Decentralized Coordination of DERs for Dynamic Economic Emission Dispatch

Abstract

This paper focuses on the dynamic economic emission dispatch (DEED) problem, to coordinate the distributed energy resources (DERs) in a power system and achieve economical and environmental operation. Distributed energy storages (ESs) are introduced into problem formulation in which charging/discharging efficiency is taken into account. By relaxing the nonconvexity induced by the charging/discharging model of ESs and network losses, we convert the non-convex DEED problem into its convex equivalency. Then, through a Lagrangian duality reformulation, an equivalent unconstrained consensus optimization model is established—a novel consensus-based decentralized algorithm, where the incremental cost is chosen as the consensus variable. At each iteration, only one primal variable requires sub-optimization, and it is completely locally updated. This is different from the well-known alternating direction method of multiplier (ADMM)-based algorithms where more than one subproblem needs to be solved at each iteration. The results of the comparative experiments also reflect the algorithm’s advantage in terms of computational efficiency. The simulation results validate the effectiveness of the proposed algorithm, achieving a balance between emissions and economic considerations.

1. Introduction

In recent years, power systems have been revolutionized due to the integration of massive DERs. These resources have gained increasing significance due to their ability to generate electricity locally and efficiently. Simultaneously, ES technologies, such as batteries and pumped hydro storage, have emerged as key components in the evolution of the power grid. They play a pivotal role in storing excess energy generated by DERs and releasing it when needed. ES systems enhance grid stability, provide backup power during outages, and facilitate the integration of intermittent renewable energy sources. This growing emphasis on DERs and ES represents a shift toward more decentralized and sustainable power generation, impacting not only the power industry but also the broader context of energy sustainability. In this context, our research seeks to address the challenges and opportunities presented by optimizing the operation of DERs and ESs in modern power systems.

1.1. Literature Review

Economic dispatch (ED) is a critical topic in power systems, which aims at seeking the optimal power generation schedule of generators under the most economical condition while satisfying supply–demand and certain constraints [1]. ED problems can be further separated into two groups, static economic dispatch (SED) and dynamic economic dispatch (DED), depending on whether predicted loads are at a time slot or over a certain period. DED is more complicated than SED since it respects ramp limits, which are significant to the lifespan of generators. A vast literature has been reported for solving ED, e.g., the linear programming method [2], the Lambda-iteration method [3], the interior point method [4], computational intelligence methods [5], etc.

As mentioned above, the objective of ED is to allocate power generation among generators to minimize the costs of power generation, but it neglects the gas produced by thermal generators. With the rising awareness of the global warming phenomenon, environmental concerns put forward requirements to reduce greenhouse gas emissions. Emission dispatching is an effective option to address this issue. However, more benefits of power generation and less emissions of greenhouse gases are mutually contradictory, which results in a sophisticated optimization problem, i.e., the combined economic emission dispatch (EED) problem. It has to achieve economical power generation and maintain a low emission level simultaneously [6].

The reported work examining emissions in the SED/DED problem can be divided into three categories in accordance with the formulation of the EED problem. The first one is to optimize the power generation cost and convert the emission into a constraint with a pre-specified limit [7]. However, this formulation hampers the trade-off between cost and emission. The second one treats power generation cost and greenhouse gas emission as competitive objectives and minimizes both of them simultaneously [8,9]. In the third one, through a linear combination of the two conflicting objectives based on the weight-sum method, a single structure of the optimization problem is formulated, where a user-defined weight factor is imposed on each objective function [10,11,12]. The weight factor can be used as a means of studying the trade-off between costs and emissions and usually refers to the importance of the objective. These methods receive meaningful results for the decision-maker by adjusting the value of the weight factor.

1.2. Motivations

1.2.1. Social Level Motivation

The EED problem holds great significance within the broader context of power systems and sustainability. By optimizing the economic operation of power generation, EED contributes to the efficient allocation of resources and the minimization of operational costs, which is of paramount importance in the energy sector. This optimization not only improves the financial performance of power generation but also reduces the environmental footprint by minimizing emissions.

The power industry plays a vital role in global efforts to reduce greenhouse gas emissions and transition toward a sustainable, low-carbon energy future. EED is a key component of this transition, as it enables power systems to operate in a manner that maximizes efficiency while minimizing environmental impact. Through EED, power generators can strike a balance between economic efficiency and environmental sustainability, aligning with the broader goals of cleaner and more sustainable energy production.

In summary, EED is a crucial problem within the power systems field, as it directly impacts the cost-effectiveness, reliability, and environmental sustainability of power generation. Our research aims to contribute to the ongoing efforts to address these challenges.

1.2.2. Technical Level Motivation

Note that all of the above-mentioned works for ED or EED problems are centralized methods, where all committed generators are regulated by the central coordinator. The future power system will be integrated with a great number of DERs, advanced communication, and control technologies [13]. Centralized methods have to confront the following challenges:

(i)

Increasing computational and communicational costs of the central coordinator.

(ii)

Scalable and flexible coordination of DERs.

(iii)

Privacy and prerogative concerns raised by different DER owners.

Hence, the pattern of dispatch methods needs to endow control systems with high flexibility, efficiency, security, and low computation and communication costs.

