Power System Reliability Assessment Considering Coal-Fired Unit Peaking Characteristics

Abstract

The large-scale integration of renewable energy and the serious safety issues faced by coal-fired units during peak regulation pose significant challenges to the reliable operation of power systems. Traditional probabilistic production simulation (PPS) methods fail to account for the fluctuations in load demand and the time-series variability of renewable energy, and they do not sufficiently consider the lifespan degradation of coal-fired units during actual peak regulation operations, which results in an inaccurate reflection of system reliability. This paper proposes an improved PPS method. First, this method rigorously considers the time-series fluctuations of load demand and renewable energy. Secondly, a multi-state model is established for coal-fired units, incorporating lifespan degradation and failure rates under different output conditions, which are dynamically updated based on load demand and operational status. An equivalent multi-state model for wind power output is also developed for different time periods. Finally, the Universal Generating Function (UGF) algorithm is used for PPS, enabling the calculation of system reliability indices, as well as dynamic costs such as unit start-up and shutdown, to assess the system’s capability to accommodate wind power. The impact of different peak regulation capabilities of units on system reliability is also studied. This paper presents the theoretical foundation, algorithm implementation, and related case studies, verifying the rationality and effectiveness of the proposed method in addressing the above-mentioned issues.

1. Introduction

With the global transformation of the energy structure and the promotion of low-carbon goals, the proportion of renewable energy in power systems is gradually increasing. However, the intermittency and uncertainty of renewable energy pose significant challenges to the stability and economy of power systems [1]. Coal-fired units, as conventional base-load power sources, possess strong regulation capabilities. They can quickly respond to load changes, thus becoming an important resource for peak regulation in power systems [2]. However, coal-fired units operate in off-design conditions during peak regulation, leading to frequent fluctuations in unit parameters. The heat surfaces in the furnace are often subjected to thermal stress, making them more prone to failure [3]. This results in unplanned outages and the loss of load resources and has a significant impact on the reliability of power system operation. Furthermore, the proportion of adjustable capacity in conventional thermal power units is too low, and the space for renewable energy absorption is severely insufficient, leading to phenomena such as wind and solar curtailment. Therefore, it is of great significance to effectively assess the impact of coal-fired units’ peak regulation characteristics on power system reliability after the integration of renewable energy. This helps to reasonably arrange generation plans and promote the integration of renewable energy.

PPS is a crucial tool for generation planning and unit operation scheduling in power systems. This method enables the calculation of unit output across different operational modes, along with system production costs and reliability indices. It is particularly useful for studies focusing on generation unit output characteristics and system load curves. PPS methods are generally classified into two main categories: simulation-based approaches [4,5] and analytical methods [6,7,8]. The simulation methods, such as Monte Carlo simulation, have the advantages of strong adaptability and simple models, but they suffer from issues related to computational efficiency and accuracy. Analytical methods, with clear physical concepts, high mathematical model accuracy, and fast computation speeds, are suitable when many related factors need to be considered. However, they involve complex mathematical modeling, and it is difficult to obtain the probability distribution and characteristic parameters of various technical and economic indices. Currently, analytical methods remain the core of research on PPS in power systems. Scholars worldwide have conducted extensive work to improve the computational efficiency and accuracy of PPS algorithms. For instance, study [9] integrates transition frequency analysis into the PPS framework, refining the equivalent electrical energy frequency method to assess the impact of wind power integration on dynamic costs. Another study [10] introduces a PPS method based on the UGF theory, effectively capturing the reliability implications of wind power integration. Furthermore, study [11] presents a structural expansion method that models state variable distributions in power systems, incorporating clean energy penetration and low-carbon benefit analysis. While extensive research has addressed the impact of renewable energy intermittency [12] on system reliability, fewer studies have examined the degradation and operational dynamics of coal-fired units during peak regulation and frequency control. A study [13] proposes a PPS-based evaluation framework for deep peak regulation of thermal power to facilitate renewable energy integration. However, it does not account for failure rates or state transition characteristics during operation. Another study [14] develops four multi-state unit models considering various operational conditions, base-load constraints, and segmented unit capacities. Nonetheless, its complex model structure limits computational efficiency and fails to accurately capture the degradation effects of peak regulation on system reliability.

UGF serves as a robust mathematical tool for modeling and analyzing multi-state systems by representing system states and their corresponding probabilities in polynomial form. The fundamental principle of UGF lies in mapping discrete system states into a generating function, where each polynomial term characterizes a potential system state with its coefficient denoting the occurrence probability of that state. In the field of reliability modeling for power systems, UGF has emerged as a critical methodology that enhances model clarity and computational efficiency. A study [15] leverages the UGF model for the reliability analysis of hybrid uncertainty structures in turbine systems, demonstrating a nearly 18-time improvement in computational efficiency compared with Monte Carlo simulations. Another study [16] integrates UGF with Bayesian methods for the reliability assessment of complex multi-unit power generation systems, achieving significant efficiency optimization. Furthermore, researchers [17] have employed UGF techniques combined with Weibull distribution analysis to evaluate the operational reliability of wind turbine components, providing a probabilistic framework for failure mode assessment in wind power generation systems.

Probabilistic models serve as critical tools for describing and analyzing system behaviors under uncertain operational conditions, particularly in evaluating key performance indicators such as operational reliability and risk profiles of power systems. The development of component-level state probabilistic models forms the foundation for such analyses. Researchers [18] have established a probabilistic reliability model for microgrids and develops an analytical framework for assessing the reliability of distribution systems with microgrids. The proposed microgrids probabilistic model can simplify and accelerate computational processes associated with reliability evaluation of distribution systems incorporating multiple microgrids. For coal-fired power plants, researchers [19] have developed accident probabilistic models to assess operational risks through dynamic likelihood evaluation of turbine generator contingencies. Additionally, in smart grid applications, an Electric Vehicle Virtual Energy Storage System charging/discharging power probabilistic model has been proposed to enhance the reliability assessment of distribution networks incorporating virtual energy storage technologies [20]. These implementations demonstrate the adaptability of probabilistic modeling in addressing diverse challenges across renewable energy integration, conventional power generation, and modern grid management.

