Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
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Received June 9, 2020, accepted July 5, 2020, date of publication July 21, 2020, date of current version August 5, 2020.
Digital Object Identifier 10.1109/ACCESS.2020.3010876
Distributed OPF Algorithm for System-Level
Control of Active Multi-Terminal DC Distribution
Grids
ASIMENIA KOROMPILI , (Student Member, IEEE), PETROS PANDIS,
AND ANTONELLO MONTI , (Senior Member, IEEE)
Institute for Automation of Complex Power Systems, E.ON Energy Research Center, RWTH Aachen University, 52074 Aachen, Germany
Corresponding author: Asimenia Korompili (akorompili@eonerc.rwth-aachen.gr)
This work was supported by the project ‘‘Forschungscampus Elektrische Netze der Zukunft (FEN)’’ of the Federal Ministry of Education
and Research of Germany under Grant FKZ: 03SF0488 and Grant FKZ: 03SF0594.
ABSTRACT This paper presents a distributed optimal power flow (OPF) algorithm for the system-level
control of multi-terminal DC (MTDC) distribution grids with distributed energy resources (DER). At each
control period, the algorithm updates the nominal voltage and power set-points of the DER-interfacing
converters, which operate according to active network management (ANM) concepts. To achieve this,
the OPF problem, in its nodal formulation, includes power dispatch strategies for diverse DER according
to their technical characteristics, which change during the system operation. This multi-objective OPF-
for-ANM problem is solved by distributed control units (DCUs) according to the distributed algorithm for
the alternating direction method of multipliers (ADMM). All DCUs have identical roles in the distributed
control structure and thus the distributed OPF-for-ANM algorithm is highly modular. Simulation results in
different IEEE standard systems and various scenarios demonstrate that the algorithm is fast and scalable,
irrespective of the number and location of integrated DER, as well as the operating condition of the system.
The convergence speed of the algorithm is analysed considering the computation and communication time
needed for its execution. The online application in a computers cluster demonstrates the fast execution of
the developed algorithm in a physically-distributed implementation. Through the proposed OPF-for-ANM
algorithm, the system-level control can dispatch fast diverse DER in different coordination approaches in a
distributed manner.
INDEX TERMS Distributed control, power system control, DC-DC converters, optimisation methods,
distributed power generation, energy resources.
I. INTRODUCTION node, which is needed for the voltage restoration (secondary
The recent advancements in power electronic converters and control objective). In addition, for active distribution grids,
the desired high integration of distributed energy resources the integrated DER should participate in the system regu-
(DER) in the power systems have stimulated the develop- lation, realising thus ‘‘connect-and-manage’’ practices and
ment of DC distribution grids. In the hierarchical control active network management (ANM) [2], [3]. For this purpose,
approach, the system-level control of such grids can com- power dispatch strategies for DER should be employed in
bine secondary and tertiary regulation objectives, similarly the optimal power flow (OPF) calculations (tertiary control
to regulation concepts in AC systems [1]. In DC systems, objective). Conventional DER, like µ-combined-heat-and-
the controllable quantity, the DC voltage, has local nature. power plants (µCHPs), are dispatched according to their
Therefore, in multi-terminal DC (MTDC) distribution grids, operational costs, forming thus the classical OPF problem.
power flow calculations in the network are required for the However, other types of DER, like renewable energy sources
determination of the nominal value of the DC voltage at each (RES), energy storage systems (ESS) and controllable loads
(CL), are dispatched according to different operational objec-
The associate editor coordinating the review of this manuscript and tives, local or global. The former refer to operational objec-
approving it for publication was Tariq Masood . tives of each DER, e.g. optimisation of the operation of each
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
136638 VOLUME 8, 2020
A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids ESS according to its state-of charge (SoC); the latter refer to Power dispatch strategies for RES, ESS and CL have operational objectives for the entire system, e.g. minimisation been integrated in centralised OPF-for-ANM problems for of total RES curtailment. These denote the principles of the scheduling the system operation [9]–[12]. Similar power dis- power dispatch strategies for DER, which should consider patch strategies for the aforementioned types of DER are technical characteristics that change during the system opera- presented in [13], where a centralised energy management for tion, like the state-of-charge (SoC) of ESS. For achieving the DC systems is proposed. Opposite to the centralised imple- ANM, power dispatch strategies for diverse DER should be mentation of these power dispatch strategies, in our work we included in the classical OPF calculations, to form the multi- implement them in a distributed manner, by integrating them objective OPF-for-ANM problem. In this context, the system- in the classical OPF problem in its nodal formulation and level control of MTDC distribution grids should determine realising them through the modified distributed algorithm. the nominal DC voltage at each node and dispatch the inte- Distributed implementation of power dispatch strategies grated DER, in the time frame of secondary control level. for DER has been presented in [14], [15] for the opera- This is achieved by solving the OPF-for-ANM problem in tion scheduling in city districts. However, the power flow short periods, to update the voltage or power set-points of constraints are not considered in these works, as in our the converters of the DC system, which interface the DER research. ANM and coordination of DER in a distributed (DER-interfacing converters) or link DC buses (interlinking manner can also be realised through transactive energy converters). [16], [17]. This is a complementary concept to our OPF- The system-level control requires fast computations to for-ANM problem, since the nodal formulation of the power accommodate the fast DC dynamics and the power fluctua- dispatch strategies for DER, presented in this paper, can be tions of RES. In addition, it should be scalable and modular, an element of the transactive energy system, which comprises reliable, as well as secure for data privacy i.e. possess the economic and control mechanisms. characteristics deemed critical for the emerging MTDC dis- System-level control with coordination of various DER tribution grids with high integration of DER. The aforemen- is achieved in [18], [19] with an hierarchical control struc- tioned requirements can be fulfilled by distributed control ture. This architecture is vulnerable to single-point fail- strategies [3]. ure, whereas our distributed control structure based on the In this paper we present a distributed algorithm to solve nodal problem formulation is not. Peer-to-peer control is a the OPF-for-ANM problem at each period of the system- distributed control strategy to coordinate the operation of level control. We capitalise on the distributed algorithm for DER, alternative to our distributed OPF-for-ANM control the classical OPF problem in its nodal formulation presented problem [20], [21]. These works on peer-to-peer control in [4]. This algorithm is reliable, since it does not include any focus on the communication network architecture and the central controller, unlike the distributed algorithms in [5], [6], achievement of the consensus of exchanged data that is nec- which solve regional OPF formulations, where each regional essary in distributed strategies. Our paper presents a com- OPF problem for a group of nodes (region) is solved by plete distributed control concept, which includes not only one computation entity (centralised structure from the per- the necessary communication tasks, but also the development spective of the region). In addition, the