamma}=0.$

Thus, there exist $\psi_{0},\psi_{1}\in H^{2}$ and $\delta>0$ such that

$S(0)>\left(\frac{\Gamma}{\gamma}+\delta\right){\rm
e}^{\frac{\Gamma}{\gamma}}\geq(\eta(t)+\delta){\rm e}^{\eta(t)},$

which implies that

$\lim_{t\to\infty}S(t)\geq\delta>0$

and completes the proof. ∎

##  4\. Numerical tests

In this section we study numerically the energy asymptotics for (1) with
$L=-\Delta,$ $p=2$ and $d=1,2.$ The undamped equation ($L=0$) has a conserved
quantity $E(\psi).$ Then, for long time simulations, one wants to construct
numerical methods that approximately conserve this energy . When using Fourier
spectral methods, we primarily need to ensure that the time discretization
preserves these property, since the spectral spatial discretization will
typically automatically satisfy it. We suggest therefore to use the time
stepping discretization

$UD_{n}:=\frac{\psi_{n+1}-2\psi_{n}+\psi_{n-1}}{(\delta
t)^{2}}+(I-\Delta)\frac{\psi_{n+1}+2\psi_{n}+\psi_{n-1}}{4}+|\psi_{n}|^{2}\psi_{n},$

where $\psi_{n}$ is the approximation of $\psi(n\,\delta t)$ and $\delta t$ is
the time step. Thus, the scheme is given by

$UD_{n}-\Delta\frac{\psi_{n+1}-\psi_{n}}{\delta t}=0.$ (22)

For the approximation of $E(\psi),$ we consider the quantity

$E_{n}(\psi):=\frac{1}{2}\int_{\mathbb{T}^{d}}\left|\frac{\psi_{n}-\psi_{n-1}}{\delta
t}\right|^{2}+\left|\frac{\psi_{n}+\psi_{n-1}}{2}\right|^{2}+\left|\nabla\frac{\psi_{n}+\psi_{n-1}}{2}\right|^{2}+\frac{1}{4}\int_{\mathbb{T}^{d}}\left|\frac{\psi_{n}+\psi_{n-1}}{2}\right|^{4}$

which is approximately conserved for the scheme

$UD_{n}=0$

discretizing the undamped equation (1) with ($L=0$). We study the long time
asymptotic behaviour of $E_{n}(\psi),$

$E_{n}(\phi):=\frac{1}{2}\int_{\mathbb{T}^{d}}\left|\frac{\phi_{n}-\phi_{n-1}}{\delta
t}\right|^{2}+\left|\frac{\phi_{n}+\phi_{n-1}}{2}\right|^{2}+\left|\nabla\frac{\phi_{n}+\phi_{n-1}}{2}\right|^{2}+\frac{1}{4}\int_{\mathbb{T}^{d}}\left|\frac{\phi_{n}+\phi_{n-1}}{2}\right|^{4}$

and

$Q_{n}(\theta):=(2\pi)^{d}\left(\frac{1}{2}|\theta_{n}|^{2}+\frac{1}{2}\left|\frac{\theta_{n}-\theta_{n-1}}{\delta
t}\right|^{2}+\frac{1}{4}|\theta_{n}|^{4}\right),$

where $\theta_{n}$ is the zero coefficient of the discrete Fourier transform
applied on $\psi_{n}$ and $\phi_{n}:=\psi_{n}-\theta_{n}.$

Figure 1. Approximate representation for long time asymptotics of
$E(\psi),E(\phi)$ and $Q$ by $E_{n}(\psi),E_{n}(\phi)$ and $Q_{n}$
respectively with $d=1,\,\,$ $\psi_{1}(x)=0$ and left:
$\psi_{0}(x)=1+3\cos(x);$ right: $\psi_{0}(x)=(1+0.5\cos(x))^{2}.$

Figure 2. Approximate representation for long time asymptotics of
$E(\psi),E(\phi)$ and $Q$ by $E_{n}(\psi),E_{n}(\phi)$ and $Q_{n}$
respectively with $d=2,\,\,$ and left:
$\psi_{0}(x,y)=1+\cos(x)+2\cos(y),\,\,\psi_{1}(x,y)=\sin(x)+2\sin(y);$ right:
$\psi_{0}(x,y)=1+0.2\cos(x)+0.5\cos(y),\,\,\psi_{1}(x,y)=0.$

Figure 3. Time decay of $E_{n}(\phi)$ when $d=1,\,\,$ $\psi_{1}(x)=0$ and
left: $\psi_{0}(x)=1+3\cos(x);$ right: $\psi_{0}(x)=(1+0.5\cos(x))^{2}.$

Figure 4. Time decay of $E_{n}(\phi)$ when $d=2,\,\,$ and left:
$\psi_{0}(x,y)=1+\cos(x)+2\cos(y),\,\,\psi_{1}(x,y)=\sin(x)+2\sin(y);$ right:
$\psi_{0}(x,y)=1+0.2\cos(x)+0.5\cos(y),\,\,\psi_{1}(x,y)=0.$

##  5\. Discussion

Our numerical results show, for the solutions defined on the one and two
dimensional spacial domains ($\mathbb{T}^{d},\,\,d=1,2$), that $E(\psi)$ and
$Q(\theta)$ converge to the same limit (Figures 1 and 2) and that $E(\phi)$
decays exponentially to zero (Figures 3 and 4) regardless of the assumptions
of Theorem 4 made on the initial data $\psi_{0},\psi_{1}.$ In other words, we
believe that the results of Theorem 4 are still valid for weaker assumptions
on the initial data and we leave the proof of such results as an open problem.
In general, the methods used in the literature (for example in [1, 3, 7]) to
study the energy decay for the weakly damped Klein-Gordon equation having the
damping operator $L=\gamma(x)I$ relay essentially on the fact that $L$ is
bounded in the given functional setting which is not the case for the strongly
damped equation with $L=-\Delta.$ Moreover, in view of Theorem 5, the energy
for the strongly damped equation doesn’t decay necessarily to zero. These two
main differences make the methods used for the weakly damped equation
unapplicable to study the energy asymptotics for the strongly damped one and
lead to think differently to address the above open question.

## Acknowledgments

The author thanks Marcel Oliver for useful discussions. The work was supported
by the German Research Foundation grant MO 4162/1-1. The author further
acknowledges support through German Research Foundations Collaborative
Research Center TRR 181.

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