To fully consider the impact of coal-fired unit peak shaving characteristics on the reliability of the power system, an improved PPS method is proposed in this study, inspired by the multi-state unit model in the literature [21]. This method first rigorously considers the temporal fluctuations of load demand and renewable energy. Secondly, a multi-state model for coal-fired units is established, incorporating their life degradation and failure rates under different output states. The transition rates between unit states are dynamically adjusted based on load demand and operational status. Similarly, a multi-period equivalent multi-state model is developed for wind power generation. Finally, the UGF algorithm is employed to perform joint calculations on the probabilistic models of different units, enabling the PPS. This approach allows for the computation of system reliability indicators, dynamic costs such as unit start-up and shut-down, and the evaluation of the system’s ability to accommodate wind power. This study investigates the influence of different peak shaving capabilities of units on the reliability of the power system. The proposed method is then verified through case analysis using the IEEE-RTS79 test system.

2. Improvement of the UGF

2.1. UGF

The core idea of the UGF algorithm is to map the sequence of a discrete structure into a polynomial function [22]. First, the combination operator of the polynomial is defined based on the operation rules of discrete random variables. Then, by recursive operations, the polynomial form of the discrete random variables can be obtained [23]. Suppose there are two independent systems, $G_1$ and $G_2$, each corresponding to states $k_1$ and $k_2$ The state arrays can be represented as follows:

$$\left\{\begin{array}{l}{G}_1=\{g_{1,1},g_{1,2},\dots ,g_{1,i},\dots ,g_{1,k_1}\}\\ G_2=\{g_{2,1},g_{2,2},\dots ,g_{2,j},\dots ,g_{2,k_2}\}\end{array}\right.$$

(1)

The probability of each state occurring is expressed as follows:

$$\left\{\begin{array}{l}{P}_1=\{p_{1,1},p_{1,2},\dots ,p_{1,i},\dots ,p_{1,k_1}\}\\ P_2=\{p_{2,1},p_{2,2},\dots ,p_{2,j},\dots ,p_{2,k_2}\}\end{array}\right.$$

(2)

where $g_{1,i}$ is the i-th state of the system $G_1$, and $p_{1,i}$ is the occurrence probability of the corresponding state. The corresponding UGF model can be expressed as follows:

$$\left\{\begin{array}{l}{u}_1(z)={\displaystyle \sum _{i=1}^{k_1}{p}_{1,i}{z}^{g_{1,i}}=p_{1,1}{z}^{g_{1,1}}+p_{1,2}{z}^{g_{1,2}}+\cdots +p_{1,k_1}{z}^{g_{1,k_1}}}\\ u_2(z)={\displaystyle \sum _{j=1}^{k_2}{p}_{2,j}{z}^{g_{1,j}}=p_{2,1}{z}^{g_{2,1}}+p_{2,2}{z}^{g_{2,2}}+\cdots +p_{2,k_2}{z}^{g_{2,k_2}}}\end{array}\right.$$

(3)

where the exponent of z represents the value of the different states of the system, and the coefficient represents the occurrence probability of the different states. The value of z itself has no real significance or value and is used solely to distinguish between the state values and the probability values of the system.

Multiple independent systems can be combined to form a new system. For example, systems $G_1$ and $G_2$ can be combined through the operation of UGF to form a new system $G_3$, as shown in the following equation.

$$u_3(z)={\Omega }_{\varphi }(u_1,u_2)={\displaystyle \sum _{i=1}^{k_1}{\displaystyle \sum _{j=1}^{k_2}{p}_{1,i}{p}_{2,j}{z}^{g_{1,i}+g_{2,j}}}}={\displaystyle \sum _{s=1}^{k_3}{p}_{3,s}{z}^{g_{3,s}}}$$

(4)

where $p_{3,s}$ and $g_{3,s}$ represent the state values and the occurrence probabilities of the system; $k_3$ represents the number of states in system $G_3$.

2.2. Improved UGF

In the process of PPS, assume that the system consists of m components, each with n states. The total number of states in the system is nm, which grows exponentially. Even if the number of states is reduced by merging states with the same probability, the growth of system states remains difficult to control, especially when the system scale is large and the precision of state values is high. In such cases, the number of state combinations may exceed the computational capacity of the system.

The average rounding method can effectively address the state explosion problem. This method draws on the common factor approach from sequence operation theory [24] and is improved through the following steps: First, an appropriate common factor Δc is selected. Then, state values are rounded to the nearest integer multiple of the common factor. Finally, states with the same value are merged. In this process, since the state values are standardized as integer multiples of the common factor, the growth of the number of states no longer follows an exponential pattern but instead depends on the specific operational method. Therefore, for large-scale systems, the UGF method based on the average rounding approach can suppress the exponential growth of the number of states, thus improving computational efficiency.

3. Multi-Stage Probabilistic Model

3.1. Multi-State Power Model of Wind Farms

In PPS, accurately modeling the randomness and time-series characteristics of wind power output is crucial for assessing the reliability and economic performance of power systems. This section establishes a multi-state power model for wind farms, which captures the stochastic nature of wind speed and the nonlinear relationship between wind speed and power output.

The short-term random variations in wind speed in nature generally follow a Weibull distribution [25], with the probability distribution given by

$$f(v)=({\displaystyle \frac{k}{c}}){({\displaystyle \frac{v}{c}})}^{k-1}\exp [-{({\displaystyle \frac{v}{c}})}^k]$$

(5)

where $v$ is the wind speed value, k is the shape parameter, and c is the scale parameter.