algorithm in [4] is of the control logic, i.e. the OPF-for-ANM problem in its suitable for any network topology, unlike other distributed nodal formulation, which is not described in the aforemen- OPF algorithms, which are appropriate only for radial net- tioned works. Distributed OPF-based system-level control works [7], [8]. We modify the objective function and the for MTDC networks is proposed in [22]. Unlike our work, constraints of this nodal classical OPF problem, to include this control does not employ power dispatch strategies for the power dispatch strategies for diverse DER in their nodal DER according to their technical characteristics. In addition, formulation (nodal OPF-for-ANM problem). Furthermore, that work demonstrates the ability of the distributed OPF we include an additional step in the initial algorithm from [4], algorithm to realise the system-level control for MTDC grids to determine at each control period the parameters of this by considering the (computational) convergence speed. In OPF-for-ANM problem (terms of objective function and con- our work, the time required for the communication steps of straints limits), which are related to the technical character- the distributed algorithm is additionally considered. This is istics of the DER that change during the system operation. included for the first time in the relevant literature of the These parameters are determined in a distributed manner distributed OPF algorithms, when the convergence speed is irrespective of the global or local principle of the power investigated [4]–[8]. On the other hand, there is research dispatch strategy. The proposed distributed algorithm for in the field of cooperative control, which coordinates the the multi-objective OPF-for-ANM problem presents bene- integrated DER in distributed or decentralised approaches, ficial characteristics: scalability and modularity, as well as but without considering the power flow constraints, as our fast convergence. The latter is demonstrated considering the control concept does [23]–[30]. In fact, those control concepts total execution time of the algorithm, which includes the apply in microgrids, where the lines are short and the voltage computation and communication time, as well as different drop is negligible. Hence, they are not applicable in large integration conditions of DER and operating conditions of MTDC distribution grids, as our control approach. Similar the system. limitations are present in the consensus-based distributed VOLUME 8, 2020 136639
A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids control methods for microgrids [31]–[34]. In addition to the the distributed algorithm for the alternating direction method limitation of power flow neglect, they do not dispatch the of multipliers (ADMM) to solve this problem [4]. In the different types of power units according to their technical next sections, we describe the modifications in the nodal characteristics, as our work does. formulation of the classical OPF problem, to form the OPF- The contribution of this paper is a distributed algorithm for for-ANM problem, and the modifications in the distributed the OPF-for-ANM problem, which realises the system-level algorithm to solve this problem. control of DER-dominated MTDC distribution grids. Oppo- site to the centralised formulations and implementations in A. DISTRIBUTED SYSTEM-LEVEL CONTROL STRUCTURE the existing literature, the OPF-for-ANM problem is formed The system-level control is performed by distributed control here in the nodal formulation and the proposed algorithm units (DCUs). Each DCU corresponds to one node of the solves this problem in a truly distributed approach. While power system, which has zero or more non-controllable and remaining scalable and modular, the proposed algorithm pro- controllable converters. The former refer to non-controllable vides fast the OPF-for-ANM solution. This is demonstrated loads; the latter refer to DER-interfacing or interlinking considering not only the computation time, as usually in the converters and can be in voltage- or power-control mode. literature of distributed OPF algorithms, but also the com- At each period Ti of the system-level control, each DCU munication time. The distributed OPF-for-ANM algorithm receives only local measurements from the presumed non- is thus suitable for the system-level control of the emerging controllable loads or DER of its node, regarding their current DER-dominated MTDC grids, irrespective of the integra- operational status, e.g. loading condition, maximum avail- tion conditions of DER (number, location of DER) and the able RES power, current state-of-charge (SoC) of ESS. Each system operating conditions. To the best of our knowledge, DCU performs local computations with these measurements such distributed system-level control for MTDC grids, which and exchanges locally computed quantities with other DCUs provides the nominal DC voltage at each node of the power through a communication network, as described in the steps system, while dispatching diverse DER according to their of the distributed algorithm in Section II.B. The execution different operational objectives, is proposed for the first time of the algorithm is synchronous, i.e. the DCUs execute each in literature. step of the algorithm in parallel and when all complete their The remainder of this paper is organised as follows: Sec- tasks at this step, they continue to the next step. In this way, tion II presents the distributed control structure and describes the DCUs solve the OPF problem and compute the nodal volt- the steps of the distributed algorithm for the nodal formu- ages and power outputs of the DER. Each DCU provides volt- lation of the classical OPF problem. Section III derives the age or power set-points to the controllable converters of its nodal formulation of the power dispatch strategies of each node. It should be mentioned that each DCU has the same role type of DER and it presents the strategis combinations to form in the solution of the problem, as described above. This allows the ANM schemes, to handle various types of DER simulta- the modularity of the algorithm. Fig. 1 depicts the physical neously in the same optimisation problem, i.e. the OPF-for- layer of an MTDC grid, where each node hosts only one ANM problem. Section IV describes the modifications in the converter for simplicity of depiction, and the parallel cyber algorithm steps, to solve the OPF-for-ANM problem and thus layer of the communication network between DCUs, which realise the power dispatch strategies for DER in a distributed realise the system-level control. The communication network manner. Section V describes the performance metrics and can be wireline or wireless [35]–[37]. The topology of the presents the simulation results. The conclusions of this work communication network can be different from the topology are presented in Section VI. of the power system and does not need to include this as With regard to notation, lowercase letters represent scalars, subgraph, given that there is a path in the communication boldface uppercase and lowercase letters represent matrices network that allows the data exchange between electrically and vectors, respectively. Sets are represented by italics and their cardinality by ||. The set of real m × n matrices is denoted by Rm×n . The aT denotes the transpose of the vector a. Comma separated elements of a list in parentheses denote column vectors. A ◦ B denotes the Hadamard product of matrices A and B. (a)n is the n-th element of the vector a. II. DISTRIBUTED OPTIMAL POWER FLOW ALGORITHM FOR SYSTEM-LEVEL CONTROL OF MTDC GRIDS In this section, we present the distributed control structure that realises the system-level control of the MTDC grid. This executes the distributed algorithm that solves the OPF problem (classical OPF or OPF-for-ANM problem). In addi- tion, we present the nodal formulation of the classical OPF FIGURE 1. System-level control structure of distributed control units problem in an MTDC system and we describe the steps of (DCUs). 136640 VOLUME 8, 2020
A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
neighbouring DCUs. Fig. 1 presents also examples of dif-
ferent converters in the grid and their interactions with their
DCUs (measurements and set-points flow).
