2023 62nd IEEE Conference on Decision and Control (CDC)

December 13-15, 2023. Marina Bay Sands, Singapore

Passivity-based economic ports for optimal operation of networked DC

microgrids
Pol Jané-Soneira, Albertus J. Malan, Ionela Prodan and Sören Hohmann

Abstract— In this paper, we introduce the novel concept important theoretical contributions, hamper the application
of economic ports, allowing modular and distributed optimal to low inertia microgrids with lossy lines, which will adopt a
operation of networked microgrids. Firstly, we design a novel crucial role in future power systems. In [8], [9], optimization-
price-based controller for optimal operation of a single mi-
crogrid and show asymptotic stability. Secondly, we define based controllers for AC and DC microgrids with droop-
novel physical and economic interconnection ports for the controllers are proposed. Although these methods are not
microgrid and study the dissipativity properties of these ports. based on passivity, asymptotic stability of an economically
Lastly, we propose an interconnection scheme for microgrids optimal steady state together with plug-and-play capabil-
via the economic ports. This interconnection scheme requires ities are shown. However, this again comes at the cost
only an exchange of the local prices and allows a globally
2023 62nd IEEE Conference on Decision and Control (CDC) | 979-8-3503-0124-3/23/$31.00 ©2023 IEEE | DOI: 10.1109/CDC49753.2023.10383694

economic optimal operation of networked microgrids at steady of considering a system model with limiting assumptions
state, while guaranteeing asymptotic stability of the networked and approximations, e.g. static lines and single capacitance
microgrids via the passivity properties of economic ports. The dynamics as DC microgrid model or a simple oscillator as
methods are demonstrated through various academic examples. AC microgrid node dynamics. In particular, dynamics of
the DGUs, transmission lines or nonlinear loads are not
I. I NTRODUCTION considered.
Contributions: We propose an optimization-based control-
Future energy systems are expected to rely on many small- ler for DC microgrids that steers a microgrid to an econom-
scale distributed generation units (DGUs) rather than on few ically optimal steady state in a distributed manner without
large-scale generators based on fossil resources. Thus, an knowledge of the loads or transmission line parameters.
optimal coordination of the DGUs while ensuring a stable Conditions for asymptotic stability are provided. We further
operation is crucial. In literature, many approaches propose leverage these results to study the interconnection of various
a passivity-based controller design for DGUs [1], [2]. These microgrids by introducing the novel concept of passivity-
regulators achieve an offset-free regulation of a given voltage based economic ports. These economic ports have a cyber-
reference and have desirable plug-and-play properties while physical nature, which differs from the typical physical inter-
guaranteeing asymptotic stability of the overall intercon- connection ports defined with physical variables like voltages
nected system via passivity. Recently, extensions have been and currents in passivity-based control. The economic port
proposed in order to achieve current- [3] or power-sharing [4] allows interconnecting microgrids on an information level
within the passivity-based framework, or approximate power- to achieve overall economic optimality while ensuring plug-
sharing considering a simultaneous voltage- and frequency and-play stability in a distributed manner.
control in AC systems [5]. Although allowing plug-and-play The remainder of this paper is structured as follows.
operation and ensuring asymptotic stability, passivity-based In Section II, the system model for a general, converter-
methods are in general purely decentralized approaches based microgrid considered in this work is presented. The
which cannot achieve an economically optimal operation or distributed optimization-based controller design is presented
steer the system to an economically optimal steady state. in Section III. In Section IV, we introduce the electric and
Addressing this issue, [6], [7] propose distributed economic ports allowing the microgrids to interconnect on a
passivity- and optimization-based controllers for a microgrid physical- and information-basis. In Section V, the perform-
in port-Hamiltonian form that is able to steer the system ance of the proposed controller for a cluster of networked
to an economically optimal steady state. The intrinsic, fa- microgrids is illustrated via simulations.
vorable passivity properties of the port-Hamiltonian system Notation: Lowercase letters x ∈ Rn represent vectors,
enables plug-and-play operation while ensuring asymptotic and uppercase letters X ∈ Rn×n represent matrices. The
stability. However, in both approaches, the whole microgrid transpose of a vector x ∈ Rn is written as x⊤ . The vector
is modeled as a synchronous generator, which is inter- x = col{xi } and matrix X = [x] = diag{xi } are the n × 1
connected with other microgrids via lossless, static lines. column vector and n × n diagonal matrix of the elements
These simplifications and assumptions, although allowing xi , i = 1, . . . , n, respectively. Let In denote the n × n
P. Jané-Soneira, A. J. Malan and S. Hohmann are with the Institute of identity matrix and 1n ∈ Rn a vector of ones. Calligraphic
Control Systems, Karlsruhe Institute of Technology, 76131, Karlsruhe, Ger- letters X represent sets, and X × I denotes the cartesian
many. Corresponding author is Pol Jané-Soneira, pol.soneira@kit.edu. product of the two sets. For vectors xmin , xmax ∈ Rn , the
I. Prodan is with the Univ. Grenoble Alpes, Grenoble INP, LCIS, F-
26000, Valence, France. I. Prodan’s research benefited from the support of set X = [xmin , xmax ] is a shorthand notation for the convex
the FMJH Program PGMO and from the support to this program from EDF. polytope X = {x ∈ Rn | xmin ≤ x ≤ xmax }, where ≤ holds

