Multi-Area Economic Dispatch Under Renewable Integration: Optimization Challenges and Research Perspectives

Abstract

The shift toward decentralized energy systems and the rapid growth of renewable integration have brought renewed attention to the Multi-Area Economic Dispatch (MAED) problem. Unlike single-area dispatch, which focuses only on local balance, MAED must also coordinate inter-area exchanges, respect regional operating limits, and ensure overall reliability. This paper reviews both MAED and its dynamic extension, the Multi-Area Dynamic Economic Dispatch (MADED). The review examines core objectives—cost minimization, emission reduction, and renewable utilization—and surveys a wide range of solution methods. These include classical mathematical programming, metaheuristic and hybrid approaches, and more recent advances based on machine learning and reinforcement learning. Special emphasis is placed on uncertainty-oriented models that address demand variability, market dynamics, and renewable fluctuations. The discussion also highlights the role of Distributed Energy Resources (DERs), Energy Storage Systems (ESSs), and Demand Response (DR) in improving system flexibility and resilience. Despite notable progress, research gaps remain, including limited treatment of uncertainty, insufficient integration of DR, oversimplified modeling of electric vehicles, and the marginal role of reliability. To address these issues, a research agenda is proposed that aims to develop more adaptive, scalable, and sustainable dispatch models. The insights provided are intended to support both academic research and practical applications in the planning and operation of interconnected grids.

1. Introduction

Modern power systems are becoming larger, more interconnected, and increasingly complex. At the same time, global calls for sustainable operation are reshaping the way electricity is produced and managed [1,2]. Conventional generation, dominated by fossil fuels, has long supported a reliable supply, yet it is costly and environmentally damaging. These drawbacks have motivated steady research into more efficient and better coordinated dispatch strategies for regional and national grids [1,2].

At the center of system operation lies the Economic Dispatch (ED) problem. Its goal is straightforward: allocate generation among available units so that overall cost is minimized, while technical requirements—such as unit operating limits, supply–demand balance, and ramping constraints—are satisfied [3,4,5,6]. In particular, references [3,4,5,6] represent the classical ED and DED formulations that establish the foundational modeling framework used throughout the study. Early formulations of ED were designed for centralized, single-area systems in which one operator controlled local resources. However, the growth of distributed architectures and stronger interconnections has rendered such models insufficient for today’s heterogeneous grids. This shift led to the development of Multi-Area Economic Dispatch (MAED), which coordinates generation and energy exchanges across areas linked by tie-lines [7,8,9,10]. References [7,8,9,10] focus on the early development of multi-area ED and highlight coordination mechanisms and tie-line constraints. Each area may face different conditions and priorities, yet inter-area cooperation is vital for maintaining balance and ensuring efficient operation at the system level.

Compared with classical ED, MAED introduces added complexity. It must respect tie-line transfer limits, adapt to regional demand variations, and ensure compatibility between distinct operating practices. Over time, MAED has moved beyond simple cost-minimization. Contemporary formulations often include multiple objectives, such as emission reduction and system robustness [11,12,13,14,15]. References [11,12,13,14,15] include studies that integrate renewable resources, demand response, and storage technologies into ED and MAED models, demonstrating how modern system components modify the objective functions and constraints. Recent work even incorporates sustainability metrics directly into the optimization, reflecting a growing emphasis on environmentally conscious operation.

Addressing these challenges requires sophisticated solution methods. Traditional approaches—including dynamic programming, Lagrangian decomposition, and sequential quadratic programming—offer solid mathematical foundations. Yet, when applied to large and nonlinear problems, they often face scalability issues and slow convergence. This limitation has spurred the adoption of metaheuristics such as Genetic Algorithms, Particle Swarm Optimization, Ant Colony Optimization, and hybrid evolutionary strategies [11,12,13,14,15,16,17,18], all of which are better suited to explore complex, high-dimensional search spaces under tight operational constraints. References [11,12,13,14,15,16,17,18] provide the main methodological advancements, including metaheuristic, hybrid, and intelligent optimization techniques that are relevant for the solution-methods section of the review. In parallel, machine learning (ML) and reinforcement learning (RL) are now being explored as adaptive tools for dispatch, capable of adjusting to uncertain and time-varying grid conditions.

The increasing role of Distributed Energy Resources (DERs) [19,20,21,22]—including photovoltaic systems, wind turbines, and micro-Combined Heat and Power (CHP) [23] units—adds variability and stochastic uncertainty to dispatch models. Energy Storage Systems (ESSs) [24], on the other hand, provide balancing and arbitrage opportunities, transforming dispatch from a static problem into a dynamic, intertemporal optimization challenge. These technologies enhance system flexibility but also bring difficulties such as intermittency, component degradation, and probabilistic operating behavior. As a result, MAED has evolved into a highly coupled, multi-objective problem. Solving it now demands adaptive methods that jointly account for cost, environmental impact, and reliability within the decentralized and uncertain landscape of modern power systems.

The primary contributions of this study include:

• System-level perspective: Unlike many earlier surveys that focus almost exclusively on optimization methods, this review looks at the MAED problem as a whole. We discuss how objectives are formulated, how operational limits are represented, and how areas are coordinated through tie-lines. Framing the problem in this broader way makes it easier to understand the systemic challenges and the interactions between different regions.
• Integration of DERs and ESSs: Particular attention is given to distributed energy resources and storage units, as their variable and uncertain behavior directly influences dispatch outcomes. Intermittency, bidirectional flows, and time-coupled effects are all considered. By emphasizing these aspects, the study highlights a modeling dimension that is still underrepresented in the literature but increasingly important in practice.
• Methodological comparison: A range of solution approaches is compared, from classical deterministic formulations to metaheuristics and recent AI-based techniques. Their performance is discussed in multi-objective and uncertainty-aware contexts, pointing out where each approach performs well, where it falls short, and the types of problems for which it is most suitable.
• Gap identification and roadmap: Beyond summarizing past work, the paper identifies several key gaps: limited use of uncertainty modeling, weak incorporation of demand response, and the absence of unified strategies for multi-area systems. Building on these gaps, a research roadmap is proposed to guide future efforts. The goal is to move toward dispatch models that are not only efficient, but also more robust and adaptable to the realities of interconnected power grids.

The structure of the paper is outlined as follows. Section 2 revisits the classical Single-Area Economic Dispatch (SAED), covering its formulation, objectives, and the traditional solution methods. Section 3 then expands the discussion to the Multi-Area Economic Dispatch (MAED) framework, highlighting its added complexity, the technical issues of inter-regional coordination, and the emerging role of DERs and ESSs. Section 4 reports a comparative analysis on three benchmark networks—a 6-bus, a 10-bus, and a 40-bus system—used to assess both the quality and efficiency of different optimization approaches under diverse operating conditions. Finally, Section 5 concludes by summarizing the main insights and suggesting directions for future research in MAED.

2. Single-Area Economic Dispatch (SAED)

In traditional power systems, a central operational task has been the economic allocation of generation resources [25,26]. When this task is carried out within a single control region, it is referred to as Single-Area Economic Dispatch (SAED). The objective is to minimize total generation cost while meeting demand and respecting technical constraints such as unit capacity limits and local balance requirements.

Most SAED models are static in nature. They treat each time interval independently and do not account for links between successive periods. With the increasing presence of distributed energy resources (DERs), higher demand uncertainty, and tighter operating margins, this static treatment has become inadequate. To address these shortcomings, researchers developed the Single-Area Dynamic Economic Dispatch (SADED) as an extension of SAED [27,28].

SADED adopts a multi-period perspective, modeling dispatch decisions across a time horizon rather than in isolation. It incorporates time-dependent constraints, including generator ramping rates, minimum up and down times, and startup or shutdown costs. By capturing these temporal interactions, SADED provides a more accurate and adaptive picture of system behavior. It shows how one scheduling choice influences the next, which makes it especially relevant for grids that experience frequent variability and require continuous re-optimization.

