Analysis of Renewable Energy Absorption Potential via Security-Constrained Power System Production Simulation

Abstract

The increasing penetration of renewable energy sources presents significant challenges for power system stability and operation. Accurately assessing renewable energy absorption capacity is essential to ensuring grid reliability while maximizing renewable integration. This paper proposes a security-constrained sequential production simulation (SPS) framework, which incorporates grid voltage and frequency support constraints to provide a more realistic evaluation of renewable energy absorption capability. Additionally, hierarchical clustering (HC) based on dynamic time warping (DTW) and min-max linkage is employed for temporal aggregation (TA), significantly reducing computational complexity while preserving key system characteristics. A case study on the IEEE 39-bus system, integrating wind and photovoltaic generation alongside high-voltage direct current (HVDC) transmission, demonstrates the effectiveness of the proposed approach. The results show that the security-constrained SPS successfully prevents overvoltage and frequency deviations by bringing additional conventional units online. The study also highlights that increasing grid demand, both locally and through HVDC export, enhances renewable energy absorption, though adequate grid support remains crucial.

1. Introduction

Driven by the global shift towards decarbonization to meet climate goals, global renewable capacity additions in 2023 reached 473 GW, accounting for 86% of all electricity capacity additions, according to IRENA [1]. The share of renewable energy in global electricity capacity rose to 43%, up from 28.2% in 2014. However, the intermittent and unpredictable nature of renewable sources makes their integration into power systems more complex. This leads to the share of renewable energy in electricity production lagging behind, standing at only 29.1%, compared to 22.2% in 2014. While progress has been made in increasing renewable capacity, the integration of intermittent sources into power systems remains a significant challenge. As such, understanding renewable energy absorption capacity is crucial for achieving the energy goals set forth at COP28.

International experience with the curtailment of wind and solar energy in bulk power systems has been reviewed in [2], highlighting that renewable energy absorption is influenced by various factors, including grid infrastructure, power system stability, market structure, and technological advancements. In [3], artificial intelligence algorithms were employed to capture the complex impact of these factors on absorption capacity. The study developed the quantum harmony search optimized modified discounted mean squared forecast error combined forecast model to predict renewable energy absorption potential under different scenarios. In [4,5], EnergyPLAN was used to simulate the energy system, incorporating technologies across the electricity, heat, and transport sectors. Three scenarios with different shares of renewable energy in electricity demand were set up to assess renewable energy absorption capacity.

The intermittent and unpredictable nature of renewable energy poses significant challenges to real-time supply–demand balancing, ultimately leading to the inability to fully absorb renewable generation. To reflect this key factor affecting renewable energy absorption, a logic-based method was proposed in [6], based on the idea that each controllable resource, such as thermal power, pumped storage, and grid interconnection, adjusts to maintain supply–demand balance. The total non-controllable imbalance was allocated to each controllable resource while ensuring compliance with their operating limits. A minute-based economic dispatch model was employed in [7] to capture the effects of solar PV intermittency. The study defines the PV hosting capacity of a transmission network as the maximum solar PV capacity that can be connected to the system without requiring significant upgrades to its circuits to ensure stable operation. However, real-time grid balancing is constrained not only by the operating limits of controllable resources but also by transmission capacity. To address this, a power flow analysis method that considers transmission congestion was proposed in [8] to determine renewable power curtailment, offering a measure of the grid’s renewable energy absorption capacity.

The integration of renewable energy, replacing conventional synchronous units, leads to a decline in system inertia and worsens frequency response under external disturbances. To address these issues, the commitment strategy of conventional synchronous units (e.g., thermal and hydro) has become crucial. Frequency-related constraints—such as the rate of change of frequency (RoCoF), frequency deviation at quasi-steady-state, and maximum transient frequency deviation—are introduced into traditional unit commitment models. These constraints are typically derived from a uniform frequency response model, which is a reduced representation of the full-order frequency response model [9,10,11,12,13]. By applying the uniform frequency response model, study [14] introduces a new transient frequency security index and establishes quantitative relationships between this index and renewable energy absorption capacity. Similarly, study [15] translates the frequency constraint into a kinetic energy constraint by analyzing active energy changes during disturbances, and derives an analytical expression for the maximum renewable energy penetration, which is used to measure renewable energy absorption capacity. The integration of renewable energy also introduces voltage stability challenges in the power grid due to its lack of reactive power support capability. Study [16] established a quantitative transient stability assessment framework for power systems with high renewable energy penetration. By adjusting the penetration level, the study evaluates the renewable energy absorption capacity that ensures voltage stability. Study [17] evaluates renewable energy absorption capacity by adjusting the dispatch power of committed synchronous generators under multiple load scenarios, considering both frequency nadir constraints and dynamic voltage security constraints. Study [18] constructs an optimization model for renewable energy absorption capacity in an AC-DC hybrid grid, aiming to maximize renewable energy penetration and minimize network losses, while considering both static and transient stability constraints. For simplification, only the typical operational scenario of the power grid was considered.

Assessment in a single scenario fails to capture the temporal characteristics of renewable generation and the regulation capability of the system, leading to inaccurate results. To address these challenges, study [19] proposes a multi-scenario renewable energy absorption capacity assessment method based on an attention-enhanced time convolutional network. Study [20] uses a multi-time-horizon optimization of unit commitment and economic dispatch to examine the hosting capacity of renewables. Specifically, one sampled typical week from each month in the horizon is modeled, and the results are then applied to the other weeks. Study [21] presents a renewable energy accommodation capability model based on standard time-series production simulation. Moreover, a week-by-week optimization strategy, combined with the hybrid particle swarm optimization algorithm, was used to achieve the rapid solution of the model. Similarly to capacity expansion planning (CEP), the assessment of renewable energy absorption capacity typically spans time periods of one year or more. Such extended time horizons substantially increase the computational complexity of the problem. To reduce the temporal resolution, time-series aggregation methods are widely employed to identify representative periods that preserve key temporal and operational characteristics. Typically, the original time series is partitioned into K operationally independent periods, and multiple similar periods are aggregated and represented by a single typical period. In recent years, clustering-based techniques—such as k-means clustering, k-medoids clustering, and hierarchical clustering—have gained popularity for this purpose [22,23,24,25,26].

