Decentralized Optimal Dispatch of Photovoltaic Inverters in Residential Distribution Systems

I Introduction

THE proliferation of residential-scale photovoltaic (PV) systems has highlighted unique challenges and concerns in the operation and control of low-voltage distribution networks. Secondary-level control of PV inverters can alleviate extenuating circumstances such as overvoltages during periods when PV generation exceeds the household demand, and voltage transients during rapidly varying atmospheric conditions . Initiatives to upgrade inverter controls and develop business models for ancillary services are currently underway in order to facilitate large-scale integration of renewables while ensuring reliable operation of existing distribution feeders .

Examples of ancillary services include reactive power compensation, which has been recognized as a viable option to effect voltage regulation at the medium-voltage distribution level . The amount of reactive power injected or absorbed by inverters can be computed based on either local droop-type proportional laws , or optimal power flow (OPF) strategies . Either way, voltage regulation with this approach comes at the expense of low power factors at the substation and high network currents, with the latter leading to high power losses in the network . Alternative approaches require inverters to operate at unity power factor and to curtail part of the available active power . For instance, heuristics based on droop-type laws are developed in to compute the active power curtailed by each inverter in a residential system. Active power curtailment strategies are particularly effective in the low-voltage portion of distribution feeders, where the high resistance-to-inductance ratio of low-voltage overhead lines renders voltage magnitudes more sensitive to variations in the active power injections .

Recently, we proposed an optimal inverter dispatch (OID) framework where the subset of critical PV-inverters that most strongly impact network performance objectives are identified and their real and reactive power setpoints are computed. This is accomplished by formulating an OPF-type problem, which encapsulates well-defined performance criteria as well as network and inverter operational constraints. By leveraging advances in sparsity-promoting regularizations and semidefinite relaxation (SDR) techniques , the problem is then solved by a centralized computational device with reduced computational burden. The proposed OID framework provides increased flexibility over Volt/VAR approaches and active power curtailment methods by: i) determining in real-time those inverters that must participate in ancillary services provisioning; and, ii) jointly optimizing both the real and reactive power produced by the participating inverters (see, e.g., Figs. 3 (c)-(d) for an illustration of the the inverters’ operating regions under OID).

As proposed originally, the OID task can be carried out on a centralized computational device which has to communicate with all inverters. In this paper, the OID problem proposed in is strategically decomposed into sub-problems that can be solved in a decentralized fashion by the utility-owned energy managers and customer-owned PV systems, with limited exchanges of information. Hereafter, this suite of decentralized optimization algorithms is referred to as decentralized optimal inverter dispatch (DOID). Building on the concept of leveraging both real and reactive power optimization , and decentralized solution approaches for OPF problems , two novel decentralized approaches are developed in this paper. In the first setup, all customer-owned PV inverters can communicate with the utility. The utility optimizes network performance (quantified in terms of, e.g., power losses and voltage regulation) while individual customers maximize their economic objectives (quantified in terms of, e.g., the amount of active power they might have to curtail). This setup provides flexibility to the customers to specify their optimization objectives since the utility has no control on customer preferences. In the spirit of the advanced metering infrastructure (AMI) paradigm, utility and customer-owned EMUs exchange relevant information to agree on the optimal PV-inverter setpoints. Once the decentralized algorithms have converged, the active and reactive setpoints are implemented by the inverter controllers. In the second DOID approach, the distribution network is partitioned into clusters, each of which contains a set of customer-owned PV inverters and a single cluster energy manager (CEM). A decentralized algorithm is then formulated such that the operation of each cluster is optimized and with a limited exchange of voltage-related messages, the interconnected clusters consent on the system-wide voltage profile. The decentralized OID frameworks are developed by leveraging the alternating direction method of multipliers (ADMM) .

Related works include , where augmented Lagrangian methods (related to ADMM) were employed to decompose non-convex OPF problems for transmission systems into per-area instances, and , where standard Lagrangian approaches were utilized in conjunction with Newton methods. ADMM was utilized in to solve non-convex OPF renditions in a decentralized fashion, and in , where successive convex approximation methods were utilized to deal with nonconvex costs and constraints. In the distribution systems context, semidefinite relaxations of the OPF problem for balanced systems were developed in , and solved via node-to-node message passing by using dual (sub-)gradient ascent-based schemes. Similar message passing is involved in the ADMM-based decentralized algorithm proposed in where a reactive power compensation problem based on approximate power flow models is solved. SDR of the OPF task in three-phase unbalanced systems was developed in ; the resultant semidefinite program was solved in a distributed fashion by using ADMM.