2.1. Objective Functions and Optimization Constraints

In the SADED problem, various objective functions have been used in previous studies. The most well-known objective functions are introduced and their formulations are presented below.

• Generation cost objective function

The objective function represents the fuel cost of power generation units, incorporating the valve-point loading effect (VPE) [28], which causes non-convexity in the cost curve. The mathematical formulation is given as follows:

$$F_1^{VPE}\sum _{i=1}^{N_g}\sum _{t=1}^T\left({\alpha }_i+{\beta }_i{P}_{it}+c_i{P}_{it}^2+\left|d_i\mathrm{s}\mathrm{i}\mathrm{n}(e_i\left(P_{it}^{min}-P_{it}\right))\right|\right)$$

(1)

The objective function ${\mathrm{F}}_1^{\mathrm{V}\mathrm{P}\mathrm{E}}$ incorporates the effect of the valve-point loading phenomenon, which causes non-convexity in the fuel cost curve. The non-smooth nature of the cost function caused by this effect is captured by the VPE formulation. For the ith generating unit, the cost function is characterized by coefficients ${\alpha }_i$, ${\beta }_i$, $c_i$, $d_i$ and $e_i$. $d_i$ and $e_i$ specifically consider the valve-point-related ripples in the cost profile. ${\mathrm{P}}_{\mathrm{i}\mathrm{t}}$ denotes the power output of unit i at time interval t. The resulting non-convex cost characteristic due to the VPE is illustrated in Figure 1.

• Emission

The emission objective function quantifies the amount of pollutants produced by each generating unit in the system [27,28]. Beyond being a mathematical component, it plays a central role in assessing and ultimately mitigating the environmental impact of electricity production. In practical optimization, this function is evaluated alongside economic and technical objectives, which means the model must not only minimize overall emissions but also respect the operating limits and reliability constraints of the system. The mathematical formulation of this objective is expressed as follows:

$$F_2=\sum _{i=1}^{N_g}\sum _{t=1}^T\left({\delta }_i+{\gamma }_i{P}_{it}+{\partial }_i{e}^{\left({\mu }_i\times P_{it}\right)}\right)$$

(2)

In this formulation, the emission coefficients for the ith generator are represented by δi, γi, ∂i, and μi. The term Pit represents the power output of generator i at time interval t. The objective function F₂ represents the total emissions from all units and is considered as the environmental criterion to be minimized.

Some equality and inequality constraints of the SADED problem are presented and formulated below.

• Power balance

At every time step, the combined generation of all units has to match the system demand. This requirement, known as the power balance constraint, forms the foundation of the dispatch problem. It is more than just an accounting condition: the balance equation connects operational feasibility with the broader economic and environmental objectives of the system. In practice, the optimization process must ensure that demand is met without compromise, while at the same time striving to reduce fuel consumption and associated emissions. The mathematical form of this constraint is expressed as:

$$\sum _{i=1}^{N_g}\sum _{t=1}^T{P}_{it}=P_{Dt}+P_{Lt}$$

(3)

Here, P_Dt denotes the total system demand at time interval ttt, while P_it denotes the active power output of the ith thermal generating unit. The term P_Lt accounts for the transmission losses during the same time period.

• Power generation capacity

Every generating unit can only operate within its technical minimum and maximum output levels, and these limits have to be respected over the entire scheduling horizon. Such capacity constraints are fundamental for ensuring the secure operation of the grid. They also shape the outcome of the optimization, since restricting output directly influences both cost and emissions. In practice, the formulation needs to make sure that the production of each generator remains within its admissible range at all times. The mathematical representation of this constraint is given as:

$$\sum _{i=1}^{N_g}{\sum }_{t=1}^T{P}_{it}^{min}\le P_{it}\le P_{it}^{max}$$

(4)

In this constraint, $P_{it}^{min}$ and $P_{it}^{max}$ represent the minimum and maximum allowable power outputs, respectively, for the ith generator at time interval t.

• Rampe-rate

Generators cannot adjust their output levels arbitrarily fast. Each unit is limited by ramp-rate constraints, which specify how much its power can increase (ramp-up) or decrease (ramp-down) from one-time step to the next [18]. These limits are essential for representing the physical capabilities of the machines and for keeping system operation realistic. They also affect economic and environmental outcomes, since tighter ramping margins can raise costs or restrict the use of certain units. The mathematical form of these constraints is expressed as:

$$\sum _{i=1}^{N_g}\sum _{t=1}^T{P}_{it}-P_{it-1}\le {UR}_{min}$$

(5)

$$\sum _{i=1}^{N_g}\sum _{t=1}^T{P}_{it-1}-P_{it}\le {DR}_{max}$$

(6)

Here, ${UR}_{min}$ and ${DR}_{max}$ represent the minimum ramp-up and maximum ramp-down limits, for the generating units, in that order.

• Tie-line capacity

For secure and reliable operation, the power that flows over transmission lines linking different areas has to stay within their allowable limits. If these limits are exceeded, the network may face thermal overload or even stability problems. To capture this restriction in the dispatch model, the condition governing inter-area transfers is written as Equation (7).

$$\left|{TL}_t\right|\le {TL}^{Max}$$

(7)

where ${TL}_t^{max}$ denotes the active power limit of the Ti-line at tth hour.

2.2. Solution Approaches for SAED

In recent decades, a broad spectrum of methods has been used to tackle ED problems. Early studies focused mainly on classical mathematical techniques such as linear and quadratic programming, whereas more recent work has increasingly turned to artificial intelligence–based approaches. When the dispatch is modeled within a single region and under fixed demand, it is referred to simply as ED. If demand varies over time; however, the problem takes the form of Dynamic Economic Dispatch (DED). This distinction will be followed in the remainder of the paper.

• Classical methods

Historically, Traditionally, ED problems were solved using quadratic cost models, which approximate the input–output characteristics of generating units in a relatively simple way. While convenient, these models often leave out important operational constraints [29]. A range of classical methods have been applied in this context—including Newton’s method [30], lambda iteration [31], Lagrangian relaxation [32], branch and bound [33], base-point and participation factor methods [34], gradient-based solvers [35], and linear or quadratic programming with interior-point techniques [36].

These solvers are fast and computationally efficient, but they perform poorly when the cost functions are non-smooth or non-convex. This is a common situation in thermal plants and becomes even more pronounced when renewable sources are integrated. In such conditions, classical approaches struggle because they are designed for smooth, monotonic, or piecewise-linear cost structures [37,38]. In large interconnected systems, they may also converge prematurely, get stuck in local minima, or fail to model nonlinear operational limits accurately, especially when many interdependent variables are involved [39].

For these reasons, research has gradually shifted toward evolutionary computation and swarm intelligence. Unlike traditional solvers, these metaheuristics are built to handle complex, multidimensional search spaces. They offer robustness, flexibility, and stronger global exploration, making them better suited for the challenges of modern ED.

• Population-based evolutionary methods

Metaheuristic optimization techniques take their inspiration from natural systems—ranging from evolutionary biology to collective behaviors found in animals and plants [40]. Over roughly the last twenty years, they have become some of the most widely used approaches for solving Economic Dispatch (ED), particularly in single-period problems. Their value goes beyond simple cost minimization: they also support the efficient use of fossil resources and help align dispatch planning with broader sustainability goals.

Most population-based methods start by creating a diverse set of candidate solutions across the search space. These candidates are then updated over multiple iterations, using operators that mimic biological or social interactions [40]. Because of this, the algorithms can strike a balance between global exploration and local refinement. That balance reduces the risk of getting stuck in weak local solutions and makes metaheuristics well-suited for ED problems with nonlinearities, complex constraints, and large solution spaces.

