An Optimal Capacity Configuration Method for a Renewable Energy Integration-Transmission System Considering Economics and Reliability

Abstract

Integrated Energy Transmission Systems (IETSs) are essential to bridge the geographical gap between where energy is produced and where it is needed, transporting power from resource-rich regions to distant load centers. The fundamental challenge is to resolve the inherent asymmetry between an intermittent power supply and distant load demand. Conventional approaches, focusing only on capacity, fail to address this issue while achieving an effective economic and reliable balance. To address the concerns above, a bilevel optimization framework is proposed to optimize the capacity configuration of IETSs, including wind power, photovoltaic (PV), thermal power, and pumped storage. The optimal capacity of wind and PV is determined by the upper-level model to minimize electricity price, whereas the lower-level model optimizes the system’s operational dispatch for given configuration to minimize operational expenses. A detailed IETS model is also developed to accurately capture the operational characteristics of diverse power sources. Furthermore, the proposed model integrates carbon emission costs and High-Voltage Direct Current (HVDC) utilization constraints, thereby allowing for a comprehensive assessment of their economic efficiency and reliability for capacity configuration. Case studies are conducted to verify the proposed method. The results show that the capacities of wind and PV are optimized, and the electricity costs of IETSs are minimized while satisfying reliability constraints.

1. Introduction

The traditional three-phase power grid is a model of engineered symmetry, with its stability rooted in the perfect balance of its symmetrical three-phase voltages. However, the large-scale integration of renewables introduces a critical spatiotemporal asymmetry. This imbalance arises because intermittent generation in resource-rich regions rarely aligns with the demand patterns of distant load centers, creating significant operational challenges. This paper proposes an optimal design for an Integrated Energy Transmission System (IETS) to act as a symmetrizing agent, restoring the crucial equilibrium between power supply and demand.

A power system dominated by renewable energy is being actively promoted in China to achieve the dual carbon goals [1,2,3,4]. Renewable sources such as wind and solar, due to their clean and energy-saving characteristics, have become key strategies for addressing challenges like fossil fuel shortages and environmental pollution [5,6,7,8]. However, the increasing penetration of renewable energy brings challenges including output intermittency, volatility, and suboptimal capacity planning, which can cause supply fluctuations, reverse power flows, and degrade power quality in power grids [9,10,11,12].

The adverse impacts of renewables mainly arise from the uncertainty and intermittency of their output. With ongoing technological advancements, energy storage systems (ESSs) with steadily decreasing costs offer broad potential to address these challenges [13,14,15]. Building on this, the IETS provides a feasible solution for accommodating renewable energy [16,17,18]. In this context, this study focuses on IETSs, considering the uncertainties of wind and solar resources, delivered electricity prices, and the carbon reduction benefits of renewables, to investigate the optimal capacity configuration of IETSs.

In multi-energy system planning, capacity configuration plays a pivotal role. Rationally allocating capacity according to the generation characteristics of each energy source helps avoid overbuilding and ensures safe, stable, and economical grid operation [19]. Appropriate renewable energy capacity configuration combined with well-designed ESS strategies can reduce line losses and grid investment costs [20]. Conversely, improper configurations may increase line losses at specific buses, cause voltage violations, and ultimately reduce the economic efficiency of the distribution system [21]. Consequently, extensive research on IETS capacity optimization has been conducted, which can be primarily categorized into two aspects: the development of optimization models and the application of solution algorithms.

In terms of model formulation, research has evolved from single-objective frameworks to multi-objective approaches. Early models typically focus on a single metric, such as voltage deviation or economic cost. In [22], a method combining subjective and objective weighting is proposed to address multiple uncertainties in decision-making, where economic, resource, transportation, environmental, and social factors are considered as evaluation criteria for optimal siting. In [23], a mathematical model for distributed generation (DG) optimal configuration is established, aiming to minimize the total voltage deviation while considering power flow, node voltage, branch current, and DG capacity constraints. In [24], an economic objective function based on life-cycle cost theory is formulated, and HOMER software is used to optimize the capacity of wind–photovoltaic (PV) storage systems. In [25], the key technical characteristics of ESSs in new power systems are summarized, and coordinated control methods are proposed to achieve a multi-timescale power–energy balance under large-scale renewable integration. In [26], a joint capacity optimization model for electricity, thermal, and hydrogen storage systems is presented, targeting the maximization of annual system value while considering renewable utilization and operational constraints. In [27], normalized economic and voltage improvement indicators are combined with the analytic hierarchy process to construct a composite objective function. In [28], the multi-objective problem of minimizing power losses and voltage deviations is transformed into a single-objective problem through weighting, and an optimal renewable capacity and siting scheme for distribution networks is proposed.

To solve these increasingly complex and nonlinear models, diverse optimization algorithms are employed. In [29], an active distribution network scheduling model is proposed with the objectives of minimizing economic–environmental cost, control cost, and node voltage deviation, and is solved using a tuna swarm optimization algorithm with a weighted average compromise decision method. In [30], a multi-strategy hybrid sparrow search algorithm is developed to optimize the economic operation of a grid-connected microgrid under carbon emission constraints, integrating PV, wind, fuel cells, micro-gas turbines, and ESSs. In [16], an optimization strategy based on symplectic geometry mode decomposition is introduced for hybrid battery–supercapacitor systems in wind applications. In [31], a hybrid particle swarm–gray wolf optimization algorithm is applied to determine the optimal capacity mix of solar, wind, biomass, diesel generators, and battery storage in a microgrid, with the objectives of minimizing power cost and supply shortages. In [32], a multi-objective model uses a weighted sum method to balance economic costs and carbon emissions, but this technique relies on subjective weightings that can obscure clear market-driven goals. In [33], a stochastic programming model effectively handles uncertainty, yet its high computational burden requires scenario reduction techniques that may filter out critical operational extremes.