The decentralized OID framework considerably broadens the setups of by accommodating different message passing strategies that are relevant in a variety of practical scenarios (e.g., customer-to-utility, customer-to-CEM and CEM-to-CEM communications). The proposed decentralized schemes offer improved optimality guarantees over , since it is grounded on an SDR technique; furthermore, ADMM enables superior convergence compared to . Finally, different from the distributed reactive compensation strategy of , the proposed framework considers the utilization of an exact AC power flow model, as well as a joint computation of active and reactive power setpoint.

For completeness, ADMM was utilized also in for decentralized multi-area state estimation in transmission systems, and in to distribute over geographical areas the distribution system reconfiguration task.

The remainder of the paper is organized as follows. Section II briefly outlines the centralized OID problem proposed in . Sections III and IV describe the two DOID problems discussed above. Case studies to validate the approach are presented in Section V. Finally, concluding remarks and directions for future work are presented in Section VI.

II Centralized optimal inverter dispatch

Consider a distribution system comprising $N+1$ nodes collected in the set ${\cal N}:=\{0,1,\ldots,N\}$ (node denotes the secondary of the step-down transformer), and lines represented by the set of edges ${\cal E}:=\{(m,n)\}\subset{\cal N}\times{\cal N}$. For simplicity of exposition, a balanced system is considered; however, both the centralized and decentralized frameworks proposed subsequently can be extended to unbalanced systems following the methods in . Subsets ${\cal U},{\cal H}\subset{\cal N}$ collect nodes corresponding to utility poles (with zero power injected or consumed), and those with installed residential PV inverters, respectively (see Fig. 1).

where ${\cal N}_{m}:=\{j\in{\cal N}:(m,j)\in{\cal E}\}$ denotes the set of nodes connected to the $m$-th one through a distribution line.

where $P_{c,h}$ is the active power curtailed, and $Q_{c,h}$ is the reactive power injected/absorbed by the inverter at node $h$. Notice that if there is no limit to the power factor, then $\theta=\pi/2$, and the operating region is given by Fig. 3(c).

II-B Centralized optimization strategy

The centralized OID framework in invokes joint optimization of active and reactive powers generated by the PV inverters, and it offers the flexibility of selecting the subset of critical PV inverters that should be dispatched in order to fulfill optimization objectives and ensure electrical network constraints. To this end, let $z_{h}$ be a binary variable indicating whether PV inverter $h$ provides ancillary services or not and assume that at most $K<|{\cal H}|$ PV inverters are allowed to provide ancillary services. Selecting a (possibly time-varying) subset of inverters promotes user fairness , prolongs inverter lifetime , and captures possible fixed-rate utility-customer pricing/rewarding strategies . Let ${\bf p}_{c}$ and ${\bf q}_{c}$ collect the active powers curtailed and the reactive powers injected/absorbed by the inverters. With these definitions, the OID problem is formulated as follows:

As with various OPF-type problem formulations, the power balance and lower bound on the voltage magnitude constraints (2b), (2c) and (2d), respectively, render the OID problem nonconvex, and thus challenging to solve optimally and efficiently. Unique to the OID formulation are the binary optimization variables $\{z_{h}\}$; finding the optimal (sub)set of inverters to dispatch involves the solution of combinatorially many subproblems. Nevertheless, a computationally-affordable convex reformulation was developed in , by leveraging sparsity-promoting regularization and semidefinite relaxation (SDR) techniques as briefly described next.

In order to bypass binary selection variables, key is to notice that that if inverter $h$ is not selected for ancillary services, then one clearly has that $P_{c,h}=Q_{c,h}=0$ [cf. (2g)]. Thus, for $K<|{\cal H}|$, one has that the $2|{\cal H}|\times 1$ real-valued vector $[P_{c,1},Q_{c,1},\ldots,P_{c,|{\cal H}|},Q_{c,|{\cal H}|}]^{\textsf{T}}$ is group sparse ; meaning that, either the $2\times 1$ sub-vectors $[P_{c,h},Q_{c,h}]^{\textsf{T}}$ equal $\mathbf{0}$ or not . This group-sparsity attribute enables discarding the binary variables and to effect PV inverter selection by regularizing the cost in (2) with the following function:

where $\lambda\geq 0$ is a tuning parameter. Specifically, the number of inverters operating under OID decreases as $\lambda$ is increased .

