# Distributed and Asynchronous Operational Optimization of Networked
Microgrids

Nima Nikmehr, Mikhail A. Bragin, Peter B. Luh, and Peng Zhang This work was
supported in part by the National Science Foundation under Grant ECCS-2018492,
CNS-2006828 and OIA-2040599.N. Nikmehr and P. Zhang are with the Department of
Electrical Engineering and Computer Science, Stony Brook University, Stony
Brook, NY 11794, USA (e-mail: p.zhang@stonybrook.edu).M. A. Bragin and P. B.
Luh are with the Department of Electrical and Computer Engineering, University
of Connecticut, Storrs, CT06269, USA.

###### Abstract

Smart programmable microgrids (SPM) is an emerging technology for making
microgrids more software-defined and less hardware-independent such that
converting distributed energy resources (DERs) to networked community
microgrids becomes affordable, autonomic, and secure. As one of the
cornerstones of SPM, this paper pioneers a concept of software-defined
operation optimization for networked microgrids, where operation objectives,
grid connection, and DER participation will be defined by software and plug-
and-play, and can be quickly reconfigured, based on the development of
modularized and tightened models and a novel asynchronous price-based
decomposition-and-coordination method. Key contributions include: (1) design
the architecture of the operational optimization of networked microgrids which
can be readily implemented to ensure the programmability of islanded
microgrids in solving the distributed optimization models, (2) realize a novel
discrete model of droop controller, and (3) introduce a powerful distributed
and asynchronous method Distributed and Asynchronous Surrogate Lagrangian
Relaxation (DA-SLR) to efficiently coordinate microgrids asynchronously. Two
case studies are tested to demonstrate the efficiency of developed DA-SLR, and
specifically, the testing results show the superiority of DA-SLR as compared
to previous methods such as ADMM.

###### Index Terms:

Networked microgrids, Droop control, Distributed optimization, Software-
Defined Networking.

##  I Introduction

The smart programmable microgrids (SPMs) is the emerging phenomenon to address
the issues associated with the existing drawbacks of microgrids’ (MGs)
structure such as the dependence on hardware, challenges in network
virtualization, and vulnerability of communication signals to cyber-attacks
[1]. To provide flexible and easily manageable distributed and asynchronous
operational optimization of islanded networked microgrids, software-defined
networking (SDN) has been used. Within SDN, the network programmability is
enabled through the use of logically centralized controllers [2]. The
programmability of SDN allows the network to efficiently manage the
communication signals, and to enable the user access to the switches to manage
the network after detecting the failures owing to the data plane and control
plane separation. Since MGs necessitate the exploitation of SDN in the
communication network, the SDN realizes software and plug-and-play-based
operation objectives, grid connection, and distributed energy resources (DER)
participation. In [3], an SDN-based MG framework is designed to manage a self-
healing network and enhance the resilience of the network. To ensure resilient
microgrids operations, SDN-based communication architecture is developed in
[4] to manage the cyber-physical disturbances.

Within the SDN infrastructure, discrete operations play an important role and
the control signals are sent through the switches as packets. Further , in
programmable MGs, the discrete operation mode can also help the network
operators to accurately understand the network dynamics even in presence of
communication delays in real applications [5]. Discrete controllers possess
the desired features such as simple programming, cost-efficiency, and digital
and analog input and outputs [6]. Discrete controllers are the prevailing
control mode in microgrids to appropriately manage the DERs dispatch,
substantially resolved the intractable efforts in guaranteeing microgrid
stability. In [7, 8], to control the frequency and voltage deviations,
distributed discrete secondary control are used.

Since the computation burden in the centralized operation of networked MGs
increases with the increase of the network size, to coordinate distributed
entities, distributed optimization methods have been used to improve
computational performance and to resolve data privacy issues of centralized
methods [9]. Within the distributed methods, the problem is decomposed into
several subproblems thereby ensuring the privacy of entities, and the
avoidance of single-point failures. The Lagrangian relaxation (LR) method is
suitable for distributed coordination. Within the method, after constraints
that couple distributed entities are relaxed, the relaxed problem is split
into several subproblems, which are coordinated by updating Lagrangian
multipliers. To accelerate the convergence of the LR method, Augmented
Lagrangian relaxation (ALR) [10] has been used by penalizing the violations
[11]. However, within the method, the problem is non-separable and nonlinear.
To overcome the non-separability issue, the alternate direction method of
multipliers (ADMM) was used [12]. While within the ADMM subproblems are
smaller in size and are easier to solve than the relaxed problem within ALR,
the objective function of each subproblem includes a quadratic penalty
entailing the decision variables from other subproblems which leads to
communication and privacy issues. To coordinate distributed entities without
spending the time for synchronization, in [13], an asynchronous ADMM algorithm
was used to allow the coupling variables to be updated in each subproblem
without getting updates from other subsystems. However, the ADMM does not
converge in the presence of binary variables.

To tackle the above issues, we contribute the following:

  * •

The architecture of software-defined networking is established to prepare the
implementation and easy management of distributed operational optimization of
programmable microgrids in future studies.

  * •

Energy management of networked microgrids is formulated as a distributed
operational optimization problem considering the operational limits and power
flow constraints. The droop controllers are discretized for simple
programmability and high compatibility in the SDN framework due to sending the
control signals as packets.