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component-wise. A directed graph is denoted by G(B, E), DGU, which is set as the first filter current without loss of
where B is the set of nodes and E ⊆ B × B the set of generality, i.e. y = if,1 .
edges. The cardinality for a set B is denoted by |B|. The In the next section, a price-based controller is designed in
incidence matrix M ∈ R|B|×|E| is defined as M = (mji ) order to determine pref such that an optimal steady state is
with mji = −1 if edge ej ∈ E leaves node vi ∈ B, mji = 1 achieved.
if edge ej ∈ E enters node vi ∈ B, and mji = 0 otherwise.
III. P RICE - BASED CONTROLLER DESIGN
II. S YSTEM MODEL We aim to design a controller which (i) steers the mi-
In this paper, we consider a set of microgrids k ∈ crogrid (1) to an (unknown) economically optimal operation
M = {1, . . . , nmg }, each comprising a set B k of nk = point where the grid-forming DGU does not inject power,
|B k | electrical buses or nodes connected via a set E k of and (ii) has distributed nature and does not require any
mk = |E k | electrical lines. Defining an arbitrary line cur- knowledge of loads or line parameters.
rent direction over the microgrid power lines, we describe
A. Controller design: Optimality model
the network topology of each microgrid with the directed
graph G k (B k , E k ), where B k is the set of nodes and E k Inspired by the Linear-Convex Optimal Steady-State Con-
of edges. We consider nodes i ∈ BL,k ⊆ Bk having only trol [12], we introduce an optimality model, which describes
a nonlinear load, and nodes i ∈ BDGU,k ⊆ B k having an optimal steady state where property (i) is fulfilled:
additionally a DGU, with BL,k ∪BDGU,k = B k . The microgrid d−1
X
index k (always displayed as superscript) is omitted for min fi (pref,i ) (2a)
pref
simplicity until further notice, since the same microgrid i=1
structure holds for all k ∈ M (microgrids may have different d−1
X
sizes, topologies and parameters). A detailed deviation of s.t. pref,i = pL . (2b)
the mathematical model of each microgrid component is i=1
derived in the extended version of this work [10] or in [11]. The function fi : R → R represents the cost of the
For reasons of space, we only present the overall microgrid power infeed of the respective grid-following DGUs, which
model in the following. is assumed to be convex and quadratic in the paper at hand,
i.e. f (pref,i ) = qi p2ref,i +ri pref,i +si with qi , ri , si ∈ R, qi > 0.
A. Microgrid Model
The variable pL comprises the sum of the power consumed by
A microgrid is composed of n = |B| electrical buses with all loads and the losses of the microgrid. Thus, (2b) ensures
d = |BDGU | ≥ 1 DGUs, interconnected by m = |E| power power balance. The KKT conditions [13] for (2) are
lines according to the graph G(B, E) with incidence matrix
M ∈ Rn×m . With respect to the d DGUs, recall that there 0 = ∇fi (pref ) + λ ∀i ∈ {1, . . . , d − 1} (3a)
is one grid-forming DGU which stabilizes the grid voltages d−1
X
and d − 1 grid-following DGUs that inject power according 0= pref,i − pL , (3b)
to an economic objective function, which will be specified i=1
in Section III. The microgrid model reads where λ ∈ R is the Lagrange multiplier for the constraint
(2b), and using a primal-dual gradient method [14] with
Cf v̇ = If if − iL (v) − M iπ (1a)
positive tuning parameters τi and κ, we get
i̇f = αIf⊤ v + βif + γe (1b)