While prior studies often address frequency or voltage constraints independently, this paper presents a unified security-constrained sequential production simulation (SPS) framework that simultaneously considers both types of security constraints. To enhance computational efficiency, the framework incorporates a hierarchical clustering (HC) method based on dynamic time warping (DTW) and min-max linkage for time-series aggregation, effectively reducing the problem scale while preserving essential temporal and operational characteristics. The main contributions of this work are as follows:

(1)

The voltage and frequency support requirements for renewable energy absorption are translated into a set of constraints on the operational status of synchronous generators.

(2)

A security-constrained SPS framework is developed to assess the renewable energy absorption capacity while considering both frequency and voltage security constraints.

(3)

An HC based on DTW and min-max linkage is employed for TA, effectively reducing the computational complexity of the security-constrained SPS while preserving key system characteristics.

2. Grid Support Requirements for Renewable Energy Absorption

Renewable energy sources (RESs) offer significant environmental and economic benefits. However, their integration into the power grid poses challenges due to their variability, intermittency, and lack of inherent stability support. Ensuring effective renewable energy absorption requires sufficient grid support, particularly in voltage and frequency regulation, to maintain system reliability and efficiency.

2.1. Voltage Support Requirements

Voltage support requirements refer to the essential grid capabilities needed to maintain stable voltage levels in a power system with a high share of renewable energy. To evaluate those grid capabilities, several key indicators are considered, among which the short-circuit ratio (SCR) is one of the most critical [27]. Accordingly, the voltage support requirement for renewable energy absorption can be expressed as the following constraint on the SCR [27]:

$$S_{\mathrm{cr},j}={\displaystyle \frac{S_{\mathrm{sc},j}}{P_{\mathrm{N},j}}}={\displaystyle \frac{V_{\mathrm{N},j}^{\ast }{I}_{\mathrm{f},j}^{\ast }{S}_{\mathrm{base}}}{P_{\mathrm{N},j}}}\ge S_{\mathrm{cr},j}^{\min }$$

(1)

where $S_{\mathrm{cr},j}$ is the short-circuit ratio at bus j; $S_{\mathrm{cr},j}^{\min }$ is the required short-circuit ratio at bus j; $S_{\mathrm{sc},j}$ represents the short-circuit capacity at bus j; $P_{\mathrm{N},j}$ is the rated capacity at bus j, which corresponds to the rated power of the renewable energy unit if bus j is a renewable energy integration bus, or the rated power of the DC system if bus j is a DC converter bus; $V_{\mathrm{N},j}^{\ast }$ denotes the per-unit rated voltage at bus j; $I_{\mathrm{f},j}^{\ast }$ is the per-unit three-phase short-circuit current at bus j; and $S_{\mathrm{base}}$ represents the base value of capacity in per-unit terms.

For a given bus j, the parameters $V_{\mathrm{N},j}^{\ast }$, $P_{\mathrm{N},j}$, and $S_{\mathrm{base}}$ are given. According to Equation (1), the SRC $S_{\mathrm{sc},j}$ is exclusively determined by the short-circuit current $I_{\mathrm{f},j}^{\ast }$. To calculate $I_{\mathrm{f},j}^{\ast }$, synchronous generators in operation are modeled using their sub-transient impedance and sub-transient electromotive force (EMF), while loads are represented as constant impedance. The corresponding equivalent circuit diagram is shown in Figure 1.

Applying the principle of superposition, the three-phase short-circuit current at bus j can be expressed as:

$$I_{\mathrm{f},j}^{\ast }={\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}{y}_i{I}_{\mathrm{f},i-j}^{\ast }}$$

(2)

where $N_{\mathrm{G}}$ represents the total number of synchronous generators; $y_i$ denotes the operating status of synchronous generator i, where $y_i$ = 1 indicates the generator is online and $y_i$ = 0 indicates it is offline; and $I_{\mathrm{f},i-j}^{\ast }$ represents the short-circuit current supplied by synchronous generator i to bus j.

The short-circuit current supplied by synchronous generator i in service to bus j can be determined by short-circuiting the sub-transient EMFs of all other operating synchronous generators in the equivalent circuit depicted above. While the sub-transient EMFs of those generators are short-circuited, their sub-transient impedances remain connected, significantly influencing the equivalent impedance between synchronous generator i and bus j.

It is evident that $I_{\mathrm{f},i-j}^{\ast }$ is closely related to the operating status of synchronous generators in the power grid, exhibiting an implicit relationship. To facilitate computation, we define ${I^{\prime }}_{\mathrm{f},i-j}^{\ast }$, the short-circuit current contribution factor, as the short-circuit current supplied by synchronous generator i to bus j when it is the only generator in operation. This can be calculated by disconnecting all other operating synchronous generators in the equivalent circuit illustrated in Figure 1. Additionally, we introduce ${\alpha }_i$, a correction factor, which accounts for the impact of other in-service synchronous generators on the equivalent impedance between synchronous generator i and bus j. The correction factor is closely related to the number of online synchronous generators and, for a fixed unit count, is further influenced by the specific combination of operating units. As shown in Figure 2, for the IEEE 39-bus system with two generators online (including Generator 1), the correction factor for the short-circuit current contribution from Generator 1 to Bus 2 remains relatively consistent across different unit combinations. Therefore, it is reasonable to approximate the correction factor as a piecewise constant with respect to the number of online synchronous generators.

Thus, Equation (2) can be approximated as:

$$I_{\mathrm{f},j}^{\ast }\approx {\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}{y}_i{\alpha }_i{I^{\prime }}_{\mathrm{f},i-j}^{\ast }}$$

(3)

By substituting Equation (3) into Equation (1), the voltage support requirement for renewable energy absorption can be formulated as a constraint on the operational status of synchronous generators:

$${\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}{y}_i{\alpha }_i{I^{\prime }}_{\mathrm{f},i-j}^{\ast }}\ge {\displaystyle \frac{S_{\mathrm{cr},j}^{\min }{P}_{\mathrm{N},j}}{V_{\mathrm{N},j}^{\ast }{S}_{\mathrm{base}}}}$$

(4)

2.2. Frequency Support Requirements

RESs such as wind and solar rely on inverter-based interfaces and inherently lack physical rotational inertia. With the increasing integration of RESs into modern power systems, this lack of inertia makes the system more vulnerable to rapid frequency deviations following sudden power imbalances. Therefore, ensuring resilient frequency support is crucial for the stable and reliable operation of systems with high shares of renewable energy. The requirements for frequency support are commonly evaluated using key indicators extracted from the system’s dynamic frequency response following a power disturbance. Those indicators include the rate of change of frequency (RoCoF), frequency deviation at quasi-steady-state, and maximum transient frequency deviation. Figure 3 illustrates the typical frequency response of a high-RES system following a fault in an export transmission corridor, which results in surplus generation and consequently causes system over-frequency. As shown in Figure 3, the RoCoF reflects how rapidly the system frequency changes during the initial seconds after a major power disturbance, the maximum transient frequency deviation corresponds to the greatest deviation from the nominal frequency during the transient period, and the frequency deviation at quasi-steady-state represents the frequency level at which the system stabilizes after the transient phase.