The above-mentioned studies have investigated capacity optimization for multi-energy systems to some extent. However, most focus on a single technical indicator as the objective function, often neglecting the market competitiveness of electricity at the receiving end. To bridge this gap, this paper proposes an optimization framework. The primary objective is to minimize the delivered electricity price at the receiving end by rationally sizing wind and PV capacity, alongside the optimal scheduling of pumped-storage plant operations and the unit commitment of thermal generators. The following research is carried out in this paper.

• A bilevel optimization framework is developed to co-optimize long-term capacity expansion and short-term operational scheduling for the IETS. This structure is computationally tractable and accommodates a full-year simulation, effectively accounting for the stochastic nature of wind and PV power. Specifically, the upper level determines the optimal capacity configuration. It achieves this with a single, market-driven objective: minimizing the delivered electricity price. This approach entirely avoids subjective trade-offs between conflicting goals. Meanwhile, the lower level minimizes the annual operational cost through the optimal dispatch of thermal units and pumped-storage operation. This is performed under system power supply reliability and renewable curtailment constraints, achieving an optimal balance between long-term market competitiveness and short-term operational security.
• A high-fidelity operational model is developed. This model precisely captures the operating characteristics of diverse power sources. This ensures that investment decisions are grounded in operational reality, not statistical simplification. The model explicitly represents the intermittency of renewables, the commitment decisions and ramping limits of thermal units, and the complex operational dynamics of the pumped-storage station.
• Carbon emission costs and HVDC utilization constraints are accounted for in the optimal capacity configuration. This approach provides a tool for assessing the interplay between decarbonization policies and the efficient utilization of transmission infrastructure. The model uses a carbon price to discourage fossil fuels. It also enforces a minimum HVDC utilization rate to guarantee transmission asset efficiency. This directly shapes the economic and operational trade-offs that determine the final optimal configuration.

The remaining parts of this paper are organized as follows. Section 2 presents the model of renewable energy generation and evaluation indices of capacity configuration. The design of capacity configuration for IETSs considering reliability and economy is provided in Section 3. Section 4 verifies the proposed control method through simulation experiments. Finally, Section 5 concludes the paper.

2. The Model of Renewable Energy Generation and Evaluation Indices of Capacity Configuration

To conduct research on the optimal capacity configuration of IETSs, it is necessary to develop wind and PV generation models and perform a comprehensive assessment. This section establishes mathematical models for wind power generation, PV, and pumped-storage systems, and characterizes their power generation behaviors. The resulting formulations provide the theoretical foundation for the subsequent construction of an optimal capacity configuration model for distributed wind–solar–pumped-storage complementary power sources.

2.1. Generation Model of the Power Supply System

2.1.1. Model of Wind Power Generation Output

Wind power generation converts natural wind energy into electrical power, with its output primarily governed by environmental factors such as wind speed and direction. To ensure both high efficiency and safe operation, different turbine types adopt distinct cut-in, cut-out, and rated wind speeds. In this study, the relationship between the power output of the wind power generation system and the wind speed is approximated by the piecewise function given in Equation (1).

$$P^{\mathrm{wind}}(v)=\left\{\begin{array}{ll}0\hfill & \left(v\le v_i \mathrm{or} v\ge v_o\right)\hfill \\ P_R{\displaystyle \frac{v^3-v_i^3}{v_R^3-v_i^3}},\hfill & \left(v_i\le v\le v_R\right)\hfill \\ P_R\hfill & \left(v_R\le v\le v_o\right)\hfill \end{array}\right.$$

(1)

where $P_R$ is the rated output power of the wind turbine, $v$ is the real-time wind speed, and $v_i$, $v_R$, and $v_o$ denote the cut-in speeds, rated speeds, and cut-out wind speeds.

As shown in Equation (1), when the wind speed exceeds the cut-in speed, the turbine’s output power rises with increasing wind speed. Once the wind speed reaches the rated speed, the turbine delivers its rated power and maintains this level until the wind speed attains the cut-out speed. When the ambient wind speed continues to rise and exceeds the turbine’s cut-out speed, power generation is halted to protect the turbine. Consequently, wind turbine output is strongly governed by ambient wind speed and is characterized by pronounced randomness and volatility.

2.1.2. Model of PV Generation Output

PV generation converts solar energy into electricity. It offers the advantages of flexibility, high efficiency, and scalability. By selecting appropriate PV module parameters and applying the corresponding correction factors, a simplified output model for a PV array under any irradiance and ambient-temperature conditions can be obtained, as expressed in Equation (2).

$$\left\{\begin{array}{l}{I}_{ac}=I_{STC}\cdot {\displaystyle \frac{G_{ac}}{G_{STC}}}\cdot \left[1+\alpha \cdot (T_{ac}-T_0)\right]\hfill \\ V_{ac}=V_{STC}\cdot \ln \left[e+\gamma \cdot (G_{ac}-G_{STC})\right]\left[1-\kappa \cdot (T_{ac}-T_0)\right]\hfill \\ P^{\mathrm{pv}}=I_{ac}\cdot V_{ac}\hfill \end{array}\right.$$

(2)

where $I_{ac}$ and $V_{ac}$ denote the actual operating current and voltage of the PV array, while $I_{STC}$ and $V_{STC}$ represent the current and voltage at the maximum power point. $G_{ac}$ is the actual solar irradiance, $G_{STC}$ is the standard irradiance, $T_{ac}$ is the actual cell temperature, and $T_0$ is the reference temperature. The parameters $\alpha$, $\gamma$, and $\kappa$ are the required correction coefficients, and $P^{\mathrm{pv}}$ is the predicted output power of the PV array.

As shown in Equation (2), at a constant temperature, the maximum output power of the PV increases with rising irradiance; conversely, when irradiance is held constant, the maximum output power rises as the ambient temperature falls. Based on these findings, the predictive output model of the PV array can be simplified as Equation (3):

$$P^{\mathrm{pv}}=P_{STC}\cdot {\displaystyle \frac{G_T}{G_{STC}}}\left[1+t_e(T_e-T_{STC})\right]$$

(3)

where $P^{\mathrm{pv}}$ denotes the maximum test output power under standard test conditions, $G_T$ is the solar irradiance incident on the PV panel, and $G_{STC}$ is the solar irradiance under standard test conditions.