Several established metaheuristics have been applied to ED, including Particle Swarm Optimization (PSO) [41,42], Firefly Algorithm (FA) [43,44], Genetic Algorithm (GA) [45,46,47,48], Cuckoo Search (CS) [49,50,51,52], Harmony Search (HS) [53,54], Differential Evolution (DE) [55], and Artificial Bee Colony (ABC) [56,57,58]. More recent contributions have introduced algorithms inspired by predator–prey dynamics, swarm intelligence, and plant reproduction strategies. Examples include Ant Lion Optimizer (ALO) [59], Grey Wolf Optimizer (GWO) [60,61], Bat Algorithm (BA) [62], Whale Optimization Algorithm (WOA) [63], Flower Pollination Algorithm (FPA) [64], Moth Flame Optimization (MFO) [65], Invasive Weed Optimization (IWO) [66], Shuffled Frog Leaping Algorithm (SFLA) [67], Salp Swarm Algorithm (SSA) [68], and Coyote Search Algorithm (CSA) [69].

Despite their diversity and proven track record, these methods still face recurring challenges. They can converge slowly, global searches sometimes lack accuracy, and performance often suffers under high uncertainty [40]. To address these weaknesses, researchers have developed hybrid and improved versions. Examples include modified PSO designs [70], adaptive GA variants [71,72], refined CS [73], enhanced ABC [74] and FA [75], revised SFLA [76], optimized DE [77], advanced HS [78], chaotic extensions of WOA [79], and improved bacterial foraging methods [80]. These refinements generally aim to accelerate convergence, boost robustness under uncertain conditions, and improve adaptability to the operational constraints of real-world power systems.

2.3. Impact of Distributed Generation and Storage Integration

Traditional ED models were originally designed for centralized grids, where predictable demand could be met with thermal units using deterministic optimization [81,82,83,84]. While this worked well in the past, it no longer matches the realities of today’s decentralized and dynamic systems. The growth of distributed generation (DG)—from rooftop solar to small wind and biomass—has introduced both spatial and temporal variability that makes real-time scheduling much harder [85,86,87].

ESSs add flexibility by storing surplus energy and releasing it later [88,89]. At the same time, they create new challenges due to their limited capacity, efficiency losses, and gradual aging. Coordinating DG, ESSs, and conventional units together requires adaptive, dynamic scheduling methods that can handle uncertainty and nonlinear behavior.

For these reasons, modern ED research has moved beyond simple cost minimization. Reliability, emission reduction, and short-term variability are now central goals. Because deterministic solvers often fall short under these conditions, researchers are turning to adaptive, data-driven approaches that can learn from changing system dynamics.

• Microgrids and renewable dominance: Gholami et al. [81] proposed a bi-objective dispatch model for renewable-rich microgrids, balancing economic and environmental goals. Kumar et al. [83] examined the role of batteries in isolated systems, showing gains in efficiency and voltage stability.
• Integrated systems with EVs: Soni and Bhattacharjee [82,84] developed strategies for networks with both renewables and electric vehicles, explicitly addressing multi-criteria trade-offs.
• Novel optimization approaches: Sakthivel et al. [85] designed a dispatch algorithm based on herd immunity principles for coordinating hydro storage, later extending it to EV integration [86]. Mishra and Shaik [87] applied a biologically inspired metaheuristic for hybrid diesel–solar–wind systems. Nagarajan et al. [88] introduced a cheetah-inspired algorithm incorporating demand-side flexibility, and Soni and Bhattacharjee [89] advanced an oppositional-based control method for joint scheduling of EVs and renewables, achieving faster convergence and higher accuracy.

Overall, these studies show that ED research is no longer limited to static, cost-focused models. The field is shifting toward broader, more flexible frameworks that take into account distributed generation, storage, and demand-side participation, all while dealing with uncertainty in real operating conditions.

2.4. Key Challenges

With the wider adoption of DG and energy storage, the ED problem has become more complex than in the past. The traditional assumptions of centralized operation, steady demand, and predictable generator behavior no longer match today’s grid conditions. Renewable sources like wind and solar, with their variable output, make fixed schedules unreliable and push operators toward real-time, adaptive coordination.

Modern systems also bring together a diverse mix of small-scale generators, different storage technologies, and an increasing number of electric vehicles. This diversity turns ED into a nonlinear, constraint-rich problem where cost minimization is only one part of the picture. Issues such as emission limits, system reliability, and the physical limits of storage—including degradation, efficiency losses, and capacity restrictions—must also be considered.

Energy storage can smooth renewable fluctuations and improve flexibility, but it also requires accurate modeling of its technical limits. At the same time, distributed control across different regions and actors makes coordination more difficult. These overlapping challenges mean that future dispatch models cannot remain static or focus on a single objective. Instead, they need to support multiple goals and adapt to changing system conditions.

Table 1. A detailed categorization of algorithmic approaches applied in resolving ED/DELD problems.

Ref.	Year	Objective Function		Method	DERs	ESSs	Test System	Problem Type
		Cost	Emission					ED	DED
[27]	2023	✓	✓	SACDE	✓	✗	10-units	✗	✓
[28]	2025	✓	✓	PSO-MSFLA	✓	✓	10-units	✗	✓
[29]	2017	✓	✗	IPSO	✗	✗	10-units	✓	✗
[38]	2017	✓	✗	PSO-BBCO	✗	✗	6-,15- and 40-units	✓	✗
[40]	2023	✓	✓	MOSHEPO	✗	✗	10-units	✓	✗
[41]	2014	✓	✗	PSO	✗	✗	10-, 40-units	✓	✗
[44]	2012	✓	✗	FA	✗	✗	10-units	✓	✗
[48]	2013	✓	✗	CSA	✗	✗	6-,10-,20- and 40-units	✓	✗
[49]	2015	✓	✗	CSA	✗	✗	6-units	✓	✗
[51]	2011	✓	✗	HBMO	✗	✗	10-units	✓	✗
[52]	2010	✓	✓	BFA	✗	✗	6-, and 40-units	✓	✗
[53]	2011	✓	✗	HSA	✗	✗	6-, and 15-units	✓	✗
[54]	2013	✓	✗	HSA	✗	✗	6-, and 10-units	✓	✗
[56]	2010	✓	✗	ABCO	✗	✗	6-, and 10-units	✓	✗
[58]	2014	✓	✓	ABCO	✗	✗	6-, and 40-units	✓	✗
[59]	2017	✓	✗	ALO	✗	✗	6-, 10-, and 40-units	✓	✗
[60]	2016	✓	✗	HGWO	✗	✗	6-, 15-, and 40-units	✓	✗
[61]	2016	✓	✗	GWO	✗	✗	6-, 15-, and 40-units	✓	✗
[62]	2016	✓	✗	CBA	✗	✗	6-, 15-, and 40-units	✓	✗
[63]	2018	✓	✗	WOA	✗	✗	6-, and 15-units	✓	✗
[66]	2015	✓	✗	OIWO	✗	✗	13-, 40-,110-, and 140-units	✓	✗
[67]	2019	✓	✓	MSFLA	✗	✗	6-, 15-, and 40-units	✓	✗
[68]	2020	✓	✗	SSA	✗	✗	15-, 40-, and 110-units	✓	✗
[71]	2019	✓	✗	IGA	✓	✗	5-units	✗	✓
[72]	2019	✓	✗	MGA-SA	✓	✗	32-units	✗	✓
[74]	2017	✓	✓	IABCO	✗	✗	6-, and 10-units	✓	✗
[76]	2018	✓	✗	CFA	✗	✗	40-, and 140-units	✓	✗
[78]	2016	✓	✗	IDE	✗	✗	6-, 13-, and 40-units	✓	✗
[79]	2018	✓	✗	IHSA	✓	✗	5-units	✗	✓
[80]	2020	✓	✗	CWOA	✓	✗	5-units	✗	✓

✓ means present/correct/done. ✗ means not present/incorrect/not done.

shows that most of the reviewed studies still emphasize minimizing generation cost, while environmental objectives—especially emission reduction—receive much less attention or are left out entirely. This imbalance suggests that economic efficiency is often prioritized at the expense of sustainability. Another important gap is the limited consideration of DERs and ESSs. Although these technologies play a central role in today’s power grids, many models either ignore them or treat them only in a simplified way. As a result, such studies are less aligned with the needs of decentralized systems that rely heavily on renewable energy.