The power output of wind turbines has a nonlinear relationship with wind speed, expressed as follows:

$$P(v)=\left\{\begin{array}{ll}0,& 0\le v\le v_{ci}\\ P_r{\displaystyle \frac{(v-v_{ci})}{(v_r-v_{ci})}},& v_{ci}\le v\le v_r\\ P_r,& v_r\le v\le v_{co}\\ 0,& v\ge v_{co}\end{array}\right.$$

(6)

where $v_{ci}$ is the cut-in wind speed, $v_r$ is the rated wind speed, $v_{co}$ is the cut-out wind speed, and $P_r$ is the rated power. The wind power curve has been linearized between the cut-in wind speed and the rated wind speed in Equation (6).

By solving Equations (5) and (6) simultaneously, the probability density function of the total wind power at a certain time under a specific scenario can be obtained. To consider the time-series characteristics of wind power output in PPS, the continuous probability distribution of wind power is discretized into multiple states. Each state represents a specific range of power output, and the probability of each state is calculated based on the Weibull distribution and the power curve, as shown in Figure 1. Based on the discretized state model, its UGF model can be established, as shown in Equation (7):

$$u_w^m(z,t)={\displaystyle \sum _{j=1}^{k_w}{p}_{w,j}^m(t)z^{P_{w,j}^m(t)}}$$

(7)

where $u_w^m(z,t)$ is the UGF model of the wind power output at time t under scenario m; $p_{w,j}^m(t)$ and $P_{w,j}^m(t)$ are the corresponding state probabilities and state values; and $k_w$ is the number of states.

3.2. Coal-Fired Unit Multi-State Probabilistic Model

In conventional PPS, coal-fired units typically use a simple two-state model, considering the forced outage state and the rated operating state of the unit [26]. However, in practice, coal-fired units perform load-following tasks, requiring dynamic adjustment of output based on load demand. The load-following operation of coal-fired units is typically divided into flexible load-following and start–stop load-following. The former allows the unit’s output to be adjusted within a certain power range to respond to load demand, while the latter involves frequent start–stop operations to meet load requirements. Additionally, coal-fired units often operate in off-design conditions during load-following, and their load change rate and depth of load-following can affect the unit’s reliability. For example, high-temperature components, such as the economizer and rotor, are more susceptible to thermal stress during load fluctuations, which may lead to failures such as tube ruptures, significantly impacting the unit’s reliability [27]. Therefore, this study focuses on the operating characteristics of coal-fired units at different load positions and examines the impact of coal-fired units on the overall power system’s reliability during load-following by approximating the failure of high-temperature equipment during load-following as equivalent to the failure of the entire coal-fired unit.

3.2.1. State-Space Model of Coal-Fired Units

In the process of PPS, three types of multi-state models for coal-fired units have been established based on the unit’s operating conditions and load positions. These include the base-load unit model, the flexible load-following unit model, and the start–stop load-following unit model. The specific characteristics of these models are shown in

Table 1. Classification and characteristics of unit model.

Model	Features
Base-load unit model	Maintaining stable operation of the power system by continuously providing a minimum supply of power to operate at base-load at rated capacity.
Flexible peaking unit model	Responding quickly to power system load demand and performing peaking operations at waist load.
Start–stop peaking unit model	Meeting short-term load demand during peak load periods, promoting wind power consumption, and operating at peak loads.

. The models take into account constraints such as the depth of load-following, load change rate, load change time, and start-up failures. The impact of load-following depth and load change rate on high-temperature components within the unit is approximated as the overall impact on the coal-fired unit. The specific correlation and relationships are based on the research results provided in reference [28], which were used for fitting.

By considering the load-following capabilities of different units, along with system constraints and the time-series load demand characteristics, the state transition of coal-fired units is updated in real time. The specific state-space diagram is shown below:

Model 1: Base-load state-space model, as shown in Figure 2.

In Figure 2, $\lambda$ and $\mu$ represent the unit’s repair rate and failure rate, respectively, which are calculated as shown in Equations (8) and (9). Here, mean time to failure (MTTF) denotes the average time between failures, and mean time to repair (MTTR) represents the average repair time. These parameters are typically obtained from historical operational data of the unit. During the base-load period, the unit needs to maintain rated operation to meet the basic load demand. Therefore, a two-state model is used, consisting of the forced outage state and the rated operating state.

$$\lambda ={\displaystyle \frac{1}{MTTR}}$$

(8)

$$\mu ={\displaystyle \frac{1}{MTTF}}$$

(9)

Model 2: Flexible load-following state-space model, as shown in Figure 3.

In Figure 3, ${\rho }^{+}(k-1,k)$ denotes the upward demand rate for peaking unit state transition from state k − 1 to state k, which quantifies the demand urgency for driving the i-th unit from state k − 1 to k due to reduced load shortage risk. Conversely, ${\rho }^{-}(k,k-1)$ represents the downward demand rate for state transition from k to k − 1, characterizing the demand urgency for output reduction from state k to k − 1 induced by increased excess output risk. These time-dependent parameters are continuously updated based on real-time load demand and unit commitment status, with the transitional probabilities calculated as the product of system risk reduction after output increase and the original state probability to reflect temporal correlations, as shown in Equations (10)–(12).

$$P_{S_{i-1},s,k-1}(t)=P_{S_{i-1},s}(t)+P_{k-1}^i$$

(10)

$${\rho }_i^{+}(k-1,k)=p_{k-1}(t-1)({\displaystyle \sum _{P_{S_i,s,k-1}(t)<L(t)}{p}_{S_i,s,k-1}(t)}-{\displaystyle \sum _{P_{S_i,s,k}(t)<L(t)}{p}_{S_i,s,k}(t)})$$