2.1.3. Model of Pumped-Storage Station Output

A pumped-storage hydropower station is an energy conversion facility that transforms low-price electricity into high-price electricity. Its operating cycle comprises pumping, storage, pressure regulation, and power generation. During a single cycle, the mathematical model of the station acting as a load under pumping mode is expressed by Equation (4).

$$P^{\mathrm{psP}}={\displaystyle \frac{\rho g{Q}_p{H}_p}{{\eta }_p}}$$

(4)

where $P^{\mathrm{psP}}$ denotes the power absorbed by the turbine–pump unit; $\rho$ is the density of water; g is gravitational acceleration, set to 9.81; ${\eta }_p$ is the pump efficiency, set to 0.87; $H_p$ is the net head of the pumped storage, defined as the effective height to which the pump can elevate a column of water; and $Q_p$ is the flow rate through the pump, defined as the volume of fluid passing through the system per unit time.

During a single cycle, the mathematical model of the pumped-storage hydropower station acting as a power source under generation mode is expressed by Equation (5).

$$P^{\mathrm{psG}}=\rho g{Q}_h{H}_h{\eta }_h$$

(5)

where $P^{\mathrm{psG}}$ denotes the electrical power generated by the pumped-storage hydropower station; ${\eta }_h$ is the turbine efficiency, set to 0.86; $H_h$ is the net head of the pumped-storage; and $Q_h$ is the flow rate through the turbine.

2.2. Evaluation Indices for Capacity Configuration of IETS

2.2.1. Power Supply Reliability Rate

The power supply guarantee rate, denoted as $f_{RPS}$, represents the probability that the IETS can fully meet the load demand; its calculation is given in Equation (6). A lower $f_{RPS}$ indicates that the system will be unable to satisfy the load for prolonged periods, reflecting a lower capability to provide continuous power supply on the demand side.

$$f_{RPS}=1-f_{LPSP}=1-{\displaystyle \frac{{\displaystyle \sum _{t=1}^T{T}_{LPSP}}}{T}}$$

(6)

where $f_{LPSP}$ denotes the power supply loss-of-load probability; T is the total number of hours in the study period; and $T_{LPSP}$ is the cumulative hours during which the actual output of the multi-energy system falls below the system load.

2.2.2. Renewable Generation Curtailment Rate

The curtailment rate, denoted as $f_{AEP}$, is defined as the proportion of surplus electricity that exceeds the load demand but cannot be utilized, relative to the total load demand; it is calculated by Equation (7). A lower $f_{AEP}$ indicates a more effective utilization of renewable generation, resulting in reduced wind and PV curtailment.

$$f_{AEP}={\displaystyle \frac{{\displaystyle \sum _{t=1}^T{P}_t^{\mathrm{cur}}}\Delta t}{{\displaystyle \sum _{t=1}^T{P}_t^{\mathrm{load}}}\Delta t}}$$

(7)

where $P_t^{\mathrm{cur}}$ denotes the curtailed renewable power from the base at time t, $P_t^{\mathrm{load}}$ represents the system load demand at time t, and $\Delta t$ is the minimum time interval, set to 1 h throughout the paper.

3. Configuration of IETS Considering Economics and Reliability

To address the common oversight in existing distributed-generation planning, where capacity configuration is decoupled from actual operation, this section develops an optimization model for wind–solar–pumped-storage capacity that explicitly incorporates both reliability and economic performance. The model builds upon the mathematical formulations for wind power generation, PV, and pumped storage established in the second section of this paper.

3.1. Framework of Bilevel Optimization Model

A bilevel optimization framework for capacity configuration is presented, accounting for both reliability and economics, as shown in Figure 1.

The framework’s overall objective is to minimize the total cost (f), which comprises the annualized investment cost ${\lambda }_{\mathrm{inv}}$, annual operational cost ${\lambda }_{\mathrm{op}}$, and transmission cost ${\lambda }_{\mathrm{trans}}$. As illustrated, the framework decomposes the problem into two levels. The upper-level model optimizes the long-term capacities of wind and PV. The lower-level model then solves a short-term operational problem to minimize the hourly operational costs for the given capacity configuration. The formula shown within the lower-level block of Figure 1, $\min f={\displaystyle \sum _T^{t=1}}[C_{\mathrm{base}}{S}_t^{\mathrm{dc}}+C_{\mathrm{fuel}}{P}_t^{\mathrm{coal}}+\dots -C_{\mathrm{dc}}{P}_t^{\mathrm{dc}}]$, is a conceptual representation of this operational cost minimization, which accounts for key economic factors like fuel expenses, start-up costs, curtailment penalties, and revenues from power export. The detailed mathematical formulations for all components of this framework are presented comprehensively in Section 3.2.

As shown in Figure 1, the upper-level optimization determines the optimal capacities of wind and PV to minimize the total system cost. The lower-level optimizes system operation under the given configuration. The operational results feed back into the upper-level objective, enabling coordinated optimization to achieve the optimal capacity configuration.