To overcome the limitations and meet the requirements, an alternative decentralized scheme has become a well-studied methodology for dealing with dispatch problems in power systems. The key is to decentralize the complex task of the central coordinator into coordinators embedded in each committed generating unit, achieving an optimal schedule through local communications and computations. We would like to emphasize that the decentralized paradigm does not mean the complete elimination of the central coordinator but a useful complement of localized manipulations to realize scalable and flexible management of distributed generators. The majority of research on decentralized methods for ED problems is consensus-based protocols, conducted on various situations, including fixed networks [14,15], dynamic networks [16,17], information losses [18,19], etc. Due to the importance of ES devices in improving the reliability of power supply, Wu et al. [20] propose a leader following a consensus algorithm for the DED problem, where the ES is considered the leader. Then, Yang et al. [21] removed the requirement of a leader and developed a fully decentralized DED algorithm over time-varying directed networks. Furthermore, the ADMM [22] and Quasi-Newton methods [23] show their effectiveness in DED. The research on decentralized algorithms for the EED problem is relatively scarce. Recently, the push-sum method [24] has been employed to address the EED problem involving energy storage devices without the consideration of charging/discharging efficiency [25]. In [26], the authors present a decentralized algorithm for the EED problem over dynamic networks, where two consensus protocols run in parallel in each generator. Nevertheless, both [24,26] are only applicable to a static case where ramp constraints are neglected.

In recent years, ADMM received popularity in decentralized optimization, due to its high adaptability, easy implementation, and fast convergence. In [27], a decentralized algorithm is developed on the basis of ADMM for DEED. The updates of primal and dual variables are decomposed into separate steps, and each of them is executed in a decentralized manner. Two primal variables are updated by solving sub-optimization problems, one uses the consensus theory, the other is further decomposed into independent sub-problems. Ref. [28] takes into account environmental cost with an E-exponential term; the formulated DEED problem is solved by an ADMM-based algorithm. By using the consensus theory and a parallel projection approach in the primal updates, the algorithm achieves decentralized implementation. In [29], a dual consensus ADMM is proposed for dynamic optimal power flow with the carbon emission trading problem. By applying ADMM to the dual problem, it reduces public information at each iteration. Communication uncertainties, including packet drops and delays, are considered in [30] for ED problems. Similar with [28], a consensus-based ADMM is proposed where the primal updates use the consensus theory and a projection operator. It is worth noting that most current distributed ADMM approaches model the optimization problem in the form of two- or multi-block joint optimization, see [27,28,30] and the references therein. Although the ADMM algorithm framework decomposition uses techniques to transform joint optimization into multiple subproblems, it often requires two or more sub-optimizations at each iteration for primal updates, undoubtedly increasing computational complexity. Meanwhile, in Ref. [29], by formulating the dual problem in the form of consensus optimization, only one subproblem is required at each iteration, the solution of the subproblem still relies on information exchange with neighboring nodes. These motivate us to investigate a new decentralized methodology for the DEED problem.

1.3. Contributions

This paper studies decentralized solutions to the DEED problem. The weight-sum method is adopted to formulate the problem, to investigate the trade-off between power generation cost and emission that aims at providing instructions to scientific decision making for power system operation. We first transform the DEED problem through a Lagrangian duality reformulation into an equivalent consensus optimization problem, where the multipliers are regarded as consensus variables. We then iterate them to the optimal solution using a distributed gradient descent algorithm. The optimal solution for the original variables is obtained through fully localized sub-optimization. Detailed contributions can be summarized as follows:

• Compared with SED, DED, and EED problems, a more general DEED problem is studied, in which outputs of generators are subject to ramp constraints. In particular, transmission losses among grids are introduced to agree more with the real-world power system. Additionally, energy storage devices are considered to improve the reliability of the power supply with charging/discharging efficiency taken into account.
• With a rigorous conversion process, the formulated DEED problem is converted into an equivalent consensus optimization problem, which is applicable to be addressed by a majority of existing decentralized consensus-based algorithms [24,31,32,33,34,35]. This result may facilitate the popularization of decentralized solutions to the DEED problem.
• A new decentralized algorithm is developed for solving the consensus optimization problem. Compared with existing common ADMM-based methods [27,28,29,30], it reduces computational costs to some extent at each iteration because only one subproblem needs to be solved, and it is entirely localized. Especially for conventional quadratic cost functions, the proposed decentralized algorithm achieves a Q-linear convergence rate.
• Compared with decentralized methods [14,15,16,17,18,19,20,21,22,26,27], the proposed algorithm is more flexible in the choice of stepsizes since nonidentical stepsizes are adopted and an explicit upper bound for nonidentical stepsizes is provided. More notably, it allows asynchronous implementation that removes the global clock coordination and thus achieves fully decentralized realization.

1.4. Organization

The rest of this paper is organized as follows: Section 2 formulates the DEED problem. Section 3 presents the conversion process of the DEED problem into an equivalent consensus optimization problem. Section 4 proposes the decentralized algorithm and demonstrates the convergence results. Simulations are shown in Section 5. Section 6 makes conclusions.

2. Problem Formulation

This section introduces the cost functions and physical constraints of DGs and ESs, and necessary grid constraints need to be considered. Then, the DEED problem is formulated based on the weight-sum method.

2.1. Objective Functions

2.1.1. Cost Functions

The conventional cost functions of thermal generators are approximated by quadratic functions:

$$C_i^{}\left(p_{i,t}\right)=a_i{p}_{i,t}^2+b_i{p}_{i,t}+c_i$$

(1)

where $p_{i,t}$ is the power generated by unit i at time slot $t\in \mathcal{T}:=\{1,\dots ,T\}$, T is the horizon of the schedule period, $C_i\left(p_{i,t}\right)$ is the power generation cost, $a_i,b_i,c_i>0$ are the cost coefficients.

In this paper, the cost functions of DGs and ESs are considered as more general functions that cover the quadratic functions, satisfying the following assumption.

Assumption 1. 