(11)

$${\rho }_i^{-}(k,k-1)=p_k(t-1)({\displaystyle \sum _{P_{S_i,s,k}(t)>L(t)}{p}_{S_i,s,k}(t)}-{\displaystyle \sum _{P_{S_i,s,k-1}(t)>L(t)}{p}_{S_i,s,k-1}(t)})$$

(12)

where $P_{S_{i-1},s}(t)$ represents the output distribution of the entire system S after the (i − 1)-th unit is put into operation; the output distribution is obtained during the PPS process and can be seen in detail in Section 4. $P_{k-1}^i$ is the output size of the (k − 1)-th state of the i-th unit, $P_{S_{i-1},s,k-1}(t)$ refers to the corrected output distribution of system S when the i-th unit is put into operation in state k − 1, where $p_{S_i,s,k-1}(t)$ denotes the corresponding output probability. ${\displaystyle \sum _{P_{S_i,s,k-1}(t)<L(t)}{p}_{S_i,s,k-1}(t)}$ represents the probability of power deficiency in the system at time t after correction for k − 1 states of i units. ${\displaystyle \sum _{P_{S_i,s,k}(t)>L(t)}{p}_{S_i,s,k}(t)}$ represents the probability of power surplus in the system at time t after correction for k states of i units. $p_k(t-1)$ is the probability of unit state k at moment t − 1.

Furthermore, once the demand rate between system states is obtained, the transition rate between these states can be computed. The transition rate represents the probability per unit time that a unit transitions from one state to another. For transitions between operational output states, the transition rate is determined by the demand rate and the transition time, where a higher demand or a shorter transition time results in a higher transition rate. The demand rate for a transition from state k − 1 to state k is calculated using Equation (13), where $T{R}_{k-1~k}$ denotes the time required for a transition from state k – 1 to state k, as derived from Equation (14). For transitions between normal operational states and failure states, the transition rate is determined by the failure rate ${\mu }_k$ and the repair rate $\lambda$. The failure rate is adjusted based on the equipment’s lifecycle conditions during operation. Further details can be found in Section 4.

$${\nu }_{k-1~k}^{+}={\displaystyle \frac{{\rho }^{+}(k-1,k)}{T{R}_{k-1~k}}}$$

(13)

$$T{R}_{k-1~k}={\displaystyle \frac{P_k-P_{k-1}}{L_{rate}}}$$

(14)

where $P_k$ represents the output corresponding to state k, $L_{rate}$ is the load change rate, and ${\nu }_{k-1~k}^{+}$ is the transition rate between states.

The flexible load-following unit typically operates in the mid-load range and adjusts the unit’s output within a certain load range to meet the load demand. A multi-state model is used, where the unit’s output is discretized between the minimum technical output and the rated output. The transition characteristics between different states are considered to determine the load-following positions of the flexible load-following unit.

Model 3: Start–stop load-following state-space model, as shown in Figure 4.

In Figure 4, compared to the flexible load-following unit model, the start–stop load-following model includes the consideration of the unit’s active shut-down state. This model is typically applied during peak-load periods to primarily satisfy short-term load demand responses of the power system. Similarly, a multi-state model is used to determine the unit’s start–stop operations and load-following positions.

3.2.2. Calculation of Unit Steady-State Probabilities

For the three multi-state unit models described above, the steady-state probabilities of each state are determined using the Markov process state transition equations:

$$\left\{\begin{array}{l}(p_1,p_2,\dots ,p_n)\cdot A=0\\ {\displaystyle {\sum }_{k=1}^n{p}_k=1}\end{array}\right.$$

(15)

where $p_k$ represents the steady-state probability of the k-th state of the unit, and A is the state transition matrix of the coal-fired unit.

Taking Model 2 as an example, suppose it is the i-th operational unit, with the corresponding load at time t being L(t). The steady-state probabilities for the states in the state space to be solved are $p_k$. According to Figure 3, the state transition matrix A for unit i is

$$\begin{array}{l}A=\\ \left(\begin{array}{cccccc}-{\displaystyle \sum _{n=2}^k\frac{{\rho }_i^{+}(1,\mathrm{n})}{T{R}_{1~n}^i}-u_1}& {\displaystyle \frac{{\rho }_i^{+}(1,2)}{T{R}_{1~2}^i}}& {\displaystyle \frac{{\rho }_i^{+}(1,3)}{T{R}_{1~3}^i}}& \dots & {\displaystyle \frac{{\rho }_i^{+}(1,\mathrm{k})}{T{R}_{1~k}^i}}& u_1\\ {\displaystyle \frac{{\rho }_i^{-}(2,1)}{T{R}_{2~1}^i}}& -{\displaystyle \frac{{\rho }_i^{-}(2,1)}{T{R}_{2~1}^i}}-{\displaystyle \sum _{n=3}^k\frac{{\rho }_i^{+}(2,\mathrm{n})}{T{R}_{1~n}^i}-u_2}& {\displaystyle \frac{{\rho }_i^{+}(2,3)}{T{R}_{2~3}^i}}& \dots & {\displaystyle \frac{{\rho }_i^{+}(2,\mathrm{k})}{T{R}_{2~k}^i}}& u_2\\ {\displaystyle \frac{{\rho }_i^{-}(3,1)}{T{R}_{3~1}^i}}& {\displaystyle \frac{{\rho }_i^{-}(3,2)}{T{R}_{3~2}^i}}& -{\displaystyle \sum _{j=1}^2\frac{{\rho }_i^{-}(3,\mathrm{j})}{T{R}_{3~j}^i}}-{\displaystyle \sum _{n=4}^k\frac{{\rho }_i^{+}(3,\mathrm{n})}{T{R}_{1~n}^i}-u_3}& \dots & {\displaystyle \frac{{\rho }_i^{+}(3,\mathrm{k})}{T{R}_{3~k}^i}}& u_3\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{{\rho }_i^{-}(\mathrm{k},1)}{T{R}_{k~1}^i}}& {\displaystyle \frac{{\rho }_i^{-}(\mathrm{k},2)}{T{R}_{k~2}^i}}& {\displaystyle \frac{{\rho }_i^{-}(\mathrm{k},\mathrm{k}-1)}{T{R}_{k~(k-1)}^i}}& \dots & -{\displaystyle \sum _{j=1}^{k-1}\frac{{\rho }_i^{-}(\mathrm{k},\mathrm{j})}{T{R}_{k~j}^i}}& u_k\\ \lambda & \lambda & \lambda & \dots & \lambda & -k\lambda \end{array}\right)\end{array}$$