3.2. Objective Function of the Model

As the electricity market structure and market-based pricing mechanisms continue to evolve towards maturity, the competition for exported power at the receiving-end market is becoming increasingly intense. Therefore, this paper takes the minimization of the delivered electricity price at the receiving end of the IETS as the objective, and investigates the optimal scale of PV that can be integrated into the system pumped-storage station under different wind power installed capacities. The objective function is formulated by Equation (8).

$$\min f={\lambda }_{\mathrm{inv}}+{\lambda }_{\mathrm{op}}+{\lambda }_{\mathrm{trans}}$$

(8)

$${\lambda }_{\mathrm{inv}}={\displaystyle \frac{C_{\mathrm{inv}}}{E_{\mathrm{dc}}}}$$

(9)

$$C_{\mathrm{inv}}=\sum _{i\in \mathcal{J}}{c}_i{S}_i{\phi }_i(r_i,n_i)$$

(10)

$$\mathcal{J}=\left\{\mathrm{wind},\mathrm{pv},\mathrm{coal},\mathrm{ps}\right\}$$

(11)

$${\phi }_i(r,n)={\displaystyle \frac{r{(1+r)}^n}{{(1+r)}^n-1}}$$

(12)

$${\lambda }_{\mathrm{op}}={\displaystyle \frac{C_{\mathrm{op}}}{E_{\mathrm{dc}}}}$$

(13)

$$\begin{array}{ll}{C}_{\mathrm{op}}& ={\displaystyle \sum _T^{t=1}}[C_{\mathrm{base}}{S}_t^{\mathrm{dc}}+C_{\mathrm{shed}}{L}_t^{\mathrm{shed}}+C_{{\mathrm{CO}}_2}{\alpha }_{{\mathrm{CO}}_2}{P}_t^{\mathrm{coal}}+C_{\mathrm{fuel}}{P}_t^{\mathrm{coal}}\\ & +C_{\mathrm{start}}{U}_t^{\mathrm{coal}}+C_{\mathrm{ps}}\left(P_t^{\mathrm{psG}}+P_t^{p\mathrm{sP}}\right)+C_{\mathrm{curt}}\left(P_t^{\mathrm{wind},\mathrm{cur}}+P_t^{\mathrm{pv},\mathrm{cur}}\right)-C_{\mathrm{dc}}{P}_t^{\mathrm{dc}}]\end{array}$$

(14)

where ${\lambda }_{\mathrm{inv}}$, ${\lambda }_{\mathrm{op}}$, and ${\lambda }_{\mathrm{trans}}$ denote investment cost, operating cost, and transmission cost, respectively; $C_{\mathrm{inv}}$ and $C_{\mathrm{op}}$ denote the annual investment cost and annual operating cost, respectively; $E_{\mathrm{dc}}$ denotes the annual DC energy target; $c_i$ denotes the unit capital cost of the power generation units i; $S_i$ denotes the installed capacity of device i; ${\phi }_i(r_i,n_i)$ denotes the capital-recovery factor of device i; $r_i$ and $n_i$ denote the discount rate and economic lifetime of device I, respectively; $\mathcal{J}$ denotes the set of power generation units; $S_t^{\mathrm{dc}}$ is the HVDC export shortfall in time slot t, defined as the gap between the actual export power and the base export load; $L_t^{\mathrm{shed}}$ represents the load-shedding amount, defined as the difference between load demand and actual supply; ${\alpha }_{{\mathrm{CO}}_2}$ is the thermal power emission factor, representing the CO₂ emissions per unit of electricity generated by thermal power plants; $P_t^{\mathrm{coal}}$ is the thermal generation output; $P_t^{\mathrm{psP}}$ and $P_t^{\mathrm{psG}}$ are the pumped-storage discharge and charge powers in time slot t, respectively; $P_t^{\mathrm{wind},\mathrm{cur}}$ and $P_t^{\mathrm{pv},\mathrm{cur}}$ denote the curtailed wind power and curtailed PV during time slot t; $P_t^{\mathrm{dc}}$ represents the HVDC export power in time slot t; and $U_t^{\mathrm{coal}}$ is the thermal-unit start-up flag, etc., denoting the reward–penalty coefficients for the corresponding costs.

In the objective function, positive costs denote system expenditures or penalties—such as fuel consumption and wind/PV curtailment—that increase total cost, whereas negative costs represent system revenues, e.g., earnings from power exports, that reduce total cost. Assigning high positive coefficients discourages undesirable operating states like load shedding and renewable spillage, while introducing negative-cost terms incentivizes the model to raise export volumes for additional revenue. By calibrating these coefficients appropriately, the optimization is steered to prioritize renewable consumption for export rather than local curtailment.

3.3. Constraints of the Model

The constraints in the capacity configuration model consist of renewable energy capacity limits, power-balance requirements, etc., as detailed below:

• The constraints of system power balance, as given in Equation (15).

  $$P_t^{\mathrm{dc}}+P_t^{\mathrm{load}}=P_t^{\mathrm{wind}}+P_t^{\mathrm{pv}}+P_t^{\mathrm{coal}}+P_t^{\mathrm{psG}}-P_t^{\mathrm{psP}}-P_t^{\mathrm{A}}$$

  (15)

  where $P_t^A$ denotes the total renewable energy curtailment at time t, comprising curtailed hydro, wind, and PV.
• The output constraints of wind power generation and PV, as given in Equation (16).

  $$\left\{\begin{array}{l}0\le P_t^{\mathrm{wind}}\le P_{\max }^{\mathrm{wind}}\\ 0\le P_t^{\mathrm{pv}}\le P_{\max }^{\mathrm{pv}}\\ P_t^{\mathrm{wind},\mathrm{cur}}=P_{\max }^{\mathrm{wind}}-P_t^{\mathrm{wind}}\\ P_t^{\mathrm{pv},\mathrm{cur}}=P_{\max }^{\mathrm{pv}}-P_t^{\mathrm{pv}}\end{array}\right.$$

  (16)

  where $P_{\max }^{\mathrm{wind}}$ and $P_{\max }^{\mathrm{pv}}$ denote the maximum active-power outputs of the wind and PV, respectively.
• The constraints of thermal-unit output and ramp-rate, as given in Equation (17).

  $$\left\{\begin{array}{c}{P}_{\min }^{\mathrm{coal}}\le P_t^{\mathrm{coal}}\le P_{\max }^{\mathrm{coal}}\\ -V_t^{\mathrm{down}}\Delta t\le P_{t+1}^{\mathrm{coal}}-P_t^{\mathrm{coal}}\le V_t^{\mathrm{up}}\Delta t\end{array}\right.$$