Key to developing a relaxation of the OID task is to express powers and voltage magnitudes as linear functions of the outer-product Hermitian matrix ${\bf V}:={\bf v}{\bf v}^{\textsf{H}}$, and to reformulate the OID problem with cost and constraints that are linear in ${\bf V}$, as well as the constraints ${\bf V}\succeq\mathbf{0}$ and ${\textrm{rank}}({\bf V})=1$ . The resultant problem is still nonconvex because of the constraint ${\textrm{rank}}({\bf V})=1$; however in the spirit of SDR, this constraint can be dropped.

If the optimal solution of the relaxed problem (4) has rank 1, then the resultant voltages, currents, and power flows are globally optimal for given inverter setpoints . Sufficient conditions for SDR to be successful in OPF-type problems are available for networks that are radial and balanced in , whereas the virtues of SDR for unbalanced medium- and low-voltage distribution systems have been demonstrated in . As for the inverter setpoints $\{(P_{h}^{\textrm{av}}-P_{c,h},Q_{c,h})\}$, those obtained from (4) may be slightly sub-optimal compared to the setpoints that would have been obtained by solving the optimization problem (2). This is mainly due to the so-called “shrinkage effect” introduced by the regularizer (3) . Unfortunately, a numerical assessment of the optimality gap is impractical, since finding the globally optimal solution of problem (2) under all setups is computationally infeasible.

To solve the OID problem, all customers’ loads and available powers $\{P_{h}^{\textrm{av}}\}$ must be gathered at a central processing unit (managed by the utility company), which subsequently dispatches the PV inverter setpoints. Next, decentralized implementations of the OID framework are presented so that the OID problem can be solved in a decentralized fashion with limited exchange of information. From a computational perspective, decentralized schemes ensure scalability of problem complexity with respect to the system size.

III DOID: utility-customer message passing

Consider decoupling the cost $C({\bf V},{\bf p}_{c})$ in (4a) as $C({\bf V},{\bf p}_{c})=C_{\textrm{utility}}({\bf V},{\bf p}_{c})+\sum_{h}R_{h}(P_{c,h})$, where $C_{\textrm{utility}}({\bf V},{\bf p}_{c})$ captures utility-oriented optimization objectives, which may include e.g., power losses in the network and voltage deviations ; and, $R_{h}(P_{c,h})$ is a convex function modeling the cost incurred by (or the reward associated with) customer $h$ when the PV inverter is required to curtail power. Without loss of generality, a quadratic function $R_{h}(P_{c,h}):=a_{h}P_{c,h}^{2}+b_{h}P_{c,h}$ is adopted here, where the choice of the coefficients is based on specific utility-customer prearrangements or customer preferences .

where constraints (5g) ensure that utility and customer agree upon the inverters’ setpoints, and $\bar{C}({\bf V},\bar{{\bf p}}_{c},\bar{{\bf q}}_{c}):=C_{\textrm{utility}}({\bf V},\bar{{\bf p}}_{c})+G(\bar{{\bf p}}_{c},\bar{{\bf q}}_{c})$ is the regularized cost function to be minimized at the utility.

The consensus constraints (5g) render problems (4) and (5) equivalent; however, the same constraints impede problem decomposability, and thus modern optimization techniques such as distributed (sub-)gradient methods and ADMM [15, Sec. 3.4] cannot be directly applied to solve (5) in a decentralized fashion. To enable problem decomposability, consider introducing the auxiliary variables $x_{h},y_{h}$ per inverter $h$. Using these auxiliary variables, (5) can be reformulated as

Problem (6) is equivalent to (4) and (5); however, compared to (4)-(5), it is amenable to a decentralized solution via ADMM [15, Sec. 3.4] as described in the remainder of this section. ADMM is preferred over distributed (sub-)gradient schemes because of its significantly faster convergence and resilience to communication errors .