  * •

A distributed and asynchronous surrogate Lagrangian relaxation method [14] is
employed to coordinate the interconnected microgrids. Within DA-SLR, each sub-
system shares data only with the coordinator without sending data to
neighboring sub-systems or MGs. This sharing policy of DA-SLR preserves
privacy. Also, compared to the classical distributed methods, our DA-SLR
method ensures the convergence in the presence of discrete variables.

##  II SDN-enabled energy management of islanded Microgrids

###  II-A System model of MGs

Software-defined networking takes the role of the data transfer between
microgrids and the coordinator. Within the SDN architecture, switches are
converted to faster and easy forwarding devices owing to the separation of
data and control planes. Besides, the control technique is the centralized
operating system [2]. As shown in Fig. 1, the SDN structure includes several
interconnected switches and only controllers with a wide view of the network
are selected to route data transfer [15].

Figure 1: Scheme of SDN-enabled operational optimization of networked MGs

In the designed model, a networked structure of islanded microgrids is
considered. Each microgrid is connected to one or a group of neighbouring MGs.
Therefore, the control center of MGs should collect data from their local
controllers to make an accurate decision about the power exchanging
possibility with neighbouring MGs. Each microgrid consists of dispatchable and
nondispatchable generators and loads. In this structure, MG control center
observes the difference between generation and load, and then make a decision
about power exchanging with neighbour MGs. In the presented networked MGs
structure, each MG is considered as an autonomous entity. To guarantee the
operational optimization of networked MGs, all MG entities should be
coordinated. Thus, a distributed algorithm is employed to coordinate the MGs
operation. In Fig. 2, a general description for optimal connection between
islanded MGs is shown. According to Fig. 1, the shared data with the
coordinator are the amount of purchased and sold real and reactive powers
which are determined after MGs optimization.

Figure 2: Networked MG-based distribution grid

###  II-B Optimization model of islanded MGs

Objective function. In this subsection, the optimization model consisting of
objective function and technical constrains is presented for the networked
microgrids.

The objective function is described as:

$\begin{split}OF=&\sum_{t\in T}\sum_{i\in
I}[C_{g}(P^{g,MT}_{t,i})+C_{g}(P^{g,FC}_{t,i})+C_{g}(P^{g,CHP}_{t,i})]\\\
&+\sum_{t\in T}\sum_{b\in
B}[C_{dch}(P^{BAT,dch}_{t,b})-C_{ch}(P^{BAT,ch}_{t,b})]\\\ &+\sum_{m\in
M}\sum_{t\in T}[\zeta_{IL}\cdot P_{t,m}^{IL}+\zeta_{IL}\cdot
Q_{t,m}^{IL}]\end{split}$ (1)

where,

$\displaystyle C_{g}(P^{g,MT})$ $\displaystyle=\alpha_{MT}^{g}\cdot P^{g,MT}$
(2a) $\displaystyle C_{g}(P^{g,FC})$ $\displaystyle=\alpha_{FC}^{g}\cdot
P^{g,FC}$ (2b) $\displaystyle C_{g}(P^{g,CHP})$
$\displaystyle=\alpha_{CHP}^{g}\cdot P^{g,CHP}$ (2c)

are generation costs of micro turbine (MT), fuel cell (FC) and combined heat
and power (CHP), respectively, with $P^{g,MT}$, $P^{g,FC}$ and $P^{g,CHP}$
being their corresponding power generation levels and $\alpha_{MT}^{g}$,
$\alpha_{FC}^{g}$ and $\alpha_{CHP}^{g}$ being their generation prices. The
costs $C_{ch}$ and $C_{dch}$ of charging and discharging level of batteries
are calculated as follows:

$\displaystyle C_{ch}(P^{BAT,ch})$ $\displaystyle=\alpha_{ch}^{BAT}\cdot
P^{BAT,ch}$ (3a) $\displaystyle C_{dch}(P^{BAT,dch})$
$\displaystyle=\alpha_{dch}^{BAT}\cdot P^{BAT,dch}$ (3b)

where, $P^{BAT,ch}$ and $P^{BAT,dch}$ are charging and discharging power of
batteries, respectively and $\alpha_{ch}^{BAT}$ and $\alpha_{dch}^{BAT}$ are
the charging and discharging prices. Decision variables $P^{IL}_{t,m}$ and
$Q^{IL}_{t,m}$ are real and reactive power load shedding in microgrid $m$ at
hour $t$. Also, $\zeta_{IL}$ is load shedding price and $m$ is used to
describe microgrid index. In the above, sets $I$, $B$, $M$ and $T$ are used to
denote sets of generation units, batteries, microgrids and time periods,
respectively.

As shown in (1), the first summation is the total generation cost, the second
summation is the total battery charging and discharging cost and the third
summation is the total real and reactive powers interruption cost.

Constraints. The objective function (1) is minimized subject to the following
technical constraints.

  * •

Generation Capacity.