A further limitation is the common assumption of static operating conditions. Many models treat both demand and generation as fixed, overlooking the time-varying nature of real power systems. This simplification makes dispatch results less accurate, especially when loads fluctuate or renewable resources are intermittent. Although dynamic formulations can better capture these changes, they remain relatively scarce in the existing literature.

On the methodological side, most studies rely on single metaheuristic algorithms, often inspired by natural or social processes—for example, Particle Swarm Optimization (PSO) or the Firefly Algorithm (FA). While these methods can perform well, they often struggle with issues such as premature convergence or limited solution diversity in complex systems. To address this, some researchers have proposed hybrid approaches that combine multiple algorithms. These strategies generally show stronger robustness, faster convergence, and better ability to handle nonlinear effects such as valve-point loading, ramp-rate limits, and prohibited operating zones. However, the majority of these methods have been tested only on small- or medium-sized benchmark systems. Their effectiveness on large, realistic networks is still unclear, raising concerns about scalability and practical relevance.

In short, Table 1 highlights that while algorithmic progress in MAED research has been considerable, there is still a significant gap between theoretical models and real-world needs. Future studies should move toward dynamic, multi-objective frameworks that integrate DERs, ESSs, and realistic operational constraints to better reflect the complexity of modern power systems.

3. Advances in Multi-Area Economic Dispatch (MAED)

With the increasing interconnection of regional power grids, generation scheduling is no longer a matter confined to individual areas. Modern systems often operate as networks of control regions, each with its own load demand, generation mix, and operating rules. These areas are linked by transmission lines that enable energy exchange, forming the basis of MAED. The objective of MAED is to coordinate generation across regions in order to minimize overall cost while maintaining secure and reliable operation [7,11,12].

Compared with single-area dispatch, MAED must satisfy a wider set of constraints. Balance must be ensured within each region and across the entire interconnected system. This requires close attention to tie-line flows, regional surpluses or shortages, and transmission limits. A scheduling decision in one area can affect neighboring regions, particularly when transmission bottlenecks or price differences are present. Figure 2 illustrates an example involving four control areas and 140 generating units, which demonstrates the scale and complexity of such systems [90].

As renewable penetration rises and market conditions become more volatile, static MAED formulations are increasingly inadequate. To address this, Multi-Area Dynamic Economic Dispatch (MADED) [8,10] has been developed. Unlike static models, MADED links decisions across multiple time intervals, considering generator ramping constraints, varying loads, and inter-period power transfers. This ensures that scheduling decisions are forward-looking and account for evolving operating conditions.

The introduction of DERs and ESSs adds further flexibility but also introduces new uncertainties due to intermittency and limited predictability. Dealing with these challenges requires optimization techniques that are adaptive and predictive, capable of managing both spatial and temporal complexity. In this respect, MADED provides a more realistic framework, one that calls for advanced solution strategies to achieve both economic efficiency and operational reliability in interconnected power systems.

3.1. Multi-Objective Functions and Optimization Constraints in MADED

MADED can be understood as a natural extension of the SADED framework, with the key distinction being its multi-regional scope. While SADED focuses on scheduling within a single area, MADED explicitly incorporates the interactions among multiple regions. This introduces additional challenges, such as tie-line flow limits, regional power balances, and unit-level constraints like generation capacities and ramping limits. Taking these factors into account expands both the scale and the detail of the optimization, giving a more faithful representation of interconnected power system operations.

The formulation also captures spatial coupling across regions, the impact of transmission bottlenecks, and the presence of shared operating boundaries—all of which influence both cost and security. These elements make MADED-based dispatch more realistic and robust, providing a stronger basis for planning and control in systems where regions retain some autonomy and the resource mix continues to diversify. The following sections present the objective functions and constraints adopted in this study, highlighting the specific features that distinguish MADED from earlier dispatch models.

• Cost function incorporating valve-point effects and multiple fuel options

Another notable characteristic of power plants is their ability to operate using various types of fuel, known as Multi-Fuel Operation (MFO). In practical scenarios, generating units may make use of multiple fuel types, as shown in Figure 3. The associated cost function for such cases can be represented by Equation (8).

$$\begin{array}{l}{F}_1^{Final}={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{n=1}^{N_m}}(a_{mn,f}{P}_{mnt}^2+b_{mn,f}{P}_{mnt}+c_{mn,f}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left|d_{mn,f}sin\left(e_{mn,f}\left(P_{mn}^{min}-P_{mnt}\right)\right)\right|)\hspace{1em}\hspace{1em}{P}_{mn,f-1}\le P_{mnt}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le P_{mn,f}^{max}\end{array}$$

(8)

Here, $f$ represents the number of fuel types; $a_{mn,f}$, $b_{mn,f}$, $c_{mn,f}$, $d_{mn,f}$ and $e_{mn,f}$ are the cost coefficients associated with the nth generator in the mth area for each fuel type f; $P_{mn}^{min}$ and $P_{mn}^{max}$ denote the minimum and maximum power output limits of that generator, respectively. Finally, $F_1^{Final}$ represents the total cost objective function accounting for both VPE and MFO [8,10].

• Emission objective function

The emission objective function closely resembles the cost function, but it focuses on capturing the various types of pollutants emitted by generating units into the environment. Its mathematical formulation is given in Equation (9).

$$\begin{gathered} F_2=\sum _{t=1}^T\sum _{m=1}^M\sum _{n=1}^{N_m}{F_2}_{mnt}\left(P_{mnt}\right)=\sum _{t=1}^T\sum _{m=1}^M\sum _{n=1}^{N_m}{\sigma }_{mn}+{\phi }_{mn}{P}_{mnt}+{\rho }_{mn}{P}_{mnt}^2 \\ +{\eta }_{mn}\times e^{\left({\zeta }_{mn}\times P_{mnt}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} } \end{gathered}$$

(9)

In this context, ${\sigma }_{mn}$, ${\phi }_{mn}$, ${\rho }_{mn}$, ${\eta }_{mn}$ and ${\zeta }_{mn}$ denote the emission coefficients of the nth generator in the mth area. ${F_2}_{mnt}\left(P_{mnt}\right)$ represents the emission output of the generator at time interval t, and F₂ refers to the overall emission objective function [8,10]. Several equality and inequality constraints associated with the MADED problem are defined and formulated in the following sections.

• Restricted operating zones

In real power systems, the cost curves of generating units are not always smooth. Many machines exhibit prohibited operating zones (POZs), which stem from issues such as mechanical vibrations or limits in auxiliary devices like pumps and boilers. These zones create discontinuities in the input–output characteristics, as illustrated in Figure 4. To capture this effect, the present study employs a customized mathematical formulation of POZs, given in [10].

$${\forall n\in N_m, \forall m\in M, \forall t\in T, \forall z\in N_{DPZ}⟹P}_{mnt}\in \left\{\begin{array}{c}{P}_{mn}^{min}\le P_{mnt}\le P_{{mn}_1}^{Low}\\ P_{{mn}_{z-1}}^{Up}\le P_{mnt}\le P_{{mn}_z}^{Low}\\ P_{{mn}_Z}^{Up}\le P_{mnt}\le P_{mn}^{max}\end{array}\right.$$

(10)

Here, $N_{DPZ}$ indicates the total number of disjoint prohibited zones, while $P_{{mn}_z}^{Low}$ and $P_{{mn}_Z}^{Up}$ represent the lower and upper limits of the zth prohibited zone for the nth generator in the mth area, respectively.