Per inverter $h$, let $\bar{\gamma}_{h},\gamma_{h}$ denote the multipliers associated with the two constraints in (23c), and $\bar{\mu}_{h},\mu_{h}$ the ones associated with (23d). Next, consider the partial quadratically-augmented Lagrangian of (6), defined as follows:

where $\bar{{\cal P}}:=\{{\bf V},\bar{{\bf p}}_{c},\bar{{\bf q}}_{c}\}$ collects the optimization variables of the utility; ${\cal P}_{h}:=\{P_{c,h},Q_{c,h}\}$ are the decision variables for customer $h$; ${\cal P}_{xy}:=\{x_{h},y_{h},\forall h\in{\cal H}\}$ is the set of auxiliary variables; ${\cal D}:=\{\bar{\gamma}_{h},\gamma_{h},\bar{\mu}_{h},\mu_{h},\forall h\in{\cal H}\}$ collects the dual variables; and $\kappa>0$ is a given constant. Based on (7), ADMM amounts to iteratively performing the steps [S1]-[S3] described next, where $i$ denotes the iteration index:

[S1] Update variables $\bar{{\cal P}}$ as follows:

Furthermore, per inverter $h$, update $P_{c,h},Q_{c,h}$ as follows:

[S2] Update auxiliary variables ${\cal P}_{xy}$:

In [S1], the primal variables $\bar{{\cal P}},\{{\cal P}_{h}\}$ are obtained by minimizing (7), where the auxiliary variables ${\cal P}_{xy}$ and the multipliers ${\cal D}$ are kept fixed to their current iteration values. Likewise, the auxiliary variables are updated in [S2] by fixing $\bar{{\cal P}},\{{\cal P}_{h}\}$ to their up-to-date values. Finally, the dual variables are updated in [S3] via dual gradient ascent.

It can be noticed that step [S2] favorably decouples into $2|{\cal H}|$ scalar and unconstrained quadratic programs, with $x_{h}[i+1]$ and $y_{h}[i+1]$ solvable in closed-form. Using this feature, the following lemma can be readily proved.

Suppose that the multipliers are initialized as $\bar{\gamma}_{h}=\gamma_{h}=\bar{\mu}_{h}=\mu_{h}=0$. Then, for all iterations $i>0$, it holds that:

iii) $x_{h}[i]=\frac{1}{2}\bar{P}_{c,h}[i]+\frac{1}{2}P_{c,h}[i]$; and,

iv) $y_{h}[i]=\frac{1}{2}\bar{Q}_{c,h}[i]+\frac{1}{2}Q_{c,h}[i]$.

Using Lemma 1, the conventional ADMM steps [S1]–[S3] can be simplified as follows.

[S1′] At the utility side, variables $\bar{{\cal P}}$ are updated by solving the following convex optimization problem:

At the customer side, the PV-inverter setpoints are updated by solving the following constrained quadratic program:

[S2′] At the utility and customer sides, the dual variables are updated as:

The resultant decentralized algorithm entails a two-way message exchange between the utility and customers of the current iterates $\bar{{\bf p}}_{c}[i],\bar{{\bf q}}_{c}[i]$ and ${\bf p}_{c}[i],{\bf q}_{c}[i]$. Specifically, at each iteration $i>0$, the utility-owned device solves the OID rendition (12) to update the desired PV-inverter setpoints based on the performance objectives described by $\bar{C}({\bf V},\bar{{\bf p}}_{c},\bar{{\bf q}}_{c})$ (which is regularized with the term $F(\bar{{\bf p}}_{c},\bar{{\bf q}}_{c},\{{\cal P}_{h}[i]\})$ enforcing consensus with the setpoints computed at the customer side), as well as the electrical network constraints (5b)-(5e); once (12) is solved, the utility relays to each customer a copy of the iterate value $(\bar{P}_{c,h}[i+1],\bar{Q}_{c,h}[i+1])$. In the meantime, the PV-inverter setpoints are simultaneously updated via (13) and subsequently sent to the utility. Once the updated local iterates are exchanged, utility and customers update the local dual variables (14).

The resultant decentralized algorithm is tabulated as Algorithm 1, illustrated in Fig. 4, and its convergence to the solution of the centralized OID problem (4) is formally stated next.

Notice that problem (12) can be conveniently reformulated in a standard SDP form (which involves the minimization of a linear function, subject to linear (in)equalities and linear matrix inequalities) by introducing pertinent auxiliary optimization variables and by using the Schur complement . Finally, for a given consensus error $0<\epsilon\ll 1$, the algorithm terminates when $\|\bar{{\bf p}}_{c}[i]-{\bf p}_{c}[i]\|_{2}^{2}+\|\bar{{\bf q}}_{c}[i]-{\bf q}_{c}[i]\|_{2}^{2}\leq\epsilon$. However, it is worth emphasizing that, at each iteration $i$, the utility company solves a consensus-enforcing regularized OID problem, which yields intermediate voltages and power flows that clearly adhere to electrical network constraints.