B. DISTRIBUTED ADMM ALGORITHM FOR NODAL OPF
PROBLEM
The original (centralised) OPF problem in MTDC sys-
tems is divided into nodal OPF sub-problems, each one
solved locally by a DCU of the abovementioned distributed
control structure. Any individual DCU k solves the sub- FIGURE 2. Individual node k and its electrically neighbouring nodes m
problem of the node k and determines the vector zk = and l , with all locally computed variables of the OPF sub-problem at DCU
k.
(pG k , ik , ikfl , pk , pkfl , vk ) of local variables, where pk
G ∈
R k|G |×1 are the power outputs of the conventional DER Gk
of the node k, ik is the injected current from the node k,
ikfl ∈ R(|Nk |−1)×1 are the current flows in the lines between described below. The terms in the objective function associ-
the node k and its neighbouring nodes, pk is the injected ated with the consistency constraint force the local copies that
power from the node k, pkfl ∈ R(|Nk |−1)×1 are the power flows correspond to the same node to become equal to the voltage
in the lines between the node k and its neighbouring nodes of this node.
and vk ∈ R|Nk |×1 denotes the vector of the voltage copies,
where the first element is the voltage of node k followed by 1
if (vk )m is ‘‘clone’’ voltage
the copies of the voltages of its neighbouring nodes computed (Ek )mn = of the net voltage variable (v)n (2)
at DCU k. Nk denotes the node k and its neighbouring nodes.
0 otherwise
Since the nodes in the power system are coupled, the nodal
OPF sub-problems are coupled. Hence, the locally computed
OPF sub-solutions have to be consistent. To achieve this, The affine constraint (1b) denotes the current ik according to
the DCUs exchange locally computed quantities with their the admittances gk ∈ R|Nk |×1 of the lines adjacent to the node
electrical neighbours according to the steps of the distributed k. The affine constraint of the current flows ikfl is represented
ADMM algorithm. In this way, the algorithm provides the by (1c), where Ck ∈ R(|Nk |−1)×|Nk | is given by:
OPF solution for the entire power system.
The nodal OPF sub-problem at any individual node k is
(gk )2 −(gk )2 ··· 0
formulated as follows: (gk )3 0 ··· 0
Ck = .. .. .. .. (3)
X
k + yk (vk − Ek v)
T
fkG pG . . . .
min L =
∀G∈Gk
ρ (gk )|Nk | 0 ... −(gk )|Nk |
+ kvk − Ek vk22 (1a)
2
T
s.t. ik = gk vk (1b) The (1d) denotes the affine constraint of the nodal power
ikfl = Ck vk (1c) balance, where pD k is the power of each load D of the loads
X X Dk connected to node k. The pk is determined by the non-
pk = pGk − pDk (1d) linear equality constraint (1e), while (1f) forms the non-
∀G∈Gk ∀D∈Dk
pk = (vk )1 ik (1e) linear equality constraint of the power flows pkfl . The (1g)-
pkfl = (vk )1 ikfl (1f) (1j) represent the linear convex inequality constraints of the
power, current and voltage limits, where pG G
k , pk and vk , vk
pG G G
k ≤ pk ≤ pk (1g) G
are the upper and lower limits of pk and vk , respectively, and
ikfl < ikfl (1h) ikfl and pkfl are the upper limits of ikfl and pkfl , respectively.
pkfl < pkfl (1i) Fig. 2 depicts the individual node k with its electrically neigh-
vk < vk < vk (1j)
bouring nodes m and l, where pG k , ik , ikfl , pk , pkfl , vk are the
local variables computed by the local OPF sub-problem at the
where the fkG (pG DCU k, with vkk , vm k , vk being the local copies of the voltages
l
k ) is the cost function of each conventional
at nodes k, m, l, respectively, pm kfl , pkfl being the power flows
l
DER G connected to node k of the system; ρ is the penalty
from node k to nodes m and l, respectively, and im kfl , ikfl being
l
parameter and yk are the dual variables of the ADMM algo-
rithm, associated with the consistency constraint vk = Ek v, the current flows from node k to nodes m and l, respectively.
where v ∈ R|N |×1 are the nodal voltages of the system and To solve the nodal OPF sub-problem (1a)-(1j) and deter-
Ek ∈ R|Nk |×N is given by (2). The nodal voltages v are mine the sub-solution zk , the non-linear, non-convex equality
known parameters for the local OPF sub-problem, which are constraints (1e) and (1f) are convexified by using first-order
determined in the steps of the iterative ADMM algorithm Taylor approximations about the voltage vsol k , which is an
VOLUME 8, 2020 136641
A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
OPF sub-solution [4]: which is the same in all electrically neighbouring DCUs, and
X by updating the nodal voltages as the average of the local
pk = A1 · (vk )1 + Aj · (vk )j + ANk +1 (1e-lin) voltage copies of the node, the nodal voltages converge to
j∈Nk the voltage copies of the same node, which is part of the
(pkfl )j = B1 · (vk )1 + B2 · (vk )j + B3 (1f-lin) OPF sub-solution. In this way, the local OPF sub-solutions
form the OPF solution for the entire system. When all voltage
where the coefficients of the Taylor series are: copies that correspond to the same node converge to the nodal
voltage, the voltage convergence is achieved at the relevant
X
A1 = (gk )j · (vsol
k )j
j∈Nk
DCU (step 5 of the algorithm). Upon the achievement of
local voltage convergence, each DCU sends its nodal injected
Aj = (gk )j · (vsol
k )1 power pk to all DCUs for the test of the total power balance.
X
(gk )j · (vsol sol When all DCUs know the pk of all DCUs, meaning that
ANk +1 = − k )1 · (vk )j
the voltage convergence at all nodes is achieved, all DCUs
j∈Nk
test locally the total power balance in the entire system,
B1 = (gk )j [2 · (vsol sol
k )1 − (vk )j ] as mentioned in the algorithm in TABLE 1. The summation
of the injected powers should be positive, to ensure that the
B2 = −(gk )j · (vsol
k )1
total generated power from the units (power variables of
B3 = (gk )j · (vsol sol sol
k )1 · [(vk )j − (vk )1 ] local OPF sub-problems) can cover the total load of the grid.
A threshold εP for the accepted losses in lines can be defined.
