Integrated Generation and Transmission Expansion Planning Through Mixed-Integer Nonlinear Programming in Dynamic Load Scenarios

Abstract

A deterministic Mixed-Integer Nonlinear Programming (MINLP) model for the Integrated Generation and Transmission Expansion Planning (IGTEP) problem is presented. The proposed framework is distinguished by its foundation on the complete AC power flow formulation, which is solved to global optimality using BARON, a deterministic MINLP solver, which ensures the identification of truly optimal expansion strategies, overcoming the limitations of heuristic approaches that may converge to local optima. This approach is employed to establish a definitive, high-fidelity economic and technical benchmark, addressing the limitations of commonly used DC approximations and metaheuristic methods that often fail to capture the nonlinearities and interdependencies inherent in power system planning. The co-optimization model is formulated to simultaneously minimize the total annualized costs, which include investment in new generation and transmission assets, the operating costs of the entire generator fleet, and the cost of unsupplied energy. The model’s effectiveness is demonstrated on the IEEE 14-bus system under various dynamic load growth scenarios and planning horizons. A key finding is the model’s ability to identify the most economic expansion pathway; for shorter horizons, the optimal solution prioritizes strategic transmission reinforcements to unlock existing generation capacity, thereby deferring capital-intensive generation investments. However, over longer horizons with higher demand growth, the model correctly identifies the necessity for combined investments in both significant new generation capacity and further network expansion. These results underscore the value of an integrated, AC-based approach, demonstrating its capacity to reveal non-intuitive, economically superior expansion strategies that would be missed by decoupled or simplified models. The framework thus provides a crucial, high-fidelity benchmark for the validation of more scalable planning tools.

1. Introduction

Electricity consumption is steadily increasing, driven by the degree of technological innovation across various commercial and industrial processes. Consequently, electricity generation sources must be sufficient and reliable to ensure sustainable and productive economic progress. While a large part of the world has access to electric power services, it cannot be guaranteed that this energy is delivered with high quality standards, due to energy crises experienced in different regions. This situation, at least partially, stems from inadequate expansion of power systems to meet future demand requirements [1].

In power systems (PSs), both the integration of various electricity generation technologies and the implementation of networks that enhance supply reliability can provide solutions to several challenges faced by modern electrical systems. These challenges include power quality issues, voltage instability, electrical losses, and environmental pollution. Such problems arise because power networks were not originally designed to accommodate the technological innovations that have emerged over time in the generation and transmission stages. Therefore, it is essential to focus on planning the expansion of generation facilities and transmission networks to enable effective adaptation of the resources available in the electrical grid [2].

Expansion problems are typically addressed in two interrelated but independent stages aimed at meeting demand and its growth over a given time horizon [3]. The first stage corresponds to Generation Expansion Planning (GEP), which determines generation requirements, and the second to Transmission Expansion Planning (TEP), which specifies network requirements or reinforcements [4,5]. Each stage considers distinct technical and economic conditions and can be formulated as an optimization problem with the objective of minimizing or maximizing a function based on a mathematical approach [6]. As energy needs become increasingly demanding, additional tools and expertise are required to resolve the challenges involved in managing the PS as a whole. This has led to the evolution of power system planning models and tools that integrate generation expansion and transmission growth to act in coordination. This integrated approach helps mitigate network issues arising from the high penetration of generation in response to dynamically growing demand [4,7].

Integrated or simultaneous Generation and Transmission Expansion Planning (IGTEP) aims to identify the most cost-effective plan for determining the power capacity to be added, the type of generation and transmission required, the timing of investments, and the optimal locations for generation plants and transmission lines. This planning ensures that projected demand is met while complying with system constraints and required reliability levels [8].

In an effort to develop IGTEP models that reflect the real conditions of power systems, several studies have proposed approaches focused on maximizing social welfare, as seen in [9]. Other works, such as [10], have considered security constraints. In [11], both investment and operating costs were minimized, while ref. [12] analyzed the effects of uncertainty in primary energy sources and their prices. In contrast, ref. [13] addressed renewable energy penetration, and ref. [12] explored losses and reactive power using AC and DC network models. These and other studies have emphasized the benefits of combining generation and transmission planning into unified objective functions under appropriately defined constraints.

Current research on IGTEP explores various methods grouped into two main categories. The first is based on mathematical optimization, where studies have formulated the problem using Mixed-Integer Linear Programming (MILP) [14,15] and other linear approaches [16]. The second category involves metaheuristic algorithms, with examples including Monte Carlo simulations [17], genetic algorithms [18], and hybrid methods [19]. Additionally, recent works have proposed robust planning frameworks to address renewable-dominant systems. A notable example is the bi-directional converter-based interconnection planning approach for hybrid AC/DC microgrids presented in [20], which incorporates dynamic converter efficiencies and a data-driven uncertainty model to enhance robustness and reduce interconnection costs. Although the focus differs from IGTEP, the methodological contributions, such as the tri-level optimization and the treatment of uncertainty via data-correlated uncertainty sets, offer valuable insights for expansion planning in complex systems. These studies present a variety of methodologies capable of solving simultaneous expansion planning problems. Such methodologies foster a large number of studies because IGTEP results tend to be satisfactory, ensuring both the overall minimization of costs and the fulfillment of projected demand within the defined planning horizon.

While these diverse methodologies have significantly advanced the field, they present a critical trade-off that defines the current research landscape. A significant portion of these studies, particularly those employing MILP, relies on simplified DC power flow approximations [14,15]. While this approach ensures computational tractability and can guarantee global optimality for the linearized model, it does so by fundamentally ignoring crucial technical aspects of power system operation. The DC model’s inherent assumptions—neglecting reactive power, assuming a flat voltage profile, and disregarding accurate network loss calculations—can yield solutions that are either suboptimal or technically infeasible in real-world operation [5]. Conversely, a second stream of research employs metaheuristic algorithms [17,18,19] to tackle the more complex and non-linear AC power flow model. These methods can accommodate a higher level of technical detail, but they do not guarantee convergence to a global optimum. While they provide valuable, high-quality solutions, they cannot establish a definitive, provably optimal benchmark. Consequently, it remains uncertain whether the expansion plans they produce represent the most economically efficient strategy, leaving a critical gap in the ability to quantify the “cost of heuristic uncertainty”.

This analysis reveals a critical research gap: the need for a framework that combines the technical fidelity of the full AC power flow model with the mathematical rigor of global optimization to establish a true, verifiable benchmark for the IGTEP problem. Such a benchmark is essential for validating the performance of faster heuristic methods and for providing planners with a definitive understanding of the minimum possible cost for system expansion under a given set of conditions.

While various studies have explored IGTEP using AC power flow, a significant challenge remains in ensuring global optimality due to the non-convex nature of AC constraints. Many existing approaches rely on simplifications or metaheuristics. This paper distinguishes itself by developing a MINLP model with a complete AC power flow formulation that is rigorously solved to global optimality. This not only provides a high-fidelity benchmark for integrated planning but also enables the discovery of economically superior and non-intuitive expansion strategies that might be overlooked by decoupled or linearized models.