The constraints restricting real and reactive power generation levels are
defined as follows:

$u^{g}_{t,i}\cdot\underline{P}^{g}\leq P^{g}_{t,i}\leq
u^{g}_{t,i}\cdot\overline{P}^{g},\quad\forall t\in T,i\in I$ (4a)
$u^{g}_{t,i}\cdot\underline{Q}^{g}\leq Q^{g}_{t,i}\leq
u^{g}_{t,i}\cdot\overline{Q}^{g},\quad\forall t\in T,i\in I$ (4b) where,
$u^{g}_{t,i}$ defines commitment status of $i^{th}$ DG at hour $t$,
$P_{t,i}^{g}$ and $Q_{t,i}^{g}$ are real and reactive power generation by unit
$i$ at hour $t$. $\underline{P}^{g}$, $\overline{P}^{g}$, $\underline{Q}^{g}$,
and $\overline{Q}^{g}$ are minimum generated real power, maximum generated
real power, minimum generated reactive power, and maximum generated reactive
power, respectively.

  * •

Charge/Discharge Power Limits.

The charging and discharging status of batteries should follow the charging
limits as below:

$0\leq P_{t,b}^{BAT,ch}\leq u_{t,b}^{ch}\cdot\overline{P}^{ch},\quad\forall
t\in T,b\in B$ (4c) $0\leq
P_{t,b}^{BAT,dch}\leq(1-u_{t,b}^{ch})\cdot\overline{P}^{dch},\quad\forall t\in
T,b\in B$ (4d)

where, $u_{t,b}^{ch}$ is a binary variable showing the status of $b^{th}$
battery which can be either charged $(u_{t,b}^{ch}=1)$ or discharged
$(u_{t,b}^{ch}=0)$ in each time slot. Besides, $\overline{P}^{ch}$,
$\overline{P}^{dch}$ are maximum allowed amount of battery charging and
discharging, respectively.

  * •

Power Flow Limits.

Each distribution line between two nodes can distribute real and reactive
power within a range between zero and $\overline{P}^{flow}$, and
$\overline{Q}^{flow}$ as follows:

$0\leq P_{t,n-k}^{flow}\leq\overline{P}^{flow},\quad\forall t\in T,(n,k)\in N$
(4e) $0\leq Q_{t,n-k}^{flow}\leq\overline{Q}^{flow},\quad\forall t\in
T,(n,k)\in N$ (4f)

  * •

Power Interruption Constraints.

The interrupted real and reactive power levels cannot exceed the limits
$\overline{P}^{IL}$ and $\overline{Q}^{IL}$:

$0\leq P_{t,m}^{IL}\leq\overline{P}^{IL},\quad\forall t\in T,m\in M$ (4g)
$0\leq Q_{t,m}^{IL}\leq\overline{Q}^{IL},\quad\forall t\in T,m\in M$ (4h)

  * •

Voltage and Frequency Restrictions.

Voltage and frequency restrictions are described as follows:

$\underline{|V_{n}|}\leq|V_{t,n}|\leq\overline{|V_{n}|},\quad\forall t\in
T,n\in N$ (4i) $59.5\leq f_{t}\leq 60.5,\quad\forall t\in T$ (4j)

where, the node voltage is restricted between $\underline{|V_{n}|}$ and
$\overline{|V_{n}|}$.

  * •

Droop Control.

Droop control is deployed to enhance the ability of MGs in ensuring the real
and reactive power balances. In this regard, the $f-P$ and $V-Q$
characteristics of a droop control can be described as follows [16]:

$\displaystyle f$ $\displaystyle=f_{ref}-m_{p}\cdot(P^{g}_{i}-P^{g}_{i,ref})$
(5a) $\displaystyle|V|$
$\displaystyle=|V_{ref}|-m_{q}\cdot(Q^{g}_{i}-Q^{g}_{i,ref})$ (5b)

where, $f$ and $f_{ref}$ are fluctuated frequency and reference frequency,
which is set to 60 Hz. Similarly, $|V|$ and $|V_{ref}|$ are bus voltage
magnitude and nominal voltage magnitude, respectively. Also, $P^{g}_{i}$ and
$Q^{g}_{i}$ are real and reactive power of DGs. Droop control coefficients for
frequency and voltage based controller are $m_{p}$ and $m_{q}$, respectively.
The limits dealing with $m_{p}$ and $m_{q}$ are described as following:

$\displaystyle\underline{m}_{p}\leq m_{p}\leq\overline{m}_{p}$ (6a)
$\displaystyle\underline{m}_{q}\leq m_{q}\leq\overline{m}_{q}$ (6b)

where, $\underline{m}_{p}$ and $\overline{m}_{p}$ are minimum and maximum
values of frequency-based droop controller respectively. Also, minimum and
maximum values of voltage-based droop controller are described by
$\underline{m}_{q}$ and $\overline{m}_{q}$ respectively.

  * •

Power flow.