The nodal OPF sub-problems of all DCUs are coordinated Upon achievement of the second stopping criterion at step
according to the ADMM algorithm to solve the OPF problem 5 of the algorithm, the algorithm stops and returns zk , which
for the entire system. The steps of the algorithm are presented includes the OPF sub-solution. The local voltage (vk )1 and
in TABLE 1. After step 1 for the initialisation of the OPF the power outputs pG k are provided by the individual DCU k
problem, the iterative steps of the algorithm start. At each to the converters of its node as voltage and power set-points,
iteration n of the algorithm, at step 2 each DCU solves its respectively.
(n+1)
own nodal OPF sub-problem, to compute zk . For this, Remark 1: For the stopping criterion of the total power
the OPF sub-problem (1a)-(1j) is convexified according to balance used in this algorithm here, data exchange from each
the Taylor series (1e-lin) and (1f-lin) about vsol
k . This voltage DCU to all DCUs is required. This necessitates an all-to-all
sub-solution is the solution of the local OPF sub-problem at communication network, in which there is a path of commu-
the previous iteration of the distributed ADMM algorithm, nication links from each DCU to all the others. Although all
(n)
vsol
k = vk . At the first iteration the voltage sub-solution vk
sol
DCUs test the second stopping criterion, which seems not
is equal to the starting point vsol
k = v (0) = 1. At step 2 each
effective, this concept avoids any central controller, which
DCU also exchanges with its electrical neighbours the voltage undertakes the duty of checking the total power balance in
(n+1)
copies vk , which are part of the locally computed OPF the entire system. This eliminates the risk of single-point-
(n+1)
sub-solution zk (primal variables of the ADMM algo- of-failure and thus ensures the reliability of the distributed
rithm). These are used to update locally the nodal voltages control structure. For our concept, all DCUs know the IP
v(n+1) by averaging the voltage copies that refer to the same address of all DCUs and which of these are their electrical
node. This averaging constitutes the consensus problem of neighbours. It should be noticed that the information of the
step 3. At the same step each DCU exchanges these updated IP addresses of all DCUs in the communication network does
nodal voltages with its electrically neighbouring DCUs. The not reflect any information of the topology of the power
(n+1)
voltage copies vk and the nodal voltages v(n+1) are used system, i.e. the connections of the nodes corresponding to
(n+1)
to update locally the dual variables of the algorithm yk these DCUs. The only information about the power system
at step 4. Until the convergence of the algorithm is achieved, that each DCU has is the electrically neighbouring DCUs
steps 2, 3 and 4 are repeated at the next iterations. At each (IP addresses), the parameters of the lines adjacent to the
subsequent iteration, the nodal sub-problem at step 2 is solved corresponding node of the DCU and the total number of nodes
again with the updated nodal voltages and dual variables. The with the IP addresses of the corresponding DCUs. Hence,
electrically neighbouring DCUs consider the same updated the information about the topology of the power system is
nodal voltage for each node needed in their local computa- only local.
tions, exchanged at step 3. Hence, the power flows, locally It should be also noticed that the communication tasks at
computed in the subsequent OPF sub-problems according steps 2 and 3 require data exchange only between electri-
to these updated nodal voltages, are consistent, following cally neighbouring DCUs, only the second criterion at step
the coupling between the nodes in the power system. The 5 requires data exchange between all DCUs. However, this
consistency constraint in the OPF sub-problem forces the is not tested at all iterations, but only upon the achieve-
newly computed local copies to become equal to the updated ment of the voltage convergence (first stopping criterion).
nodal voltages. By forcing the local copies vk that refer to the This can allow the selection of different communication
same node to converge to the corresponding nodal voltage, technologies for different communication links, where faster
136642 VOLUME 8, 2020
A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
TABLE 1. Distributed ADMM algorithm for nodal OPF problem. Remark 3: Failures in the communication network,
in terms of data packet drops or failures of the communication
resources, can affect the performance of the distributed OPF
algorithm and thus the distributed system-level control. Such
issues are not considered here and in the following sections
we assume that the communication network in the cyber
layer operates normally. Modifications in the distributed OPF
algorithm, which are needed for its robustness against such
issues, are out of the scope of this paper.
III. POWER DISPATCH STRATEGIES FOR DER
In this section, we present the modifications in the nodal for-
mulation of the classical OPF problem presented in Section II,
to form the nodal OPF-for-ANM sub-problem. We develop
various power dispatch strategies for each type of DER in
their nodal formulation. We combine these strategies in ANM
schemes, which dispatch simultaneously diverse DER. Each
ANM scheme forms an OPF-for-ANM problem in its nodal
formulation.
A. LOCAL POWER DISPATCH STRATEGY FOR RES
(STRATEGY: RES-1)
This strategy dispatches the RES of the system according to
the maximum available power of each RES. This is a local
operational objective. We introduce the power output variable
pRES
k of each RES at the individual node k into the nodal OPF
sub-problem, to form the nodal sub-problem for this strategy.
The power output variables pRESk of the RES at node k are
limited by their maximum available powers p̄RESk , which are
local instantaneous power measurement:
0 ≤ pRES
k ≤ p̄RES
k (4)
The pRES
k are introduced in (1a) and (1d), to form the objective
function and the local power balance constraint of the nodal
sub-problem for this strategy:
X X
L= fkG pG
k − priorRESk ◦pk
RES
∀G∈Gk ∀RES∈RES k
ρ
k (vk − Ek v) +
+ yT kvk − Ek vk22 (1a-1)
2
X X X
pk = pG
k + pRES
k − pD
k (1d-1)
∀G∈Gk ∀RES∈RES k ∀D∈Dk
where RES k is the set of all RES connected to node k and
communication technologies are implemented in the most priorRES
k are the priority factors of the RES at node k in
frequently used communication links. the nodal power dispatch problem. The larger this factor is,
Remark 2: The communication of the nodal injected the larger pRES
k in the minimisation problem of the power
powers to all DCUs does not violate the data privacy dispatch in (1a-1) becomes. This means that with larger
requirement. This power quantity refers to the injected prior RES
k the RES gains priority over the conventional DER
power from the node to the system, without any informa- to supply the load and less RES power is curtailed. The
tion about the nodal generation or demand, or the num- nodal sub-problem for this strategy is then determined by the
ber and types of the individual generators and loads of aforementioned formulas additionally to the constraints (1b),
the node. (1c) and (1e)-(1j).