Consider a large-scale power system where most conventional generating units are reheating steam turbines. Assuming a uniform network frequency, the system’s frequency response can be aggregated and modeled as an equivalent single-machine system with a centralized load [28,29], as illustrated in Figure 4.

From Figure 4, the transfer function between the frequency deviation $\mathrm{\Delta }f$ and the disturbance power $\mathrm{\Delta }{P}_{\mathrm{L}}$ can be derived as:

$${\displaystyle \frac{\mathrm{\Delta }f\left(s\right)}{\mathrm{\Delta }{P}_{\mathrm{L}}\left(s\right)}}=-{\displaystyle \frac{1+T_{\mathrm{R}}s}{2H{T}_{\mathrm{R}}{s}^2+\left(2H+D{T}_{\mathrm{R}}\right)s+D+\frac{1}{R}}}$$

(5)

where

$$H={\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}{y}_i{S}_i{H}_i}$$

(5a)

$$D={\displaystyle \sum _{j=1}^{N_{\mathrm{L}}}{D}_j{P}_j^{\mathrm{L}}}$$

(5b)

$${\displaystyle \frac{1}{R}}={\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}\frac{y_i{S}_i}{R_i}}$$

(5c)

In the equations: T_R is the reheating time constant; D denotes the system load damping factor; H is the system equivalent inertia constant; and R represents the system governor regulation coefficient. $N_{\mathrm{G}}$ denotes the total number of synchronous generators in the system, where y_i, S_i, H_i, and R_i correspond to the operational status, capacity, inertia, and governor regulation coefficient of the ith synchronous generator, respectively. $N_{\mathrm{L}}$ represents the set of system loads, where D_j and $P_j^{\mathrm{L}}$ denote the damping coefficient and active power of the jth load, respectively.

Sudden power disturbances can be approximated as a step input. By applying the Laplace transform, the time-domain expression of the frequency deviation can be obtained. Subsequently, RoCoF, frequency deviation at quasi-steady-state, and maximum transient frequency deviation can be formulated as (6) and (7). The detailed derivations can be found in references [28,29].

$$R_{\mathrm{CoF}}={\left.{\displaystyle \frac{\mathrm{d}\mathrm{\Delta }f\left(t\right)}{\mathrm{d}t}}\right|}_{t=0}=-{\displaystyle \frac{\mathrm{\Delta }{P}_{\mathrm{L}}}{2H}}\le R_{\mathrm{CoF}}^{\max }$$

(6)

$$\Delta f_{\mathrm{s}}=-{\displaystyle \frac{\mathrm{\Delta }{P}_{\mathrm{L}}}{D+1/R}}\le \mathrm{\Delta }{f}_{\mathrm{s}}^{\max }$$

(7)

$$\mathrm{\Delta }{f}_{\mathrm{peak}}=-{\displaystyle \frac{\mathrm{\Delta }{P}_{\mathrm{L}}}{D+1/R}}\left(1+\sqrt{1-{\zeta }^2}\beta {\mathrm{e}}^{-\zeta {\omega }_{\mathrm{n}}{t}_{\mathrm{peak}}}\right)\le \mathrm{\Delta }{f}_{\mathrm{peak}}^{\max }$$

(8)

where

$$\beta =\sqrt{{\displaystyle \frac{1-2T_{\mathrm{R}}\zeta {\omega }_{\mathrm{n}}+T_{\mathrm{R}}^2{\omega }_{\mathrm{n}}^2}{1-{\zeta }^2}}}$$

(8a)

$${\omega }_{\mathrm{n}}=\sqrt{{\displaystyle \frac{DR+1}{2H{T}_{\mathrm{R}}R}}}$$

(8b)

$$\zeta ={\displaystyle \frac{1}{2{\omega }_{\mathrm{n}}}}\left({\displaystyle \frac{1}{T_{\mathrm{R}}}}+{\displaystyle \frac{D}{2H}}\right)$$

(8c)

In the equations: $t_{\mathrm{peak}}$ is the time corresponding to the peak transient frequency deviation.

The expression of the maximum transient frequency deviation is inherently nonlinear. Several studies [9,10,11,12,13] have proposed approximate linear methods to address this complexity. In this work, we adopt the approach presented in [11], which assumes linear governor power regulation and derives an approximate analytical expression for the maximum transient frequency deviation constraint, as given in Equations (9) and (10). The detailed derivation and theoretical proof can be found in [11].

$$H/R\ge k_{\mathrm{peak}}$$

(9)

where $k_{\mathrm{peak}}$ is the unique solution derived from the following equation:

$$\mathrm{\Delta }{f}_{\mathrm{peak}}^{\max }=-{\displaystyle \frac{\mathrm{\Delta }{P}_{\mathrm{L}}}{D}}+{\displaystyle \frac{2k_{\mathrm{peak}}}{T_{\mathrm{R}}{D}^2}}\log \left({\displaystyle \frac{2k_{\mathrm{peak}}}{2k_{\mathrm{peak}}-T_{\mathrm{R}}D\mathrm{\Delta }{P}_{\mathrm{L}}}}\right)$$

(10)

By substituting Equation (5a) into Equation (6), Equation (5b,c) into Equation (7), and Equation (5a,c) into Equation (8), the frequency support requirement for renewable energy accommodation can be formulated as a set of constraints on the operational status of synchronous generators:

$${\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}{y}_i{S}_i{H}_i}\ge -{\displaystyle \frac{\mathrm{\Delta }{P}_{\mathrm{L}}}{2R_{\mathrm{CoF}}^{\max }}}$$