  (17)

  where $V_t^{up}$ and $V_t^{down}$ are the maximum upward and downward ramp rates of thermal at time t, respectively.
• The output constraints of the pumped-storage unit, as given in Equation (18).

  $$\left\{\begin{array}{l}{B}_t^{\mathrm{psG}}{P}_{\min }^{\mathrm{psG}}\le P_t^{\mathrm{psG}}\le B_t^{\mathrm{psG}}{P}_{\max }^{\mathrm{psG}}\\ B_t^{\mathrm{psP}}{P}_{\min }^{psP}\le P_t^{\mathrm{psP}}\le B_t^{\mathrm{psP}}{P}_{\max }^{\mathrm{psP}}\\ B_t^{\mathrm{psG}}+B_t^{\mathrm{psP}}\le 1\end{array}\right.,B_t^{\mathrm{psG}},B_t^{\mathrm{psP}}\in \left\{0,1\right\}$$

  (18)

  where $B_t^{psG}$ and $B_t^{psP}$ are binary variables, taking values of 0 or 1, which indicate whether the pumped-storage unit is operating in generation or pumping mode. These two operational states are mutually exclusive. The unit cannot simultaneously generate and absorb power. When $B_t^{psG}$ = 1 and $B_t^{psP}$ = 0, the unit is in the generation mode and discharges stored energy to the grid; conversely, when $B_t^{psG}$ = 0 and B = 1, it operates in the pumping mode, consuming electricity to store potential energy by elevating water. The combination $B_t^{psG}$ = $B_t^{psP}$ = 0 represents the idle state, while $B_t^{psG}$ = $B_t^{psP}$ = 1 is physically infeasible and therefore restricted by the constraint $B_t^{\mathrm{psG}}+B_t^{\mathrm{psP}}\le 1$. $P_{\min }^{psG}$ and $P_{\max }^{psG}$ are the minimum and maximum power outputs in generation mode; $P_{\min }^{psP}$ and $P_{\max }^{psP}$ are the minimum and maximum power inputs in pumping mode;
• The water-balance constraints of the pumped-storage unit, as given in Equation (19).

  $$\left\{\begin{array}{c}{V}_{t+1}^u=V_t^u+(Q_t^{\mathrm{psP}}-Q_t^{\mathrm{psG}})\Delta t\\ V_{t+1}^d=V_t^d+(Q_t^{\mathrm{psG}}-Q_t^{\mathrm{psP}})\Delta t\end{array}\right.$$

  (19)

  where $V_t^u$ and $V_{t+1}^u$ are the upper-reservoir storage volumes at time t and t + 1, respectively; $V_t^d$ and $V_{t+1}^d$ are the lower-reservoir storage volumes at time t and t + 1, respectively; $Q_t^{\mathrm{psG}}$ and $Q_t^{\mathrm{psP}}$ are the flow rates through the pumped-storage units in generation and pumping modes at time t, respectively.
• The initial–final water-level balance constraints of the pumped-storage unit, as given in Equation (20).

  $$\left\{\begin{array}{l}{V}_{\min }^u\le V_t^u\le V_{\max }^u\hfill \\ V_{\min }^d\le V_t^d\le V_{\max }^d\\ V_0\le V_T\end{array}\right.$$

  (20)

  where $V_{\min }^u$ and $V_{\max }^u$ are predefined parameters representing the minimum and maximum operational storage capacity of the upper reservoir. They define the lower and upper bounds for $V_t^u$ and are measured in cubic meters. $V_{\min }^d$ and $V_{\max }^d$ are predefined parameters representing the minimum and maximum operational storage capacity of the downstream reservoir. To ensure the physical safety and operational feasibility of the pumped-storage hydropower station, the stored volume cannot exceed the maximum effective capacity, nor can it fall below the minimum level required for safe operation. $V_0$ is the initial volume at hour 0 and $V_T$ is the water volume in the reservoir at the final hour of the simulation period. This constraint ensures that the water volume at the end of the scheduling horizon is close to its initial level, thereby maintaining the continuity of reservoir operation and avoiding long-term bias in storage utilization.
• The output-to-water conversion efficiency constraints of the pumped-storage unit, as given in Equation (21).

  $$\left\{\begin{array}{c}{P}_t^{\mathrm{psG}}=\rho g{H}_{h,t}{Q}_{h,t}{\eta }_h\\ P_t^{\mathrm{psP}}=\rho g{H}_{p,t}{Q}_{p,t}/{\eta }_p\end{array}\right.$$

  (21)

  where ${\eta }^{psG}$ and ${\eta }^{psP}$ are the generation and pumping efficiencies of pumped-storage, respectively; ${\rho }^{\mathrm{w}}$ is the density of water; and h is the average hydraulic head of the pumped-storage plant.
• The constraints of HVDC export power, as given in Equation (22).

  $$0\le P^{\mathrm{dc}}\le P_{\max }^{\mathrm{dc}}$$

  (22)

  where $P_{\max }^{dc}$ denotes the maximum transmission capacity of the HVDC transmission. In this paper, the power exported via HVDC is delivered in a stepped manner.
• The constraints of HVDC utilization duration, as given in Equation (23).

  $$\left\{\begin{array}{c}\begin{array}{l}{\displaystyle \sum _t^{d=t-\tau }}{v}_d^{\mathrm{dc}}\le 1,\forall t>\tau \\ {\nu }_t^{\mathrm{dc}}=0,\forall t\le \tau \end{array}\\ \begin{array}{l}\mid P_t^{\mathrm{dc}}-P_{t-1}^{\mathrm{dc}}\mid \le {\nu }_t^{\mathrm{dc}}{P}_{\max }^{\mathrm{dc}},\forall t\\ (\sum P_t^{\mathrm{dc}}/P_{\max }^{\mathrm{dc}})\times 8760\ge h_{\min }\end{array}\end{array}\right.$$