VOLUME 8, 2020 136643A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
B. GLOBAL POWER DISPATCH STRATEGY FOR RES known to the individual DCU k.
(STRATEGY: RES-2) (
ESS 0 if EkESS,Ti > EkESS,thr (ESSdischarges)
This strategy follows the shared percentage principle: all RES dk = (7)
of the system curtail the same percentage B of their maximum 1 if EkESS,Ti < EkESS,thr (ESScharges)
available power [9]. In our concept, B is determined by (5), The parameters dESS of all ESS of node k determine the
k
so that the total RES power injected to the system supplies upper limits of their power variables pch disch , making
k and pk
a share A of the total load of the system. The parameter A is them mutually exclusive:
set by the distribution system operator (DSO) and is known
to the individual DCU k at each period Ti of the system-level 0 ≤ pch ESS
k ≤ dk ◦ p̄ch
k (8)
control. This principle denotes a global operational objective, 0≤ pdisch (1 − dESS disch
k ≤ k ) ◦ p̄k (9)
which relates all RES of the entire system.
where p̄ch disch are the nominal charging and discharging
k and p̄k
X X X X
pRES = A· pD powers of the ESS, respectively. The power variables of each
k k
∀k∈N ∀RES∈RES k ∀k∈N ∀D∈Dk ESS of node k are introduced in the local power balance
X X constraint of the nodal OPF sub-problem:
= (1 − B) · p̄RES
k (5) X X
∀k∈N ∀RES∈RES k pk = pGk + pdisch
k
∀G∈Gk ∀ESS∈ESS k
The parameter B is calculated through (5), according to the X X
total load and total RES available power. The parameter B is − pch
k − pD
k (1d-2)
∀ESS∈ESS k ∀D∈Dk
then used to determine the power outputs pRES k of the RES
of the individual node k according to the principle of this where ESS k are all ESS connected to the node k. We introduce
strategy: also the variables of the instantaneous stored energy EESS
k for
the ESS of node k, and we formulate the following constraints
pRES
k = (1 − B) · p̄RES
k (6) [9]:
EESS = EESS,T i
+ nch ◦pch 1
k − ndisch ◦pk
disch
(10)
The power output variables pRES k are introduced as fixed k k
ESS
generation into the nodal OPF sub-problem. The nodal sub- EESS
k ≤ EESS
k ≤ Ek (11)
problem for this strategy is thus formed by (1a)-(1c), (1d-1) ESS
and (1e)-(1j). This strategy dispatches the RES according to where EESS
k and Ek are the lower and upper limits of
total RES available power and total load in the entire system. the stored energies of the ESS of node k, respectively. The
However, it is realised in a distributed approach as described parameters nch and ndisch are the charging and discharging
in Section IV. It should be mentioned that the maximum avail- efficiencies of the ESS of node k, respectively. The EESS
k and
able power from the RES in both aforementioned strategies is the dESS
k form a new term in (1a):
considered known (instantaneous power measurement) in the X
time frame of the system-level control. Therefore, the RES L= fkG pG
k
power output is not a stochastic variable for the optimisation ∀G∈Gk
X
problem studied here. + priorESS
k ◦ ((1 − 2 · dESS ESS
k )◦Ek )
∀ESS∈ESS k
C. LOCAL POWER DISPATCH STRATEGY FOR ESS ρ
k (vk − Ek v) +
+ yT kvk − Ek vk22 (1a-2)
(STRATEGY: ESS-1) 2
This strategy dispatches the ESS of the system according to where priorESS are the priority factors of the ESS of node k.
k
the current SoC of each ESS, denoting thus a local operational According to (1a-2) and (10), larger prior ESS of an ESS
k
objective. We introduce in the nodal OPF sub-problem two enforces larger charged or discharged power from this ESS
power variables for each ESS connected to the individual in the minimisation problem of the power dispatch. In this
node k, pchk and pk
disch for charging and discharging modes,
way, a large prior ESS gives priority to the ESS over the
k
respectively, which are mutually exclusive. The decision for conventional DER to participate in the power dispatch of
the operating mode of each ESS of node k is taken locally the system according to their current SoC. This can facili-
according to its current SoC EkESS,Ti , which is a local mea- tate the integration of RES. The nodal sub-problem for this
surement taken at the beginning of Ti . EkESS,Ti above the strategy is determined by the aforementioned formulas addi-
SoC threshold EkESS,thr enforces discharge of the ESS and tionally to the constraints (1b), (1c) and (1e)-(1j).
the decision parameter dkESS takes value 0, whereas EkESS,Ti
below this threshold enforces charge of the ESS and the D. LOCAL POWER DISPATCH STRATEGY FOR ESS WITH
dkESS takes value 1, as presented in (7). The SoC threshold STORAGE VIRTUAL COSTS (STRATEGY: ESS-2)
EkESS,thr is determined either from the DSO for all ESS of the In this strategy, the operating mode and dispatched power
system or from the owner of each ESS individually, and it is of each ESS at the individual node k are determined locally
136644 VOLUME 8, 2020A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
according to its current SoC EkESS,Ti and the nodal electricity In this strategy, the limits of pESS
k and the term in the objective
price pr k , which is known to the DCU k at each control period function are determined according to the svck :
Ti . These two factors determine the parameters techk and
econk for the ESS at node k: − |svck | ◦ p̄ESS
k ≤ pESS ≤ + |svck | ◦ p̄ESS (17)
X k k
G G
EESS,Ti
− EESS,md L= fk pk
k k
techk = ESS
(12) ∀G∈Gk
Ek − EESS,Ti X
k − priorESS
k ◦ (|svck | ◦ pESS
k )
pr − pr md
econk = k k
(13) ∀ESS∈ESS k
pr k − pr k ρ
k (vk − Ek v) +
+ yT kvk − Ek vk22 (1a-3)
2
where EESS,md
k is the vector of the median values of the range
ESS ESS
[E k , E k ] for each ESS of the node k and pr md where p̄ESS are the nominal power outputs of the ESS of node
k is the k
median value of the range [pr k , pr k ], with pr k , pr k being the k. The priority factors priorESS
k of the ESS have here the same
lower and upper limits of the nodal electricity price. Through functionality as in the previous strategy. However, the power
(12) and (13) techk and econk are scaled in the range [−1, 1]. dispatch between ESS depends on the SoC of each ESS and
We combine these parameters into the storage virtual costs the price at the node, denoted through the storage virtual cost.
svck calculated locally for the ESS of node k as follows: Hence, ESS with same nominal power but different storage
virtual costs are dispatched with different power outputs.
|techk | + |econk |
sign (techk ) ◦ This is different from the previous strategy, where the SoC
2
determines only the operating mode and the dispatched power
if sign (techk ) = sign (econk )
of each ESS can have any value within the nominal range.
sign (techk ) ◦ |techk | ◦ (1 − |econk |)
The nodal sub-problem for this strategy is determined by the
if sign (techk ) 6 = sign (econk ) (14)
svck = formulas above, additionally to the constraints (1b), (1c) and
and |techk | ≥ |econk | (1e)-(1j).