For each $i=1,\dots ,m+s$ and $t\in \mathcal{T}$, the cost function $C_i\left(p_{i,t}\right):R\to R^{+}$ is proper, closed, and ${\mu }_i$-strongly convex with ${\mu }_i>0$.

2.1.2. Emission Functions

Emissions of greenhouse gases, i.e., SO${}_2$, CO${}_2$, and NO${}_x$, are the most significant factors considered in emission dispatch. The emissions can be expressed respectively or conjunctively as [10]

$$E_i^{emi}\left(p_{i,t}\right)={\alpha }_i+{\beta }_i{p}_{i,t}+{\gamma }_i{p}_{i,t}^2+{\xi }_{i}exp\left({\sigma }_i{p}_{i,t}\right)$$

(2)

where $E_i^{emi}\left(p_{i,t}\right)$ is the amount of emissions from DG at time slot t, and ${\alpha }_i,{\beta }_i,{\gamma }_i,{\xi }_i,{\sigma }_i$ are the emission coefficients.

2.2. Constraints

The constraints of the studied DEED problem include the power balance constraint, power output limits, ramping limits for DGS, and state-of-charge (SOC) limits of ESs.

2.2.1. Power Balance

At each time slot t, the total power generated by all committed generating units equals the sum of the total load demand and power losses. That is,

$$\sum _{i=1}^{m+s}{p}_{i,t}-P_{loss,t}=D_t$$

(3)

where $D_t$ and $P_{loss,t}$ are the demands and network losses, respectively. The losses $P_{loss,t}$ are inevitable among the grid. The network loss could be rigorously assessed by integrating the power flow equations into the equality constraints [36]. Although this approach allows the analysis to gain insight into the power system operation, it demands high computational resources that could inhibit its deployment on distributed and cooperative computing entities. Therefore, a simplified B-coefficient approximation approach is adopted to compute the network loss in this paper [37,38]. In detail, according to the micro-incremental transmission losses of each generator, the amount of transmission losses induced by each generator can be represented as a simple quadratic function. The B-coefficients can provide a sufficiently accurate estimation of the total transmission losses requiring only off-line computations. Using the micro-incremental transmission losses of each generator, the network losses can be estimated by:

$$\begin{array}{c}\hfill P_{loss,t}=\sum _{i=1}^{m+s}{p}_{i,t}^l=\sum _{i=1}^{m+s}{B}_i{p}_{i,t}^2\end{array}$$

(4)

where $p_{i,t}^l$ denotes the losses induced by DG or ES i, and $B_i>0$ is the loss coefficient.

2.2.2. Constraints for DG

For each DG $i\in \mathcal{M}:=\{1,\cdots ,m\}$, two physical constraints are considered. The first is the power generation limit reflecting the generation capacity of DG i, denoted by

$$p_i^{min}\le p_{i,t}\le p_i^{max},i\in \mathcal{M}$$

(5)

where $p_i^{min}$ and $p_i^{max}$ are, respectively, the lower and upper bound of the power generation capacity of DG i.

The second is the ramping limit for DG i as follows

$$p_i^{down}\le p_{i,t}-p_{i,t-1}\le p_i^{up},i\in \mathcal{M}$$

(6)

where $p_i^{down}$ and $p_i^{up}$ are the maximum ramp up/down rates for DG i.

2.2.3. Constraints for ES

Similar to DGs, each ES has bounded power exchange with the grid due to its power-limited charging/discharging capacity, i.e.,

$$p_i^{min}\le p_{i,t}\le p_i^{max},\forall t\in T,i\in \mathcal{S}$$

(7)

We introduce the charging/discharging efficiency model of ES $i\in \mathcal{S}:=$$\{m+1,\cdots ,m+s\}$ as below

$$\begin{array}{c}\hfill p_{i,t}^{batt}=\left\{\begin{array}{c}\hfill \frac{1}{{\eta }_i^{+}}{p}_{i,t},\mathrm{if}{p}_{i,t}\ge 0\hfill \\ \hfill {\eta }_i^{-}{p}_{i,t},\mathrm{if}{p}_{i,t}\le 0\hfill \end{array}\right.\forall t\in T\end{array}$$

(8)

where $p_{i,t}^{batt}$ is the rate of change of energy stored in ES i at time slot t, and ${\eta }_i^{+},{\eta }_i^{-}\in (0,1)$ are the discharging and charging efficiencies of ES i, respectively. Hence, the SOC of ES i across the time horizon can be expressed as

$$E_{i,t}=E_{i,t-1}-p_{i,t}^{batt}\nabla T,\forall t\in T$$

(9)

where $\nabla T$ is the size of the period and $E_{i,t}$ is the energy stored in ES i at time slot t, which should satisfy the storage capacity, i.e.,

$$0\le E_{i,t}\le E_i^{max},\forall t\in T$$

(10)

Now, we proceed to the formulation of the DEED problem.

2.3. DEED Problem

Considering a power system consisting of m DGs and s ESs, the DEED problem is described as below

$$\begin{array}{c}minF\left(p_{i,t}\right)=\omega \sum _{t=1}^T\sum _{i=1}^{m+s}{C}_i\left(p_{i,t}\right)+(1-\omega )\sum _{t=1}^T\sum _{i=1}^m{E}_i^{emi}\left(p_{i,t}\right)\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{constraints}\left(3)–(10\right)\hfill \end{array}$$

(11)

where $w\in [0,1]$ is a user-defined weight to study the trade-off between the two conflicting objectives. Note that when $w=1$, problem (11) is a typical DED problem, and a pure dynamic emission dispatch problem when $w=0$. However, due to the nonlinearity of the B-coefficient approximation, constraint (3) results in a non-convex feasible set.