This research addresses the aforementioned gap by proposing a novel framework for the simultaneous expansion of the generation park and the transmission network, founded on the complete AC power flow model to ensure technical fidelity. The expansion is formulated as a deterministic MINLP problem and solved to global optimality. While the integration of AC power flow constraints within a mixed-integer framework has been explored in the literature, the primary contribution of this work lies in establishing a definitive, globally optimal benchmark for the integrated, dynamic-load Generation and Transmission Expansion Planning (GTEP) problem. By solving the full, non-convex AC-OPF-based model without resorting to linearizations (e.g., DC power flow) or heuristics, this study provides a high-fidelity solution that serves as a crucial reference point for both economic and technical analysis. This benchmark is essential for evaluating the quality and assessing the “optimality gap” of more computationally tractable methods, such as heuristics or decomposition techniques, which are often necessary for larger-scale systems.

The innovation of the proposed model also lies in the continuous and intensive interaction involved in solving n nonlinear programming problems for the modeled system. These problems, governed by the electrical laws of AC power flows, determine the inclusion of new generation units and possible new links in the network. The AC power flows, which include both active and reactive power, as well as voltage levels, are treated as variables that must remain within technical limits to guarantee reliability, continuity, and quality in steady-state operation of the system over the medium and long term.

The objective of this work is to minimize the investment costs of new generators and newly added transmission elements, the operating costs of both existing and new generation units, and the Cost of Energy Not Supplied (CENS). The formulated model considered multiple dynamic load scenarios to evaluate the technical and economic aspects associated with the Integrated Generation and Transmission Expansion Planning in a test network. It is noted that this study is intentionally focused on a deterministic framework, where parameters such as load growth ($f_{Lg}$) and plant factors of the generators in operation ($f_{P{L}_{G_o}}$) are treated as fixed inputs for given scenarios. This approach was adopted to first establish a foundational, globally optimal benchmark for the complex, non-linear, and mixed-integer nature of the integrated planning problem. Therefore, the proposed model provides optimal investment strategies for sizing, siting, and scheduling new investments in generation units and interconnection networks to meet the forecasted demand.

This article is organized as follows: Section 2 analyzes the criteria for integrated expansion planning, along with the model and solution methods used for IGTEP. Section 3 formulates the objective function and constraints of the proposed IGTEP methodology. Section 4 presents the case studies. Section 5 discusses the results, and, finally, Section 6 provides the conclusions of this research.

2. Integrated Planning for Power System Expansion

Expansion planning in power systems is a complex study aimed at determining the optimal way to expand generation, distribution, and transmission in order to meet consumers’ energy needs at the lowest possible cost over a defined time horizon [21]. While the primary objective is to meet demand, the delivery of electricity must also be reliable, secure, and economical, considering the technical, economic, and political aspects involved in the power grid [22]. Traditionally, expansion planning has focused on generation, due to the high investment costs it entails compared to transmission network expansion. It is worth noting that distribution also requires considerable investment, but in large part this is the responsibility of entities other than those involved in production and transmission [23,24].

The relationship between producer and consumer needs results in combined variables that highlight the necessity for tools that integrate the apparent separation between generation and transmission planning [25]. However, in practice, generation is typically planned and scheduled based on demand growth, thereby providing guidelines for the transmission network’s expansion plan. Generation planning is based on assumptions about load growth at the distribution stage, which means that each planning process is treated separately, due to the fragmented decisions made by investors and operators—ultimately leading to disorder in system expansion and reduced overall economic benefits [26].

Therefore, considering the interdependence between generation and transmission expansion—and the significant investments both require—it is essential to adopt an integrated planning approach that justifies the new infrastructure from both a technical and economic standpoint, since such developments have a direct impact on all users of the system. This issue has been the subject of limited research and is known as Integrated Generation and Transmission Expansion Planning [27]. IGTEP is a process that employs various methodologies and models to coordinate expansion plans simultaneously, with the goal of determining the location, quantity, and type of generation and transmission units to be added to the power system at the appropriate time, ensuring that projected demand is met at minimum cost and under reliable conditions [28].

2.1. Criteria for Integrated Expansion Planning

The formulation of the IGTEP problem involves several criteria distributed across different stages, enabling effective system integration for electricity supply. Moreover, these criteria impact the complexity of the modeling process, due to the number of variables and data involved, which can lead to challenges in data processing and computation. Therefore, robust computational tools are required to handle this type of analysis. The following section presents these criteria in a general manner.

2.1.1. Demand Growth

Electric power systems evolve in efficiency and structure according to demand behavior. Their adaptation depends on energy consumption, which is complex to measure, due to its high level of uncertainty [29]. For this reason, over time, various methods have been developed and refined to forecast the dynamic behavior of demand and to establish corresponding growth rates. These forecasts serve as the foundation for power system studies within their respective time frames. Therefore, demand contributes to expansion planning by incorporating load growth rates, particularly exponential growth rates applicable at the continental (American) level [30].

2.1.2. Generation Expansion Planning

GEP generally determines the entry, size, and optimal location of new power generation units while minimizing total cost over a medium- or long-term planning horizon. Generation expansion typically adopts an energy-focused approach and disregards transmission network constraints [31]. Therefore, to address the IGTEP problem, it is essential to consider the active and reactive power contributions of the generators, plant capacity factors, investment and operational costs of generation facilities, and the location and capacity of the units involved in the expansion plan.

2.1.3. Transmission Expansion Planning

TEP refers to the installation of new transmission lines or the expansion of the capacity of existing lines within a power system [10]. Transmission Expansion Planning identifies the network reinforcements required to ensure energy delivery to system users at minimum cost, while also pinpointing the supply points established by GEP. The criteria that TEP must consider for integrated planning focus on link-related characteristics, such as line investment and operating costs, network topology, line loading capacity, operating voltage levels, and active and reactive power flow.

2.1.4. Unsupplied Energy

This criterion is incorporated when the generation resources or transmission infrastructure are unable to meet the demand during the period in which the expansion is evaluated.

2.2. Model and Solution Method for IGTEP

To perform a combined expansion study, it is necessary to define a mathematical model capable of incorporating the previously mentioned criteria. For this study, the model considered the expansion of both the generation and transmission stages, while also applying AC power flow equations. Solving these equations enables a highly accurate approximation of the real operation of power systems, despite the mathematical complexity introduced by the nonlinearities of the load flow equations [32]. These equations are characterized by voltage, angle, active power, and reactive power variables, which determine the flows in the transmission network based on generation and demand levels [27,33].

The power flow model in this work is based on the following assumptions: the technical and economic characteristics of generators and transmission lines are known; new transmission lines have the same characteristics as existing ones; growing demand—which drives the need for system expansion—is represented through dynamic load scenarios based on growth rates; and Energy Not Supplied (ENS) is assigned a value when generation resources or transmission infrastructure are insufficient to meet demand over a given period.

For the formulation of load flow studies within the integrated expansion plan, it is essential to define a methodology that satisfies the objective function of minimizing costs (investment, operating, and ENS) by optimally deploying new generation and reinforcing the transmission network according to demand growth. A variety of solution methods exist, ranging from metaheuristic algorithms to classical mathematical optimization methods, each with its own advantages and limitations [34]. Considering the challenges posed by Generation and Transmission Expansion Planning, this work used a mathematical co-optimization algorithm based on MINLP. This approach was suitable, due to the nonlinear nature of the equations, the non-convexity arising from decision variables inherent to electrical laws, and the large scale of the problem. The selected algebraic modeling approach enables the simultaneous identification of the best expansion alternatives for generation units and the transmission network to meet future demand, using a minimum-cost criterion for both operation and investment.