(econk ) ◦ |econk | ◦ (1 − |techk |)
sign
E. LOCAL POWER DISPATCH STRATEGY FOR CL
if sign (techk ) 6 = sign (econk )
(STRATEGY: CL-1)
and |econk | ≥ |techk |
This strategy handles the loads that accept to curtail their
where the vector econk includes identical elements all equal power absorption, when the operating condition of the sys-
to econk , as this parameter is determined by the nodal price tem requires such control action. Hence, the CL become
and is the same for all ESS of the node k. According to this dispatchable units according to the cost factors for the load
strategy, discharging of an ESS is enforced when both EkESS,Ti curtailment of each CL (local operational objective). The
and pr k are higher than the median values of their ranges, power variable pCL
k of each CL connected to the individual
whereas charging is enforced in the opposite case. For the node k is introduced to the power balance constraint of the
rest combinations of SoC and price, (14) determines the most local OPF sub-problem:
dominant factor, to decide the operating mode. For each ESS X X X
of the node k, positive svck from (14) means discharging pk = pG
k − pCL
k − pD
k (1d-4)
mode, whereas negative svck means charging mode. ∀G∈Gk ∀CL∈CL k ∀D∈Dk
Here we use an alternative ESS model than in the previous where CL k are the CL connected to the node k. Their limits
strategy, by introducing in the local OPF sub-problem only are:
one power variable for each ESS, pESS k , which takes posi-
k < pk < p̄k
tive or negative values for discharging or charging modes, cCL ◦ p̄CL CL CL
(18)
respectively. The two ESS models are interchangeable in the
two strategies. The constraint (11) of stored energy variables with p̄CL
k the nominal powers of the CL of node k and
EESS
k remains the same, whereas (1d) and (10) are reformu- cCL their parameters of maximum accepted load curtailment,
lated as: provided by the CL owners to the individual DCU k at each
X X X control period Ti . We introduce also a term in the objective
pk = pG
k + pESS
k − pDk (1d-3) function of the local OPF sub-problem:
∀G∈Gk ∀ESS∈ESS k ∀D∈Dk
= EkESS,Ti
X
EESS
k − n◦pESS
k (15) L= fkG pG
k
∀G∈Gk
where n includes the charging/discharging efficiencies of the X 2
ESS of node k: − aCL · (pCL
k ) +b
CL
· pCL
k
( ∀CL∈CL k
(nch )m if ESS m charges ρ
k (vk − Ek v) +
+ yT kvk − Ek vk22
(n)m = 1 (16) (1a-4)
/(ndisch ) if ESSm discharges 2
m
VOLUME 8, 2020 136645A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
where aCL , bCL are the positive curtailment cost factors of be implemented and realised in a distributed approach. This
each CL. The nodal formulation for this local strategy is means that the relevant OPF-for-ANM problem can be solved
determined by the aforementioned formulas additionally to through the distributed ADMM algorithm as described in the
the constraints (1b), (1c) and (1e)-(1j). next section.
F. ANM SCHEMES FOR POWER DISPATCH OF DIVERSE IV. DISTRIBUTED ALGORITHM FOR OPF-FOR-ANM
DER PROBLEM
The strategies presented above for each individual type of In order to realise the ANM schemes presented in their nodal
DER are combined to ANM schemes for the power dispatch formulation in Section III.F, we modify the steps of the OPF
of the diverse DER of the system in one optimisation prob- algorithm of Section II.B. We introduce an additional step,
lem: which computes parameters needed for the OPF-for-ANM
• ANM scheme 1: RES-2 – ESS-1 – CL-1 problem of each ANM scheme. These refer to constraint
• ANM scheme 2: RES-2 – ESS-2 – CL-1 limits and factors of the objective function of the ANM
• ANM scheme 3: RES-1 – ESS-1 – CL-1 scheme, as presented in TABLE 2. This step is necessary,
• ANM scheme 4: RES-1 – ESS-2 – CL-1 since these parameters are related to technical characteristics
of DER or system operational factors that change during
To develop these combinations, the nodal sub-problems of
the system operation. Therefore, these parameters should be
the strategies are now merged to an OPF-for-ANM nodal sub-
computed at every control period according to measurements,
problem that dispatches all types of DER. This means that
e.g. current SoC of ESS, or decided factors provided by
the nodal OPF-for-ANM sub-problem of each ANM scheme
the DSO or the DER owners at each control period Ti , e.g.
consists of the merged objective functions of the strategies
nodal price or accepted load curtailment, as described in
that participate in the scheme, and all the constraints of these
the strategies in Section III. The parameters of the OPF-for-
strategies, as presented in paragraphs A-E. TABLE 2 lists
ANM problem should be determined as initialisation of the
the formulas that constitute the nodal OPF-for-ANM sub-
optimisation problem. This is different from the parameters
problem of each ANM scheme. The OPF-for-ANM prob-
of the classical OPF problem, which remain fixed during the
lem of each ANM scheme is a multi-objective optimisation
system operation, e.g. power flow or voltage limits. In order
problem, which dispatches diverse DER according to differ-
to maintain the scalability and modularity of the system-
ent operational objectives. It should be mentioned that the
level control structure, the determination of these parameters
aforementioned strategies and ANM schemes are examples of
should be performed in a distributed approach, without the
power dispatch principles for diverse types of DER. Different
need for a central coordinator/controller, irrespective of the
strategies, which include different DER models and dispatch
nature of the power dispatch strategies for DER, i.e. global or
them according to different operational objectives, can be
local strategies. To achieve this, the parameters are computed
developed and derived in the nodal formulation similar to the
through only local calculations and data exchange between
aforementioned strategies. It should be noticed that the nodal
all DCUs, when total electrical quantities referring to the
OPF-for-ANM sub-problem is formed in a modular manner.
entire system have to be considered in the power dispatch
The sub-problem module can be replicated in any DCU of
strategy. It should be mentioned that the exchanged data
the distributed control structure, to deal with the nodal power
refer to aggregated nodal quantities, without disseminating
units of any number and type. The power dispatch strategies
the number or type of DER of the individual node k, or any
and their combination (multi-objective ANM schemes) can
technical characteristics or power profiles of individual DER.