Due to the non-convex constraint (8), the set (15) is non-convex. Thus, problem (11) is a non-convex optimization problem that is hard to address.

3. Problem Reformulation

Next, to solve the non-convex problem (11) in a decentralized manner, before presenting the algorithm, we first convert the DEED problem into its convex equivalency.

3.1. Convex Equivalency of DEED

We first address the nonconvexity of the charging/discharging efficiency model (8). Define $p_{i,t}^{+}\in [0,p_i^{max}]$ and $p_{i,t}^{-}\in [0,-p_i^{min}]$, indicating that ES i charges and discharges at time slot t, respectively. We have $p_{i,t}=p_{i,t}^{+}-p_{i,t}^{-},\forall t\in T$. Then, $p_{i,t}^{batt}$ (8) can be replaced by

$$\begin{array}{c}\hfill p_{i,t}^{batt}=\frac{1}{{\eta }_i^{+}}{p}_{i,t}^{+}-{\eta }_i^{-}{p}_{i,t}^{-},\forall t\in T,i\in \mathcal{S}\end{array}$$

(12)

Thus, the nonconvex problem (11) is equivalent to

$$\begin{array}{c}minF\left(p_{i,t}\right)=\omega \sum _{t=1}^T\sum _{i=1}^{m+s}{C}_i\left(p_{i,t}\right)+(1-\omega )\sum _{t=1}^T\sum _{i=1}^m{E}_i^{emi}\left(p_{i,t}\right)\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{constraints}\left(3)–(7),\right(9),\left(10),\mathrm{and}\right(12)\hfill \end{array}$$

(13)

Since storages cannot be charged and discharged simultaneously, either $p_{i,t}^{+}$ or $p_{i,t}^{-}$ needs to be zero, i.e.,

$$\begin{array}{c}\hfill p_{i,t}^{+}{p}_{i,t}^{-}=0,\forall t\in T,i\in \mathcal{S}\end{array}$$

(14)

According to Theorem 1 in [20], since ${\eta }_i^{+}$ and ${\eta }_i^{-}$ are strictly less than 1, they can be neglected.

For simplicity of expression, we rewrite problem (13) in a compact form. For each ES, $i\in \mathcal{S},\forall t\in T,$ we define the constraint set

$$J_i^{\mathcal{S}}=\left\{p_i\in R^T\left|p_{i,t}\mathrm{satisfies}\left(9\right),\left(10\right),\mathrm{and}\left(12\right)\right.\right\}$$

(15)

where $p_i={[p_{i,1},p_{i,2},\cdots ,p_{i,T}]}^{\prime }$. For each DG, $i\in \mathcal{M},\forall t\in T,$ we define the constraint set

$$\begin{array}{c}\hfill J_i^{\mathcal{M}}=\left\{p_i\in R^T\left|p_{i,t}\right.\mathrm{satisfies}\left(5\right)\mathrm{and}\left(6\right)\right\}\end{array}$$

(16)

Define $J_i=\left\{p_i\left|J_i^{\mathcal{M}}\cup J_i^{\mathcal{S}},i\in \mathcal{M}\cup \mathcal{S}\right.\right\}$, $D={[D_1,D_2,\cdots ,D_T]}^{\prime }$, and $P_{loss}=[P_{loss,1}$, $P_{loss,2},\cdots ,P_{loss,T}{]}^{\prime }$, $\forall i\in \mathcal{M}\cup \mathcal{S},t\in \mathcal{T}$. By supplementing the emission functions for ESs with $E_i^{emi}\left(p_{i,t}\right)=0,i\in \mathcal{S}$, problem (13) can be rewritten as follows,

$$\begin{array}{cc}\hfill \underset{p_1,\cdots ,p_{m+s}}{min}& \hfill \sum _{i=1}^{m+s}{\tilde{C}}_i\left(p_i\right)\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \hfill \sum _{i=1}^{m+s}{p}_i-P_{loss}=D\hfill \end{array}$$

(17b)

$$\begin{array}{cc}\hfill & \hfill p_i\in J_i\hfill \end{array}$$

(17c)

where ${\tilde{C}}_i\left(p_i\right)={\sum }_{t=1}^T\omega C_i\left(p_{i,t}\right)+(1-\omega ){\sum }_{t=1}^T{E}_i^{\mathrm{emi}}\left(p_{i,t}\right)$. Note that the set $J_i$ is convex polyhedra since all constraints therein are affine. Then, we proceed to convert problem (17) to an equivalent convex one by relaxing the equality constraint (17b) to an inequality one, as follows

$$\begin{array}{cc}\hfill \underset{p_1,\cdots ,p_{m+s}}{min}& \hfill \sum _{i=1}^{m+s}{\tilde{C}}_i\left(p_i\right)\hfill \end{array}$$

(18a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \hfill \sum _{i=1}^{m+s}{p}_i-P_{loss}\ge D\hfill \end{array}$$

(18b)

$$\begin{array}{cc}\hfill & \hfill p_i\in J_i\hfill \end{array}$$

(18c)

The following lemma provides a sufficient condition to ensure that problem (17) and (18) have the same solution.

Lemma 1 

([37]). Suppose that $D_t\ge {\sum }_{i=1}^{m+s}\left(p_i^{min}-B_{i}p{{}_i^{min}}^2\right)$. Then, the optimal solution to problem (18) is the same as problem (17).

Remark 1. 