(11)

$${\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}\frac{y_i{S}_i}{R_i}}\ge -{\displaystyle \frac{\mathrm{\Delta }{P}_{\mathrm{L}}}{\mathrm{\Delta }{f}_{\mathrm{s}}^{\max }}}-D_{\mathrm{L}}{\displaystyle \sum _{j=1}^{N_{\mathrm{L}}}{P}_j^{\mathrm{L}}}$$

(12)

$$\left({\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}{y}_i{S}_i{H}_i}\right)\left({\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}\frac{y_i{S}_i}{R_i}}\right)\ge k_{\mathrm{peak}}$$

(13)

3. Security-Constrained Sequential Production Simulation for Assessing Renewable Energy Absorption Capacity

Standard SPS primarily focuses on power and energy balance constraints, ensuring that generation meets demand while optimizing economic and operational efficiency over a given time horizon. However, with the large-scale development and integration of renewable energy, it fails to explicitly account for the critical role of synchronous generators in maintaining grid security, particularly in terms of voltage and frequency support. To more accurately evaluate the renewable energy absorption capacity, this paper proposes a production simulation model incorporating voltage and frequency support constraints. Figure 5 illustrates the structure of the proposed security-constrained sequential production simulation model. The objective function minimizes both renewable energy curtailment cost and conventional generation cost. The basic constraints include conventional unit, renewable curtailment, and system operation constraints, as commonly formulated in the literature (see [21] for details). The proposed constraints, derived in Section 2.1 and Section 2.2, are newly introduced in this study.

3.1. Objective Function

For assessing renewable energy absorption capacity, the objective of the security-constrained SPS aims to maximize the utilization of renewable energy while ensuring system reliability and economic efficiency. This is achieved by minimizing the cost of renewable energy curtailment, along with reducing conventional generator production costs over the operational period. Specifically, the optimization objective is:

$$\min {\displaystyle \sum _{t=1}^{T_{\mathrm{simu}}}\left\{{\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}\left[a_i{\left(P_{i,t}^{\mathrm{G}}\right)}^2+b_i{P}_{i,t}^{\mathrm{G}}+c_i+C_{i,t}^{\mathrm{SU}}+C_{i,t}^{\mathrm{SD}}\right]}\right.}\left.+{\displaystyle \sum _{i=1}^{N_{\mathrm{R}}}\eta P_{i,t}^{\mathrm{R},\mathrm{CUR}}}\right\}$$

(14)

where $T_{\mathrm{simu}}$ represents the operational period for production simulation, which is typically set to one year (8760 h). $a_i$, $b_i$, and $c_i$ are the operation cost coefficients of conventional generator i; $P_{i,t}^{\mathrm{G}}$ is the power output of conventional generator i at time t; and $C_{i,t}^{\mathrm{SU}}$ and $C_{i,t}^{\mathrm{SD}}$ are auxiliary variables for the startup/shutdown costs of conventional generator i at time t. $N_{\mathrm{R}}$ represents the total number of renewable energy power stations, $\eta$ is the penalty cost per unit curtailed power, and $P_{i,t}^{\mathrm{R},\mathrm{CUR}}$ is the curtailed power of renewable energy power station i at time t.

3.2. Startup and Shutdown Cost Constraints

In the objective function, both startup and shutdown costs are included to capture the operational economics of conventional generators. However, these two cost terms are mutually exclusive at any given time step, as a generator cannot be started up and shut down simultaneously. To ensure that this exclusivity is properly enforced, the following constraints are introduced:

$$\left\{\begin{array}{l}{C}_{i,t}^{\mathrm{SU}}\ge c_i^{\mathrm{SU}}\left(y_{i,t}-y_{i,t-1}\right)\\ C_{i,t}^{\mathrm{SU}}\ge 0\end{array}\right.$$

(15)

$$\left\{\begin{array}{l}{C}_{i,t}^{\mathrm{SD}}\ge c_i^{\mathrm{SD}}\left(y_{i,t-1}-y_{i,t}\right)\\ C_{i,t}^{\mathrm{SD}}\ge 0\end{array}\right.$$

(16)

where $c_i^{\mathrm{SU}}$ and $c_i^{\mathrm{SD}}$ represent the startup cost and shutdown cost of conventional generator i, respectively; and $y_{i,t-1}$ and $y_{i,t}$ denote the operating status of synchronous generator i at time t − 1 and time t, with 1 signifying the generator is online and 0 indicating it is offline.

3.3. Conventional Generator Unit Constraints

(1)

Power Output Limits

Each generating unit has a limit on the power it can generate:

$$y_{i,t}{P}_i^{\mathrm{G},\min }\le P_{i,t}^{\mathrm{G}}\le y_{i,t}{P}_i^{\mathrm{G},\max }$$

(17)

where $P_i^{\mathrm{G},\min }$ and $P_i^{\mathrm{G},\max }$ are the minimum and maximum power limits of generator i, respectively.

(2)

Ramp Rate Limits

Generally, a power turbine cannot adjust its output instantaneously. Generating units have limits on how quickly they can increase or decrease output:

$$\left\{\begin{array}{l}{P}_{i,t+1}^{\mathrm{G}}-P_{i,t}^{\mathrm{G}}\le P_i^{\mathrm{G},\mathrm{UP}}\Delta t\\ P_{i,t}^{\mathrm{G}}-P_{i,t+1}^{\mathrm{G}}\le P_i^{\mathrm{G},\mathrm{DN}}\Delta t\end{array}\right.$$

(18)

where $P_{i,t}^{\mathrm{G}}$ and $P_{i,t+1}^{\mathrm{G}}$ are the power output of generator i at time t and t + 1; $P_i^{\mathrm{G},\mathrm{UP}}$ and $P_i^{\mathrm{G},\mathrm{DN}}$ are the ramp-up rate and ramp-down rate of generator i; and $\Delta t$ is the time step duration.