TABLE 2. Nodal OPF-for-ANM sub-problem for each ANM scheme.
Hence, the data exchange for the initialisation of the OPF-for-
ANM problem does not violate the data privacy of the DER
owners. The computations and communication needed for
this additional step are presented in TABLE 3. These are exe-
cuted as step 1a added to the initial algorithm after step 1, to
initialise the OPF-for-ANM problem. The step 2 of the initial
algorithm is substituted by the step 20 to solve the new nodal
sub-problem (nodal OPF-for-ANM sub-problem) presented
in Table 2 according to the selected ANM scheme. The steps
3-5 are executed as in the initial algorithm in TABLE 1 for the
coordination of the OPF-for-ANM sub-problems according
to the distributed ADMM technique. In this way, the OPF-for-
ANM problem for the entire system is solved in a distributed
manner and thus the power dispatched strategies for DER are
implemented and realised in a distributed approach.
Remark 4: The developed distributed algorithm can solve
equivalent OPF-for-ANM problems also for AC distribution
136646 VOLUME 8, 2020A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
grids. For this, similar power dispatch strategies can be devel- A. PERFORMANCE METRICS
oped for the inverter-interfaced DER. These strategies can be Since the distributed OPF-for-ANM algorithm realises a
combined to ANM schemes for the diverse DER, which form system-level control, the convergence speed is the most sig-
the OPF-for-ANM problems in the AC grid. These constitute nificant performance metric. The number of iterations that
modifications of the classical OPF problem in AC grids, the algorithm needs to solve the problem is commonly used
which is solved by the distributed ADMM algorithm in [4]. as indicator of the convergence speed. However, in order to
assess the feasibility of the algorithm for system-level control,
TABLE 3. Modifications of distributed ADMM algorithm for OPF-for-ANM we provide also results of its convergence speed in terms
problem.
of the needed computation and communication time, which
determine the total execution time of the algorithm. These
time results can be compared with the desired time frames
for system-level control in DC grids, to determine the ability
of the distributed algorithm to realise such control functions.
The parallel computation time TP is the computation time
that each DCU needs for the OPF problem in the distributed
(parallel) execution of the algorithm. Since in our work the
algorithm is executed sequentially in one computer, the par-
allel computation time TP is theoretically calculated by:
TP = sequential computation time/nodes (19)
The time TC required for all communication tasks of the
algorithm, presented in Sections II.B and IV, is theoretically
calculated by:
TC = 3 · iterations · latency (20)
for the communication tasks at steps 2, 3 and 5 of each
iteration of the algorithm. It should be mentioned that this
is a conservative approximation of the communication time
required by the algorithm. Only the communication at steps
2 and 3 takes place at every iteration. The dissemination of the
nodal injected power pk from each DCU to all DCUs at step
5 takes place only upon the achievement of the local voltage
convergence and thus at fewer iterations than the total num-
ber of iterations until convergence. In the case of the ANM
schemes that include the global RES strategy, one additional
communication task is required according to the initialisation
step 1a presented in Section IV. However, this is performed
V. SIMULATION RESULTS only once, for the computation of the parameters of the opti-
In this section, we test the performance of the distributed misation problem, and thus it increases minimally the TC . The
algorithm for the OPF-for-ANM problem in DC systems. For latency in (20) refers to the technology of the communication
this purpose, we modify various standard IEEE AC networks network between DCUs. For our calculations, we assume
to model DC systems, by setting zero reactive power in that the data to be transmitted between two DCUs can form
generation and load and zero inductance in lines. We integrate one message, considering the data size of our problem [38].
in these networks several RES, ESS and CL, with different We also assume that all parallel message transmissions of
ESS,Ti
maximum available powers p̄RES k , current SoC Ek and the communication tasks need the same time, determined by
CL
nominal powers p̄k , respectively, as shown in Appendix A. the latency of the communication technology. We neglect any
The algorithm is implemented in MATLAB. The quadprog data transmission for other purposes and data congestions in
solver is applied to solve the optimisation problem, i.e. the the communication network. It should be mentioned that the
local OPF problem at step 2. The algorithm is executed in time for any transformation of data models and communica-
one PC. This means that in our simulations each step of the tion protocols, as well as for memory access at each DCU, is
algorithm is executed sequentially by the DCUs. Our simu- also neglected, since it is much smaller than the time needed
lations focus on the verification of the distributed algorithm for the data transmission.
to solve fast the OPF-for-ANM problem and thus realise Moreover, we investigate the scalability of the developed
the system-level control of the DER-dominated MTDC algorithm, indicated by (21). For a scalable algorithm this
grids. performance metric remains at similar values irrespective of
VOLUME 8, 2020 136647A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
the network size and topology [4]. TABLE 4. Parallel computation time TP [s] and number of iterations (in
parentheses).
TSC = TP iterations (21)
It should be mentioned that the TP in (21) includes the compu-
tation time for the initialisation of the OPF-for-ANM problem
at step 1a, which is not executed at every iteration, like the
steps 2-5. However, the calculations at step 1a are simple and
thus take less time than the calculations at the other steps,
esp. considering the local optimisation sub-problem at step
2. Hence, the computation time of step 1a can be assumed as
negligible part of TP in (21). tions) for the schemes are mentioned in Appendix B. TABLE
For comparison purposes, we provide also the aforemen- 4 presents the TP and the number of iterations for each
tioned computation metrics in the classical OPF problem, scenario. It can be observed that for each network the TP and
where only conventional DER are dispatched according to the number of iterations remain in the same order of magni-
their operating costs. This aims to demonstrate that the pro- tude for all ANM schemes with regard to the classical OPF
posed distributed algorithm can solve the multi-objective problem, in spite of the fact that the integration of additional
OPF-for-ANM problem as fast as the known classical OPF DER creates additional variables and constraints for the local
problem, while being modular and scalable. Hence, it can sub-problems and the incorporation of the ANM schemes
realise the system-level control for MTDC grids, to dispatch implies the calculation of more parameters at the initialisation
fast several diverse DER according to various operational step 1a. Although the different ANM schemes mean different
objectives and their technical characteristics. It should be optimisation problems with different multi-objective func-
mentioned that this comparison can be performed between tions, variables and constraints, all present similar results of
the order of magnitude of the results, not their absolute values, convergence speed. It should be noticed that the presented
as the different problems mean different optimisation objec- TP for all ANM schemes is very short. Hence, the presented
tives with different variables and different constraints. There- OPF-for-ANM algorithm can dispatch fast diverse DER in the
fore, the convergence speed (in computation time or number MTDC grid, being thus suitable for fast system-level control.
of iterations) of the algorithm for these problems is differ-
ent. Moreover, this comparison eliminates the effect of the
TABLE 5. Scalability metric TSC [s].
computation resources on the number of iterations and the
computation time, since in all OPF problems (classical OPF
and OPF-for-ANM problems of different ANM schemes) the
corresponding algorithm is implemented in the same pro-
gram, solved by the same solver and executed in the same
computer.