(16)

In this equation, the off-diagonal elements a_ij of matrix A represent the transition rate from state i to state j, while the diagonal elements a_ii represent the negative of the exit rate from state i. Eventually, the steady-state probability of the unit state can be obtained after repeated amendments.

3.2.3. Coal-Fired Unit Equivalent Three-State Model

Considering that the multi-state model may result in an explosion of states during PPSs, which can reduce computational efficiency, it is proposed to equivalently simplify the load-following unit multi-state model into a three-state model to improve computational efficiency [29]. After obtaining the multi-state probabilistic model of the load-following unit based on the current load demand and the operational status of the unit, all lower output states are equivalently merged into a single forced output state. This results in the equivalent three-state model, as shown in Figure 5.

Taking Model 3 as an example, the multi-state model of the start–stop load-following unit is equivalently simplified into a three-state model, where the three states include the rated output $P_{rate}$, equivalent derated output $P_e$, and zero output $P_{zero}$, as calculated by Equation (17). The corresponding state probabilities are equivalently calculated according to Equation (18).

$$\left\{\begin{array}{l}{P}_{rate}=P_r\\ P_e={\displaystyle \frac{{\displaystyle \sum _{k=1}^n{p}_k{P}_k}}{{\displaystyle \sum _{k=1}^n{p}_k}}}\\ P_{zero}=0\end{array}\right.$$

(17)

$$\left\{\begin{array}{l}{p}_{rate}={\displaystyle \frac{p_r}{{\displaystyle \sum _{k=1}^n{p}_k}+p_o}}\\ p_e={\displaystyle \frac{{\displaystyle \sum _{k=1}^n{p}_k{P}_k}}{{\displaystyle \sum _{k=1}^n{p}_k}+p_o}}\\ p_{zero}={\displaystyle \frac{p_{FOR}}{{\displaystyle \sum _{k=1}^n{p}_k}+p_o}}\end{array}\right.$$

(18)

When converting a multi-state unit model into an equivalent unit model, the states of the equivalent unit model are subsets of the states in the original model. The following section further explains the calculation of the transition rates in the equivalent unit model.

$$\left\{\begin{array}{l}{\lambda }_{er}={\displaystyle \sum _{k=1}^n\frac{{\rho }_i^{+}(\mathrm{k},\mathrm{r})}{T{R}_{k~r}^i}}{p}_k^i\\ {\lambda }_{re}={\displaystyle \sum _{k=1}^n\frac{{\rho }_i^{-}(\mathrm{r},\mathrm{k})}{T{R}_{r~k}^i}}\\ {\lambda }_{0e}={\mu }_e\\ {\lambda }_{e0}=\lambda \\ {\lambda }_{r0}={\displaystyle \sum _{k=1}^n{\mu }_i}{p}_k^i\\ {\lambda }_{0r}={\displaystyle \sum _{k=1}^n{\mu }_i}\end{array}\right.$$

(19)

where ${\lambda }_{er}$, ${\lambda }_{0e}$, and ${\lambda }_{0r}$ represent the equivalent transition rates between the states of the unit.

4. PPS Considering the Load-Following Characteristics of Coal-Fired Units

PPS in power systems is an effective tool for calculating the generation output of each unit, system generation costs, and reliability indicators. In this section, by considering the different state transition characteristics of units under varying load-following capabilities, the principle of prioritizing the operation of thermal power units with lower unit fuel costs is applied to arrange the operation of thermal power plants sequentially. Then, using the multi-state unit models established earlier, a joint calculation is performed using the universal production function. After all units are put into operation, the relevant technical and economic indicators are further calculated.

4.1. Load-Following Capability Constraints for Coal-Fired Units

This paper primarily focuses on the limitations of a unit’s load-following capacity when responding to load demand, studying the state transition characteristics under different load-following capabilities. Considering that the equipment of the unit is often subjected to high-temperature environments during actual operation, the reliability of high-temperature heat surfaces generally represents the reliability of the entire unit [30]. Therefore, this paper derives the relationship between load change rate, peak shaving depth, and the lifetime of high-temperature heat surface pipelines and uses the remaining life of high-temperature equipment to correct the unit’s mean time between failures, thus calculating the failure rate of coal-fired units under different load-following depths and change rates.

4.1.1. Fault Probabilistic Model of Coal-Fired Units

This paper will use a multilayer perceptron (MLP) to fit the relationship between the variable load rate, peak shaving depth, and the remaining life of the high-temperature heat-exposed surfaces in coal-fired units. MLP is a feedforward neural network model, and its hidden layers are capable of extracting nonlinear features from the input data and performing representation learning [31]. Let the variable load rate and peak shaving depth be denoted as $x_1$ and $x_2$, respectively. The fitting process is as follows:

$$Z_{n,h(n)}=f(W_{h(n),h(n-1)}{Z}_{n-1,h(n-1)}+B_{h(n)})$$

(20)

$$\stackrel{\wedge }{y}=f(W_{h(n)}{Z}_{n,h(n)}+b)$$

(21)

where $Z_{n-1,h(n-1)}$ is the input matrix of the n-th layer network with a dimension of $h(n-1)\times 1$, where n is the number of layers in the network; $h(n)$ is the number of neurons in the hidden layer of the n-th network. Specifically, for the first layer, the input is ${[x_1,x_2]}^T$; $W_{h(n),h(n-1)}$ is the weight matrix of the n-th layer with a dimension of $h(n)\times h(n-1)$; $B_{h(n)}$ is the bias matrix with a dimension of $h(n)\times 1$; the actual predicted value of the network is denoted as $\stackrel{\wedge }{y}$; $f$ is the activation function, and $W_{h(n)}$ is the weight matrix of $1\times h(n)$.