• Load demand balance

The active power balance relationship is expressed mathematically in Equation (11).

$$\forall m\in M, \forall t\in T⟹\sum _{n=1}^{N_m}{P}_{mnt}=P_{{Demand}_{m,t}}+P_{{Loss}_{m,t}}+\sum _{k\ne m}{P_{Ti-Line}}_{m,h,t}$$

(11)

Here, $P_{{Demand}_{m,t}}$ represents the system load or power demand in area m at time t (in MW); $P_{{Loss}_{m,t}}$ refers to the power losses in area m at time t (MW); and ${P_{Ti-Line}}_{m,h,t}$ denotes the active power transferred through the tie-line from area m to area h at time t (MW). A positive value indicates power flow from area m to h, while a negative value implies the opposite direction.

• Active power transmission losses

Transmission power losses are calculated using the B-coefficient method, as shown in Equation (12), where the loss coefficient matrix has dimensions of $\left(N_m\times N_m\right)$.

$${ m\in M, t\in T⟹P}_{{Loss}_{m,t}}=\sum _{k=1}^{N_m}\sum _{n=1}^{N_m}\left(P_{mnt}{\times B}_{mkn}{\times P}_{mkt}\right)+\sum _{n=1}^{N_m}\left(B_{o_{mn}}{\times P}_{mnt}\right)+B_{{oo}_m}$$

(12)

Here, $B_{{oo}_m}$ represents the constant loss coefficient in the mth area; $B_{o_{mn}}$ refers to the nth element of the loss coefficient vector with a dimension of $N_m$ in area m; and $B_{mkn}$ denotes the $\left(k,n\right)$th element of the loss coefficient matrix.

• Capacity of tie-lines

To ensure system security, the active power transferred across tie-lines must remain within specified limits. This constraint is mathematically expressed as follows.

$$\left|{P_{Ti-Line}}_{m,h,t}\right|\le {P_{Ti-Line}}_{m,h}^{max}$$

(13)

Here, ${P_{Ti-Line}}_{m,h}^{max}$ represents the maximum active power capacity of the Ti-Lines connecting area m to area h.

• Generator ramp rate limits

The constraints on the Ramp Rate (RR) are defined as follows [8].

$${\forall n\in N_m, \forall m\in M,\forall t\in T⟹P}_{mnt}-P_{mn(t-1)}\le {UR}_{mn}$$

(14)

$${\forall n\in N_m, \forall m\in M,\forall t\in T⟹P}_{mn(t-1)}-P_{mnt}\le {DR}_{mn}$$

(15)

where ${UR}_{mn}$ and ${DR}_{mn}$ denote the ramp-up and ramp-down rate limits, respectively.

3.2. Solution Strategies for MAED Problems

Recent advances in MAED and MADED optimization reflect the growing challenge of managing systems that are increasingly complex and uncertain. Researchers have explored a variety of heuristic and stochastic methods to address these problems. For example, the Flower Pollination Algorithm (FPA) has attracted attention for its balance of fast convergence and reliable accuracy. Other algorithms, such as Binary Whale Optimization (BWO) [91] and Simulated Annealing (SA) [92], have been tested in dual-objective settings, targeting both cost and emissions. A broader set of heuristics—including ABCO [57], TLBO [93], GSO [94], DE [95], and SSO [96]—have also been applied, each offering distinct strengths for handling nonlinearities and complex search spaces.

To cope with renewable variability and system nonlinearities, researchers have developed enhanced or hybrid methods. Examples include modified PSO variants [97], chaos-enhanced ABC models [98], and refined Grasshopper Optimizer (GOA) approaches [8,99], sometimes combined with Evolutionary Memetic Algorithms (EMA) [100]. Hybrid strategies such as chaotic PSO with GA-based tuning [101], bi-level SBO frameworks [102], improved Fireworks Algorithms (FWA) [103], GWO–PSO combinations [10], and decentralized methods like DDCCO [104] have also been reported. Other approaches, including ICS [105] and QOGS [106], have been applied to improve convergence in MADED contexts.

Stochastic and analytical models are another active direction, especially under high renewable penetration. Examples include the Unscented Transform (UT) [107] and robust optimization (RO) [108,109,110] for wind and demand uncertainty, as well as probabilistic modeling of solar and wind resources using SS methods [111,112,113] and Weibull or log-normal distributions [114]. Parallel efforts have focused on emission-conscious dispatch, with methods such as CSO [115] designed to optimize both cost and environmental objectives.

The increasing role of EVs and storage has opened new avenues of research. Approaches include MPSO with Monte Carlo–based uncertainty modeling [116,117], decentralized agent-based strategies [118], semi- and fully distributed scheduling [119], and PSO–WOA hybrids for renewable–storage coordination [23]. Unified dispatch frameworks with economic and environmental goals [120], scenario-based decomposition strategies [121], NSGA-II for hydro–wind–thermal coordination [22], and adaptive RO for unit commitment [122] further expand this scope. Demand response has also begun to appear in MAED, from CS-based dispatch [123] to probabilistic models that jointly consider load and renewable variability [124,125].

Finally, learning-based techniques are gaining influence. Deep learning (DL) has been applied to model time-varying demand [126], while deep reinforcement learning (DRL) has been used in distributed, multi-agent contexts [127]. Other work explores consensus-based optimization [128], hierarchical RL [129], and model-free DRL for CHP-integrated systems [130]. These methods are particularly valuable because they can adapt to dynamic conditions without relying on detailed physical models.

Overall, this body of work signals a transition from traditional heuristic algorithms toward more intelligent, adaptive, and decentralized frameworks. Such approaches are better aligned with the realities of modern MAED and MADED, where uncertainty, renewable integration, and multi-objective trade-offs shape everyday operation.

Table 2. A comprehensive taxonomy of algorithms utilized in addressing MAED/MADED formulations.

Ref.	Year	Objective Function		Method	DERs	Uncertainty	DR	Problem Type
		Cost	Emission					MAED	MADED
[91]	2023	✓	✓	BWO	✗	✗	✗	✓	✗
[92]	2020	✓	✗	SA	✗	✗	✗	✓	✗
[93]	2014	✓	✗	TLBO	✗	✗	✗	✓	✗
[94]	2018	✓	✗	GSO	✗	✗	✗	✓	✗
[95]	2014	✓	✗	DE	✗	✗	✗	✓	✗
[96]	2021	✓	✗	SSO	✗	✗	✗	✓	✗
[57]	2013	✓	✗	ABCO	✗	✗	✗	✓	✗
[97]	2018	✓	✗	EPSO	✗	✗	✗	✓	✗
[98]	2015	✓	✗	ABCO	✗	✗	✗	✓	✗
[99]	2021	✓	✗	IGO	✗	✗	✗	✓	✗
[100]	2023	✓	✗	HGO	✗	✗	✗	✓	✗
[101]	2020	✓	✗	CPSO	✗	✗	✗	✓	✗
[102]	2024	✓	✓	SBO	✗	✗	✗	✓	✗
[103]	2021	✓	✗	IFA	✗	✗	✗	✗	✓
[105]	2022	✓	✗	ICS	✗	✗	✗	✓	✗
[106]	2016	✓	✗	QOGS	✗	✗	✗	✗	✓
[113]	2020	✓	✗	SS	✗	✓	✗	✓	✗
[115]	2022	✓	✗	CSO	✗	✗	✗	✓	✗
[116]	2020	✓	✗	RO	✓	✓	✗	✗	✓
[117]	2022	✓	✓	MPSO	✓	✓	✗	✗	✓
[23]	2024	✓	✓	PSO-WOA	✓	✓	✗	✗	✓
[22]	2019	✓	✓	NSGA-II	✓	✗	✗	✗	✓
[121]	2013	✓	✗	RO	✓	✓	✗	✓	✗
[122]	2015	✓	✗	RO	✓	✓	✗	✓	✗
[123]	2025	✓	✗	CSA	✗	✓	✓	✗	✓
[124]	2018	✓	✗	Lagrangian relaxation	✓	✓	✗	✓	✗
[125]	2017	✓	✓	PSO-SFLA	✗	✓	✗	✓	✗
[126]	2024	✓	✗	DL	✗	✗	✗	✓	✗
[127]	2024	✓	✗	RL	✗	✗	✗	✓	✗
[129]	2023	✓	✗	HRL	✗	✗	✗	✓	✗

✓ means present/correct/done. ✗ means not present/incorrect/not done.

shows that most early studies on MAED concentrated almost exclusively on minimizing generation cost. In more recent years, however, several works—such as [23,91,102,117]—have expanded this view by introducing dual objectives that combine economic performance with environmental goals, signaling a stronger emphasis on sustainability in power system planning.