3. Methodology for the Simultaneous Expansion of Generation and Transmission

The simultaneous expansion of the generation and transmission stages of the power system is obtained by solving a MINLP model, which involves binary decision variables and mixed-type variables. Their interaction enables the determination of the new infrastructure required in the aforementioned segments of the power sector.

3.1. Modeling Approach: Justification for AC Power Flow

A foundational decision in this research was the use of the complete AC power flow model as the basis for the optimization problem. This choice was made to ensure that the resulting expansion plans were not only economically optimal but also technically viable and operationally secure.

In the literature, many planning models employ a simplified DC power flow approximation to maintain linearity and reduce computational burden [14,15]. However, the DC model operates on several limiting assumptions, such as neglecting reactive power, assuming a flat voltage profile (1.0 p.u.) across all buses, and disregarding network losses. Consequently, expansion plans derived from DC models may appear optimal but can be infeasible in practice, potentially leading to voltage instability or requiring significant unplanned investments in reactive power compensation [1,5].

By contrast, the proposed AC-based MINLP model provides a high-fidelity representation of the power system. It endogenously manages reactive power balance (Equation (17)), treats nodal voltage magnitudes as optimization variables constrained within secure operational limits (Equation (25)), and accurately accounts for both active and reactive power losses through the non-linear power flow equations (Equations (7)–(10)). Although this approach introduces significant non-convexity and computational complexity, it is deemed essential for generating robust and realistic expansion strategies. The resulting plan is guaranteed to be operationally feasible from a steady-state perspective, thereby providing a more reliable and valuable benchmark for long-term power system planning.

3.2. Objective Function

The objective function is divided into three components: the first corresponds to the costs associated with the generation park, the second relates to the costs of the new transmission system infrastructure, and the third evaluates the ENS.

3.2.1. Generation Park Costs

For generators, operational costs are modeled in (1), and the annualized investment costs of new generators incorporated in the medium term are represented in (2) and (3):

$$COG=Tt\left(\sum _{G_o}{P}_{G_o}·C{O}_{G_o}+\sum _{G_n}{P}_{G_n}·C{O}_{G_n}\right)$$

(1)

$$CAIG=\sum _{G_n}{C}_{G_n}·fc{r}_{G_n}·C{I}_{G_n}$$

(2)

$$fc{r}_{G_n}={\displaystyle \frac{t{d}_{G_n}{\left(1+t{d}_{G_n}\right)}^{V{u}_{G_n}}}{{\left(1+t{d}_{G_n}\right)}^{V{u}_{G_n}}-1}}$$

(3)

3.2.2. Transmission Infrastructure Costs

The new transmission infrastructure implemented through system expansion results in annualized investment costs, as modeled in (4) and (5):

$$CAIT={\displaystyle \frac{1}{2}}\sum _N\sum _M\sum _{k}C{I}_{n,m}·fc{r}_{n,m}·{\alpha }_{n,m,k}$$

(4)

$$fc{r}_{n,m}={\displaystyle \frac{t{d}_{n,m}{\left(1+t{d}_{n,m}\right)}^{V{u}_{n,m}}}{{\left(1+t{d}_{n,m}\right)}^{V{u}_{n,m}}-1}}$$

(5)

3.2.3. Costs of Energy Not Supplied

ENS occurs when the generation system is unable to meet demand. This shortage results in a significant cost to consumers and is modeled in (6):

$$CENS=Tt·CE·\sum _{n}Pen{s}_n$$

(6)

3.3. Constraints

The constraints in the model correspond to the technical aspects of the power system, including the modeling of AC power flows, operational characteristics of generators, interconnection link capacity limits, and their implications for the associated variables.

3.3.1. Flows in Transmission Network Links

This constraint associates the variable ${\alpha }_{n,m,k}$ with the AC power flow equations. In this case, both active and reactive power must be modeled. For this purpose, the Big M method is used [35,36], which is based on associating constraints with large negative constants that would not be part of any optimal solution. When applying the Big M method in constraints, it ensures that variable enforcement occurs only when a defined binary variable takes on a specific value, while leaving the variables “open” if the binary variable takes the opposite value:

$$\begin{array}{cc}\hfill P_{n,m,k}-\left[\sum _{m=1}^M\left|V_n\right|\left|V_m\right|(g_{nm}cos{\theta }_{nm}+b_{nm}sen{\theta }_{nm})\right]& \le M·(1-{\alpha }_{n,m,k})\hfill \end{array}$$

(7)

$$\begin{array}{c}\hfill P_{n,m,k}-\left[\sum _{m=1}^M\left|V_n\right|\left|V_m\right|(g_{nm}cos{\theta }_{nm}+b_{nm}sen{\theta }_{nm})\right]\ge -M·(1-{\alpha }_{n,m,k})\end{array}$$

(8)

$$\begin{array}{c}\hfill Q_{n,m,k}-\left[\sum _{m=1}^M\left|V_n\right|\left|V_m\right|(g_{nm}sin{\theta }_{nm}-b_{nm}cos{\theta }_{nm})\right]\le M·(1-{\alpha }_{n,m,k})\end{array}$$

(9)

$$\begin{array}{c}\hfill Q_{n,m,k}-\left[\sum _{m=1}^M\left|V_n\right|\left|V_m\right|(g_{nm}sin{\theta }_{nm}-b_{nm}cos{\theta }_{nm})\right]\ge -M·(1-{\alpha }_{n,m,k})\end{array}$$

(10)

3.3.2. Power Transfer Limits of the Links

The transfer limits on the links are related to the maximum or minimum amount of active or reactive power that can be transferred through a given link, whether it is an existing or a new one. The transferred power must not exceed the thermal or loading limit of the link:

$$P_{n,m,k}\le {LTP}_{n,m}·{\alpha }_{n,m,k}$$

(11)

$$P_{n,m,k}\ge -{LTP}_{n,m}·{\alpha }_{n,m,k}$$

(12)

$$Q_{n,m,k}\le {LTQ}_{n,m}·{\alpha }_{n,m,k}$$

(13)

$$Q_{n,m,k}\ge -{LTQ}_{n,m}·{\alpha }_{n,m,k}$$

(14)

3.3.3. Bidirectionality of Decision Variables for the New Link

This constraint ensures that the decision to incorporate a new link between two nodes is not duplicated and does not have an additional impact on the investment cost of the new transmission infrastructure:

$${\alpha }_{n,m,k}={\alpha }_{m,n,k}$$

(15)

3.3.4. Nodal Balance

The formulation of the nodal balance is applicable to the active and reactive power at each node of the power system and must comply with the concept established by Kirchhoff’s First Law:

$$\begin{array}{cc}\hfill {Pens}_n+\sum _{G_o\in n}{P}_{G_o}+\sum _{G_n\in n}{P}_{G_n}-\left({P_L}_n·f_{Lg}\right)& =\sum _m\sum _k{P}_{n,m,k}\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill \sum _{G_o\in n}{Q}_{G_o}+\sum _{G_n\in n}{Q}_{G_n}-\left({Q_L}_n·f_{Lg}\right)=\sum _m\sum _k{Q}_{n,m,k}\end{array}$$