vref
 ṗref,i = −τi (∇fi (pref,i ) − λ) ∀i ∈ {1, . . . , d − 1} (4a)
ė = Iv v + Ip [If⊤ v]if + (1c) d−1
pref X
λ̇ = κ(pL − pref,i ). (4b)
Lπ i̇π = −Rπ iπ + M T v, (1d) i=1

where α = [αi ], β = [βi ] and γ = [γi ] contain the control The multiplier λ can be interpreted as the electrical power
parameters, Cf = [Cf,i ], Rf = [Rf,i ], Lf = [Lf,i ], iL = [iL,i ], price; if the load pL is greater than the power supplied by the
Rπ = [Rπ,j ] and Lπ = [Lπ,j ] are the filter, load and line grid-following DGUs, the price in (4b) increases and vice-
parameters, and v = col{vi } ∈ Rn , if = col{if,i } ∈ Rd , versa. Equation (4a) means that the grid-following DGUs
e = col{ei } ∈ Rd and iπ = col{iπ,j } are the stacked states inject power such that their marginal costs equal the power
of the DGUs i ∈ B and power lines j ∈ E. The voltage price. This is the best solution for rational decision-makers,
and power references vref ∈ R>0 and pref ∈ Rd−1 are inputs, since feeding in more power would lead to less economic
where d defines the number of inputs of the microgrid. The benefit per kW.
matrix If ∈ Rn×d is a permutation matrix assigning the filter With controller (4), property (i) is fulfilled, since at steady
currents of d DGUs to the correct n ≥ d nodes. The matrices state, every grid-following DGU produces at marginal cost
Iv = diag{1, 0, . . . , 0} ∈ Rd×d and Ip = diag{0, 1, . . . , 1} ∈ and the grid-forming DGUs inject no power. However, prop-
Rd×d are diagonal matrices such that the correct error signals erty (ii) requires more attention. Even if (4a) can be com-
are induced for the integrator states [10]. The output y of puted by every grid-following DGU in distributed manner,
the system is defined as the filter current of the grid-forming the price-forming (4b) uses the load pL and the sum of the

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power injection pref,i of the grid-following DGUs, both being section, the interconnection ports for considering networked
system-wide knowledge. To circumvent that, we present the microgrids are defined, and the dissipativity properties are
following proposition. studied.
Proposition 1: Let vref ∈ R>0 . Let all the load in mi-
crogrid (1), including the loads and transmission losses, be IV. I NTERCONNECTION OF MICROGRIDS WITH
PRICE - BASED CONTROLLERS
denoted by pL . Then, at steady state, we have y = 0 iff
d−1 The following definitions lay the foundation for analysing
the stability and optimality of a set of networked mi-
X
pref,i = pL . (5)
i=1
crogrids (1), each controlled with (6). First, we define novel
Proof: See our extended version [10], page 4, column 1. physical and economic interconnection ports and study their
dissipativity properties. Then, we propose an interconnection
Applying Proposition 1 to (4), we can use the grid-forming scheme for the microgrids k ∈ M such that the networked
DGU current y = if,1 for the price-forming mechanism, since microgrids are asymptotically stable and operate at a globally
it is a measure for the unmet power demand in the microgrid, economically optimal steady state.
i.e. Since an interconnection of multiple microgrids is con-
sidered, the microgrid index k ∈ M is not further omitted.
ṗref,i = −τi (∇fi (pref ) − λ) ∀i ∈ {1, . . . , d − 1} (6a)
λ̇ = −κy. (6b) A. Microgrid ports: Definition and dissipativity
The following electric port defines an interface for inter-
This way, the price-forming mechanism does not need
connecting microgrids via electric lines.
system-wide knowledge; the price is formed solely by the
Definition 2 (electric ports): Let ikelec,i be an external cur-
grid-forming DGU and forwarded to the grid-following
rent injected at a node i ∈ B and vik the voltage at that node
DGUs within the microgrid (it may be also forwarded
for microgrid k ∈ M. The input-output pair (ikelec,i , vik ) is
in a distributed manner). Thus, the DGU responsible for
stabilizing the grid (grid-forming) is the price-making entity, called an electric port1 for that microgrid.
and the grid-following DGUs are the price-taking agents. The electric port is interfaced with system (7) through
For the sake of simplicity, we represent the the vectors bkelec = col{ti , 03d+m } and ckelec = bk⊤ elec , where