  (23)

  where ${\nu }_t^{dc}$ is a binary variable where ${\nu }_t^{\mathrm{dc}}$ = 1 if the HVDC transmission power is adjusted at hour t, and ${\nu }_t^{\mathrm{dc}}$ = 0 otherwise. $\tau$ denotes the minimum required duration for which the HVDC power must remain constant. These variables are introduced to represent the operational constraint that large-capacity HVDC systems cannot frequently adjust their power setpoints due to stability and coordination requirements. Equation (22) ensures that the power schedule is allowed to change at most once during the specified minimum adjustment period. Additionally, according to relevant national policies and existing research, the annual utilization duration of the HVDC corridor is set to 4400 h. This value corresponds to an annual utilization rate of approximately 50% (8760 × 0.5 ≈ 4380 h) and is consistent with planning targets reported in the recent literature.
• The constraint of HVDC power ramping, as given in Equation (24).

  $$R_{\mathrm{D}}^{\mathrm{dc}}\le P_t^{\mathrm{dc}}-P_{t-1}^{\mathrm{dc}}\le R_{\mathrm{U}}^{\mathrm{dc}},\forall t$$

  (24)

  where $R_{\mathrm{D}}^{dc}$ and $R_{\mathrm{U}}^{dc}$ denote the maximum upward and downward ramp rates of the HVDC export power, respectively.
• The constraints of $f_{RPS}$ and $f_{AEP}$, as given in Equation (25).

  $$\left\{\begin{array}{l}{f}_{RPS,\min }\le f_{RPS}\le 1\\ 0\le f_{AEP}\le f_{AEP,\max }\end{array}\right.$$

  (25)

  where $f_{RPS,\min }$ denotes the minimum acceptable reliability level, which is set to 0.99 in this study; this constraint ensures that the system’s supplied energy satisfies a minimum reliability requirement. $f_{AEP,\max }$ denotes the maximum allowable renewable generation curtailment rate, which is set to 0.05 in this study; this constraint limits the renewable energy curtailment within an acceptable range to enhance energy utilization efficiency.

3.4. Solution Method

The proposed capacity configuration framework is formulated as a bilevel optimization model. The lower-level operational optimization model is characterized by high dimensionality and multiple integer decision variables. Its objective function and operational constraints are linear in form, including generation dispatch, unit commitment, and the water-balance equations of the pumped-storage system. Therefore, the lower-level problem can be categorized as a typical mixed-integer linear programming (MILP) problem.

In contrast, the upper-level planning model introduces nonlinear components. In particular, the investment cost calculation involves a nonlinear capital recovery factor φ (r, n), which makes the overall framework a mixed-integer nonlinear programming (MINLP) problem. To facilitate computation, it is assumed that the output of each unit remains constant within each time interval (1 h), and the start-up and shut-down transitions are instantaneous.

For the solution strategy, the problem is addressed through a decomposition-based approach, where the upper-level nonlinear problem is solved via a Grid Search procedure that iteratively proposes candidate capacity configurations. For each candidate configuration, the lower-level MILP is solved exactly using the commercial solver Gurobi, yielding the optimal annual operational cost. This cost is then passed back to the upper level as a fitness value, which guides the heuristic search process toward the global optimum.

The optimization models were implemented in MATLAB R2020a using YALMIP and solved by Gurobi 11.0. Solver parameters were configured with a 3600 s time limit, MIP gap = 0.03, MIPGapAbs = 1000, Method = 2, MIPFocus = 1, Cuts = 1, Presolve = 2, and Heuristics = 0.2. These settings ensured a good trade-off between accuracy and computational efficiency for hourly resolution MILP problems. All simulations were performed on an Intel r7-8845H CPU with 16 GB RAM.

4. Case Study

The data used in this case study, including the wind/PV generation profiles, load curves, and economic parameters, are configured to represent the typical characteristics of energy systems in the Shago desert region in Northwest China. The renewable generation series are simulated based on public meteorological data to reflect the region’s resource abundance and variability, while load and cost data are compiled from official statistics, industry reports, and the relevant literature. This approach ensures the representativeness of the case study data and the general applicability of our findings.

To verify the effectiveness of the proposed capacity configuration method, simulations are carried out in MATLAB. The main parameters of the capacity configuration model are shown in

Table 1. Main parameters of the capacity configuration model.

Items	Values
Thermal power installations	1320 MW
Thermal minimum technical output	10%
Thermal unit ramp rate	80% h−1
Pumped-storage installed installations	1400 MW
HVDC transmission capacity	4000 MW
Unit investment cost ci (wind/PV/thermal/pumped-storage)	4.00/3.30/4.00/7.85 million CNY MW−1
Discount rate r (wind–solar–thermal/pumped-storage)	8%/6.5%
Lifetime n (wind–solar–thermal/pumped-storage)	20 years/40 years
Coal fuel cost Cfuel	130 CNY·MWh−1
Curtailment penalty cost Ccurt	200 CNY·MWh−1
Carbon cost per ton $C_{{\mathrm{CO}}_2}$	70 CNY·tCO2−1
Carbon emission factor ${\alpha }_{{\mathrm{CO}}_2}$	0.5 tCO2·MWh−1

.

Accounting for the uncertainties of wind and solar resources, this paper develops a 600 MW wind turbine model and a 2510 MW PV model that emulate real-world wind and PV output. The wind–solar capacity is then determined by varying a scaling factor. The wind and solar power output in this paper is shown in Figure 2.

Figure 2a shows the annual generation profile of a wind farm. Notable output gaps occur in the second and third quarters, demonstrating that wind power exhibits not only short-term variability but also sustained, cyclical shortfalls. Figure 2b,c present the annual and weekly generation patterns of PV. PV output is strongly influenced by seasonal variations, with notable shortfalls in the first and fourth quarters. At the same time, PV generation follows the day–night cycle, displaying pronounced intermittency and limited stability.