(3)

Minimum Up/Down Time Limits

Before shutting down, a generating unit has to stay online for a minimum duration $T_i^{\mathrm{G},\mathrm{ON}}$. Similarly, once shut down, it must stay offline for at least $T_i^{\mathrm{G},\mathrm{OFF}}$ consecutive time periods:

$${\displaystyle \sum _{k=t}^{t+T_i^{\mathrm{G},\mathrm{ON}}-1}\left(1-y_{i,k}\right)}\ge T_i^{\mathrm{G},\mathrm{ON}}\left(y_{i,t-1}-y_{i,t}\right)$$

(19)

$${\displaystyle \sum _{k=t}^{t+T_i^{\mathrm{G},\mathrm{OFF}}-1}{y}_{i,k}}\ge T_i^{\mathrm{G},\mathrm{OFF}}\left(y_{i,t}-y_{i,t-1}\right)$$

(20)

3.4. Renewable Energy Curtailment Constraints

To ensure that curtailment does not exceed available generation:

$$0\le P_{i,t}^{\mathrm{R},\mathrm{CUR}}\le P_{i,t}^{\mathrm{R},\mathrm{FORE}}$$

(21)

where $P_{i,t}^{\mathrm{R},\mathrm{FORE}}$ is the available generation power of renewable energy power station i at time t.

3.5. System Constraints

(1)

Spinning Reserve Constraint

To ensure system reliability, a certain amount of additional generation capacity must be maintained:

$${\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}\left(y_{i,t}{P}_i^{\mathrm{G},\max }-P_{i,t}^{\mathrm{G}}\right)}\ge \rho {\displaystyle \sum _{i=1}^{N_{\mathrm{L}}}{P}_{i,t}^{\mathrm{L}}}$$

(22)

where $\rho$ is the spinning reserve factor.

(2)

Power Balance Constraints

At each bus i, the net injected power must equal the sum of power flows on all lines connected to the bus. Using DC power flow:

$$P_{i,t}^{\mathrm{B},\mathrm{G}}-P_{i,t}^{\mathrm{B},\mathrm{L}}={\displaystyle \sum _{j\in {\mathrm{\Omega }}_{\mathrm{B}}}{B}_{ij}\left({\theta }_{i,t}-{\theta }_{j,t}\right)}$$

(23)

where $P_{i,t}^{\mathrm{B},\mathrm{G}}$ and $P_{i,t}^{\mathrm{B},\mathrm{L}}$ represent generated power and load demand at bus i at time t. $B_{ij}$ is the susceptance of the transmission line between buses i and j, ${\mathrm{\Omega }}_{\mathrm{B}}$ denotes the set of buses connected to bus i, and ${\theta }_i$ and ${\theta }_j$ are the voltage phase angles at buses i and j at time t.

(3)

Transmission Line Flow Limits

The absolute value of the power flow on each transmission line must not exceed its thermal capacity:

$$-P_{ij}^{\max }\le B_{ij}\left({\theta }_{i,t}-{\theta }_{j,t}\right)\le P_{ij}^{\max }$$

(24)

where $P_{ij}^{\max }$ is the maximum permissible power flow on the line between buses i and j.

(4)

Voltage Support Constraints

The voltage support constraints can be formulated by Equation (4), which has already been discussed in Section 2.1.

(5)

Frequency Support Constraints

As discussed in Section 2.2, the frequency support constraints can be described by Equations (11)–(13).

4. Temporal Aggregation-Based Solution for Security-Constrained Sequential Production Simulation

Although the security-constrained SPS accurately captures time-series characteristics of real-world power system operation, its long time horizon significantly expands the optimization model size, leading to high computational complexity. A natural solution to this challenge is to reduce the dimensionality in the time domain, such as through TA.

4.1. Hierarchical Clustering-Based TA

TA compresses long time-series data into a few typical periods. It reduces computation time significantly while preserving the model’s core time-varying characteristics. To achieve a more structured and efficient representation of time-series data, clustering is often employed. Compared to traditional feature-based classification methods, clustering does not rely on subjective experience. Instead, it uses mathematical techniques to identify similarities or differences in data. This enables natural grouping.

Among various clustering algorithms, k-means clustering and HC are two of the most widely used methods for TA. Unlike k-means, HC requires neither initial cluster centers nor a pre-specified number of clusters. Instead, it determines cluster merging or splitting based on distance metrics between data points or clusters, forming a clear hierarchical structure. This structure enables multi-scale analysis. It allows patterns to be examined at different levels of granularity, making it particularly useful for capturing complex temporal dynamics.

Similarity measurement is the foundation of HC, as well as all other clustering algorithms. For time-series data, Euclidean distance measurement may result in poor similarity assessments due to temporal misalignment. To address this issue, we consider DTW in this paper. DTW measures the similarity by computing the minimum cumulative distance between elements of two time series [25]. Given two time series $X=\left\{x_1,x_2,\dots ,x_m\right\}$ and $Z=\left\{z_1,z_2,\dots ,z_n\right\}$, let $\mathrm{\Delta }\left(x_i,z_j\right)$ represent the distance between element $x_i$ in X and element $z_j$ in Z. The cumulative distance matrix $D_{\mathrm{ist}}$ is constructed based on $\mathrm{\Delta }\left(x_i,z_j\right)$ as follows:

$$D_{\mathrm{ist}}\left(i,j\right)=\mathrm{\Delta }\left(x_i,z_j\right)+\min \left\{D_{\mathrm{ist}}\left(i-1,j\right),\right.\left.D_{\mathrm{ist}}\left(i,j-1\right),D_{\mathrm{ist}}\left(i-1,j-1\right)\right\}$$

(25)

The second term of Equation (25) accounts for the “warping” of the time axis, ensuring that similar patterns are aligned even if they occur at different time steps. The final DTW distance is the value at $D_{\mathrm{ist}}\left(m,n\right)$.

Besides that, another essential requirement of HC is determining how to measure the distance between clusters, known as the linkage criterion. In this paper, we employ the min-max linkage criterion, which identifies a point referred to as the cluster prototype [30]. Having a cluster prototype is beneficial as it aligns with the purpose of temporal aggregation—that is, selecting representative periods to replace the full time series.