The detailed proof of Lemma 1 refers to [37]. It is shown that if the total demand is larger than or equal to the total minimal power generation subtracting power losses at each time slot t, the optimal solution to problem (18) satisfies $D={\sum }_{i=1}^N{p}_i^{★}-P_{loss}$, where $p_i^{★}$ is the optimal power generation schedule for each committed units $i\in \mathcal{M}\cup \mathcal{S}$. That is, problem (18) is an equivalent convex of problem (17). Although each DG has a lower bound of power generation $p_i^{min}>0,i\in \mathcal{M}$, due to the introduction of ESs, the sufficient condition is easy to satisfy since each ES can charge power with $p_i^{min}<0,i\in \mathcal{S}$ to relax the sufficient condition.

Next, we will transform the DEED problem (18) through a Lagrangian duality reformulation into an equivalent consensus optimization problem.

3.2. Lagrange Dual Problem

Note that both the constraint sets (18b) and (18c) are convex. Recall Assumption 1 and the definition of the emission function. Hence, problem (18) has a unique minimizer denoted  $p_i^{★}$. Moreover, there is a zero duality gap, which allows us to convert the original problem (18) to its Lagrange dual form.

To facilitate the design of decentralized algorithms, we introduce $V_i\in R^T$ with no physical meanings, representing virtual local demands at unit i for all the time slots such that ${\sum }_{i=1}^{m+s}{V}_i=D$. The virtual demand for each unit i can choose $V_i=D/(m+s)$ for convenience. Define the indicator function ${\delta }_{J_i}\left(p_i\right)$ as ${\delta }_{J_i}\left(p_i\right)=0$ if $p_i\in {\delta }_{J_i}$ and ${\delta }_{J_i}\left(p_i\right)=\infty$, otherwise. Then, problem (18) can be written as the following equivalent formulation

$$\begin{array}{cc}\hfill \underset{p_1,\cdots ,p_{m+s}}{min}& \hfill \sum _{i=1}^{m+s}{\tilde{C}}_i\left(p_i\right)+{\delta }_{J_i}\left(p_i\right)\hfill \end{array}$$

(19a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \hfill \sum _{i=1}^{m+s}(p_i-p_i^l)\ge \sum _{i=1}^{m+s}{V}_i\hfill \end{array}$$

(19b)

Define the Lagrangian function associated with problem (19) as follows:

$$\Phi (p,\lambda )=\sum _{i=1}^{m+s}{\tilde{C}}_i\left(p_i\right)+{\delta }_{J_i}\left(p_i\right)+{\lambda }^{\prime }\left(\sum _{i=1}^{m+s}(p_i-p_i^l-V_i)\right)$$

(20)

where $p={[{p_1}^{\prime },\dots ,{p_{m+s}}^{\prime }]}^{\prime }\in R^{(m+s)\times T}$, and $\lambda \in R^T$ is the Lagrangian multiplier corresponding to constraint (19b). Then, the corresponding Lagrangian dual problem of (19) is defined as

$$\underset{\lambda \in R^T}{max}\underset{p\in R^{(m+s)\times T}}{inf}\Phi \left(p,\lambda \right)$$

(21)

where

$$\begin{array}{cc}\hfill \underset{p\in R^{(m+s)\times T}}{inf}\Phi \left(p,\lambda \right)=& \hfill \underset{p\in R^{(m+s)\times T}}{inf}\left\{\sum _{i=1}^{m+s}{\tilde{C}}_i\left(p_i\right)+{\delta }_{J_i}\left(p_i\right)+{\lambda }^{\prime }\left(\sum _{i=1}^{m+s}(p_i-p_i^l-V_i)\right)\right\}\hfill \\ \hfill =& \hfill -\sum _{i=1}^{m+s}\underset{p_i\in R^T}{sup}\left(-{\tilde{C}}_i\left(p_i\right)-{\delta }_{J_i}\left(p_i\right)-{\lambda }^{\prime }(p_i-p_i^l)\right)-\sum _{i=1}^{m+s}{\lambda }^{\prime }{V}_i\hfill \end{array}$$

(22)

Define the conjugate function $F_i^{\otimes }\left(\lambda \right)={sup}_{p_i\in R^T}\{{\lambda }^{\prime }(p_i-p_i^l)-{\tilde{C}}_i\left(p_i\right)-{\delta }_{J_i}\left(p_i\right)\}$. Then, we have

$$\underset{p\in {}^{(m+s)\times T}}{inf}\Phi \left(p,\lambda \right)=-\sum _{i=1}^{m+s}{F}_i^{\otimes }\left(-\lambda \right)-\sum _{i=1}^{m+s}{\lambda }^{\prime }{V}_i$$

(23)

Hence, the dual problem (21) can be rewritten as

$$-\underset{\lambda \in {}^T}{min}\left(\sum _{i=1}^{m+s}{F}_i^{\otimes }\left(-\lambda \right)+\sum _{i=1}^{m+s}{\lambda }^{\prime }{V}_i\right)$$

(24)

which is equivalent to the following consensus optimization problem

$$\begin{array}{c}min\sum _{i=1}^{m+s}{\varphi }_i\left({\lambda }_i\right)\hfill \\ \mathrm{s}.\mathrm{t}.{\lambda }_i={\lambda }_j,(i,j)\in \mathcal{E}\hfill \end{array}$$

(25)

where ${\varphi }_i\left(\lambda \right)\stackrel{\Delta }{=}{F}_i^{\otimes }\left(-\lambda \right)+{\lambda }^{\prime }{V}_i$ is convex, continuously differentiable, and $1/{\mu }_i$-Lipschitz smooth (see Theorem 4.2.1, [39]). For any given $\lambda \in R^T$, the dual function ${\varphi }_i\left(\lambda \right)$ is differentiable at $\lambda$ and its gradient is $\nabla {\varphi }_i\left(\lambda \right)=-{argmin}_{p_i\in R^T}\left(\lambda (p_i-p_i^l)+{\tilde{C}}_i\left(p_i\right)+{\delta }_{J_i}\left(p_i\right)\right)+V_i$.