closed-loop system defining the state variable ti ∈ Rn has a 1 at the i-th element and zero elsewhere, since
x = col{v, if , e, iπ , pref , λ} ∈ Rncl with ncl = n + 3d + m an external current drawn to a node i ∈ B acts on the voltage
the number of states. Let x̄ be an equilibrium point of the dynamics (1a) of node i ∈ B. Note that a microgrid may
closed-loop system for a constant vref . In shifted coordinates contain an arbitrary number z ∈ R of electric ports, yielding
k
x̃ = x − x̄, the nonlinear closed-loop system reads matrices Belec = [bkelec,1 , . . . , bkelec,z ] and Celec
k k⊤
= Belec .
The following economic port defines an interface for
x̃˙ = A(x̃, x̄, P )x̃ (7) interconnecting microgrids economically.
Definition 3 (economic ports): Let λkext ∈ R denote an
with
external electric power price and λkloc ∈ R the local price
A(x̃, x̄, P ) = (8) for a certain microgrid. The input-output pair (λkext , λkloc ) is

−Cf−1 Y + P [v̄]−1 [v]−1 Cf−1 If 0 −Cf−1 M 0 0
 called the economic port for microgrid k ∈ M.
 αIf⊤ β γ 0 0 0  When the economic port (λkext , λkloc ) is connected, we
replace the price used for the grid-following DGUs in (6a)
 
 −Iv − Ip [if ] −Ip [v̄] 0 0 Ip 0 
,
with the input λkext , yielding


 L−1
π M 0 0 −L−1
π Rπ 0 0 

0 0 0 0 −Q −τ 
ṗkref,i = −τik (∇fik (pkref ) − λkext )

0 −κ 0 0 0 0
(11a)
(9) λ̇kloc = −κk ikf,1 . (11b)

where v = v̄ + ṽ, if = īf + ĩf , Q = [τi qi ], τ = col{τi }, The local price λkloc (output of economic port) is still determ-
Y = [yi ] and P = [pi ]. ined by the current of the grid-forming DGU, but is no longer
Definition 1: The feasible subspace of the state space used directly in the local microgrid. Splitting the price in a
X ⊂ Rncl for safe operation is defined as microgrid into local and external prices allows, using a spe-
cial interconnection structure for economic ports as proposed
X = V × I × R2d+m , (10) in Section IV-B, the local price λkloc to contribute towards
where V = [vmin , vmax ] ⊂ Rn and I = [if,min , if,max ] ⊂ a (global) external price. The external price then already
Rd are polytopic sets describing maximum and minimum implicitly contains a coordination between microgrids, and is
feasible node voltages and filter currents. In addition, define used by the grid-following DGUs in order to achieve global
X̃ analogously for error variables and P = [pmin , pmax ]. optimal dispatch. Note that only a single economic port per
The stability proof of the closed-loop system is included 1 Note that electric ports have been used in the literature for intercon-
in the extended version [10], page 5, column 1, and left necting DGUs and lines [1] within a microgrid. Definition 2 can hence be
out from this paper due to space constraints. In the next understood as leveraging these ports between microgrids.

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microgrid is allowed in this work, since we have a single In this subsection, the interconnection ports have been
local price per microgrid. defined and their passivity properties studied. Next, an in-
The economic port thus interfaces with the system (7) terconnection scheme for the electric and economic ports of
through the vectors bkecon = col{0n+2d+m , τ, 0} and ckecon = networked microgrids ensuring global economic optimality
col{0n+3d+m−1 , 1}. System (7) with electric and economic and asymptotic stability is proposed.
ports reads then
B. Networked microgrid operation
x̃˙ k = Akλ (x̃k , x̄k , P k )x̃k + Belec
k k
ielec + bkecon λkext (12a) We now consider a set M of microgrids interconnected
k
ỹelec = k
Celec x̃k k
= velec (12b) electrically and economically, using the port properties es-
k tablished in Theorem 1. To this end, we first characterize a
ỹecon = ckecon x̃k = λkloc . (12c) globally optimal steady state for economically interconnected
The matrix Akλ (·) is the same as A(·) in (7) except that (11a) microgrids.
replaces (6a), where the input λkext is used instead of the state Proposition 3: If the external electric power price λkext ∈
from (11b). Vector ikelec = col{ikelec,z } is the input for all R is equal for all microgrids, i.e.
electric ports z. λkext = λ̄, ∀k ∈ M (15)
We are now interested in the dissipativity properties of
the interconnection ports, in order to analyse microgrids with constant λ̄ ∈ R>0 , we have global optimal dispatch and
interconnected via physical-electric or information-economic all DGUs inject power at marginal cost at any steady state.
ports. First, we analyse the dissipativity properties of the Proof: See our extended version [10].
electric port; thereafter, the properties of the economic port. To achieve λkext = λ̄ ∀k as in Proposition 3, we propose
Proposition 2: Let a microgrid self-close its economic a consensus-based algorithm with which the microgrids per-
port with λkext = yecon k
= λkloc , i.e. without interconnecting form a distributed dynamic averaging of the local price λkloc
economically with other microgrids. System (12) is then (output of the economic port). The output of the distributed
equilibrium independent passive (EIP) w.r.t. the electric port dynamic averaging is used as the external price λkext (input
k
(velec , ikelec ) if a there exists a symmetric S ∈ Rncl ×ncl solving of the economic port). Then, at steady state, the external
prices λkext of all nmg microgrids taking part in the P
distributed
nmg
S>0 (13a) dynamic averaging are equal, i.e. λkext = n1mg k=1 λkloc ,
 k ⊤ k k k⊤