Let $r\left({\mathrm{\Omega }}_i\right)$ denote the radius of the cluster Ω_i, defined as:

$$r\left({\mathrm{\Omega }}_i\right)=\underset{X\in {\mathrm{\Omega }}_i}{\min }\left(\underset{\begin{array}{l}Z\in {\mathrm{\Omega }}_i\\ Z\ne X\end{array}}{\max \mathrm{\Delta }\left(X,Z\right)}\right)$$

(26)

where $\mathrm{\Delta }\left(X,Z\right)$ represents the distance between point X and point Z within cluster Ω_i. The point X that minimizes (26) is designated as the prototype of cluster Ω_i. Consequently, the min-max linkage can be expressed as follows:

$$L_{\mathrm{inkage}}\left({\mathrm{\Omega }}_i,{\mathrm{\Omega }}_j\right)=r\left({\mathrm{\Omega }}_i\cup {\mathrm{\Omega }}_j\right)$$

(27)

Since HC does not require a predefined number of clusters, the optimal number of clusters can be determined by cutting the dendrogram using methods such as the Elbow Method. Once the final set of clusters is obtained, each cluster is represented by its cluster prototype, which serves as a representative day. The weight assigned to each representative day is equal to the number of points within the corresponding cluster.

4.2. Solution Framework

Based on HC with DTW and min-max linkage discussed above, the solution process for security-constrained SPS using TA is illustrated in Figure 6. The key steps are as follows:

(1)

Set wind and solar installed capacity.

(2)

Generate 8760 h time-series power output based on annual wind and solar generation characteristics.

(3)

Organize full-year data by structuring wind, solar, and load demand in a daily format.

(4)

Select representative days using HC with DTW and min-max linkage.

(5)

Perform security-constrained SPS for each representative day. If no renewable energy curtailment occurs, increase the renewable energy installation ratio and return to Step 2. Otherwise, statistically analyze renewable energy absorption.

(6)

Estimate annual renewable energy absorption by aggregating the weighted results of representative days.

5. Case Study

5.1. Case1: The IEEE 39-Bus System

The IEEE 39-bus system is a widely used benchmark model proposed by the IEEE Task Force on Benchmark Systems for Stability Controls [31]. This system was modified to test the proposed security-constrained SPS for assessing renewable energy absorption capability. Wind farms were integrated at Buses 3, 8, and 18, while photovoltaic (PV) stations were connected at Buses 4 and 16. Bus 2 was modified to include an HVDC transmission system for power export to external grids. The annual 8760 h time-series load data, coupled with the time-series output data for wind and photovoltaic generation, were sourced from [32] and are illustrated in Figure 7, Figure 8 and Figure 9. The dynamic simulation parameters used in this paper—such as generator and governor models—are derived from the official MATLAB (since R2024b) example [33], which is based on the IEEE Task Force specification.

(1)

Representative Day Selection

The proposed HC with DTW and min-max linkage was applied for temporal aggregation. Using the Elbow Method, the optimal number of clusters is determined as 18. Figure 10 shows the prototypes of each cluster and their assigned days. Different colors represent different clusters. Solid dots indicate the clustered days, and black-edged hollow dots represent the prototypes.

The core of temporal aggregation is to preserve the time-dependent features of the original time series. For renewable energy, this involves maintaining the distribution of generation levels and their corresponding frequencies of occurrence. This is typically represented by the duration curve of renewable energy output [26]. Figure 11 illustrates the wind duration curve for both original and clustered data. As seen in the figure, the proposed method (HC with DTW and min-max linkage) produces a wind duration curve that closely aligns with the original wind power output duration curve, especially in the lower output range. This is crucial for the accuracy of renewable energy absorption assessments, as it directly impacts the demand for grid reserve capacity.

To further evaluate the performance of each clustering method, the root mean square deviation (RMSE) between the clustered duration curves and the original duration curve was calculated, as shown in

Table 1. RMSE for each clustering method.

Methods	RMSE
HC with DTW and min-max linkage	0.0238
HC with ED and ward linkage	0.0486
k-means	0.0317

. The proposed method outperforms HC using Euclidean distance and ward linkage, as well as k-means, indicating that the proposed TA provides a better approximation.

(2)

Renewable Energy Absorption Assessment

After selecting the representative days, security-constrained sequential production simulations were performed for each typical day to determine renewable energy absorption. Then, the annual renewable energy absorption was calculated by weighting each typical day, as illustrated in Figure 12.

Table 2. RE absorption (p.u.) under different methods in the IEEE 39-bus system.

Month	The Available Generation Power of RES	Renewable Energy Absorption Using Standard SPS	Renewable Energy Absorption
			Security-Constrained SPS Without TA	Security-Constrained SPS with TA	Related Error
January	4955.22	4955.22	4939.04	4954.66	0.32%
February	5182.58	5182.58	5167.06	5174.17	0.14%
March	8094.10	8094.10	7835.93	8024.94	2.41%
April	7526.10	7526.10	7225.50	7441.94	2.99%
May	8143.92	8143.92	7976.64	8084.04	1.35%
June	8420.04	8420.04	8188.83	8329.42	1.72%
July	9297.89	9297.89	8874.79	9068.78	2.19%
August	7874.09	7874.09	7701.22	7860.44	2.07%
September	7557.15	7557.15	7262.77	7472.09	2.88%
October	4479.67	4479.67	4455.42	4479.67	0.55%
November	5083.13	5083.13	5077.67	5080.60	0.06%
December	5042.09	5042.09	5035.53	5042.09	0.13%
Annual	81,655.99	81,655.99	79,740.40	81,012.84	1.60%

presents a detailed comparison of renewable energy absorption results obtained from the security-constrained SPS with TA and without TA, covering both monthly and annual data. The relative errors between the two methods are also analyzed. For the case without TA, the security-constrained SPS over the full 8760 h horizon requires 7050.52 s of computation time and yields an annual renewable energy absorption of 79,740.40 p.u. In contrast, the TA-based approach completes in only 13.19 s, achieving a comparable annual absorption of 81,012.84 p.u. Calculations were performed using GUROBI 10.0.1 on a PC with Intel Core i5-13400 2.5 GHz CPU and 16 GB RAM. The relative error between the two methods is merely 1.60%, indicating that the TA-based method significantly enhances computational efficiency while preserving a high level of accuracy in the assessment results.

The security-constrained SPS entails additional conventional units online to fulfill the system’s voltage and frequency support requirements. This reduces the space available for renewable energy absorption, leading to curtailment. On the other hand, the standard SPS that only considers power and energy balance may fail to ensure the grid’s voltage and frequency safety, potentially resulting in an inaccurate reflection of renewable energy absorption.