The relationship between the variable load rate, peak shaving depth, and the remaining life of high-temperature equipment is fitted into a nonlinear function using MLP, as shown in Equation (22). Based on the remaining life of the high-temperature equipment, the MTTF of the unit is then corrected. Assume that MTTF of unit i is $M_{TTF,i}$, and its correction is given by Equation (23).

$$N_{life}=F(x_1,x_2)$$

(22)

$$M_{TTF,i}^{*}(x_1,x_2)=F(x_1,x_2){\displaystyle \frac{M_{TTF,i}}{N_{life,d}}}$$

(23)

where $N_{life,d}$ is the design remaining life, $M_{TTF,i}^{*}(x_1,x_2)$ is the corrected MTTF.

4.1.2. Unit Start-Up and Shut-Down Time Constraints

Assume that the system’s normal operation time and shut-down duration follow exponential distributions with the parameters ${\rho }^{-}(k,o)$ and ${\rho }^{+}(o,k)$, respectively. The minimum continuous operation time and minimum economic shut-down time for a generator unit are denoted as $T_U$ and $T_M$, respectively. Then,

$$p_U^n={\mathrm{e}}^{-{\rho }^{-}(k,o)T_U}={\displaystyle {\int }_{T_U}^{\infty }{\rho }^{-}{(\mathrm{k},\mathrm{o})\mathrm{e}}^{-{\rho }^{-}(k,o)t}dt}$$

(24)

$$p_M^n={\mathrm{e}}^{-{\rho }^{+}(o.k)T_M}={\displaystyle {\int }_{T_M}^{\infty }{\rho }^{+}{(\mathrm{o},\mathrm{k})\mathrm{e}}^{-{\rho }^{+}(o,k)t}dt}$$

(25)

where $p_U^n$ is the probability that the operation time in state k exceeds the minimum continuous operation time $T_U$, and $p_M^n$ is the probability that the shut-down time of the generating unit moving to state k exceeds the minimum economic shutdown time $T_M$.

Considering the start-up and shut-down time constraints of the unit, the demand ${\rho }^{+}(0,k)$ and non-demand rates ${\rho }^{-}(k,0)$ in the flexible peak shaving models should be corrected as follows:

$${\rho }_{+}^{*}(o,k)={\rho }^{+}(o,k)\cdot p_U^n\cdot p_M^n$$

(26)

$${\rho }_{-}^{*}(\mathrm{k},o)={\rho }^{-}(k,o)\cdot p_U^n\cdot p_M^n$$

(27)

4.1.3. Unit Variable Load Time Constraints

The ability of a unit to transition from one state to another during its operation depends on its ramp rate and variable load time. Given that the simulation time step is 1 h, the equivalent output of unit i at different times during the simulation period must satisfy the following constraint:

$$P_i^e(t-1)-v_i\cdot 60\le P_i^e(t)\le P_i^e(t-1)+v_i\cdot 60$$

(28)

where $P_i^e(t)$ is the equivalent output of unit i at time t, and $v_i$ is the ramp rate of unit i, expressed in %/min.

Based on this constraint, the equivalent output of unit i is corrected as follows:

$$\left\{\begin{array}{ll}{P}_i^{e*}(t)=P_i^e(t)& P_i^e(t-1)-v_i\cdot 60\le P_i^e(t)\le P_i^e(t-1)+v_i\cdot 60\\ P_i^{e*}(t)=P_i^e(t-1)-v_i\cdot 60& P_i^e(t)\le P_i^e(t-1)-v_i\cdot 60\\ P_i^{e*}(t)=P_i^e(t-1)+v_i\cdot 60& P_i^e(t)\ge P_i^e(t-1)+v_i\cdot 60\end{array}\right.$$

(29)

4.2. UGF Model of Coal-Fired Unit

Based on the three unit models described in Section 2.2, in actual power system operation, units must flexibly adjust their output levels in response to load demand to accommodate instantaneous load fluctuations while meeting generation requirements as well as facilitating the integration of wind and photovoltaic power. This process requires careful consideration of the unit’s peak shaving capacity and output range to ensure the stable operation of the system.

Furthermore, to ensure the long-term reliability of the units, maintenance personnel typically develop detailed maintenance plans based on the operational status of the units, which is also one of the key factors in ensuring stable unit operation. Therefore, when performing system simulations, the maintenance schedule must be incorporated into the model to enhance the accuracy of the simulation results.

Based on the above factors, assume that coal-fired unit i has k states, and its UGF model is as follows:

$$u_{i,k}(z,t)=(1-M_i(t)){\displaystyle \sum _{j_i=1}^k{p}_{j_i}^i(t)z^{P_{j_i}^i}}+M_i(t)z^0$$

(30)

where $u_{i,k}(z,t)$ represents the UGF model of unit i, $P_{j_i}^i(t)$ and $p_{j_i}^i(t)$ represent the output and corresponding probability of unit i at time t, respectively. $M_i(t)$ is the parameter reflecting the planned maintenance of the unit. If the unit is in maintenance at time t, then $M_i(t)=1$; otherwise, $M_i(t)=0$.