Once the decentralized algorithm has converged, the real and reactive setpoints are implemented in the PV inverters. Notice however that Algorithm 1 affords an online implementation; that is, the intermediate PV-inverter setpoints $\bar{{\bf p}}_{c}[i],\bar{{\bf q}}_{c}[i]$ are dispatched (and set at the customer side) as and when they become available, rather than waiting for the algorithm to converge.

IV DOID: Network cluster partitions

Based on this network partitioning, consider decoupling the network-related cost $\bar{C}({\bf V},\bar{{\bf p}}_{c},\bar{{\bf q}}_{c})$ in (5a) as

where $N_{a}$ is the number of clusters, $C^{a}({\bf V}^{a},\bar{{\bf p}}_{c}^{a})$ captures optimization objectives of the $a$-th cluster (e.g., power losses within the cluster ), and the sparsity-promoting regularization function is used to determine which PV inverters in ${\cal H}^{a}$ provide ancillary services. Further, per-cluster $a=1,\ldots,N_{a}$, define the region of feasible power flows as [cf. (5b)-(5e)]:

Notice that, similar to (5g), constraints (21d) ensure that the CEM and customer-owned PV systems consent on the optimal PV-inverter setpoints. Formulation (21) effectively decouples cost, power flow constraints, and PV-related consensus constraints (21d) on a per-cluster basis. The main challenge towards solving (21) in a decentralized fashion lies in the positive semidefinite (PSD) constraint ${\bf V}\succeq\mathbf{0}$, which clearly couples the matrices $\{{\bf V}^{a}\}$. To address this challenge, results on completing partial Hermitian matrices from will be leveraged to identify partitions of the distribution network in clusters for which the PSD constraint on ${\bf V}$ would decouple to ${\bf V}^{a}\succeq\mathbf{0}$, $\forall a$. This decoupling would clearly facilitate the decomposability of (21) in per-cluster sub-problems .

Under (i)–(ii), there exists a rank-$1$ matrix ${\bf V}^{\textrm{opt}}$ solving (21) optimally if and only if ${\textrm{rank}}\{{\bf V}^{a}\}=1,\,\forall a=1,\ldots,N_{a}$. $\Box$

Similar to (6), auxiliary variables are introduced to enable decomposability of (22) in per-cluster subproblems. With variables $x_{h},y_{h}$ associated with inverter $h$, and ${\bf W}^{a,j},{\bf Q}^{a,j}$ with neighboring clusters $a$ and $j$, (22) is reformulated as:

This problem can be solved across clusters by resorting to ADMM. To this end, a partial quadratically-augmented Lagrangian, obtained by dualizing constraints $\Re\{{\bf V}^{a}_{j}\}={\bf W}^{a,j}$, $\Im\{{\bf V}^{a}_{j}\}={\bf Q}^{a,j}$, $\bar{P}_{c,h}=x_{h}$, and $\bar{Q}_{c,h}=y_{h}$ is defined first; then, the standard ADMM steps involve a cyclic minimization of the resultant Lagrangian with respect to $\{{\bf V}^{a},{\bf p}_{c}^{a},{\bf q}_{c}^{a},\bar{{\bf p}}_{c}^{a},\bar{{\bf q}}_{c}^{a}\}$ (by keeping the remaining variables fixed); the auxiliary variables $\{{\bf W}^{a,j},{\bf Q}^{a,j},x_{h},y_{h}\}$; and, finally, a dual ascent step [15, Sec. 3.4]. It turns out that Lemma 1 still holds in the present case. Thus, using this lemma, along with the result in [12, Lemma 3], it can be shown that the ADMM steps can be simplified as described next (the derivation is omitted due to space limitations):

[S1′′] Each PV system updates the local copy ${\cal P}_{h}[i+1]$ via (13); while, each CEM updates the voltage profile of its cluster, and the local copies of the setpoints of inverters ${\cal H}^{a}$ by solving the following convex problem:

[S2′′] Update dual variables $\{\gamma_{h},\mu_{h}\}$ via (14) at both, the customer and the CEMs; variables \{{\mbox{\boldmath\Upsilon}}_{{a,i}},{\mbox{\boldmath\Psi}}_{{a,i}}\} are updated locally per cluster $a=1,\ldots,N_{a}$ as:

The resultant decentralized algorithm is tabulated as Algorithm 2, illustrated in Fig. 5, and it involves an exchange of: (i) the local submatrices $\{{\bf V}^{a}_{j}[i+1]\}$ among neighboring CEMs to agree upon the voltages on lines connecting clusters; and, (ii) the local copies of the PV inverter setpoints between the CEM and customer-owned PV systems. Using arguments similar to Proposition 3, convergence of the algorithm can be readily established.