Remark 2. 

Note that the dual problem (25) is a typical consensus-based optimization problem that has been intensively investigated in the field of decentralized optimization on multi-agent systems. Such a formulation caters to a majority of existing decentralized consensus-based algorithms. For conventional undirected communication networks, one may resort to DGD [24], EXTRA [31], AugDGD [34], etc., while for directed or dynamic communication networks, the insightful works like Push-DIGing [33], Frost [32], etc., are worth adoption.

4. Decentralized Algorithm

In this section, we propose a decentralized consensus-based algorithm to address the dual problem (25) on the basis of [35]. The underlying communication network is featured by an undirected graph $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, where $\mathcal{V}=\{1,\dots ,N\}$ is the set of committed generating units and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ denotes the edge set.

4.1. Algorithm Development

To clearly illustrate the difference between our work and [35], defining $U_{ij}=\left\{\begin{array}{c}\hfill I,i<j\hfill \\ \hfill -I,i>j\hfill \end{array}\right.$, problem (25) is equivalent to the following form

$$\begin{array}{c}min\sum _{i=1}^N{\varphi }_i\left({\lambda }_i\right)\hfill \\ \mathrm{s}.\mathrm{t}.U_{ij}{\lambda }_i+U_{ji}{\lambda }_j=0,(i,j)\in \mathcal{E}\hfill \end{array}$$

(26)

Thus, we convert the DEED problem into a simplified form that is applicable to using a modified TriPD algorithm. The detailed iterative steps are presented in Algorithm 1. Figure 1 provides an illustration of the algorithm’s iterative process.

Algorithm 1 decentralized algorithm for DEED problem

1: Initialization: Each committed generating unit $i\in \mathcal{M}\cup \mathcal{S}$ sets ${\lambda }_i^0\in R^T$, $w_{(i,j),i}^0\in R^T,j\in N_i$, and $p_i^0={argmin}_{p_i\in J_i}\left({\left({\lambda }_i^0\right)}^{\prime }{p}_i+{\tilde{C}}_i\left(p_i\right)\right)$ 2: For $k=0,1,\cdots ,K$  do 3: For $\forall j\in N_i$, each unit $i\in \mathcal{M}\cup \mathcal{S}$ excuete $$\begin{array}{cc}\hfill {\overline{w}}_{\left(i,j\right),i}^k& \hfill =\frac{1}{2}\left(w_{\left(i,j\right),i}^k+w_{\left(i,j\right),j}^k\right)+\frac{{\kappa }_{\left(i,j\right)}}{2}\left(U_{ij}{\lambda }_i^k+U_{ji}{\lambda }_j^k\right)\hfill \end{array}$$ (27) $$\begin{array}{cc}\hfill {\lambda }_i^{k+1}& \hfill ={\lambda }_i^k-{\tau }_i\left(-p_i^k+V_i+\sum _{j\in N_i}{U}_{ij}^{\mathrm{T}}{\overline{w}}_{\left(i,j\right),i}^k\right)\hfill \end{array}$$ (28) $$\begin{array}{cc}\hfill w_{\left(i,j\right),i}^{k+1}& \hfill ={\overline{w}}_{\left(i,j\right),i}^k+{\kappa }_{\left(i,j\right)}{U}_{ij}\left({\lambda }_i^{k+1}-{\lambda }_i^k\right)\hfill \end{array}$$ (29) $$\begin{array}{cc}\hfill p_i^{k+1}& \hfill =\underset{p_i\in J_i}{argmin}\left({\left({\lambda }_i^{k+1}\right)}^{\prime }{p}_i+{\tilde{C}}_i\left(p_i\right)\right)\hfill \end{array}$$ (30) 4: Unit i sends $U_{ij}{\lambda }_i^{k+1}$ and $w_{(i,j),i}^k$ to its neighbours. 5: End

In Algorithm 1, for $i=1,\dots ,N$, $k>0$, and $j\in N_i$, where $N_i$ represents the neighbors of i, ${\overline{w}}_{(i,j),i}$ is an auxiliary variable, $w_{(i,j)}$ is the edge variable corresponding to coupling edge constraint (26), and ${\tau }_i>0$ is the stepsize; the edge weight/stepsize ${\kappa }_{(i,j)}$ can be alternatively interpreted as an inherent parameter of the underlying graph.

4.2. Convergence Results

Next, we will show that Algorithm 1 with properly chosen step sizes is capable of solving the DEED problem. Due to the limited space, we only provide the main convergence result and a brief proof outline.

Theorem 1. 

Suppose Assumption 1 holds. When the stepsizes satisfy the following local condition

$${\tau }_i<\frac{1}{\frac{1}{2{\mu }_i}+{\displaystyle \sum _{j\in N_i}}{\kappa }_{(i,j)}}$$

(28)

Algorithm 1 solves the DEED problem (11). That is, the sequences ${\lambda }_i^k$ and $p_i^k$ generated by Algorithm 1 converge to some ${\lambda }^{★}$ and $p_i^{★}$, where ${\lambda }^{★}$ is the optimal incremental cost, and $p_i^{★}$ is the optimal power generation schedule for each committed units $i\in \mathcal{M}\cup \mathcal{S}$. Furthermore, if the cost functions $C_i$ are quadratic, $\forall i\in \mathcal{M}\cup \mathcal{S}$, then ${\lambda }^k$ converges R-linearly to ${\lambda }^{★}$.