Researchers have employed a wide variety of heuristic and metaheuristic algorithms to address MAED. Early contributions typically relied on methods such as DE, TLBO, and PSO. Later studies refined these techniques, introducing variants like CPSO, MPSO, and PSO–WOA, which showed improvements in both convergence speed and solution accuracy. For instance, Refs. [91,100] reported that incorporating chaotic dynamics or combining different algorithms could significantly enhance optimization outcomes.

Alongside these developments, machine learning has begun to play a more prominent role. Reinforcement learning (RL), hierarchical RL (HRL), and deep learning (DL) [125,126,127,128,129] have all been applied to multi-area dispatch problems. RL adapts by continuously interacting with the system environment, while DL helps identify hidden relationships in large datasets. These methods point to a gradual shift toward more adaptive, data-driven approaches to dispatch optimization.

Nevertheless, several important limitations remain. Only a handful of studies [23,113,116,117] explicitly consider renewable DERs such as wind and solar, and many still assume a centralized system structure. Uncertainty is often simplified or ignored: while some works [113,116,117,121,122,123] adopt stochastic or robust models, deterministic formulations continue to dominate. Demand response has received even less attention, with [123] standing out as one of the few cases where it is explicitly modeled. Moreover, although MADED provides a more realistic dynamic representation of system behavior, static MAED models are still far more common in the literature. Table 3 summarizes these contributions, grouping the reviewed studies according to optimization method, modeling approach, and the degree to which they incorporate flexibility-enhancing elements such as DERs, ESSs, EVs, and DR.

3.3. Impact of Spinning Reserve on Multi-Area Economic Dispatch

Spinning reserve plays a central role in maintaining system reliability in multi-area dispatch. In MAED systems, where regions are interconnected through tie-lines, unexpected demand spikes or generator outages can easily destabilize operations if adequate reserve is not available. Having spare capacity that can be activated on short notice improves resilience, but it also introduces scheduling challenges, since the reserve must be allocated alongside normal generation. This makes the joint treatment of cost and reliability an essential requirement for MAED optimization.

Different approaches in the literature show how spinning reserve can be incorporated. For example, Zare et al. [131] proposed a modified Fireworks Algorithm for dynamic MAED that explicitly included spinning reserve and achieved both lower cost and faster convergence while preserving reliability. Chen et al. [132] analyzed large-scale wind integration and applied stochastic co-optimization of generation and reserves, highlighting the benefits of probabilistic modeling under renewable variability. Similarly, Ref. [133] developed a multi-objective evolutionary framework that optimized cost, emissions, and reliability simultaneously, while [134] introduced an adaptive differential evolution method with dynamically tuned mutation rates to improve efficiency and avoid premature convergence.

Overall, these studies demonstrate that spinning reserve should not be treated merely as a secondary safeguard. Instead, it is a fundamental component of multi-area scheduling and needs to be built directly into optimization frameworks to ensure both economic efficiency and operational resilience.

Table 3. Extended Conceptual Classification of MAED/MADED Studies.

Category Type	Features	Ref.
Single-objective deterministic	Cost-only, no uncertainty, static models	[30,36,57,93,135]
Multi-objective deterministic	Cost & Emission, no uncertainty	[38,91,100,115,117]
Multi-objective stochastic	Cost & Emission + uncertainty in RES/load	[116,121,123,124,125,132]
DER-integrated models	Include RES/ESS in formulation	[23,79,113,116,117]
DR-integrated models	Consumer-side flexibility modeled	[118,119,123]
EV-aware dispatch models	Electric vehicles as storage/loads	[82,86,89,116,130]
MADED with dynamic constraints	Ramping, intertemporal, spinning reserve	[22,88,131,132,134]

Table 3. Extended Conceptual Classification of MAED/MADED Studies.

Category Type	Features	Ref.
Single-objective deterministic	Cost-only, no uncertainty, static models	[30,36,57,93,135]
Multi-objective deterministic	Cost & Emission, no uncertainty	[38,91,100,115,117]
Multi-objective stochastic	Cost & Emission + uncertainty in RES/load	[116,121,123,124,125,132]
DER-integrated models	Include RES/ESS in formulation	[23,79,113,116,117]
DR-integrated models	Consumer-side flexibility modeled	[118,119,123]
EV-aware dispatch models	Electric vehicles as storage/loads	[82,86,89,116,130]
MADED with dynamic constraints	Ramping, intertemporal, spinning reserve	[22,88,131,132,134]

3.4. Incorporation of Demand Response and Uncertainty Modeling in MAED Frameworks

While research on MAED and its dynamic counterpart MADED has advanced significantly in terms of optimization techniques and multi-objective modeling, two aspects remain largely underexplored: uncertainty and demand response (DR). In most formulations, demand, generation, and market prices are still assumed to be fixed and fully known. This simplification makes the models easier to solve mathematically, but it does not reflect real operating conditions, where renewable variability, volatile prices, and shifting consumption patterns are the norm. Only a limited number of studies—such as [113,115,124,125]—have attempted to capture these uncertainties using scenario-based or robust methods, leaving a noticeable gap in the literature.

In addition to demand response, some recent MAED studies emphasize the importance of uncertainty, particularly the variability in demand, market prices, and renewable generation [124,125]. To model these sources of uncertainty, researchers typically rely on historical time-series measurements, short-term forecasting errors, or probability distributions derived from system operators. In some cases, synthetic scenarios are generated using simple statistical or Monte-Carlo-based techniques to capture typical fluctuations over different time intervals.

Once these data sources are defined, several modeling approaches are used to incorporate uncertainty into the MAED framework. Scenario-based stochastic modeling is the most common, where multiple possible realizations of demand or renewable output are considered simultaneously. Other studies use probabilistic models that directly integrate distributions of forecast errors, while robust optimization approaches define upper and lower bounds for uncertain parameters such as loads, prices, or renewable output.

These uncertainty representations influence both the objectives and operational constraints of the dispatch problem. Demand variability changes the expected load profiles and affects generation schedules; market-price uncertainty shifts the cost structure and may alter inter-area power exchanges; and renewable fluctuations impact balancing requirements, reserves, and tie-line flows. By integrating these elements, recent MAED formulations provide a more realistic picture of system behavior and allow the dispatch to remain feasible and efficient under a wider range of operating conditions.

A similar challenge arises with DR. Despite its well-documented potential for enhancing flexibility, reducing peak loads, and improving stability, DR remains absent in most MAED formulations. Demand is often modeled as static and unresponsive, overlooking the role of consumers in shaping grid behavior. Even in dynamic MADED studies, where time-varying conditions are explicitly considered, the integration of uncertainty and DR is uncommon. Notable exceptions include [22,23,116,117,123], with the latter explicitly embedding DR into the dispatch framework.

Overall, these gaps highlight the need for more comprehensive MAED and MADED frameworks that move beyond cost and emissions alone. Future models should integrate uncertainty and consumer-side flexibility as central components, while also addressing the multi-area and time-dependent nature of modern power systems.

Table 4. Extended Quantitative Analysis of Reviewed MAED/MADED Literature.

Feature	Percentage of Studies Covering
Deterministic formulation	~70%
Stochastic/robust approaches	~30%
Integration of DERs	~30%
Integration of DR mechanisms	~10%
Consideration of EVs	~18%
Use of MADED (dynamic) models	~25%
Inclusion of reliability metrics	~18%

provides a summary of 50 reviewed studies, contrasting deterministic and stochastic approaches and documenting the extent to which DERs, DR, and EVs have been included.