(17)

3.3.5. Capacity of New Generators

This constraint allows the incorporation of a generation plant based on demand requirements and its growth through the use of a binary decision variable:

$${Cap}_{G_n}\le {Cmax}_{G_n}·{\beta }_{G_n}$$

(18)

3.3.6. Generation Output Limits

This constraint ensures that the active and reactive power dispatched by any generator does not exceed the limits defined by the generator capability curve:

$${Pmin}_{G_o}\le P_{G_o}\le {Cmax}_{G_o}$$

(19)

$${Pmin}_{G_n}\le P_{G_n}\le {Cmax}_{G_n}$$

(20)

$$Q_{G_o}\le {Cmax}_{G_o}·tan\left[{cos}^{-1}\left({f_{PL}}_{G_o}\right)\right]$$

(21)

$$Q_{G_n}\le {Cmax}_{G_n}·tan\left[{cos}^{-1}\left({f_{PL}}_{G_n}\right)\right]$$

(22)

$$Q_{G_o}\ge -{Cmax}_{G_o}·tan\left[{cos}^{-1}\left({f_{PL}}_{G_o}\right)\right]$$

(23)

$$Q_{G_n}\ge -{Cmax}_{G_n}·tan\left[{cos}^{-1}\left({f_{PL}}_{G_n}\right)\right]$$

(24)

3.3.7. Nodal Voltage Limits

This constraint ensures that the nodal voltages in per unit remain within the defined range to guarantee voltage stability in the system:

$${Vmin}_n\le V_n\le {Vmax}_n$$

(25)

3.3.8. Nodal Angle Limits

This constraint ensures that the voltage angles at the nodes remain within the specified range, thereby maintaining angular stability in the system:

$${\theta min}_n\le {\theta }_n\le {\theta max}_n$$

(26)

$${\theta }_{slack}=0$$

(27)

3.4. Optimization Model Pseudocode and Workflow

The fundamental steps of the proposed heuristic are initially outlined in

Table 1. Pseudocode of the optimization model.

Integrated Generation and Transmission Expansion Planning through Mixed-Integer Nonlinear Programming
Step 1	Acquisition of the power system data.
Step 2	Determination of the factors associated with the expansion of the generation and transmission stages:
	- Potential generators to be incorporated;
	- Discount rate;
	- Useful life;
	- Demand growth rate;
	- Cost of energy not supplied;
	- Analysis period.
Step 3	Definition of variables
	Continuous:
	$P_{G_o},Q_{G_o},P_{G_n},Q_{G_n},Ca{p}_{G_n},V_n,{\theta }_n,{Pens}_n,P_{n,m,k},Q_{n,m,k}$
	Binary:
	${\alpha }_{n,m,k},{\beta }_{n,m,k}$
Step 4	Optimization model
	Objective function:
	Minimization of generation and transmission stage costs, including the cost of energy not supplied.
	Constraints:
	- Decision to implement new links;
	- Power transfer limits of the links;
	- Bidirectionality of decision variables for the new link;
	- Nodal balance;
	- Capacity of new generators;
	- Generation output limits;
	- Nodal voltage limits;
	- Nodal angle limits.
Step 5	Application of the model to the case studies.
Step 6	Analysis of results.
Step 7	End.

, where the pseudocode of the optimization model is presented. For a more comprehensive and intuitive understanding of the methodology’s implementation, a detailed workflow diagram is subsequently provided. Figure 1, presented below, is employed to illustrate the entire process, encompassing aspects from data acquisition to the final output of the expansion plan.

The methodology for Integrated Generation and Transmission Expansion Planning is implemented through the computational workflow visually depicted in Figure 1. This structured process ensures that all relevant technical, economic, and operational aspects are systematically considered, thereby facilitating the determination of a globally optimal expansion plan.

The workflow is initiated through the compilation of necessary inputs, which are systematically categorized into three principal groups. Firstly, power system data are collected, encompassing the network topology, electrical parameters for existing lines and generators, and bus data, which can be exemplified by a specific test system such as the IEEE 14-bus system. Secondly, economic parameters are incorporated, including investment costs for candidate generators and transmission lines, unit operating costs, applicable discount rates, and the CENS. Thirdly, scenario parameters are defined, which establish the planning horizon and the dynamic load growth rates intended for evaluation.

These compiled inputs are subsequently introduced into the core of the methodology: the proposed MINLP model. This mathematical model is formulated with an objective function directed at minimizing the total annualized costs of investment and operation. This minimization is subjected to a comprehensive set of constraints, which includes AC power flow equations, nodal power balances, and technical limits pertinent to all system components. The model’s decision variables are instrumental in determining the optimal selection, precise location, and appropriate sizing of new assets $({\beta }_{G_n},{\alpha }_{n,m,k})$, in addition to governing the optimal dispatch of all generating units.

The MINLP problem is implemented in the GAMS environment and is subsequently solved through the application of the BARON global optimization solver. The selection of this solver is attributable to its demonstrated capability in handling the non-convexities inherently present in AC power flow formulation. The solver processes the formulated model to systematically identify the optimal expansion strategy, ensuring that all defined constraints are satisfied at a minimum cost.

Ultimately, the optimization process yields a comprehensive set of outputs, which collectively constitute the optimal expansion plan for the given scenario. These results encompass specific investment decisions pertaining to new generation and transmission infrastructure, detailed economic outcomes (including total, investment, and operational costs), and a full technical description of the expanded system’s steady-state operation. This technical description includes power flows, system losses, and nodal voltage profiles. The entire workflow is capable of being iteratively repeated for diverse scenarios, thereby facilitating a thorough analysis of the impact associated with varying load growth rates or planning horizons.

4. Case Studies

The proposed Mixed-Integer Nonlinear Programming mathematical model was applied to the IEEE 14-bus system, as illustrated in Figure 2. To evaluate the mathematical model, the following case studies were considered:

• $\mathrm{Case}1$: Expansion of the model power system in the generation and transmission stages for a period of 2 years.
• $\mathrm{Case}2$: Expansion of the model power system in the generation and transmission stages for a period of 5 years.

To assess the model’s performance under different future conditions, each case was evaluated against three distinct load scenarios. These scenarios were characterized by uniform exponential demand growth, with annual rates of 5%, 7%, and 9% applied over the planning horizon. Accordingly, three analyses were conducted for each case described.

Table 2. Parameters of the 14-bus network.

${\mathit{N}}_{\mathit{i}}$	${\mathit{N}}_{\mathit{j}}$	R [pu]	X [pu]	$\mathit{B}\mathit{l}$ [pu]	Limit [MVA]	Cost [MMUSD]
1	2	0.01938	0.05917	0.0528	120	66.63
1	5	0.05403	0.22304	0.0492	65	36.12
2	3	0.04699	0.19797	0.0438	36	20.00
2	4	0.05811	0.17632	0.0374	65	36.18
2	5	0.05695	0.17388	0.0340	50	27.76
3	4	0.06701	0.17103	0.0346	65	36.12
4	5	0.01335	0.04211	0.0128	45	25.01
4	7	0	0.55618	0	32	12.00
4	9	0	0.25202	0	45	14.00
5	6	0.09498	0.19890	0	18	26.99
6	11	0.12291	0.25581	0	32	47.96
6	12	0.06615	0.13027	0	32	48.10
6	13	0	0.17615	0	32	8.00
7	8	0.09711	0.11038	0	32	44.12
7	9	0.03181	0.08450	0	32	48.00
9	10	0.12711	0.27038	0	32	48.07
9	14	0.08205	0.19207	0	12	18.01
10	11	0.22092	0.19988	0	12	17.99
13	14	0.17093	0.34802	0	12	18.00

presents the characteristics of the power system network used as a model [37].