Aλ (·) S + SAλ (·) SBelec − Celec and Proposition 3 is fulfilled. There exist many dynamic
k⊤ k ≤0 (13b) consensus algorithms, see [15] for a survey. In this work,
Belec S − Celec 0
a dynamic consensus algorithm that enjoys an excess of
for all x̄k ∈ X , x̃k ∈ X̃ and P k ∈ P, with X , X̃ and P passivity is needed to compensate the lack of passivity of the
defined as in Definition 1. economic port (characterized in Theorem 1 via ν k and ρk ).
Proof: See our extended version [10], page 6, column 1. We therefore use the proportional dynamic consensus [15]
ẇ = −(µInmg + L)w − Lλloc (16a)
Proposition 2 ensures stability of a scenario where differ-
ent microgrids are interconnected via electric ports. Since an λext = w + λloc , (16b)
economic port is not considered, the electric power prices in where L ∈ R nmg ×nmg
is the Laplacian matrix of an arbit-
the microgrids are independent, yielding optimal operation rary but connected topology describing the communication
in each microgrid but suboptimal operation of the networked between the microgrids via economic ports, µ ∈ R>0 a
microgrids as is shown in Section V. tuning parameter and w ∈ Rnmg auxiliary states. Note that
In order to interconnect the microgrids via the economic the input λloc = col{λkloc } and output λext = col{λkext } of
port and achieve an economic cooperation, we study the consensus algorithm (16) correspond to the economic port
dissipativity properties of both port types simultaneously. as described. All local prices thus contribute to the global,
Theorem 1: System (12) is input-feedforward and output- external price. This consensus protocol is chosen because it
feedback equilibrium independent passive w.r.t. the electric exhibits an excess of input and output passivity (it is input-
k
(velec , ikelec ) and economic ports (λkext , λkloc ) if there exists a to-state stable [16, Theorem 3] and has feedthrough, both
symmetric S ∈ Rncl ×ncl , and indices ν k ∈ R, ρk ∈ R such related to an excess of passivity [17]). If (16) is designed such
that that its excess of passivity is greater than the lack of passivity
S > 0 (14a) of the economic ports obtained in Theorem 1, the feedback
X(x̃k , x̄k , ρk ) SBelec
k k⊤
Sbkecon − ck⊤
 
− Celec econ interconnection is asymptotically stable [18, Theorem 6.2].
k⊤ k  ≤ 0 (14b)
Belec S − Celec 0 0 Note that a formal proof using the methodology described
k⊤ k
becon S − cecon 0 νk above is omitted due to space constraints.

holds for all x̄k ∈ X̄ , x̃k ∈ X̃ and P k ∈ P, where V. S IMULATION RESULTS

X(x̃k , x̄k , ρk ) = Akλ (·)⊤ S + SAkλ (·) + ρk ck⊤ k
econ cecon . In this section, the proposed methods are illustrated on
Proof: See our extended version [10], page 6, column 1. different scenarios with networked microgrids. First, we
study the operation of networked microgrids interconnected

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 Microgrid 1 Microgrid 2 8000

4 1 5 1
6000
3 4
3 2 6 2
4000
0 10 20 30 40 50 60
7 8
λ2loc λ2ext
Fig. 2. DGU injected power in both microgrids
λ1loc λ1ext