We also discuss factors that affect these performance met-
rics, namely the integration conditions of DER (number and
location of DER), the system operating conditions (loading
condition, operating point of DER) and the communication TABLE 5 presents the TSC values in IEEE standard net-
technology used in the communication network. The latter works of 14, 30, 57 and 118 nodes for the scenarios of the
affects the latency of the communication and thus the relevant classical OPF problem and the OPF-for-ANM problems of
time needed by the algorithm. The other two factors change two ANM schemes. It can be observed that for each network
the local optimisation sub-problem and might affect the con- the time that each DCU needs for one iteration remains the
vergence speed of the algorithm. In addition to time metrics, same for the three scenarios, although these represent dif-
voltage profiles and total power quantities are provided to ferent optimisation problems. In addition, for each problem
demonstrate the effectiveness of the OPF-for-ANM algorithm the TSC presents similar values for all networks. This means
to dispatch and thus coordinate diverse DER through the real- that each DCU needs the same time to compute the local
isation of various power dispatch strategies in the distributed sub-problem, irrespective of the size and the topology of the
control. network. This indicates the scalability of the proposed OPF-
for-ANM algorithm.
B. PARALLEL COMPUTATION TIME AND SCALABILITY OF
DISTRIBUTED OPF-FOR-ANM ALGORITHM C. COMMUNICATION TIME OF DISTRIBUTED
We simulate the classical OPF problem and the four OPF-FOR-ANM ALGORITHM
OPF-for-ANM problems for the four ANM schemes of TABLE 6 shows the values of TC required from the OPF-for-
the Section III.F in the IEEE standard networks of 14, ANM algorithm in different networks in the case of the ANM
30 and 57 nodes with the integrated DER as presented in scheme 4, which needs the largest number of iterations among
Appendix A. The scenarios specifications (operational condi- the other ANM schemes. Four different communication
136648 VOLUME 8, 2020A. Korompili et al.: Distributed OPF Algorithm for System-Level Control of Active Multi-Terminal DC Distribution Grids
TABLE 6. Communication time TC [s] for ANM scheme 4. TABLE 7. Voltage profile of node 9 in IEEE 57 network.
E. POWER DISPATCH OF DER THROUGH PRIORITY
FACTORS
To demonstrate the effect of the priority factors of different
technologies are considered, which are characterised by DER on the power dispatch among them, we vary these
different ranges of latency. The latency value used for parameters for RES and ESS. TABLE 8 presents the power
each technology is mentioned in TABLE [22]. It can be dispatch among the different DER of the IEEE 57 network of
observed that the communication tasks of the OPF-for-ANM Appendix A under the ANM scheme 4 in two cases of priority
algorithm require several minutes when 2G and 3G tech- factors. In Case 1, the prior RES
k and prior ESS
k of each RES
nologies are used. In the majority of network cases these and ESS, respectively, at all nodes of the system are equal to
communication technologies require 2.5-10 min. Technolo- the penalty parameter of ADMM. This value is much higher
gies of 4G need 0.5-2 min for the same tasks, whereas than the cost factors of the conventional DER, offering higher
the needed time for 5G technologies is less than 1 s for priority to the RES and ESS at the power dispatch. In Case 2,
the majority of network cases. Since TP is always very these parameters, given in Appendix D, are at the same scale
short, the total execution time of the algorithm depends of the cost factors of the conventional DER. The system con-
on TC . Considering the time frames of the control levels ditions remain the same in both cases. TABLE 8 shows that
in AC systems, the period of the system-level control in the total RES production and the total power from discharged
DC grids should be at the range of a few minutes [1]. ESS are higher in the first case, since these units have priority
Hence, the majority of existing communication technologies to feed the system and supply the load. The conventional
can be used to realise the proposed distributed system-level DER do not need to produce in this case. On the contrary,
control. in the second case, the load is supplied from all types of DER,
since all present similar factors in the objective function of the
D. VOLTAGE PROFILE optimisation problem. Consequently, the RES produce less,
To demonstrate the effectiveness of the algorithm, we present the ESS inject less power and the conventional DER now
in TABLE 7 the voltage profile of node 9 of the IEEE participate in the power dispatch. In this way, we demonstrate
57 network at different periods of the system-level control. an example on how the proposed OPF-for-ANM algorithm
At each period the system operates at different levels of dispatches the diverse DER of the system. Through the differ-
the total non-controllable load, in the range of [0.55, 2.5] ent ANM schemes with different parameters, realised by this
p.u., distributed among all load units of the system. With algorithm, the system-level control offers the flexibility to the
regard to load profiles of AC systems available from his- DSO, to dispatch the integrated DER in different coordination
torical data [39], these load changes are large. In this way, approaches, in a distributed manner and by following the data
we can demonstrate that the algorithm can work in a wide privacy of the DER owners.
range of loading conditions of the future DC distribution
grids. In addition, at each period of the system-level con- TABLE 8. Power dispatch of DER with different priority factors in ANM
scheme 4 in IEEE 57 network.
trol we apply different ANM scheme, since periodically the
DSO can decide to change the operational objectives, i.e. the
approach of dispatching the DER of the system. The parame-
ter values of the OPF-for-ANM problem for the realisation
of the schemes vary in the ranges shown in Appendix C.
By simulating these scenarios, we test the performance of
the algorithm under different operating points of DER and
loading condition. As it can be seen in TABLE 7, the nodal
voltage remains in the acceptable range, as determined in the F. EFFECT OF NUMBER AND LOCATION OF DER ON
constraints of the optimisation problem, at all periods of the CONVERGENCE SPEED AND SCALABILITY OF
system-level control. The proposed OPF-for-ANM algorithm DISTRIBUTED OPF-FOR-ANM ALGORITHM
can realise the system-level control and determine the nom- To analyse further the performance of the developed algo-
inal voltage at the nodes of the system for different power rithm, we investigate the impact of the number and location
dispatch approaches (ANM schemes), under different system of the additional DER on the convergence speed. TABLE 9
conditions. presents the number of iterations, the TP and TSC in three
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