The data associated with each of the buses are presented in

Table 3. Bus data in the 14-bus network.

Bus	$\mathit{Pd}$ [MW]	$\mathit{Qd}$ [Mvar]	Area	Voltage [kV]
1	0	0	1	138.0
2	21.7	12.7	1	138.0
3	94.2	19.0	1	138.0
4	47.8	−3.9	1	138.0
5	7.6	1.6	1	138.0
6	11.2	7.5	1	138.0
7	0	0	2	69.0
8	0	0	3	13.8
9	29.5	16.6	2	69.0
10	9.0	5.8	2	69.0
11	3.5	1.8	2	69.0
12	6.1	1.6	2	69.0
13	13.5	5.8	2	69.0
14	14.9	5.0	2	69.0

, while the parameters characterizing the generators located in the electrical system are listed in

Table 4. Data of the generators in the 14-bus system.

Bus	${\mathit{P}}_{\mathit{max}}$ [MW]	${\mathit{P}}_{\mathit{min}}$ [MW]	Cost [USD/MWh]	${\mathit{Q}}_{\mathit{max}}$ [Mvar]	${\mathit{Q}}_{\mathit{min}}$ [Mvar]
1	160	10	20	100	−100
2	80	20	20	50	−42
3	50	20	40	40	0
6	30	0	40	24	−24
8	30	0	40	24	−24

.

Table 5. Technical data of the potential generators.

	Bus	${\mathit{C}}_{\mathit{max}}$ [MW]	${\mathit{P}}_{\mathit{min}}$ [MW]	${\mathit{Q}}_{\mathit{max}}$ [Mvar]	${\mathit{Q}}_{\mathit{min}}$ [Mvar]
$G{n}_1$	2	150	0	70	−70
$G{n}_2$	6	100	0	60	−60
$G{n}_3$	14	200	0	80	−85
$G{n}_4$	3	100	0	60	−60
$G{n}_5$	4	100	0	60	−60

presents the technical parameters of the potential generators to be incorporated into the reference power system, while

Table 6. Economic data of the potential generators.

	Bus	Investment [USD/kW]	Operating Cost [USD/MWh]	Lifetime Years	$\mathit{pf}$
$G{n}_1$	2	1200	20	50	0.80
$G{n}_2$	6	600	50	20	0.94
$G{n}_3$	13	1000	30	50	0.82
$G{n}_4$	3	500	80	25	0.93
$G{n}_5$	4	350	120	25	0.96

shows the economic parameters of these generators.

5. Results Analysis

Given the described cases and the dynamic load scenarios applied to each one, the results were evaluated from both technical and economic perspectives.

5.1. Case 1

By applying the mathematical model over a 2-year analysis period for each dynamic load scenario, the results of the electrical and economic variables were obtained. Generation and demand were evaluated, as presented in

Table 7. Active power results by load scenario, Case 1.

	Load Scenarios
Parameter [MW]	5%	7%	9%
Active power from existing generation	293.84	304.95	316.26
New generation power	-	-	-
Load	285.55	296.53	307.72
Losses	8.29	8.43	8.54

. Figure 3 illustrates the results of active generation and supplied demand, including the corresponding losses.

The results show that for each analyzed load scenario the operating generation fleet—with a nominal capacity of 350 MW—was sufficiently robust to supply the demand and meet the network’s operational requirements. Therefore, the model indicates that no generation expansion was required.

On the other hand, when evaluating active power losses in relation to the supplied demand for each load scenario, it was concluded that the average percentage of losses amounted to 2.84%.

Table 8. Reactive power results by load scenario, Case 1.

	Load Scenarios
Parameter [Mvar]	5%	7%	9%
Reactive power from existing generation	85.700	90.394	93.046
Reactive power from new generation	-	-	-
Load	81.034	84.150	87.325
Losses	4.667	6.243	5.721

presents the results for reactive power. The analysis reveals that the total reactive power capacity produced by the generators was 238 Mvar, which supplied the increased load and compensates for network losses without the need to increase generation. The results are illustrated in Figure 4.

An evaluation of reactive power losses relative to the supplied demand in each load scenario shows that the average loss percentage amounted to 6.59%. Given that the reactive power production is directly related to voltage levels, the voltage profiles at the end of the analysis period were verified for each dynamic load scenario, as illustrated in Figure 5.

Figure 5 shows that for all demand growth scenarios the voltage levels remained within the operational band, ensuring voltage stability across the system. To complement the technical analysis, the transmission network expansion was also examined. The results of this verification are presented in

Table 9. Activated transmission links by load scenario, Case 1.

	Load Scenarios
Link	5%	7%	9%
Link $N_4$–$N_9$	✓	✓	✓
Link $N_2$–$N_3$	✓	✓	✓

.

Based on Table 9, it is concluded that although no expansion of the generation system is required—since the installed capacity is sufficient to supply the demand—the model indicates the need to reinforce the transmission system. For this case, regardless of the load scenario, the link connecting node 4 to node 9 is activated, corresponding to the expansion of a substation, and a new transmission line is added between nodes 2 and 3. Figure 6 shows the single-line diagram of the expanded electrical system for all load scenarios.

The resulting voltages for each load scenario are listed in

Table 10. Voltages by load scenario, Case 1.

	5%		7%		9%
Node	Voltage [pu]	Angle [deg]	Voltage [pu]	Angle [deg]	Voltage [pu]	Angle [deg]
1	1.020	0	1.020	0	1.020	0
2	1.011	−3.53	1.011	−3.53	1.012	−3.54
3	0.997	−5.32	0.998	−7.54	0.998	−7.56
4	0.986	−5.68	0.986	−5.69	0.986	−5.66
5	0.991	−7.21	0.991	−7.23	0.991	−7.18
6	1.002	−13.05	1.000	−13.66	0.999	−13.23
7	0.989	−13.20	0.989	−14.46	0.988	−14.23
8	0.998	−12.58	1.001	−13.42	1.002	−13.04
9	0.977	−13.29	0.971	−14.72	0.968	−14.57
10	0.973	−13.61	0.967	−14.91	0.964	−14.73
11	0.983	−13.49	0.979	−14.45	0.976	−14.15
12	0.989	−14.18	0.986	−14.89	0.984	−14.52
13	0.987	−14.43	0.983	−15.20	0.981	−14.85
14	0.998	−16.08	0.993	−17.29	0.989	−17.12

, while the power flows through each link of the simulated network for every load scenario are presented in Appendix A.

Based on the results of the electrical variables, the operating costs of energy production from the operating generation, as well as the annualized investment costs of the incorporated links, were determined. The costs for each load scenario in Case 1 are presented in

Table 11. Costs by load scenario, Case 1.