4.3. Power System Output State Adjustment

In the PPS of the power system, the entire system’s generation capacity is treated as a multi-state system. By introducing the UGF algorithm into the simulation process, the system’s output state is continuously adjusted as units are put into operation to reflect the simulation process. In particular, wind power, being a clean energy source, is considered with limitations in the PPS, while the remaining conventional units are arranged for operation based on their unit generation cost. Specifically, this paper primarily investigates the impact of coal-fired unit peak shaving characteristics on the reliability of the power system, focusing on the operation of coal-fired units.

Assuming that the unit commissioned in the i-th operation is unit i, after it is brought online at time t, the system output model $u_{S_i,k_{S_i}}$ is adjusted as per Equation (31).

$$u_{S_i,k_{S_i}}(z,t)={\Omega }_{\varphi }(u_{1,k_1},u_{2,k_2},\dots u_{i,k_i})={\displaystyle \sum _{s=1}^{k_{S_i}}{p}_{S_{\mathrm{i}},s}(t)z^{P_{S_{\mathrm{i}},s}(t)}}$$

(31)

where $k_{S_i}$ represents the number of output states of the power system after the commissioning of unit i, $P_{S_{\mathrm{i}},s}(t)$ and $p_{S_{\mathrm{i}},s}(t)$ represent the output magnitude and output probability of the system’s s-th state at time t after the commissioning of unit i, respectively.

The PPS based on the UGF model can calculate the reliability indices at different times during the simulation period as well as after the commissioning of different units. Assuming the load demand at time t is $L(t)$, the loss of load probability (LOLP) and expected loss of load (EENS) for the entire power system after the commissioning of the i-th unit at time t can be calculated, along with the expected power generation of the currently operating units. The specific calculations are as follows:

$$LOL{P}_i(t)={\displaystyle \sum _{P_{S_i,s}<L(t)}{p}_{S_i,s}(t)}$$

(32)

$$EEN{S}_i(t)={\displaystyle \sum _{P_{S_i,s}<L(t)}{p}_{S_i,s}(t)}(L(t)-P_{S_i,s}(t))$$

(33)

$$E_i(t)=EEN{S}_i(t)-EEN{S}_{i-1}(t)$$

(34)

where $LOL{P}_i(t)$ represents the probability that the system output is less than the load demand on a given day, and $EEN{S}_i(t)$ represents the expected value of the system output falling below the load demand. $E_i(t)$ is the expected generation of unit i.

After the commissioning of generator i, if the power supply cannot meet the demand, generator i + 1 will be commissioned. Therefore, the demand rate for the commissioning of unit i + 1 $LOL{P}_i(t)$ can be represented as ${\rho }^{+}$:

$${\rho }^{+}=LOL{P}_i(t)$$

(35)

4.4. PPS Process

A flowchart of the method is shown below. Wind power units are prioritized for commissioning, followed by the sequential commissioning of coal-fired units. The constraints and correction methods required in the simulation process are described in the previous sections. Unlike traditional convolution-based calculations, the simulation method in this paper employs a joint distribution calculation based on the UGF model. The specific process is shown in Algorithm 1.

Algorithm 1: PPS considering coal-fired unit peaking characteristics.
Input: Number of rounds T, number of unit I, time-series load data L, wind resource data W, MTTF, MTTR, Ramp rate $x_1$, peak shaving depth $x_2$ Initialization:
1.	Model wind power output using a multi-state approach	//as per Equation (7)
2.	Rank units by generation cost and set commissioning order
3.	Obtain the remaining lifetime fitting model.	//as per Equation (22)Start
4.	for t = 1 … T do
5.	Initialize system capacity based on wind power state at t
6.	for i = 1 … I do
7.	if unit i under maintenance then	//as per Equation (30)
8.	skip unit, set i = i + 1
9.	else continue
10.	end if
11.	if unit i in baseload operation then
12.	use two-state mode
13.	else use multi-state model
14.	update coal-fired unit state space $A$	//as per Equation (13)
15.	compute unit steady-state probability $p_k$	//as per Equation (16)
16.	compute equivalent output model	//as per Equations (17)–(19)
17.	end if
18.	while unit output does not meet the ramping constraints
19.	adjust the unit output
20.	end while	//as per Equations (28)–(29)
21.	update the system output distribution	//as per Equation (31)
22	compute the expected power generation of the i-th unit//as per Equation (34)
23..	end for
24.	compute the reliability indices at time t.	//as per Equations (32)–(33)
25.	end for
26.	compute the system reliability indices.	//as per Equations (36)–(47)
End

5. System Operation Parameter Calculation Method

The multi-state unit PPS based on the UGF model proposed in this paper allows for the calculation of system reliability indices, system start-up and shut-down counts, unit start-up costs, individual generator electricity generation, and generation costs. Additionally, considering the integration of wind power units, it can calculate wind power absorption capacity, wind curtailment rate, and expected curtailed wind energy.

5.1. Technical Index Calculation Methods

5.1.1. System Reliability Index Calculation

Within the simulation period T, the system’s LOLP and EENS are calculated as follows:

$$LOLP={\displaystyle \frac{{\displaystyle \sum _{t=1}^{T}LOLP(t)}}{T}}$$

(36)

$$EENS={\displaystyle \sum _{t=1}^{T}EENS(t)}$$

(37)

5.1.2. Start-Up and Shut-Down Frequency of Peaking Units

The expected number of start-up and shut-down events, $N$ and $V$, for a unit during the simulation period T is given by

$$N={\displaystyle \sum _{k=1}^n({\displaystyle \sum _{t=1}^T{p}_o(t){\rho }_t^{+}(o,k)}+{\displaystyle \sum _{{\rho }_{t-1}^{+}(o,1)<{\rho }_t^{+}(o,1)}{\rho }_t^{+}(o,k)-}{\rho }_{t-1}^{+}(o,k)})$$

(38)

$$V={\displaystyle \sum _{k=1}^n(}{\displaystyle \sum _{t=1}^T{p}_k(t)}{\rho }_t^{-}(k,o)+{\displaystyle \sum _{{\rho }_{t-1}^{-}(o,1)<{\rho }_t^{-}(o,1)}{\rho }_t^{-}(k,o)-}{\rho }_{t-1}^{-}(k,o))$$

(39)

where $p_o(t)$ represents the probability of the unit being in the off state at moment t. The first term reflects the impact of the unit’s own failures and other factors on the start-up and shut-down frequency, while the second term reflects the influence of load fluctuations and the impact of the preceding unit’s operation on the start-up and shut-down of the current unit.