Remark 3. 

Note that the dual problem (26) is a consensus optimization problem which can be seen as a special case of the decentralized optimization problem considered in [35]. Although we introduce two different steps to compute the power outputs and transmission losses, the two variables are computed locally and have no effect on the convergence of the dual variable. Hence, the convergence results follow from Theorem 5.1 in [35] where the sufficient conditions are indeed satisfied.

Remark 4. 

The implementation of Algorithm 1 requires a global clock at each iteration to coordinate communication and computation of all units. This is a rather common precondition for existing decentralized algorithms, see all of the above-mentioned decentralized works. However, such a synchronous scheme may compromise the performance of decentralized algorithms since unnecessary idle time remains in units with a fast processing capacity. From [35], we can resort to a randomized block-coordinate method to remove the global clock requirement; thus, Algorithm 1 can be executed asynchronously. Specifically, at each iteration, each unit $i\in \mathcal{M}\cup \mathcal{S}$ is activated independently with probability $P_i$; then, it proceeds to execute Steps 3 and 4 in Algorithm 1. Meanwhile, for unit $i\in \mathcal{M}\cup \mathcal{S}$, which is not activated at each iteration, it keeps the states unchanged. Moreover, the condition (31) remains unchanged for both synchronous and asynchronous implementations of Algorithm 1.

5. Simulations

In this section, we show the performance of the proposed decentralized algorithm for the DEED problem based on a test system modified from [40], where we add two ESs into the original ten-unit system. The cost function of DG is quadratic, i.e., $C_i\left(p_i\right)=a_i{p}_i^2+b_i{p}_i+c_i,\forall i\in \{1,2,\cdots ,10\}),t\in \{1,2,\cdots ,24\}$. The cost coefficients and emission coefficients are provided in

Table 1. Parameters of DGs [40].

Unit	${\mathit{a}}_{\mathit{i}}(\mathit{\$}/\mathbf{M}{\mathbf{W}}^2\mathbf{h})$	${\mathit{b}}_{\mathit{i}}(\mathit{\$}/\mathbf{M}\mathbf{W}\mathbf{h})$	${\mathit{c}}_{\mathit{i}}(\mathit{\$}/\mathbf{h})$	${\mathit{\alpha }}_{\mathit{i}}\left(\mathbf{lb}/\mathbf{h}\right)$	${\mathit{\beta }}_{\mathit{i}}\left(\mathbf{lb}/\mathbf{MWh}\right)$	${\mathit{\gamma }}_{\mathit{i}}\left({\mathbf{lb}/\left(\mathbf{MW}\right)}^2\mathbf{h}\right)$	${\mathit{\xi }}_{\mathit{i}}\left(\mathbf{lb}/\mathbf{h}\right)$	${\mathit{\sigma }}_{\mathit{i}}\left(\mathbf{l}/\mathbf{MW}\right)$	$\left[{\mathit{p}}_{\mathit{i}}^{min},{\mathit{p}}_{\mathit{i}}^{max}\right]$	${\mathit{B}}_{\mathit{i}}$
1	$0.1524$	$38.5397$	$786.7988$	$103.3908$	−2.444	$0.0312$	$0.5035$	0.0207	[150, 470]	0.000049
2	0.1058	46.1591	451.3251	103.3908	−2.4444	0.0312	0.5035	0.0207	[135, 470]	0.000014
3	0.0280	40.3965	1049.9977	300.3910	−4.0695	0.0509	0.4968	0.0202	[73, 340]	0.000015
4	0.0354	38.3055	1243.5311	300.3910	−4.0695	0.0509	0.4968	0.0202	[60, 300]	0.000015
5	0.0211	36.3278	1658.5696	320.0006	−3.8132	0.0344	0.4972	0.0200	[73, 243]	0.000016
6	0.0179	38.2704	1356.6592	320.0006	−3.8132	0.0344	0.4972	0.0200	[57, 160]	0.000017
7	0.0121	36.5104	1450.7045	330.0056	−3.9023	0.0465	0.5163	0.0214	[20, 130]	0.000017
8	0.0121	36.5104	1450.7045	330.0056	−3.9023	0.0465	0.5163	0.0214	[47,120]	0.000018
9	0.1090	39.5804	1455.6056	350.0056	−3.9524	0.0465	0.5475	0.0234	[20, 80]	0.000019
10	0.1295	40.5407	1469.4026	360.0012	−3.9864	0.0470	0.5475	0.0234	[10, 55]	0.000020

. The cost function of the ES is $C_i\left(p_i\right)=d_i{p}_i^2,\forall i\in \{11,12\}),t\in \{1,2,\cdots ,24\}$. The parameters of the ESs are provided in

Table 2. Parameters of ESs.

Unit	${\mathit{d}}_{\mathit{i}}(\mathit{\$}/\mathbf{M}{\mathbf{W}}^2\mathbf{h})$	${\mathit{E}}_{\mathit{i}}^{max}$	$\left[{\mathit{p}}_{\mathit{i}}^{min},{\mathit{p}}_{\mathit{i}}^{max}\right]$	${\mathit{\eta }}_{\mathit{i}}^{+}$	${\mathit{\eta }}_{\mathit{i}}^{-}$	${\mathit{B}}_{\mathit{i}}$
11	0.1	500	[−50, 50]	0.8	0.8	0.000015
12	0.1	400	[−40, 40]	0.88	0.88	0.000015

. The demand during a 24 h horizon is plotted in blue in Figure 2c. For comparison purposes, we first solve the DEED problem with $w=1$, i.e., a DED problem. Then, we adjust the value of the weight factor from 1 to 0 to study the tradeoff between cost and emission.