3.5. Research Gaps

Despite notable progress in MAED and MADED research, several key challenges continue to limit their large-scale applicability. One of the most persistent problems is the heavy reliance on deterministic assumptions. In many models, demand, renewable output, and market prices are treated as fixed, even though in reality they are highly uncertain. Only a small number of studies have adopted robust or probabilistic approaches, leaving uncertainty modeling relatively underdeveloped.

Another major limitation is the underutilization of demand response (DR). Although DR is widely recognized for its ability to enhance flexibility and reduce peak demand at low cost, most dispatch frameworks still treat demand as static and unresponsive, overlooking the active role consumers can play in system operation. Electric vehicles (EVs) face a similar issue: they are rarely represented in detail, even though their dual role as mobile loads and distributed storage makes them an important factor in future grids [136]. Oversimplifying EV behavior reduces the accuracy of dispatch models, particularly in urban networks where penetration is expected to rise sharply.

Reliability and resilience also remain secondary concerns. While most optimization work prioritizes cost minimization, aspects such as spinning reserve allocation, contingency management, and robustness to disturbances are less frequently addressed, which weakens the operational realism of proposed solutions.

Perhaps the most critical shortcoming is the absence of a unified framework. Current research often isolates specific aspects—such as cost and emissions, or cost and uncertainty—without simultaneously incorporating DR, EVs, and reliability. As a result, dispatch strategies tend to remain fragmented and less applicable in real-world contexts. Moving forward, progress will require integrated models that bring these dimensions together and provide solutions that are both technically rigorous and practically deployable.

4. Performance Evaluation on Different Test Systems

To enable fair comparison across different methods, many MAED studies rely on a standard set of benchmark systems with 6, 10, and 40 generating units. The smallest case models two areas with three units each and incorporates prohibited operating zones, making it suitable for testing algorithms in non-convex conditions. The 10-unit benchmark spans three interconnected regions, which introduces moderate coordination challenges. The largest system, with 40 units across four areas, reflects large-scale dispatch conditions with higher dimensionality, tighter tie-line limits, and greater demand.

Together, these benchmarks provide a common ground for evaluating optimization techniques. They make it possible to study how methods scale with system size, how effectively they handle operational constraints, and how they perform under both single- and multi-objective formulations.

4.1. 6-Unit Test System

The first benchmark case involves two interconnected regions, each operating three generating units. To capture more realistic conditions, the generators are modeled with prohibited operating zones, and transmission losses are included. Generator specifications are adapted from [137]. The total demand is 1263 MW, of which 60% is assigned to the first area and 40% to the second. Power exchange between the regions is restricted to 100 MW.

This configuration has been widely applied to evaluate optimization methods such as DE, EP, RCGA, and SA [93]. The objective is to test how effectively these algorithms handle the constraints while minimizing operating cost.

Table 5. Multi-area dispatch results related to minimizing cost using a 6-unit test system.

Control Variables	Methods
	DE [93]	SA [93]	EP [93]	RCGA [93]
P1,1	500.00	500.00	500.00	500.00
P1,2	200.00	200.00	200.00	200.00
P1,3	150.00	150.00	149.98	149.63
P2,1	204.33	204.21	206.44	205.93
P2,2	154.70	155.05	154.89	155.83
P2,3	67.55	67.53	65.22	65.35
T2,1	82.773	82.773	82.761	82.410
Cost ($)	12,255.39	12,255.39	12,255.43	12,256.23

presents the corresponding control variables and costs for each method, enabling a clear comparison of their performance.

4.2. 10-Unit Test System

The 10-unit benchmark system consists of three interconnected areas, each supplied by its own set of generators. Area 1 includes units G₁–G₄, area 2 contains G₅–G₇, and area 3 operates G₈–G₁₀. Together, they must meet a total demand of 2700 MW, distributed unevenly: 1350 MW in the first area and 675 MW in each of the other two. To reflect real operating conditions, valve-point effects (VPE), multiple fuel options (MFO), and transmission losses are explicitly modeled. Energy exchanges between regions are limited by a tie-line capacity of 100 MW, which imposes a significant operational constraint.

The three regions are connected through tie-lines that enable coordinated power sharing. Detailed system data, including cost coefficients and B-coefficients for losses, are provided in [135].

Table 6. Comparative MAED outcomes for minimizing cost using the 10-unit system.

Control Variables	Methods
	PSO-SFLA [135]	TLBO [93]	DE [93]	EP [93]	RCGA [93]	ABCO [57]	SA [93]
Output Active Power of Generators	Power (kW)
P1,1	225.297361	224.308	225.444	223.8491	239.095	225.943	228.17
P1,2	211.803184	210.664	210.166	209.5759	216.116	211.159	213.34
P1,3	489.740555	491.699	491.284	496.068	484.150	489.921	482.82
P1,4	241.433389	240.624	240.895	237.9954	240.622	240.623	242.64
P2,1	246.242359	249.564	251.004	259.4299	259.663	254.039	253.50
P2,2	232.565497	235.897	238.860	228.9422	219.910	235.492	236.57
P2,3	260.429189	263.741	264.090	264.1133	254.514	263.883	266.63
P3,1	239.008128	237.132	236.998	238.228	231.356	237.000	234.31
P3,2	336.091	332.591	326.539	331.2982	341.962	328.737	325.95
P3,3	252.9628	249.462	250.333	246.6025	248.278	248.860	251.45
Active power flows related to Ti-lines
T21	100	100	99.468	100	93.17	99.8288	100
T31	98.9523326	100	100	100	93.8739	99.7334	98.80
T32	45.1750355	35.4599	30.281	32.5231	43.7824	31.2615	28.38
Cost ($/h)	653.86223	653.9977	654.0184	655.1716	657.3325	653.9995	654.9016

summarizes the performance of several optimization methods, such as PSO-SFLA [135], SA [93], ABCO [57], TLBO [93], DE [93], EP [93], and RCGA [93]. Some of these approaches achieve noticeably lower generation costs, indicating their potential for solving multi-area dispatch problems. Beyond cost, the results also show how different methods handle inter-area transfers and generator load distribution, providing a more practical view of their strengths and limitations.

4.3. 40-Unit Test System

The third benchmark involves a 40-unit generation system spread across four interconnected areas. As with the smaller test cases, valve-point effects and transmission losses are modeled to reflect practical operating conditions. The total demand is 10,500 MW, allocated unevenly: 1575 MW in area 1, 4200 MW in area 2, 3150 MW in area 3, and 1575 MW in area 4. Each region contains 10 generators, with units G₁–G₁₀ assigned to area 1, G₁₁–G₂₀ to area 2, G₂₁–G₃₀ to area 3, and G₃₁–G₄₀ to area 4.

Inter-area exchanges are limited by tie-line capacities. Transfers of up to 200 MW are allowed between areas 1–2, 1–3, and 2–3, while stricter limits of 100 MW apply between areas 1–4 and 2–4. These constraints increase the realism of the test system and make it a demanding case for evaluating optimization strategies. Detailed generator data, including cost functions, loss coefficients, and unit parameters, are provided in [93,98].

Table 7. Comparative MAED outcomes for minimizing cost using the 40-unit system.