Distributed
electric ports are considered in order to highlight the plug-
Consensus-based Algorithm
and-play stability of microgrids stated in Proposition 2.
Fig. 1. Microgrids with grid-forming DGUs (turquoise), grid-following B. Networked microgrids through electric ports
DGUs (red), and nonlinear load (black) nodes, interconnected through the
electric ports with electric lines (purple) and economic ports (green) In this section, the microgrids interconnected via the
electric lines (purple) as shown in Figure 1 are considered.
TABLE I For both microgrids, Proposition 2 is verified for the subset
T OTAL MICROGRID LOAD of the state space defined via vmin = 950 V, vmax = 1050 V
0 s − 20 s 20 s − 40 s 40 s − 60 s for all nodes and if,min = −15 A, if,max = 15 A for all filter
currents.
Microgrid 1 16 kW 20.4 kW 9.5 kW
Microgrid 2 8 kW 8 kW 11 kW The power injection of the grid-following DGUs and the
Sum 24 kW 28.4 kW 20.5 kW price in both microgrids are shown in Figures 2 and 3,
respectively. When the load step at t1 = 20 s occurs, the
DGUs in Microgrid 1 increase their power injection, while
solely through the electric ports. Thereafter, we show that the DGUs in Microgrid 2 decrease their power injection, even
global optimality is obtained for the networked microgrids if the sum of the loads increased (see Table I). At t2 = 40 s,
with an interconnection as proposed in Section IV-B for the the power injection in Microgrid 1 decreases and increases
economic ports. in Microgrid 2, following the trend of their own load (see
Table I).
A. Proposed scenario Similar dynamics are also shown by the price in Figure 3.
We consider a meshed DC microgrid with 8 nodes and 9 In particular, the price dynamics in both microgrids depend
lines interconnected via electric lines (purple) to a second on the local microgrid load changes rather than on the sum of
meshed DC microgrid with 4 nodes and 4 lines shown in the loads across all microgrids. The microgrids are thus not
Figure 1, which was adapted from the scenario in [9]. In cooperating, since the economic ports are not considered.
Microgrid 1, Node 4 (turquoise) is equipped with a grid- As can be observed, the prices in both microgrids are not
forming DGU, which stabilizes the grid dynamics and acts equal at any steady-state, which entails suboptimal economic
here as price-forming entity. Nodes 2, 3 and 4 (red) have dispatch. Furthermore, the steady-state prices cannot be
a grid-following DGU, which are the price-taking feeders. directly influenced, since it results from the grid-forming
The cost of the power injection is set to f11 (p1 ) = 1.2p21 , controllers in each microgrid, which may not have identical
f21 (p2 ) = 1.3p22 and f31 (p3 ) = 1.4p23 . All other nodes (black) parameters.
consist only of nonlinear loads. In Microgrid 2, the grid- In order to influence the prices in each microgrid and
forming DGU is located at Node 3. Grid-following DGUs obtain global economic optimal dispatch, an interconnection
are located at Node 1 and 4, while Nodes 2 consists only of via the economic ports as described in Section IV-B is
a nonlinear load. The cost of the power injection is set for the employed in the next subsection.
DGUs in microgrid 2 to f12 (p1 ) = 1.4p21 and f42 (p4 ) = 1.5p24 .
Load steps occur at time t1 = 20 s and t2 = 40 s. The C. Economically interconnected microgrids
total load of the microgrids after the load steps is shown in In this subsection, the networked microgrids are equipped
Table I. At t1 = 20 s, we have only an increase of the load in with the economic ports and the dynamic averaging as in
Microgrid 1 and hence an increase in the sum of the loads Section IV-B. Theorem 1 is verified for the same subset of
in both microgrids. At t2 = 40 s, the load in Microgrid 1 the state space as in Section V-B.
decreases but increases in Microgrid 2, such that the sum The power injection of the grid-forming DGUs and the
of the loads decreases. The reference voltage vref is set to price in both microgrids are shown in Figures 4 and 5,
1000 V. Typical parameter values for the lines and DGU respectively. When a load step occurs, the prices in both
parameters are taken from [19]. microgrids vary. However, after a transient period, the prices
In the next section, two microgrids interconnected through in both microgrids are equal. This is due to the consensus

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 9000 8000

8000
7000
7000

6000
6000
5000
0 10 20 30 40 50 60 0 10 20 30 40 50 60 70

Fig. 3. Local price λloc of both microgrids Fig. 5. Local price λloc of both microgrids interconnected economically

7000
[3] P. Nahata, M. S. Turan and G. Ferrari-Trecate, “Consensus-
6000 based current sharing and voltage balancing in dc microgrids
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