For the power flow study, the linearized DistFlow model is deployed. The model
has been utilized and justified in distribution systems and microgrids studies
[17]-[18]. An exact approximation of AC power flow is considered in linearized
DistFlow model [19]:

$\begin{split}P_{t,n-k}^{flow}=\sum_{(\kappa:(k,\kappa))\in
N}P_{t,k-\kappa}^{flow}-(P_{t,k}^{g}-P_{t,k}^{L}),\\\ \quad\forall t\in T,\
n,k,\kappa\in N\end{split}$ (7a)
$\begin{split}Q_{t,n-k}^{flow}=\sum_{(\kappa:(k,\kappa))\in
N}Q_{t,k-\kappa}^{flow}-(Q_{t,k}^{g}-Q_{t,k}^{L}),\\\ \quad\forall t\in T,\
n,k,\kappa\in N\end{split}$ (7b)
$\begin{split}|V_{n,t}|=|V_{k,t}|-\frac{r_{n,k}\cdot
P_{t,n-k}^{flow}+\chi_{n,k}\cdot Q_{t,n-k}^{flow}}{|V_{0}|}\\\ \quad\forall
t\in T,\ n,k\in N\end{split}$ (7c)

where, the bus index is described by $n$ and $k$ and the set $N$ denotes a set
of nodes. The real and reactive power flow between sending node $n$ and
receiving node $k$ at hour $t$ are defined by $P_{t,n-k}^{flow}$ and
$Q_{t,n-k}^{flow}$, respectively. Also, real and reactive loads are denoted by
$P_{t,k}^{L}$ and $Q_{t,k}^{L}$, respectively, and $|V_{n,t}|$ and $|V_{0}|$
are voltage magnitudes at bus $n$ and the point of common coupling in an MG,
respectively. The resistance and reactance between lines $n$ and $k$ are
determined by $r_{n,k}$ and $\chi_{n,k}$.

  * •

Real power balance.

The real power balance ensures the balance between the real power consumption
and real power generation and exchanging real power through the distribution
lines which is formulated as follows:

$\begin{split}&\sum_{i\in I}(s_{n}^{g}\cdot P_{t,i}^{g})+s_{n}^{WT}\cdot
P_{t}^{g,WT}+s_{n}^{PV}\cdot P_{t}^{g,PV}\\\
&+\frac{1}{m_{p}}(f_{ref,m}-f_{t,m})+s_{n}^{BAT}(P_{t}^{BAT,dch}-P_{t}^{BAT,ch})\\\
&+P_{n,t}^{IL}+(P_{t,k-n}^{flow}-\sum_{(k:(n,k))\in N}P_{t,n-k}^{flow})+\\\
&+(\sum_{m\in M}(s_{n-m}^{Tran}\cdot P_{t,n-m}^{buy}-s_{n-m}^{Tran}\cdot
P_{t,n-m}^{sell}))=P_{t,n}^{L},\\\ &\quad\forall t\in T,(n,k)\in N,m\in
M\end{split}$ (8)

where, $P_{t}^{g,WT}$ and $P_{t}^{g,PV}$ are real power generation by WT and
PV panel. Also, $s_{n}^{g}$, $s_{n}^{WT}$, $s_{n}^{PV}$, $s_{n}^{BAT}$ and
$s_{n-m}^{Tran}$ are connection indicator of a generator, WT, PV, battery, and
transacted power of node $n$ to a node in $m^{th}$ microgrid, respectively.

  * •

Reactive power balance.

The reactive power balance denotes the balance between the reactive power
generation and loads considering the reactive power flow and voltage-based
droop control impact as follows:

$\begin{split}&\sum_{i\in I}(s_{n}^{g}\cdot Q_{t,i}^{g})+s_{n}^{WT}\cdot
Q_{t}^{g,WT}+s_{n}^{PV}\cdot Q_{t}^{g,PV}\\\
&+\frac{1}{m_{q}}(|V|_{n}^{ref}-|V_{n},t|)+(Q_{t,k-n}^{flow}-\sum_{(k:(n,k))\in
N}Q_{t,n-k}^{flow})\\\ &+(\sum_{m\in M}(s_{n-m}^{Tran}\cdot
Q_{t,n-m}^{buy}-s_{n-m}^{Tran}\cdot Q_{t,n-m}^{sell}))+Q_{n,t}^{IL}\\\
&=Q_{t,n}^{L},\quad\forall t\in T,(n,k)\in N,m\in M\end{split}$ (9)

where, $Q_{t}^{g,WT}$ and $Q_{t}^{g,PV}$ are reactive power generation by WT
and PV panel.

  * •

Interface Power Exchange Limits.

The amount of exchanging power by each MG is limited as follows:

$0\leq P_{t,m-w}^{buy}\leq u_{t,m-w}^{buy}\cdot\overline{P}^{buy},\forall t\in
T,\\{m,w\\}\in M$ (10) $0\leq
P_{t,m-w}^{sell}\leq(1-u_{t,m-w}^{buy})\cdot\overline{P}^{sell},\forall t\in
T,\\{m,w\\}\in M$ (11) $0\leq Q_{t,m-w}^{buy}\leq
u_{t,m-w}^{buy}\cdot\overline{Q}^{buy},\forall t\in T,\\{m,w\\}\in M$ (12)
$0\leq Q_{t,m-w}^{sell}\leq(1-u_{t,m-w}^{buy})\cdot\overline{Q}^{sell},\forall
t\in T,\\{m,w\\}\in M$ (13)

where, $P_{t,m-w}^{buy}$ and $P_{t,m-w}^{sell}$ are purchased/sold power
levels by $m^{th}$ microgrid at hour $t$ from/to $w^{th}$ microgrid.
Similarly, $Q_{t,m-w}^{buy}$ and $Q_{t,m-w}^{sell}$ are purchased/sold
reactive power levels, respectively. Also, $u_{t,m-w}^{buy}$ is the binary
variables denoting the status of purchasing power by $m^{th}$ MG from $w^{th}$
MG. The maximum amount of purchased and sold real and reactive powers are
illustrated by $\overline{P}^{buy}$, $\overline{P}^{sell}$,
$\overline{Q}^{buy}$ and $\overline{Q}^{sell}$, respectively.