For any $\kappa>0$, the iterates $\{\bar{{\cal P}}^{a}[i]\},\{{\cal P}_{h}[i]\},{\cal D}[i]$ produced by [S1′′]–[S2′′] are convergent, and they converge to a solution of the OID problems (4) and (21). $\Box$

Once the decentralized algorithm has converged, the real and reactive setpoints are implemented by the PV inverter controllers.

Finally, notice that the worst case complexity of an SDP is on the order $\mathcal{O}(\max\{N_{c},N_{v}\}^{4}\sqrt{N_{v}}\log(1/\epsilon))$ for general purpose solvers, with $N_{c}$ denoting the total number of constraints, $N_{v}$ the total number of variables, and $\epsilon>0$ a given solution accuracy . It follows that the worst case complexity of (24) is markedly lower than the one of the centralized problem (4). Further, the sparsity of $\{{\bf A}_{n},{\bf B}_{n},{\bf M}_{n}\}$ and the so-called chordal structure of the underlying electrical graph matrix can be exploited to obtain substantial computational savings; see e.g., .

V Case Studies

Consider the distribution network in Fig. 1, which is adopted from . The simulation parameters are set as in to check the consistency between the results of centralized and decentralized schemes. Specifically, the pole-pole distance is set to $50$ m; lengths of the drop lines are set to $20$ m; and voltage limits $V^{\textrm{min}},V^{\textrm{max}}$ are set to 0.917 pu and 1.042 pu, respectively (see e.g., ). The optimization package CVX[Online] Available: http://cvxr.com/cvx/ is employed to solve relevant optimization problems in MATLAB. In all the conducted numerical tests, the rank of matrices ${\bf V}$ and $\{{\bf V}^{a}\}$ was always $1$, meaning that globally optimal power flow solutions were obtained for given inverter setpoints.

Assume that the objective of the utility company is to minimize the power losses in the network; that is, upon defining the symmetric matrix ${\bf L}_{mn}:=\Re\{y_{mn}\}({\bf e}_{m}-{\bf e}_{n})({\bf e}_{m}-{\bf e}_{n})^{\textsf{T}}$ per distribution line $(m,n)\in{\cal E}$, function $C_{\textrm{utility}}({\bf V},\bar{{\bf p}}_{c})$ is set to $C_{\textrm{utility}}({\bf V},\bar{{\bf p}}_{c})={\textrm{Tr}}({\bf L}{\bf V})$, with ${\bf L}:=\sum_{(m,n)\in{\cal E}}{\bf L}_{mn}$ (see for more details). At the customer side, function $R_{h}(P_{c,h})$ is set to $R_{h}(P_{c,h})=0.1P_{c,h}$. The impact of varying the parameter $\lambda$ is investigated in detail in , and further illustrated in Fig. 7, where the solution of the centralized OID problem (4) is reported for different values of the parameter $\lambda$ [cf. (3)]. Specifically, Fig. 7(a) illustrates the active power curtailed from each inverter during the course of the day when $\lambda=0.8$, whereas the result in Fig. 7(b) were obtained by setting $\lambda=0$. It is clearly seen that in the second case all inverters are controlled; in fact, they all curtail active power from 8:00 to 18:00. When $\lambda=0.8$, the OID seeks a trade off between achievable objective and number of controlled inverters. It is clearly seen that the number of participating inverters grows with increasing solar irradiation, with a maximum of $7$ inverters operating away from the business-as-usual point at 13:00.

Figure 9 represents the discrepancies between local voltages on the line $(8,11)$; specifically, the trajectories of the voltage errors $|V_{8}^{1}[i]-V_{8}^{2}[i]|$ and $|V_{11}^{1}[i]-V_{11}^{2}[i]|$ are reported as a function of the ADMM iteration index $i$. The results indicate that the two CEMs consent on the voltage of the branch that connects the two clusters. The “bumpy” trend is typical of the ADMM (see e.g., ). Similar trajectories were obtained for the inverter setpoints.

VI Concluding Remarks

A suite of decentralized approaches for computing optimal real and reactive power setpoints for residential photovoltaic (PV) inverters were developed. The proposed decentralized optimal inverter dispatch strategy offers a comprehensive framework to share computational burden and optimization objectives across the distribution network, while highlighting future business models that will enable customers to actively participate in distribution-system markets.

References