Taking day 239 as an example, the load, available wind power, and photovoltaic output for that day are shown in Figure 13. The 24 h operating and output schedules for the conventional units, as determined by the standard SPS and the security-constrained SPS, are shown in Figure 14 and Figure 15, respectively. In Figure 14, solid dots indicate generator startups, while hollow dots represent shutdowns. A detailed numerical comparison of the conventional unit outputs under the standard and security-constrained SPS methods is provided in

Table 3. The 24 h unit output (p.u.) under standard SPS and security-constrained SPS.

t	Standard SPS						Security-Constrained SPS
	Unit 1	Unit 3	Unit 6	Unit 7	Unit 9	Unit 10	Unit 1	Unit 3	Unit 6	Unit 7	Unit 9	Unit 10
1	5.08	7.25	off	6.38	8.65	11.00	5.08	7.25	3.26	3.12	8.65	11.00
2	5.08	7.25	off	4.77	8.65	11.00	5.08	6.16	3.21	3.12	8.19	11.00
3	5.08	7.25	off	3.46	8.65	11.00	5.08	5.12	3.18	3.12	7.95	11.00
4	5.08	7.25	off	3.61	8.65	11.00	5.08	5.86	2.74	3.12	7.79	11.00
5	5.08	7.25	off	5.82	8.65	11.00	5.08	6.53	3.48	3.12	8.59	11.00
6	5.08	7.25	off	3.33	8.65	11.00	3.24	2.18	7.01	3.22	8.65	11.00
7	5.08	4.71	off	3.78	8.65	11.00	1.52	2.18	7.01	4.68	8.65	11.00
8	5.08	off	off	7.30	8.65	11.00	1.52	2.18	7.01	3.85	8.65	11.00
9	5.08	off	off	7.78	8.65	11.00	1.52	2.18	7.01	4.52	8.65	11.00
10	5.08	off	off	5.52	8.65	11.00	1.52	2.18	7.01	3.16	8.65	11.00
11	5.08	off	off	7.10	8.65	11.00	2.92	2.18	3.96	3.12	8.65	11.00
12	5.08	off	off	7.99	8.65	11.00	5.08	2.32	2.55	3.12	8.65	11.00
13	5.08	off	off	3.12	8.28	11.00	3.40	2.18	2.55	3.12	5.24	11.00
14	5.08	off	off	3.12	8.38	11.00	2.58	2.18	2.55	3.12	6.15	11.00
15	5.08	off	off	6.13	8.65	11.00	4.34	2.18	2.55	3.12	7.68	11.00
16	5.08	4.71	off	4.06	8.65	11.00	5.08	3.10	2.55	3.12	8.65	11.00
17	5.08	7.25	off	3.18	8.65	11.00	4.07	2.67	5.65	3.12	8.65	11.00
18	5.08	7.25	5.53	3.96	8.65	11.00	5.08	7.02	6.59	3.12	8.65	11.00
19	5.08	7.25	8.50	5.81	8.65	11.00	5.08	7.25	8.50	5.81	8.65	11.00
20	5.08	7.25	7.01	4.23	8.65	11.00	5.08	6.22	7.33	4.94	8.65	11.00
21	5.08	7.25	7.15	4.94	8.65	11.00	5.08	7.25	7.15	4.94	8.65	11.00
22	5.08	7.25	5.53	4.55	8.65	11.00	5.08	7.03	7.01	3.28	8.65	11.00
23	5.08	7.25	off	7.48	8.65	11.00	5.08	5.67	5.94	3.12	8.65	11.00
24	5.08	7.25	off	4.68	8.65	11.00	5.08	2.99	5.82	3.12	8.65	11.00

.

The standard SPS only considers power and energy balance constraints. Its goal is to minimize operational costs by activating the fewest units. However, this may lead to insufficient voltage and frequency support for the system. In contrast, the security-constrained SPS ensures that additional units are brought online to meet the system’s voltage and frequency support needs. For instance, at hour 14, when solar output was high and the load was relatively low, renewable energy accounted for 44.5% of the electricity demand. Under the standard SPS, the number of online units is reduced, and the operating units are kept at their economic load levels. On the other hand, the security-constrained SPS brings Unit 3 and Unit 6 online to provide adequate voltage and frequency support for the system.

For the system operating condition at hour 14, a dynamic simulation of the grid response to an HVDC lockout fault was conducted. Figure 16 and Figure 17 illustrate the system’s frequency response curve and DC bus voltage response curve, respectively. Under the unit startup schedule determined by the standard SPS, the maximum frequency deviation reaches approximately 0.67 Hz. According to the National Standards of the People’s Republic of China (GB standards) [34,35], if the frequency deviation exceeds 0.5 Hz, the dispatching agency may issue high-frequency disconnection commands to renewable energy units. Such disconnections can potentially lead to a large-scale blackout in the grid. Consequently, the actual renewable energy absorption is reduced.

Additionally, the maximum DC bus voltage reaches approximately 1.21 p.u., which poses a potential risk of overvoltage. According to the GB standards, if the voltage exceeds 1.3 p.u., renewable energy sources may be disconnected from the grid. Such overvoltage-related disconnections near the DC bus could further reduce the actual renewable energy absorption.

In contrast, the unit startup schedule determined by the security-constrained sequential production simulation maintains grid voltage and frequency safety under the same fault condition, thereby better representing renewable energy absorption under practical system conditions.

The same holds true for moments when the share of renewable energy output is relatively small. At hour 17, the photovoltaic output sharply decreases as the sun sets, while the load begins to increase with the approaching evening peak. The standard SPS brings Unit 3 online to balance the reduction in renewable energy output and the increase in load. Figure 18 and Figure 19 show the system’s frequency response curve and DC bus voltage response curve under the same HVDC lockout fault for the system operating condition at hour 17. Under the unit startup schedule determined by the standard SPS, the steady-state frequency deviation exceeds the allowable threshold of 0.2 Hz (GB standards [36]). The standard SPS fails to maintain the system’s frequency safety, whereas the security-constrained simulation ensures the system remains within acceptable frequency limits.