  * •

Interface Power Flow Constraints.

The following interface power flow constraints are considered to ensure that
the power bought by microgrid $m$ from microgrid $w$ equals to the power sold
by microgrid $w$ to microgrid $m$:

$P_{t,m-w}^{buy}=P_{t,w-m}^{sell},\quad\forall t\in T,\\{m,w\\}\in M$ (14)
$Q_{t,m-w}^{buy}=Q_{t,w-m}^{sell},\quad\forall t\in T,\\{m,w\\}\in M$ (15)

These constraints are coupling with respect to the microgrids.

###  II-C Problem linearization

The optimization solvers such as CPLEX and Gurobi cannot solve nonlinear
problems due to cross products of variables. In (8), frequency $f_{t,m}$ is a
continuous variable while $\frac{1}{m_{p}}$ is discrete. Similarly, in (9),
voltage magnitude $|V_{n,t}|$ and $\frac{1}{m_{q}}$ are continuous and
discrete variables, respectively. Therefore, the droop control terms make the
problem constraints nonlinear, which results in a nonlinear optimization
problem [20]. Nevertheless, the presented terms are linearized as described
below.

Generic Linearization Procedure. The linerization procedure will be explained
by using an expression $z=A\cdot d$, where $A$ is a continuous variable and
$d$ is a binary variable. If $A$ has bounds $[\underaccent{\bar}{A},\bar{A}]$,
then the exact form of the linearized inequalities are as following for
expression $d$ [20]:

$min\\{0,\underaccent{\bar}{A}\\}\leq z\leq\bar{A}$ (16a)
$\underaccent{\bar}{A}\cdot d\leq z\leq\bar{A}\cdot d$ (16b)
$A-(1-d)\cdot\bar{A}\leq z\leq A-(1-d)\cdot\underaccent{\bar}{A}$ (16c) $z\leq
A+(1-d)\cdot\bar{A}$ (16d)

Therefore, products of binary variable and continuous variables is linearized
following the procedure (16a)-(16d) through the introduction of new continuous
variables ($z$). However, in the designed optimization model, $d$ is a
discrete variable (integer non-negative variable). If $d_{j}$ is bounded by
positive integer $D$, so that $d_{j}\in\\{0,1,...,D\\}$, we can introduce
binary variables $w_{0},w_{1},...,w_{D}$ and add the following constraints:

$\sum_{l=0}^{D}w_{lj}=1,\quad\forall j\in J$ (17a) $d_{j}=\sum_{l=0}^{D}l\cdot
w_{lj},\quad\forall j\in J$ (17b)

Therefore:

$z_{j}=\sum_{l=0}^{D}l\cdot w_{lj}\cdot A,\quad\forall j\in J$ (17c)

Now, we can linearize each of the products of $w_{lj}\cdot A$ based on
inequalities (16a)-(16d). In $f-P$ droop control, $m_{p}$ is a discrete value,
while frequency ($f$) is a continuous variable. Since $m_{p}$ is discrete,
therefore, $k_{p}=\frac{1}{m_{p}}$ is also discrete. Therefore,

$w_{p,j}=k_{p,j}\cdot f_{m},\quad\forall j\in J$ (18)

Following (16a)-(16d), to resolve the non-linearity difficulty, we replace
(18) by the following constraints:

$z_{p,j}=\sum_{l=0}^{D}l\cdot w_{p,lj}\cdot f_{m},\quad\forall j\in J$ (19)

Similarly, by introducing $k_{q}=\frac{1}{m_{q}}$, the nonlinear terms

$w_{q,j}=k_{q,j}\cdot|V_{n}|,\quad\forall j\in J$ (20)

are linearized as follows:

$z_{q,j}=\sum_{l=0}^{D}l\cdot w_{q,lj}\cdot|V_{n}|,\quad\forall j\in J$ (21)

##  III Distributed and Asynchronous Surrogate Lagrangian Relaxation Model for
Networked MGs

###  III-A Distributed model of Surrogate Lagrangian Relaxation

In the networked microgrids structure, assume there are $m$ MGs conndected to
several neighbouring MGs. To coordinate the MGs, Lagrangian multipliers are
introduced first to relax constraints (14) and (15) that couple MGs. The
Lagrangian function then becomes:

$\footnotesize\begin{split}L(P,Q,\lambda)=&\sum_{t\in T}\sum_{i\in
I}[C_{g}(P^{g,MT}_{t,i})+C_{g}(P^{g,FC}_{t,i})+C_{g}(P^{g,CHP}_{t,i})]\\\
&+\sum_{t\in T}\sum_{b\in
B}[C_{dch}(P^{BAT,dch}_{t,b})-C_{ch}(P^{BAT,ch}_{t,b})]\\\ &+\sum_{m\in
M}\sum_{t\in T}[\zeta_{IL}\cdot P_{t,m}^{IL}+\zeta_{IL}\cdot Q_{t,m}^{IL}]\\\
&+\sum_{t\in T,\\{m,w\\}\in
M}\lambda_{p,t,m-w}\cdot(P_{t,m-w}^{buy}-P_{t,w-m}^{sell})\\\ &+\sum_{t\in
T,\\{m,w\\}\in M}\lambda_{q,t,m-w}\cdot(Q_{t,m-w}^{buy}-Q_{t,w-m}^{sell}),\\\
&\quad\forall t\in T,i\in I,b\in B,\end{split}$ (22)

where, $\lambda_{p,t,m-w}$ and $\lambda_{q,t,m-w}$ are Lagrange multipliers
associated with real and reactive power levels.

The relaxed problem is then decomposed into individual subproblems. The
formulation for “buying” MG $m$ is formulated as:

$\begin{split}\underset{MG_{m}}{min}\\{&\sum_{i\in
I}[C_{g}(P^{g,MT}_{t,m,i})+C_{g}(P^{g,FC}_{t,m,i})+C_{g}(P^{g,CHP}_{t,m,i})]\\\
&+\sum_{b\in B}[C_{dch}(P^{BAT,dch}_{t,m,b})-C_{ch}(P^{BAT,ch}_{t,m,b})]\\\
&+[\zeta_{IL}\cdot P_{(t,m)}^{IL}+\zeta_{IL}\cdot Q_{t,m}^{IL}]\\\
&+\sum_{t\in T,\\{m,w\\}\in M}\lambda_{p,t,m-w}\cdot(P_{t,m-w}^{buy})\\\
&+\sum_{t\in T,\\{m,w\\}\in M}\lambda_{q,t,m-w}\cdot(Q_{t,m-w}^{buy}),\\\
&\forall i\in I,b\in B,m\in M\end{split}$ (23)

Note that decision variables $P_{t,w-m}^{sell}$ and $Q_{t,w-m}^{sell}$ within
(22) belong to subproblem $w$ and are thus not included in (23). While
subproblem formulation $m$ includes variables $P_{t,m-w}^{buy}$ and
$Q_{t,m-w}^{buy}$ indicating power bought, the microgrid can in fact sell
power, in which case the corresponding optimized values will be negative. The
formulation for “selling” MG $w$ is formulated using the same logic with the
exception that variables $P_{t,w-m}^{sell}$ and $Q_{t,w-m}^{sell}$ will appear
within the objective function with the negative sign.

The MG subproblems are coordinated by Lagrangian multipliers, which are
updated based on violation of relaxed constraints as:

$\displaystyle\lambda_{p,t,m-w}^{r+1}=\lambda_{p,t,m-w}^{r}+e^{r}\cdot\big{(}P_{t,m-w}^{buy,r}-P_{t,w-m}^{sell,r}\big{)},$
(24a)
$\displaystyle\lambda_{q,t,m-w}^{r+1}=\lambda_{q,t,m-w}^{r}+e^{r}\cdot\big{(}Q_{t,m-w}^{buy,r}-Q_{t,w-m}^{sell,r}\big{)},$
(24b)

where, $r$ is the coordinator iteration number, $P_{t,m-w}^{buy,r}$ is the
most recent value available of active power at iteration $r$ that MG $m$ has
the intention to buy and $e^{r}$ is the stepsize. In [14], the contraction
mapping concept is employed to derive $e^{r}$ in the following way:

$\begin{split}&e^{r}=\gamma^{r}\frac{e^{r-1}||g(P^{r-1},Q^{r-1})||}{||g(P^{r},Q^{r})||},0<\gamma^{r}<1,r=1,2,...\end{split}$
(25)

where, $g(P^{r},Q^{r})$ is the vector of constraint violations, $\gamma^{r}$
is stepsize-setting parameter which can be updated in each iteration. An exact
formula to determine $\gamma^{r}$ can be found in [14].

As mentioned in Section II and as shown in (23), MG subproblems do not involve
decision variables from other subproblems, thus, unlike ADMM, the DA-SLR
method does not require data exchange among MGs thereby preserving privacy.

###  III-B SDN-enabled DA-SLR solution steps

According to Fig. 1, each core is devoted to a microgrid and the multipliers
are asynchronously updated using plug and play property of DA-SLR method
without waiting for all subproblem solutions to arrive at the coordinator. MGs
share their purchased and sold power levels with the coordinator, the
coordinator updates the multipliers and then broadcasts the multipliers to all
MGs.

Feasible Cost Search. Feasible solutions are searched by using heuristics.
Heuristics are operationalized by solving the original problem (1)-(15) with
decision variables fixed at the most recent values obtained by solving MG
subproblems. If the feasible solution to the original problem is not found,
the multipliers are updated for several iterations before feasible solutions
are searched again.

Dual value of DA-SLR. Dual values provide the lower bound for the feasible
cost to quantify its quality, and the dual values are obtained by minimizing
each MG subproblem by using the most recent available values of multipliers.