	Costs in Million USD [MUSD]
Scenario	Network	Operation of ${\mathit{G}}_{\mathit{o}}$	Total
5%	3.22	120.16	123.38
7%	2.10	138.10	140.19
9%	2.58	141.17	143.75

.

5.2. Case 2

For a 5-year analysis period under each dynamic load scenario, the results of the electrical and economic variables were obtained. Analogous to Case 1, production and demand were evaluated, and the results are presented in

Table 12. Active power results by load scenario, Case 2.

	Load Scenarios
Parameter [MW]	5%	7%	9%
Active power from operating generation	249.13	290.00	296.32
Active power from new generation	89.89	82.80	112.71
Load	330.56	363.26	398.50
Losses	8.46	9.53	10.54

. Figure 7 illustrates the results of active power generation and supplied demand, including the corresponding losses.

From the results for each analyzed load scenario, it was concluded that the existing generation fleet had to be expanded, in order to meet the demand and fulfill the network’s operational requirements. Therefore, the results of the generation expansion are presented in

Table 13. Active power from new generation by load scenario, Case 2.

Active Power [MW]
	Load Scenarios
Location	5%	7%	9%
Node 2	-	8.073	18.25
Node 3	35.081	17.000	32.41
Node 14	54.804	57.723	62.05

.

Considering the analysis period and the associated increase in load, it can be stated that the new generation along with the existing generation enabled the system to meet the demand and compensate for active power losses. In this context, the average percentage of active power losses with respect to the supplied demand across the load scenarios reached a value of 2.61%.

Similarly, a reactive power analysis was carried out. The results are shown in

Table 14. Reactive power results by load scenario—Case 2.

	Load Scenarios
Parameter [Mvar]	5%	7%	9%
Reactive power from existing generation	109.254	105.155	110.678
Reactive power from new generation	−8.252	6.515	14.508
Load	93.807	103.088	113.089
Losses	7.195	8.582	12.097

, from which it can be observed that the model performed an allocation of reactive power between the new and existing generators. This allocation not only supplied the increased load but also compensated for losses occurring in the network. The results are illustrated in Figure 8.

Based on the results obtained for each analyzed load scenario, as previously indicated, the existing generation fleet had to be expanded to meet demand and satisfy the network’s operational requirements. Accordingly, the results of the generation expansion are presented in

Table 15. Reactive power from new generation by load scenario, Case 2.

Reactive Power [Mvar]
	Load Scenarios
Location	5%	7%	9%
Node 2	-	3.495	13.687
Node 3	−13.327	−6.719	−12.809
Node 4	5.076	9.738	13.630

.

As a complement, by evaluating the reactive power losses relative to the supplied demand for each load scenario, it is observed that the average percentage of losses amounted to 8.90%. In this regard, since reactive power production is directly related to voltage levels, the voltage profiles were verified at the end of the analysis period for each load scenario, as illustrated in Figure 9.

From Figure 9, it is evident that in all demand growth scenarios the voltage levels remained within the operational band. This ensured that the system maintained its voltage stability. To complement the technical analysis, the network expansions were verified; the verification results are presented in

Table 16. Network link activation by load scenario.

	Load Scenarios
Link	5%	7%	9%
Link $N_2$–$N_3$	-	✓	✓

.

From Table 16, it is concluded that, except for the 5% demand growth scenario, network expansions were implemented, even though generation expanded in all the demand scenarios. In this case, the model indicated the need to reinforce the transmission system, activating the new link connecting node 2 and node 3. Figure 10 and Figure 11 show the one-line diagrams of the expanded power system.

The voltage results for each load scenario are shown in

Table 17. Voltages per load scenario, Case 2.

	5%		7%		9%
Node	Voltage [pu]	Angle [degree]	Voltage [pu]	Angle [degree]	Voltage [pu]	Angle [degree]
1	1.020	0	1.020	0	1.020	0
2	1.011	−3.53	1.011	−3.53	1.012	−3.54
3	0.997	−5.32	0.998	−7.54	0.998	−7.56
4	0.986	−5.68	0.986	−5.69	0.986	−5.66
5	0.991	−7.21	0.991	−7.23	0.991	−7.18
6	1.002	−13.05	1	−13.66	0.999	−13.23
7	0.989	−13.20	0.989	−14.46	0.988	−14.23
8	0.998	−12.58	1.001	−13.42	1.002	−13.04
9	0.977	−13.29	0.971	−14.72	0.968	−14.57
10	0.973	−13.61	0.967	−14.91	0.964	−14.73
11	0.983	−13.49	0.979	−14.45	0.976	−14.15
12	0.989	−14.18	0.986	−14.89	0.984	−14.52
13	0.987	−14.43	0.983	−15.20	0.981	−14.85
14	0.998	−16.08	0.993	−17.29	0.989	−17.12

, and the power flows for each network link in the simulated grid are presented in Appendix B.

Taking into account the results of the electrical variables, the operating costs of the power generation from the existing generators were determined, along with the annualized investment costs of the incorporated links. These results are presented in

Table 18. Costs per load scenario.

	Costs in Million US Dollars [MUSD]
Scenario	Network	Operation of ${\mathit{G}}_{\mathit{o}}$	Operation of ${\mathit{G}}_{\mathit{n}}$
5%	-	262.04	194.94
7%	1.61	297.84	142.49
9%	1.61	308.92	211.09
Scenario		Investment in $G_n$	Total
5%		7.46	464.44
7%		7.74	449.67
9%		10.25	531.88

.

5.3. Reactive Power Management and Voltage Stability Assurance

A key advantage of the proposed IGTEP model is its formulation based on the complete AC OPF equations, which inherently addresses the management of reactive power and ensures voltage stability. This section explicitly discusses how the model handles these critical aspects, a point of concern when planning models employ simplifications.

The model ensures that all reactive power demands are met through two primary mechanisms. First, the nodal reactive power balance, formulated in Equation (17), is a strict constraint that must be satisfied at every bus in the system. It enforces that the sum of the reactive power injected by all existing and new generation units equals the local demand plus the net reactive power flowing into the transmission lines. Second, the reactive power output from each generator $(Q_{G_o}$ and $Q_{G_n})$ is an optimization variable, constrained only by its capability curve as defined in Equations (21)–(24), allowing for a fully flexible and optimal dispatch to meet system needs.

More importantly, the model guarantees that meeting this reactive demand does not lead to voltage instability. This is accomplished by treating the voltage magnitude at each bus, $V_n$, as an optimization variable that is explicitly constrained to remain within secure operational limits, as shown in Equation (25): ${Vmin}_n\le V_n\le {Vmax}_n$. By including these voltage constraints directly within the optimization problem, any feasible solution found by the solver inherently represents an expansion plan that is not only economically optimal but also technically viable from a voltage stability perspective. The model must find a coordinated dispatch of reactive power from all available sources to satisfy the demand while simultaneously respecting the voltage limits at every bus.

This integrated approach is a significant improvement over simplified DC models, which neglect reactive power and voltage considerations entirely, often yielding solutions that are not operationally feasible without further analysis and adjustments. The results presented for both case studies confirm the effectiveness of this formulation. The voltage profiles shown in Figure 5 and Figure 9 demonstrate that for all the analyzed dynamic load scenarios the nodal voltages were successfully maintained within the acceptable operational band, thereby confirming that the resulting system operated in a stable state.