From the above analysis, it is evident that the renewable energy absorption evaluated by the proposed security-constrained sequential production simulation provides a representation that is more consistent with the grid’s actual operating conditions, offering a more realistic assessment.

(3)

Factors Improving Renewable Energy Absorption

Reducing the output or shutting down conventional units does not necessarily improve renewable energy absorption, as fewer units cannot ensure system safety. The key to enhancing absorption is increased demand, whether from local load or export demand. Based on the proposed security-constrained SPS, the impact of load scale and HVDC transmission capacity on renewable energy absorption was analyzed by varying either the load scale or the HVDC transmission capacity.

Using day 267 as an example, Figure 20, Figure 21 and Figure 22 show the relationship between load scale, HVDC transmission capacity, and renewable energy absorption. In these figures, the vertical axis represents the renewable energy utilization, while the horizontal axes represent total system load and HVDC transmission capacity, respectively. It can be seen that increasing both the load scale and HVDC transmission capacity can improve renewable energy absorption—one for local absorption and the other for export absorption. However, both require sufficient regulation support from the power grid. Figure 18 shows how renewable energy absorption changes as the share of renewable energy capacity is increased. If the grid’s regulation support capacity remains unchanged, increasing renewable energy capacity will eventually exceed the grid’s ability to absorb it, leading to a decline in absorption.

5.2. Case2: The IEEE 68-Bus System

The IEEE 68-bus system, one of the six benchmark systems recommended by the IEEE Task Force on Benchmark Systems for Stability Controls, includes 68 buses, 16 generators, and 86 transmission lines. The system topology can be found in [37], and the corresponding dynamic simulation parameters are also derived from the same source. Wind farms were integrated at Buses 39, 51, and 56, while photovoltaic plants were connected at Buses 44 and 60. Bus 17 was further modified to act as the sending terminal of an HVDC transmission system. The 8760 h annual time-series data for load, wind, and PV generation were also obtained from [32].

TA was performed using the proposed HC with DTW and min–max linkage. A total of 25 clusters are obtained. The number of days in each cluster and their corresponding representative days are shown in

Table 4. Representative days and corresponding cluster assignments.

No.	Representative Day	Days Included
1	148	63
2	330	25
3	55	2
4	168	13
5	146	3
6	167	9
7	290	21
8	186	9
9	177	2
10	154	9
11	316	4
12	179	5
13	212	1
14	299	6
15	352	5
16	38	17
17	357	7
18	294	8
19	268	9
20	335	8
21	286	73
22	111	24
23	235	24
24	78	9
25	207	9

. The cluster representatives and their assigned days are visualized in Figure 23. Dots in different colors represent different clusters. The solid dots of the same color indicate the days assigned to each cluster, while the black-edged hollow dots represent the prototype of each cluster.

Following the selection of representative days, security-constrained SPSs were performed for each day to evaluate renewable energy absorption. The annual absorption was then calculated by aggregating these results according to the occurrence frequency of each representative day. Figure 24 illustrates the renewable energy absorption outcomes from security-constrained SPSs both with and without TA, while

Table 5. RE absorption (p.u.) under different methods in the IEEE 68-bus system.

Month	The Available Generation Power of RES	Renewable Energy Absorption
		Security-Constrained SPS Without TA	Security-Constrained SPS with TA	Related Error
January	4731.35	4588.71	4594.22	0.12%
February	7077.31	6590.42	6533.33	0.81%
March	9006.41	8269.12	8342.46	0.81%
April	10,210.21	9308.10	9362.00	0.53%
May	9334.78	8664.07	8669.02	0.05%
June	11,696.22	10,775.16	10,765.99	0.08%
July	9747.33	9204.93	9148.44	0.58%
August	9127.13	8430.92	8505.85	0.82%
September	9137.34	8413.78	8399.01	0.16%
October	8388.05	7876.97	7764.96	1.34%
November	6034.20	5792.50	5704.11	1.46%
December	7112.83	6979.10	6869.06	1.55%
Annual	101,603.18	94,893.77	94,658.44	0.23%

provides detailed numerical data. The relative errors between the two approaches are also analyzed.

Without TA, conducting the security-constrained SPS over the full 8760 h requires 24,682.95 s and results in an annual renewable energy absorption of 94,893.77 p.u. In contrast, the TA-based approach reduces the computation time to 621.79 s, while yielding a comparable annual absorption of 94,658.44 p.u. All simulations were carried out using GUROBI 10.0.1 on a PC equipped with an Intel Core i5-13400 2.5 GHz CPU and 16 GB of RAM. The resulting relative error is only 0.23%, demonstrating that the TA-based method offers a substantial reduction in computational burden while maintaining high accuracy in the renewable energy absorption estimates.

The case study on the IEEE 68-bus system demonstrates that the proposed method can be effectively applied to more complex power systems, confirming its scalability and robustness for practical renewable energy absorption assessment in large-scale networks.

6. Conclusions

This paper presents an assessment of renewable energy absorption capacity using a security-constrained SPS, incorporating grid voltage and frequency support constraints. Through a case study of the IEEE 39-bus system, which integrated wind and photovoltaic generation alongside HVDC transmission for power export, the proposed approach was demonstrated to provide a more realistic evaluation of renewable energy absorption capacity. This was particularly evident when evaluating the system’s performance under varying load and renewable energy conditions. For instance, during periods of high renewable output or low load, the security-constrained SPS successfully brought additional conventional units online, helping maintain grid stability. It prevented overvoltage and frequency deviations that could otherwise lead to renewable energy curtailment.

Furthermore, this case study analyzed the factors that improve renewable energy absorption. It revealed that increasing grid demand—both locally and through export via HVDC transmission—is a key driver of enhanced renewable energy absorption. However, this requires the grid to provide sufficient support. To address this challenge, advanced grid technologies such as energy storage and demand response have been developed. Future research should focus on how to effectively coordinate these measures to better accommodate the growing share of renewable energy in power systems.

It is worth noting that while the proposed TA method improves computational efficiency with acceptable accuracy under typical conditions, it does not explicitly account for rare but high-impact extreme weather events. Incorporating such scenarios in future studies will help enhance the method’s robustness. Also, future work will explore integrating AI-based techniques to improve the representation of security and stability constraints. AI models trained on operational data may help approximate complex dynamics, enabling faster and more accurate assessments.