Stopping criteria. The stopping criteria is the difference between the
feasible cost and the dual value:

$\begin{split}&OG=\frac{cost^{feasible}-cost^{dual}}{cost^{feasible}}\end{split}$
(26)

The steps of distributed and asynchronous of surrogate Lagrangian relaxation
is described as Algorithm 1.

Result: Exchanging powers
($P^{buy}_{m-w}$/$P^{sell}_{m-w}$,$Q^{buy}_{m-w}$/$Q^{sell}_{m-w}$) between
microgrids

1 initialization;

$M$ microgrids’ exchanged powers $P^{0,buy}_{m-w}$/$P^{0,sell}_{m-w}$,
$Q^{0,buy}_{m-w}$/$Q^{0,sell}_{m-w}$, Lagrange multipliers $\lambda^{0}$,
updating stepsize $e^{0}$

2 iteration $r\leftarrow 0$

3 Coordinator receives MGs subproblem solutions and updates multipliers per
(24a)-(24b) without waiting for all solutions to arrive. Coordinator then
broadcasts multipliers to all MGs

4 Each idle MG starts solving its subproblem by using the latest available
multipliers and sends its solution to the coordinator

5 if  _criteria to search to feasible solution are satisfied_ then

search for a feasible solution

6 if  _the feasible solution is obtained_ then

7 obtain the dual value and calculate the gap (26)

else

go to step 3;

end if

else

go to step 3;

end if

8 if  _the stopping criteria is satisfied_ then

the optimization process is finished

else

go to step 3;

end if

Algorithm 1 Execution of DA-SLR on operational optimization of networked MGs

##  IV Simulation results

In this section, the efficacy and efficiency of the DA-SLR based operational
optimization method are tested and validated using two case studies. In case
study 1, a four-MG networked microgrid system based on a modified IEEE 33-bus
distribution network [21] is considered. In the second case study, a nine-MG
networked microgrid system based on a modified IEEE 123-bus system [22] is
tested. In both studies, the range of frequency-based droop coefficients
$m_{p}$ is assumed to fall in $[0.02,0.2]$ with a discrete step of $m_{p}$
considered as 0.0018. Similarly, the range of voltage-based droop coefficients
$m_{q}$ is set as $[0.05,0.5]$ with a discrete step of 0.0045.

###  IV-A Test results on a 33-bus networked microgrid system

In Fig. 3, the single line diagram of the 33-bus networked microgrids system
with four MGs is shown, and in Table I, DERs information is described. The
power base of the system is set to be 10 MVA. The line resistance and
reactance of the rest of the network for added buses are set to be 0.006 and
0.01 p.u., respectively.

Figure 3: Modified IEEE 33-bus distribution system with four MGs TABLE I: DERs information in networked islanded MGs structure DER type | Bus No. |  Max. real power (kW) |  Max. reactive power (kVAR)
---|---|---|---
PV |  $2,34,45,53$ | $200$ | $80$
WT |  $5,7,14,40$ | $150$ | $60$
MT |  $24,32,47,51$ | $400$ | $200$
FC |  $19,39,49$ | $300$ | $-$
CHP |  $6,16,21,37,44$ | $400$ | $300$

As demonstrated in Fig. 4, the operation cost of the system obtained by the
new method decreases fast thereby reaching the gap of less than 0.2% within 20
iterations.

Figure 4: The DA-SLR method in tracking the feasible solution in modified IEEE
33-bus system

Performance of the new method is compared against that of ADMM in Table II.

TABLE II: Comparison of feasible costs using DA-SLR and ADMM Method | Objective function ($\$$)
---|---
|  MG1 |  MG2 |  MG3 |  MG4 |  Total
DA-SLR |  $2119$ |  $5231$ |  $3150$ |  $3062$ |  $13562$
ADMM |  $4621$ |  $5759$ |  $16680$ |  $4742$ |  $31802$

The DA-SLR and ADMM methods take 16 and 102 seconds per iteration to solve
MGs’ subproblem, respectively. This comparison clearly shows the advantage of
DA-SLR as compared to the ADMM. As reviewed in Introduction, ADMM diverges in
the presence of discrete variables; and as demonstrated in Table II, DA-SLR
obtains the total feasible cost of $13,562, which is $18,240 less (or 57.3%)
than that obtained by using ADMM after 20 iterations. In this case study, each
iteration includes a total of four subproblems, and after finishing the
optimization process of those subproblems, the next iteration starts.

The contribution of active and reactive power levels within each DER obtained
by DA-SLR are shown in Fig. 5 and Fig. 6, respectively.

Figure 5: Generated real power by DERs in each microgrid obtained by DA-SLR
method Figure 6: Generated reactive power by DERs in each microgrid obtained
by DA-SLR method

According to Fig. 5 and Fig. 6, CHP and MT units have a significant
contribution in providing the real and reactive loads in each MG. Since
dispatchable DERs are equipped with droop controllers, MT, FC, and CHP
generators contribute to frequency-based droop control, in which frequency
variations can cause changes in generation level of DERs, while MT and CHP
generators are used for voltage-based droop control to adjust the reactive
power in a predetermined range. In Fig. 7, the contribution of droop control
of DERs in each MG is illustrated.