5.4. Computational Performance and Scalability

The proposed IGTEP model is formulated as a MINLP problem. It is well-established that MINLP models that embed non-convex AC power flow equations are NP-hard, and their computational complexity can pose a significant challenge, particularly as the scale of the system increases. This section analyzes the computational performance of our model to provide transparency regarding its practical application and to justify the selection of the IEEE 14-bus system for this study.

All the simulations were executed on a workstation equipped with an Intel Xeon W3-2525 CPU and 64 GB of DDR4 RAM. The model was implemented in GAMS 49.1.0 and solved using the BARON 25.3.19 global optimization solver. To accommodate BARON’s requirements, the AC power flow equations, which natively include trigonometric functions, were reformulated into their rectangular coordinate equivalents. This adaptation allows the model to retain the non-linear accuracy of the AC power flow without the simplifications of a DC model, which would compromise the validity of the analysis concerning reactive power and voltage stability.

A critical aspect of the formulation is the judicious selection of the Big-M parameter, employed within the disjunctive constraints (Equations (7)–(10)) that govern the activation and deactivation of power flow equations for candidate transmission lines. The choice of M inherently involves a delicate trade-off: its value must be sufficiently large to confidently encompass the maximum possible magnitude of the expression it constrains, thereby ensuring the logical integrity of the “either–or” conditions. However, an excessively large M value can significantly deteriorate the numerical conditioning of the optimization model, leading to numerical instability, a weakened continuous relaxation of the objective function, and, consequently, a severe degradation in the performance of the chosen solver. This often manifests as slower convergence, increased computational burden, or even solver failure.

To rigorously address this, a comprehensive sensitivity analysis was performed to identify the most suitable value for M that balanced model correctness with numerical robustness. The initial computational experiments with arbitrarily large values ($M\ge 100$) indeed resulted in significant numerical challenges for the BARON solver. These challenges included the frequent occurrence of infeasibility errors in the Nonlinear Programming (NLP) subproblems and a noticeable deterioration in the quality of their objective function evaluations, severely impeding the overall global optimization process.

The value of M was progressively reduced, and through systematic testing it was unequivocally determined that M = 25 provided the optimal balance between these conflicting requirements. This value proved to be sufficiently large to correctly enforce the logical conditions of the constraints—being over an order of magnitude larger than the maximum expected power flows on a 100 MVA base—yet simultaneously small enough to maintain a well-conditioned numerical model. This meticulous selection was paramount in enabling BARON to reliably converge to a feasible, globally optimal solution within an acceptable computational time frame, as further detailed in

Table 19. Sensitivity analysis of the Big-M parameter used in the transmission line activation constraints. Results correspond to Case 2 with a 7% load growth rate.

M	Lines Activated	Solver Status	Execution Time	Observed Solver Behavior
5	Incomplete	Infeasible/Suboptimal	∼6 h:12 m	Constraints became overly tight, preventing activation of necessary candidate lines; premature termination due to infeasibility.
10	Partial	Local Optima	∼9 h:40 m	NLP subproblems exhibited early deterioration; a solution was found but lacked the guarantee of global optimality.
25	Consistent	Globally Optimal (0.0% gap)	17 h:46 m:05 s	Demonstrated stable and robust convergence; Case 2 with 7% load growth successfully solved to verified global optimality; selected as the baseline value for all subsequent analyses.
50	Consistent	Globally Optimal	∼21 h:20 m	Solver required relaxed tolerances to proceed, likely due to large second derivatives (Hessians) leading to slower, less efficient convergence.
100	Consistent	Solver Failure (77.5% gap)	∼4 h:10 m	Experienced an access violation error and catastrophic solver behavior; persistent infeasible subproblems; loss of global convergence capabilities.

.

For the IEEE 14-bus system, the final model comprised 745 variables (including 435 discrete variables) and 1095 constraints, with 9812 non-linearities. The average computation time to solve each of the six scenarios to global optimality (0.0% optimality gap) was approximately 17 h, 40 min, and 54 s. This significant computational time, required even for a relatively small and well-behaved 14-bus system, underscores the scalability challenge of solving AC-based IGTEP problems to proven optimality, validating the need for the rigorous model implementation choices described, and directly informing the decision to use this test case. Applying this exact model to large-scale power grids (e.g., systems with 100 or more buses) would likely be computationally prohibitive for current deterministic global solvers.

The primary objective of this research was to establish a high-fidelity, globally optimal benchmark, prioritizing mathematical rigor and technical accuracy over scalability to larger systems. This benchmark is essential for validating the performance of faster methods (e.g., heuristics, decomposition techniques), which are necessary for planning large-scale systems. To explore the trade-off between solution time and optimality, we also tested the model using the COUENNE solver, another open-source global optimizer. COUENNE found a feasible solution in a fraction of the time—approximately 6 h and 1 min per scenario. However, this speed came at a substantial cost to solution quality. The solutions obtained by COUENNE were not proven to be globally optimal and yielded an average optimality gap of 74.13%, indicating that the best-found integer solution was significantly inferior to the global optimum identified by BARON.

This comparison validates our choice of BARON for this study, as our objective was to establish a definitive, globally optimal benchmark for the IGTEP problem under the proposed formulation. For practical applications on large-scale systems, future research should explore decomposition techniques, such as Benders decomposition, or advanced heuristic and metaheuristic algorithms. These methods could leverage the detailed formulation presented here to find high-quality solutions in a more reasonable time frame, using the results from this paper as a benchmark for accuracy and performance evaluation.

5.5. Discussion on the Value of an Integrated AC-Based Planning Framework

A foundational decision in this research was the use of a globally-optimized MINLP model founded on the complete AC power flow equations. To contextualize this choice, the proposed framework is contrasted here with two common simplifications in expansion planning: the sequential (or decoupled) paradigm and the use of DC power flow approximations.

The traditional approach of planning generation and transmission in separate, sequential stages fails to capture the crucial economic trade-offs between these investments. A GEP model, for instance, might trigger investment in new generation to serve a growing load, whereas the globally optimal solution could be a more affordable transmission upgrade that unlocks stranded capacity from existing, cheaper generation resources. It has been demonstrated in the literature that co-optimizing generation and transmission yields a lower total system cost by identifying precisely these types of investment synergies [26]. The results from Case 1—where transmission reinforcements were selected to defer capital-intensive generation investment—perfectly exemplify a non-intuitive, synergistic decision that a sequential planning approach would be structurally unable to identify.

Furthermore, a significant portion of the planning literature utilizes the DC power flow approximation to maintain model linearity and ensure computational tractability. This simplification comes at the cost of ignoring critical network characteristics. The DC model’s inherent assumptions—neglecting reactive power, assuming a flat voltage profile, and disregarding accurate network losses—can yield solutions that are suboptimal or technically infeasible in real-world operation. Research has explicitly underscored that failing to model these nonlinearities can lead to an incorrect assessment of system congestion and, consequently, suboptimal investment decisions [5]. The inability of a DC model to “see” voltage or reactive power constraints means it cannot identify transmission bottlenecks caused by these issues. The findings in Case 1 strongly suggest that such issues were the primary drivers for the required network reinforcement, a critical nuance that would be invisible to a DC-based model.