5.1. DED Problem ($w=1$)

The key premise of EED and DEED is that power outputs should satisfy the power balance constraint, that is, meet the demand. As shown in Figure 2a, to meet the total demand, committed generating units produced more power to offset the transmission losses. Figure 2b presents the detailed power output of each committed generating unit over 24 h. Since EGs are introduced into the power system, we illustrate their performance in improving the reliability of the power supply. In Figure 2c, the red curve is the resulting netload, which reflects the function of ESs to cut the peak and fill the valley. From the SOC results in Figure 2d, it can be found that ESs discharge during peak hours and charge during off-peak hours. Additionally, since the charging/discharging efficiency of ES2 is higher than ES1, as depicted in Figure 2e, ES2 is engaged more frequently than ES1. To show the convergence more clearly, the evaluations of estimates ${\lambda }_i$ at Hour 18 are shown in Figure 2f, where all ${\lambda }_i$ converge to 76.39 $MWh. The convergence of ${\lambda }_i$ at 24 h is presented in Figure 2g, where all ${\lambda }_i$ at different hours converge to each optimal incremental cost, and the curve of the optimal incremental cost shows the same tendency with the total demand.

5.2. DEED Problem ($w\in$ (0, 1))

In this subsection, to investigate the influence of the weight factor w on the tradeoff between costs and emissions, we adjust the value of w from 0 to 1, and then compare the cumulative costs and emissions over 24 h. The results shown in Figure 3a,b indicate that with the increase in w, the cumulative costs decrease while the cumulative emissions increase. The pareto-optimal front of the DEED problem is shown in Figure 3c after running the Algorithm 1 10 times by varying the weight factor w.

5.3. Comparison Study

In this section, we focus on the performance comparison of the algorithms, covering aspects of the number of iterations and execution time. The comparison counterparts include the centralized version of Algorithm 1, termed centralized, and the widely used decentralized ADMM algorithm, termed DADMM. We utilized a modified IEEE 118-bus system with 54 generators. To facilitate the performance comparison, we focused on a single time period dispatch problem, considering only generators with quadratic cost functions while omitting factors such as power losses. In the following simulations, the residual is defined as $(1/N){\sum }_{i=1}^N∥{\lambda }_i^k-{\lambda }^{*}∥$ to reflect the gap between the optimal result and the one obtained by the three methods. We employed the same communication network for Algorithm 1 and DADMM, ensuring a fair comparison. In this network, each edge has a probability of existence of 0.5. Both algorithms were manually adjusted to achieve optimal convergence performance.

Figure 3 and Figure 4 respectively illustrate the convergence with respect to the number of iterations and time for these three algorithms. The stopping criteria were set to reach a residual of ${10}^{-5}$. In Figure 4, we observe that to achieve the same level of accuracy, the centralized algorithm requires the fewest iterations, while within the decentralized algorithms, the DADMM algorithm converges in fewer iterations compared to Algorithm 1. Figure 5 indicates that for achieving the same level of accuracy, the centralized algorithm still outperforms in terms of time, but Algorithm 1 is more time-efficient compared to the DADMM algorithm.

Table 3. Comparative results of different algorithms.

Algorithm	Number of Iterations	Time (s)
Algorithm 1	652	30.3203
Centralized	249	12.6807
DADMM	344	35.2371

provides the specific number of iterations and time required for the three algorithms to reach a residual of ${10}^{-5}$.

6. Discussion

The simulation results confirm the effectiveness of the algorithm proposed in this paper for solving the DEED problem. In the comparative experiments, we observed that whether in terms of the number of iterations or computational time, the centralized algorithm still maintains an advantage. It is understandable that decentralized methods require more iterations to achieve the same level of accuracy compared with their centralized counterparts. This is due to the need for local communication and operations to ensure robustness against single-point failures. To achieve global objective optimization, decentralized algorithms require more iterations to propagate local information throughout the network. This necessity is also influenced by the assumption of network connectivity. Additionally, inherent characteristics of communication transmission, such as bandwidth and latency, also impact the operational efficiency of decentralized algorithms in practical applications. Therefore, while offering flexibility and robustness, enhancing the operational efficiency of decentralized algorithms remains a valuable research area.

In comparison to the DADMM algorithm, the results indicate that while our algorithm has a slower iteration speed, it holds an advantage in terms of computational complexity. This is due to the fact that ADMM-based algorithms require inner loops within each iteration to separately solve the sub-optimization problems for the primal and dual variables. In contrast, by transforming the constrained optimization problem into a consensus optimization problem, Algorithm 1 only needs to solve the sub-optimization problem for the primal variables at each iteration, and this is completed entirely locally. The dual variables do not require an inner loop for updates, significantly reducing the computational cost.

7. Conclusions

In this paper, the DEED problem with ESs was investigated. The key innovation of this work involved a Lagrangian duality transformation process of the DEED problem into an equivalent consensus optimization problem. By choosing the incremental cost as the consensus variable, a decentralized consensus-based algorithm was developed where only one sub-optimization of the primal update was required at each iteration. Thus, compared with existing ADMM-based counterparts, it reduced computational costs to some extent, a point that was corroborated by experimental results. The algorithm proposed in this paper still offers room for further exploration in terms of improving computational accuracy, such as the introduction of acceleration terms. Additionally, the dispatch model presented in this paper is relatively simple and does not delve into the detailed study of renewable energy factors. We have noted the emergence of novel multi-stage optimization models [41,42] aimed at achieving the optimal operation of renewable resources and energy storage. This is an avenue worthy of our further exploration.