Control Variables	Methods
	PSO-SFLA [134]	TLBO [93]	DE [93]	EP [93]	RCGA [93]	ABCO [57]
Output Active Power of Generators	Power (kW)
P1,1	108.0679321	110.8791	111.5448	107.6644	95.7552	111.102
P1,2	110.1474658	112.955	111.7092	112.0673	88.5828	109.9774
P1,3	94.58036466	97.4151	98.2429	91.8132	97.6063	100.9238
P1,4	177.2563282	179.9466	179.8834	175.3171	126.4966	190.0000
P1,5	86.64690215	89.4955	95.95	92.4242	71.0127	96.9390
P1,6	137.1333194	139.8937	139.3533	112.5634	116.3866	96.9675
P1,7	257.1831881	259.7338	259.3395	257.537	244.5857	259.6950
P1,8	282.1316818	284.6387	285.3569	297.3619	210.692	276.8725
P1,9	282.2345616	284.7414	284.9627	285.2035	236.1685	300.0000
P1,10	130.2275526	130.1151	130.2217	134.5862	130.1286	130.6977
P2,1	171.6969240	168.8311	243.6005	162.4313	367.4862	245.1007
P2,2	171.6872316	168.8214	95.389	217.8387	297.9501	94.0000
P2,3	127.9623013	125.0623	214.5171	125.0000	394.9246	125.0000
P2,4	396.9696801	394.2799	394.0808	384.0187	370.3473	434.8062
P2,5	396.9427012	394.2529	394.2481	397.6902	455.7123	390.6743
P2,6	486.6625878	484.0429	394.436	407.4993	393.9673	395.0043
P2,7	491.8995953	489.284	489.9552	500.0000	424.1994	500.0000
P2,8	491.8859060	489.2703	488.8885	480.8874	484.5498	500.0000
P2,9	513.9330768	511.3347	511.4713	524.8487	528.4148	530.7889
P2,10	514.053083	511.4548	511.4125	499.7857	511.3403	514.4090
P3,1	521.1923035	523.2816	523.2896	523.4522	525.4497	527.1989
P3,2	521.3430669	523.4321	523.295	526.5051	510.7391	502.0795
P3,3	521.2878705	523.377	523.4129	537.3675	533.6399	530.3657
P3,4	521.5086563	523.5974	523.4073	525.7752	518.112	542.3424
P3,5	521.4604721	523.5493	523.7703	531.2092	538.1994	520.2448
P3,6	521.1879960	523.2773	523.5424	513.5659	527.4775	533.6389
P3,7	10.01750405	10.1442	10.1621	11.3612	24.4133	10.0000
P3,8	10.01750405	10.0248	10.1326	10.0000	28.9856	10.0000
P3,9	10.01750405	10.0862	10.6366	10.0000	28.8571	10.0000
P3,10	85.38459647	88.2354	88.1189	78.3523	87.9016	96.7699
P4,1	190.0000000	189.919	161.222	162.448	159.7482	190.0000
P4,2	189.8075671	189.9718	189.5668	166.3508	153.6255	168.6841
P4,3	190.0000000	190.0000	189.924	190.0000	160.4706	173.6165
P4,4	167.8927000	164.8927	165.6621	178.4541	169.9359	186.374
P4,5	168.1343000	165.1343	165.4321	168.0752	168.522	200.0000
P4,6	168.2322000	165.2322	164.9868	174.4529	172.2638	164.957
P4,7	93.27580000	90.2758	109.8137	77.3875	91.2423	92.5627
P4,8	110.0000000	109.9813	109.7935	90.1059	86.4778	96.9911
P4,9	93.20190000	90.2019	90.1543	109.5654	88.3627	109.8153
P4,10	456.7356754	458.9376	459.114	549.0335	279.2691	431.4011
Active power flows related to Ti-lines
T12	−200.000000	185.5862	172.0652	200	−71.7855	191.7078
T31	−86.45568516	23.6686	−36.306	17.5885	161.9336	6.674
T32	91.48638895	183.0863	191.1128	200	95.2833	183.1852
T41	195.8463887	47.1037	86.807	90.8733	−76.134	86.859
T42	144.8205240	94.6933	98.8231	100	−52.39	95.3237
T43	193.9153037	97.7497	45.0391	100	83.4418	57.2192
Total cost ($/h)	121,619.8377	121,760.50	121,794.80	123,591.90	128,046.50	124,009.4

compares the performance of several algorithms, including PSO-SFLA [135], ABCO [57], TLBO [93], DE [93], EP [93], and RCGA [93]. Results indicate that hybrid and nature-inspired approaches often achieve lower costs and faster convergence, highlighting their potential for large-scale, interconnected dispatch problems.

5. Future Research and Conclusions

5.1. Future Directions

Looking ahead, several areas in MAED research remain underexplored despite their clear relevance. Perhaps the most urgent is the systematic handling of uncertainty. As renewable energy penetration increases, dispatch models must be able to respond to fluctuations in demand, variable generation, and shifting market signals. Without such capability, both system reliability and economic efficiency are at risk.

Another important direction is the integration of EVs. These mobile resources, with unpredictable charging patterns and their potential for vehicle-to-grid (V2G) support, are reshaping load profiles and creating new opportunities for ancillary services. Yet, in most studies, EVs are treated only at a conceptual level, with little attention to their dynamic, real-time effects on dispatch strategies. DR faces a similar situation. While it is often mentioned as a source of flexibility, its practical integration into MAED formulations remains limited. Future models should therefore consider adaptive and real-time DR schemes that better reflect consumer behavior and allow for more effective coordination across regions.

From a computational perspective, hybrid metaheuristic approaches appear especially promising, particularly when combined with machine learning for adaptive parameter control. These intelligent methods are capable of dealing with nonlinear constraints, large-scale systems, and multi-objective trade-offs, and they provide a practical route toward more flexible dispatch frameworks.

Finally, as power networks evolve into multi-layered structures with distributed resources and microgrids, hierarchical control frameworks will become increasingly important. Such frameworks need to coordinate across multiple system levels while also addressing real-world concerns such as communication delays and cybersecurity risks. For this reason, future MAED formulations should move beyond their traditional focus on cost minimization and emission reduction. Reliability, resilience, and long-term sustainability should be treated as core optimization objectives. To support this transition,

Table 8. Research Roadmap for Future MAED/MADED Studies.

Stage	Research Focus	Objective	Suggested Methods
Stage 1	Uncertainty modeling in RES and demand	Improve realism of dispatch models	Stochastic/Robust Optimization
Stage 2	Demand Response integration	Enable flexible, consumer-driven dispatch	Real-time DR with adaptive control
Stage 3	EV and V2G modeling	Incorporate mobile storage and dynamic load	Multi-agent and probabilistic models
Stage 4	Hybrid optimization methods	Enhance convergence and robustness	MH-AI hybrids with dynamic tuning
Stage 5	Hierarchical & resilient architectures	Align with decentralized grids	Multi-layer optimization + cybersecurity modeling

outlines a research roadmap that prioritizes five key directions for future studies in both MAED and MADED.

5.2. Conclusions

The review presented in this paper brings together the main strands of research on ED, DED, MAED, and MADED, and attempts to place them within a single, coherent structure. Rather than looking at these formulations only from the viewpoint of optimization methods, the review adopts a system-level perspective that connects the objectives, constraints, coordination requirements, and operational behaviors of multi-area power systems. This broader viewpoint helps clarify how the different models relate to each other and how they evolve when new components and practical considerations are introduced.

A central contribution of this review is the feature-based comparison of existing studies. By examining how different works incorporate elements such as distributed energy resources, storage systems, demand response, electric vehicles, uncertainty, and reliability, the review provides a clearer picture of how the field has progressed and where current efforts are concentrated. The comparison tables summarize these characteristics in a way that allows readers to see the diversity of existing approaches and the modeling gaps that still remain.

Through this analysis, several gaps in the literature become evident. These include the relatively limited use of demand response in multi-area settings, incomplete modeling of DER variability in dynamic formulations, the scarcity of multi-objective frameworks that combine cost, emissions, and flexibility, and the lack of detailed numerical reporting in many recent studies, which makes comparison more difficult. Addressing these gaps could significantly improve the realism and usefulness of dispatch models for modern power systems.

Based on the insights gathered, the review also outlines a number of directions for future research. These include developing uncertainty-aware and renewable-responsive multi-area models, improving the representation of interactions between areas, expanding the role of environmental and flexibility objectives, and exploring advanced solution techniques such as reinforcement learning and hybrid data-driven optimization. Progress in these areas will help move multi-area dispatch toward more robust, flexible, and sustainable operational strategies.

In summary, this review provides an updated and structured overview of the field, highlights the scientific developments that have shaped recent work, and offers a practical roadmap to guide future research.