In summary, while simplified models offer computational advantages, they risk producing economically inefficient and technically vulnerable expansion plans. The proposed integrated framework, based on a full AC power flow and solved to global optimality, is therefore considered essential for identifying strategies that are both cost-effective and operationally robust.

5.6. Discussion: On the Value of Integrated Planning and Interpretation of Results

A crucial aspect of this study is the interpretation of the results within the strategic context of integrated versus sequential planning approaches. The outcomes, particularly derived from Case 1, provide profound insights into the complex economic and technical trade-offs that an integrated modeling framework, especially one founded on a comprehensive AC power flow formulation and global optimality, is uniquely positioned to capture.

The findings from Case 1, where no generation expansion was triggered despite the dynamically increasing demand, represent a significant and non-obvious result that necessitates detailed interpretation. This outcome is not an anomaly but rather a fundamental insight into the underlying economics and physical constraints of the power system. It unequivocally demonstrates that for the 2-year planning horizon the primary limiting factor was not a deficiency in total generation capacity but rather a critical bottleneck within the transmission network. The existing generation fleet of 350 MW was indeed sufficient to meet the increased demand; however, the prevailing transmission infrastructure was inadequate to reliably deliver that power from generation hubs to load centers while strictly adhering to operational limits, such as line thermal limits and maintaining voltage stability.

An optimization model predicated on a sequential or decoupled planning paradigm (e.g., Generation Expansion Planning followed by Transmission Expansion Planning) would almost certainly have failed to identify this optimal strategy. A GEP-only model, by neglecting explicit network constraints, would have observed sufficient aggregate generation and might have erroneously concluded that no investment was immediately necessary. Such a conclusion would inevitably lead to future operational problems, including severe congestion and an inability to serve crucial energy demand. Conversely, if it had arbitrarily mandated new generation capacity, it would have resulted in a suboptimal and economically inefficient solution.

The proposed integrated MINLP model, by simultaneously considering generation and transmission expansion under rigorous AC power flow constraints, correctly identifies that the most cost-effective and system-beneficial strategy is to invest in targeted transmission reinforcement. Specifically, this involved activating the link from node 4 to 9 and adding a new line between nodes 2 and 3. This strategic investment effectively unlocks the stranded capacity of the existing, more economically viable generation resources.

This finding critically underscores the fundamental value and unique capability of the integrated planning approach: it intelligently avoids premature and potentially unnecessary capital expenditure on generation by strategically channeling investments into transmission, thereby leading to a globally optimal economic solution that a decoupled approach would inherently miss. Case 2 further strengthens this argument by showcasing that as demand continues its projected growth over a longer horizon, a critical juncture is reached where concurrent investments in both significant new generation capacity and further network expansion become simultaneously necessary and optimal.

Therefore, a direct quantitative comparison with a DC-based model was considered less insightful for the primary objectives of this paper. A DC model, due to its inherent linearization and decoupling of active and reactive power flows, fundamentally cannot capture the intricate voltage and reactive power dependencies that are frequently the driving factors for transmission expansion and system stability. Consequently, it would yield a solution that is technically incomplete and potentially misleading for real-world application. Instead, the compelling comparison between Case 1 (demonstrating transmission-led expansion) and Case 2 (illustrating combined generation and transmission expansion) serves as a more powerful and internally consistent validation of the model’s sophisticated ability to accurately discern and optimally react to the evolving economic and technical needs of the power system.

For Case 1, corresponding to a shorter planning horizon and initial load growth, the optimal solution indicates no new generation capacity additions. This outcome, initially counter-intuitive, is a key insight of our integrated model, directly enabled by its high-fidelity AC representation and global optimality guarantees. The results (Table 9, Figure 6, and Appendix A) unequivocally demonstrate that under these specific conditions the most economically efficient strategy is to prioritize strategic transmission reinforcements. These transmission investments effectively alleviate congestion and unlock existing, underutilized generation capacity, thereby deferring the need for capital-intensive new generation investments. This highlights the model’s ability to identify the most cost-effective solution, which involves leveraging existing infrastructure rather than immediately investing in new generation, thus providing a valuable benchmark for planners.

6. Conclusions

The proposed nonlinear mixed-integer optimization model enables the simultaneous expansion of both the generation stage and the transmission system in order to meet demand using optimal AC power flows. This approach makes it possible to determine the power delivered by new generation units and new network links—two components which, when interacting, ensure economic supply of demand over the analysis period. All of this is achieved while considering the constraints associated with the operational characteristics of the PS.

Based on the results obtained for each case study and considering the dynamic load scenarios, it can be stated that when the analysis period is shorter the model prioritizes network investments and optimization of the existing generation resources to minimize associated costs while still complying with network constraints, as evidenced in the results of Case 1. Likewise, as the analysis period increases and demand grows significantly the model introduces new generation and transmission links simultaneously into the electric system—an outcome reflected in the results of Case 2. Therefore, it is confirmed that the model achieves simultaneous expansion of the generation park and transmission system, while also modeling network-specific constraints.

The decisions to expand generation capacity or to include new transmission links consider the system’s technical constraints but also, more importantly, take into account the associated costs. Consequently, it is concluded that careful incorporation of unit cost data is essential to ensure that the model produces consistent results and accurately determines required investments. Otherwise, poor planning could lead to overcosts that, depending on the market structure in each country, must be compensated—typically resulting in increased tariffs for the end user.

It is acknowledged that the planning framework presented in this paper is deterministic. The dynamic load scenarios are based on predefined annual growth rates $\left(f_{Lg}\right)$, and the availability of generation units is modeled through fixed plant factors $\left(f_{P{L}_{G_o}}\right)$. However, real-world power systems are subject to significant uncertainties. These arise not only from load forecasting errors but, more critically, from the variable and intermittent nature of RESs like solar and wind, which are becoming prevalent in modern grids. The deterministic model established here provides a fundamental and computationally exact benchmark, which is considered an essential prerequisite before introducing more complex uncertainty models. Therefore, a natural and necessary extension of this research is the explicit incorporation of uncertainty. Methodologies such as stochastic programming or robust optimization are well-suited for this purpose. A stochastic formulation could be developed to find an expansion plan that minimizes the expected total cost over a multitude of uncertain scenarios. Alternatively, a robust optimization approach could be employed to derive a plan that remains technically feasible and economically efficient under the most adverse realization of uncertainties, thereby enhancing the system’s resilience. Future work will be directed toward extending the current MINLP model into such a stochastic or robust framework to produce expansion plans that are not only cost-effective but also reliable under uncertainty.

The proposed mathematical model integrates the nonlinear conditions of AC power flow with binary variables that define investment decisions in both generation and transmission. Furthermore, its formulation is scalable and adaptable to various electric systems, making it a valuable tool for short-, medium-, and long-term planning. While the computational effort required to guarantee global optimality is significant, as demonstrated in our performance analysis, this model serves as a robust benchmark against which more computationally efficient methods, such as decomposition techniques or heuristics, can be validated. Since the model proposes a novel methodology for the simultaneous expansion of generation and transmission stages, it is recommended that future work includes a stochastic analysis comparing the classical methodology with the approach presented herein, in order to establish the technical and economic parameters that will allow the electric system to optimally serve demand over the medium and long term.