Figure 7: Contribution of droop control on real and reactive power

In Fig. 7, generally, the contribution of frequency-based droop control in
generating real power for a DER is higher than the contribution of voltage-
based droop control in reactive power generation for the same DER. The reason
is that the maximum real power generation capacity is more than the maximum
capacity of DERs in reactive power generation (see Table I). Precisely, the
real and reactive power balances are deciding factors of the contribution of
droop controllers in the power generation of DERs. The frequency fluctuation
of each DER is caused by frequency-based droop control is shown in Fig. 8.

Figure 8: Frequency fluctuation due to droop control performance in each DER

Comparing Fig. 7 and Fig. 8 shows that the DER decreases its frequency to
enhance its droop control contribution in real power generation and vice
versa. Take the CHP unit in MG1 as an example. As shown in Fig. 7, CHP
contribution using frequency-based droop control is in the maximum level after
hour $7$, which results in the lowest frequency level comparing to previous
hours according to Fig. 8. Also, as the frequency and voltage droop
coefficients are considered discrete variables, the optimized values for droop
coefficients is shown in Figs. 9 and 10.

Figure 9: Optimized values for real power droop control coefficients obtained
by DA-SLR method

A droop controller prefers to adjust the frequency and voltage droop
coefficients in the higher or lower levels or step to keep the power balance
guaranteed. Comparing the Fig. 7 and Fig. 9 for frequency-based droop control
shows that when the real power generation of a DER increases the frequency-
based droop control coefficient increases compared to the hours that the
generated real power is lower. Similarly, the reactive power generation level
of a DER in Fig. 6 has a direct relationship with the contribution of voltage-
based droop control in Fig. 7 thereby the droop control coefficient dealing
with reactive power will be changed to guarantee the reactive power balance.
Take MT(24) of MG1 as an example, where a direct relationship between the
reactive power generation and corresponding droop control coefficient is
observable.

Figure 10: Optimized values for reactive power droop control coefficients
obtained by DA-SLR method

Moreover, to show the impact of droop controller contribution in this study,
the applied technical limit on droop control coefficients is enhanced from 20
percent to 30 percent of the total generation of each DER in the next
scenario. In Table III, this impact is demonstrated.

TABLE III: Impact of droop control contribution increment on the operation cost using DA-SLR | Entity
---|---
|  MG1 |  MG2 |  MG3 |  MG4 |  Total
Objective function ($) |  $1711$ |  $4063$ |  $2563$ |  $2410$ |  $10747$
Reduction (%) |  $19.2$ |  $22.3$ |  $18.6$ |  $21.3$ |  $20.7$

In Table III, the importance of droop control coefficients and consequently
the optimized values are shown. In the optimization process, changing droop
control coefficients limits can have a profound impact on the operation cost
of microgrids.

###  IV-B Test results on a 123-bus networked microgrids system

To demonstrate the scalability of DA-SLR, a larger network is considered. Fig.
11 shows the single line diagram of a 123-bus networked microgrids system with
nine interconnected MGs.

Figure 11: Modified IEEE 123-bus distribution system with nine MGs

The results obtained by the designed DA-SLR method are analyzed upon
approaching the feasible cost (see Fig. 12). The gap reaches 0.3% and 0.01%
after 10 and 20 iterations respectively. In Table IV, the operation cost of
MGs is analyzed using three methods, DA-SLR, sequential SLR, and ADMM.

Figure 12: Tracking the feasible solution using SLR and DA-SLR in modified IEEE 123-bus system TABLE IV: Impact of droop control contribution increment on the operation cost using DA-SLR Method | Objective function ($)
---|---
|  MG1 |  MG2 |  MG3 |  MG4 |  MG5
DA-SLR |  $4109$ |  $950$ |  $4925$ |  $677$ |  $3677$
ADMM |  $7289$ |  $-$ |  $5537$ |  $5799$ |  $-$
| Objective function ($)
|  MG6 |  MG7 |  MG8 |  MG9 |  Total
DA-SLR |  $4452$ |  $2668$ |  $6357$ |  $2744$ |  $30559$
ADMM |  $-$ |  $-$ |  $7001$ |  $9555$ |  $-$

According to the results of Table IV, the DA-SLR has a better performance of
optimization compared to sequential SLR and ADMM. It is seen that the ADMM
method suffers from divergence problems in solving some subproblems.

The DA-SLR and ADMM spend 35 and 210 seconds per iteration to solve MGs’
subproblems, respectively. This comparison shows the definite advantage of DA-
SLR as compared to the ADMM.

##  V Conclusion

In this paper, the distributed and asynchronous surrogate Lagrangian
relaxation method is used to coordinate networked microgrids to schedule the
islanded MGs in a distributed manner. The new method efficiently handles
binary decisions and excellent performance, as well as scalability, is
demonstrated. The method paves the way to facilitate software-defined
networking in enabling efficient coordination of distributed entities
asynchronously. It was shown that the DA-SLR method can achieve acceptable
convergence in fewer iterations and guarantee a minimum amount of operation
cost. The classical method ADMM, in contrast, could not efficiently manage a
distributed problem when the number of microgrids or subproblems are growing.
Moreover, discrete variables have no impact on the convergence of the DA-SLR
method, while classical methods face difficulties in solving problems with
discrete decision variables. The next step is to discuss the potential of the
DA-SLR method on stability issues of operational optimization of programmable
microgrids.

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