4.1. Analysis of the Lower-Level Optimization Results

To verify the effectiveness of the proposed lower-level optimization model, a baseline case with fixed wind, PV, thermal, and pumped-storage capacities was simulated. The model was run for a full year with an hourly resolution, and four typical days representing spring, summer, autumn, and winter were selected to reflect seasonal differences in load demand and renewable generation patterns. The results of typical days are shown in Figure 3.

Figure 3 shows the typical daily dispatch results for spring, summer, autumn, and winter. The outcomes indicate that, due to significant variations in wind and PV output across different seasons, thermal units and energy storage effectively serve as supporting power sources during periods of low renewable generation to meet the delivery requirements of the transmission profile. The transmission curves derived from the proposed method can accommodate the HVDC export needs of renewable energy bases under diverse wind and PV output fluctuations, achieving the desired performance and proving suitable for HVDC export planning of renewable energy bases in various environments.

As shown in Figure 3, the charging and discharging states of the pumped-storage system are strictly mutually exclusive—at any given moment, it operates solely in either pumping or generating mode. Over the entire cycle, the total volume of water pumped equals that released, maintaining overall water balance. Regarding renewable energy output, PV generation peaks at noon, accurately reflecting the “high at midday, low in morning and evening” resource characteristic of Shago desert regions. Wind power, on the other hand, concentrates its high output during night-time and early morning hours, forming a day–night complementarity with PV and jointly ensuring a smooth external transmission profile.

4.2. Capacity Configuration Results for IETS Under the Lowest Delivered-Electricity Prices Comparative Analysis

To rigorously evaluate the effectiveness of the proposed bilevel optimization framework (M1), a widely recognized benchmark method, the Typical Day Clustering Method (M2) [34], is introduced for comparison. This method simplifies the annual operational profile into 50 representative days selected via K-means clustering, and co-optimizes capacity and operation within a single-level model. The performance of both methods is evaluated across four wind-capacity scenarios: 1200 MW, 1500 MW, 1800 MW, and 2100 MW. The key results are summarized and compared in

Table 2. Comparison of Optimization Results from the Proposed Method and the Typical Day Method.

Method	Wind Power Capacity (MW)	PV Capacity (MW)	Lowest Price (CNY MWh−1)	HVDC-Exported Energy (Billion kWh)	Thermal Output (MW)	fRPS	fAEP
M1	1200	3865	0.3788	17.577	896	99.29%	1.24%
M2		4279	0.3831	18.214	931	99.34%	1.71%
M1	1500	3580	0.3826	17.592	942	99.32%	1.62%
M2		4064	0.3881	18.230	967	99.38%	2.13%
M1	1800	3265	0.3866	17.606	987	99.35%	2.07%
M2		3874	0.3939	18.244	1012	99.39%	2.39%
M1	2100	3000	0.3908	17.622	1035	99.39%	2.23%
M2		3713	0.4001	18.261	1052	99.43%	2.51%

.

The comparison results presented in the revised Table 2, clearly show that our proposed method achieves lower levelized electricity costs across all test scenarios. This advantage arises from the higher temporal fidelity of our model, which preserves the full 8760 h chronological resolution and thus captures the realistic weather patterns and long-term energy storage dynamics that are typically lost in clustering-based simplifications. Consequently, the proposed framework enables a more accurate and economically efficient system design than the benchmark method.

Table 3. Delivered-electricity price under different wind–PV capacity.

Wind Power Capacity (MW)	PV Capacity (MW)	λinv (CNY MWh−1)	λop (CNY MWh−1)	λtrans (CNY MWh−1)	λtotal (CNY MWh−1)
1200	3865	0.2564	0.0587	0.0637	0.3788
	3580	0.2485	0.0672	0.0637	0.3794
	3265	0.2398	0.0771	0.0637	0.3806
	3000	0.2324	0.0860	0.0637	0.3921
1500	3865	0.2665	0.0546	0.0637	0.3848
	3580	0.2586	0.0603	0.0637	0.3826
	3265	0.2499	0.0700	0.0637	0.3835
	3000	0.2425	0.0786	0.0637	0.3848
1800	3865	0.2766	0.0534	0.0637	0.3937
	3580	0.2687	0.0565	0.0637	0.3889
	3265	0.2600	0.0629	0.0637	0.3866
	3000	0.2526	0.0714	0.0637	0.3877
2100	3865	0.2867	0.0523	0.0637	0.4027
	3580	0.2788	0.0555	0.0637	0.3980
	3265	0.2701	0.0588	0.0637	0.3926
	3000	0.2627	0.0644	0.0637	0.3908

summarizes the delivered-electricity price results obtained from the capacity optimization model under different wind and PV capacity combinations. The matrix provides the precise numerical outcomes that form the basis for further visualization.

As shown in Figure 4, on the high-PV side, excessive PV deployment leads to increased renewable curtailment and stranded investment, increasing the delivered-electricity price. On the low-PV side, inadequate PV capacity forces the system to rely heavily on thermal generation, raising fuel and carbon costs. Consequently, there exists an optimal solar capacity range where investment and fuel costs are balanced: it avoids the resource waste of overinvestment while cutting the operating and carbon costs tied to thermal dependence.

4.3. Impact of HVDC Transmission Hours on Optimal Capacity Configuration and Electricity Price

To investigate the impact of HVDC transmission utilization constraints on system optimization, this study examines the results by varying the annual utilization hours from 4000 h to 5000 h in 200 h intervals, with the wind capacity held constant at 1800 MW. The optimization results, including optimal PV capacity, minimum delivered-electricity price, HVDC-exported energy, power supply reliability rate, and renewable generation curtailment rate are detailed in

Table 4. Optimization Results under Different HVDC Utilization Constraints (Wind Power: 1800 MW).

HVDC Transmission Utilization (h)	PV Capacity (MW)	Lowest Price (CNY MWh−1)	HVDC-Exported Energy (Billion kWh)	fRPS	fAEP
4000	2850	0.3752	16.650	99.34%	2.12%
4200	3080	0.3813	17.128	99.35%	2.09%
4400	3265	0.3866	17.606	99.35%	2.07%
4600	3520	0.3955	18.084	99.37%	2.04%
4800	3785	0.4038	18.562	99.38%	2.00%
5000	3975	0.4090	19.871	99.38%	1.98%

.

The data presented in Table 4 and Figure 5 demonstrates a key trade-off within the IETS. As the HVDC transmission annual utilization requirement is progressively increased from 4000 h to 5000 h, the optimal PV capacity expands significantly from 2850 MW to 3975 MW. This is because the stricter utilization mandate necessitates a larger renewable energy base to guarantee sufficient export power. Concurrently, the minimum delivered-electricity price rises steadily from 0.3750 CNY MWh⁻¹ to 0.4090 CNY MWh⁻¹. This price increase is primarily driven by the higher annualized investment cost associated with the expanded PV capacity.

4.4. Carbon-Reduction Benefits Arising from the Imposition of Penalty Costs on Thermal Power Generation

To investigate the impact of carbon pricing under different wind installation levels, this paper compares the thermal output and the PV capacity between the “with-carbon-price” and “no-carbon-price” scenarios, assuming HVDC transmission hours of 4400 h and wind capacities of 1200 MW, 1500 MW, 1800 MW, and 2100 MW. Results without carbon price are given in Table 1, those with carbon price in

Table 5. Optimization results of capacity configuration model (with-carbon).

Wind Power Capacity (MW)	PV Capacity (MW)	Lowest Price (CNY MWh−1)	HVDC-Exported Energy (Billion kWh)	Thermal Output (MW)	fRPS	fAEP
1200	3935	0.3933	17.478	843	99.24%	0.97%
1500	3780	0.3973	17.493	902	99.28%	1.16%
1800	3480	0.4014	17.507	937	99.35%	1.43%
2100	3185	0.4056	17.521	964	99.37%	1.72%

, and the comparative overview is depicted in Figure 6.

As shown in Figure 6, under the same wind power capacity, the introduction of a carbon price can further suppress thermal power output. For example, with a wind power installed capacity of 2100 MW, the annual thermal power output under the with-carbon scenario decreases by 0.1% compared to under the no-carbon scenario. Moreover, the substitution effect of carbon pricing on thermal power becomes more pronounced at high-wind power-penetration rates. This indicates that carbon pricing can create synergies with renewable energy development, jointly driving the system toward low-carbon transformation.

To evaluate the long-term impact and stability of the carbon pricing mechanism on the system’s optimal configuration, a detailed sensitivity analysis was conducted, focusing on the optimal capacity configuration and the delivered-electricity price.

Initially, the resulting optimal PV capacity, thermal output, and minimum electricity price were investigated across varying levels of carbon cost per ton. These results are presented in

Table 6. Optimization Results of Capacity Configuration under Different Carbon Cost per Ton $C_{{\mathrm{CO}}_2}$.

${\mathit{C}}_{\mathbf{C}{\mathbf{O}}_2}$ (CNY·tCO2−1)	PV Capacity (MW)	Lowest Price (CNY MWh−1)	Thermal Output (MW)	fRPS	fAEP
50	3420	0.3972	951	99.35%	1.62%
60	3455	0.3994	944	99.35%	1.52%
70	3480	0.4014	937	99.35%	1.43%
80	3510	0.4035	929	99.34%	1.33%
90	3540	0.4053	922	99.34%	1.24%
100	3565	0.4075	915	99.34%	1.15%

. Subsequently, a further sensitivity analysis was performed to examine the influence of the carbon emission factor on the overall optimal system configuration, with the outcomes detailed in

Table 7. Optimization results of capacity configuration under different carbon emission factors ${\alpha }_{{\mathrm{CO}}_2}$.

${\mathit{\alpha }}_{\mathbf{C}{\mathbf{O}}_2}$ (tCO2·MWh−1)	PV Capacity (MW)	Lowest Price (CNY MWh−1)	Thermal Output (MW)	fRPS	fAEP
0.3	3395	967	0.3950	99.35%	1.75%
0.4	3440	944	0.3985	99.35%	1.52%
0.5	3480	937	0.4014	99.35%	1.43%
0.6	3520	925	0.4042	99.34%	1.38%
0.7	3565	921	0.4071	99.34%	1.26%

.

The results from the sensitivity analyses in Table 6 and Table 7 collectively demonstrate the stability and active response of the optimal capacity configuration to carbon pricing. As shown in Table 6, when $C_{{\mathrm{CO}}_2}$ increases by 100% (from 50 to 100 CNY·tCO₂⁻¹), the optimal PV capacity only increases by 4.24%, and the lowest delivered-electricity price increases by a marginal 2.59%. As shown in Table 7, when the carbon emission factor ${\alpha }_{{\mathrm{CO}}_2}$ increases by 133.3% (from 0.3 to 0.7 tCO₂·MWh⁻¹), the optimal PV capacity increases by only 5.01%, and the lowest delivered-electricity price increases by 3.06%. The low sensitivity indicates that the capacity configuration is stable and robust in the face of changes in carbon pricing policies, demonstrating that investment decisions are not dramatically affected by such variations.

5. Discussion

This paper presents an optimal capacity configuration framework for IETSs and the main findings are summarized as follows.

A bilevel framework is presented for co-optimizing wind–PV capacity and dispatch under uncertainty. It aims to minimize the electricity price and operating cost while satisfying curtailment and security limits, and captures plant flexibility through thermal commitment and pumped-storage bidirectionality. Moreover, carbon pricing and minimum HVDC utilization are incorporated to align generation and transmission policies.

Case studies show that the proposed strategy successfully identifies the optimal PV capacity for given wind power levels. Simulation results further indicate that increasing the annual utilization hours of the HVDC transmission allows a higher renewable capacity while reducing curtailment. Moreover, introducing a carbon pricing policy increases thermal costs, limits thermal output, and thereby promotes renewable integration.