5.2. Economic Index Calculation Methods

5.2.1. Unit Generation and Fuel Costs

Each time a unit is commissioned, a joint calculation is performed for the system’s output. Therefore, the expected generation of unit i during the period $E_i$ is given by

$$E_i={\displaystyle \sum _{t=1}^T{E}_i(t)}$$

(40)

If the unit i’s unit generation cost is $Q_{fuel,i}$, then the fuel cost $C_{fuel}$ is:

$$C_{fuel}={\displaystyle \sum _{i=1}^m{E}_i{Q}_{fuel,i}}$$

(41)

5.2.2. Dynamic Costs

The dynamic cost of peaking units $C_d$ can be calculated as follows:

$$C_d={\displaystyle \sum _{j=1}^q{Q}_{sta,j}{N}_j+}{Q}_{sto,j}{V}_j$$

(42)

where $Q_{sta,j}$ and $Q_{sto,j}$ are the start-up and shut-down costs of the j-th unit, respectively.

5.2.3. Total Production Cost Calculation

The total production cost includes fuel costs, environmental costs, and dynamic costs:

$$C=C_{fuel}+C_d+C_{env}$$

(43)

where the environmental cost $C_{env}$ is calculated according to Equation (41):

$$C_{env}={\displaystyle \sum _{i=1}^m{E}_i}{Q}_{env,i}$$

(44)

where $E_i$ is the generation of unit i during the period, and $Q_{env,i}$ is the unit environmental cost for unit i, expressed in USD/(MW·h).

5.3. Wind Power Absorption Index Calculation

5.3.1. Wind Power Absorption Capacity

The wind power absorption capacity is the difference between the daily load demand and the minimum output of the generating units in the system. It is calculated as follows:

$$S_{wind}(t)=L(t)-{\displaystyle \sum _{i=1}^m{\rho }_{i,t}^{+}(0,1)p_{i\min }(t)P_{i\min }(t)}$$

(45)

where $p_{i\min }(t)$ and $P_{i\min }(t)$ are the minimum technical output and the probability of the minimum technical output for unit i, respectively. ${\rho }_{i,t}^{+}(0,1)$ is the load demand at time t.

5.3.2. Wind Curtailment Rate

The wind curtailment rate can assess the probability that, when the wind power output exceeds the wind absorption capacity, the excess wind power needs to be curtailed. The wind curtailment rate at time t in the system $p_a(t)$ is given by

$$p_a(t)={\displaystyle \sum _{P_j^q(t)>S_{wind}(t)}{p}_j^q(t)}$$

(46)

where $P_j^q(t)$ and $p_j^q(t)$ are the corresponding output state and probability of wind power.

5.3.3. Expected Wind Curtailment Energy

The expected wind curtailment energy at time t in the system $W_a(t)$ is given by

$$W_a(t)={\displaystyle \sum _{P_j^q(t)>S_{wind}(t)}{p}_j^q(t)}(P_j^q(t)-S_{wind}(t))$$

(47)

6. Case Study

To further verify the effectiveness of the algorithm in simulating the peak shaving characteristics of coal-fired units and the integration of wind power in large-scale systems, the case study uses the IEEE-RTS79 test system, which consists of 32 units with a total installed capacity of 3405 MW [32]. The operating parameters and corresponding costs of the generators are shown in

Table 2. Unit operation parameters.

Number	Capacity (MW)	Number of Units	MTTR (h)	MTTF (h)	Qfuel (USD/MW)	Qenv (USD/MW)	Scheduled Maintenance (Week/Year)
1	12	5	60	2940	27.6	5	2
2	20	4	50	450	43.5	2.5	2
3	50	6	20	1980	12.4	12	2
4	76	4	40	1960	14.4	12	3
5	100	3	50	1200	23	5	3
6	155	4	40	960	11.64	12	4
7	197	3	50	950	22.08	5	4
8	350	1	100	1150	11.4	12	5
9	400	2	150	1100	6	0.4	6

. All power generation units, except for nuclear plants, are replaced with thermal power units. The maximum annual load is 2850 MW.

6.1. Coal-Fired Unit Peak Shaving Fault Model

The relationship between peak shaving depth, load variation rate, and unit lifespan is fitted using the MLP algorithm. The hyperparameters are set as follows: two hidden layers are set, each with 20 neurons; the activation function is rectified linear unit; the initial learning rate is 0.01; the optimizer is Adam optimizer; the maximum number of iterations is 100; and the batch size is 200. Using these hyperparameters, the simulation data from the literature are fitted, with a final fitting degree of 0.9971.

6.2. Algorithm Verification

When the peak shaving depth of the unit model is set to 0%, the unit model becomes a two-state model. To verify the rationality of the proposed method, three methods are compared for the original system and the original load curve: (1) the equivalent load duration curve method; (2) the simplified model of the proposed method (0% peak shaving depth); and (3) the proposed method (2% ramp rate, 60% peak shaving depth). The simulation results are shown in

Table 3. Comparison of random production simulation results using different methods.

Number	1	2	3	4	5	6	7	8	9	LOLP	EENS
Method 1 (GWh)	1.15	0.89	2594.59	680.45	18.64	3002.4	333.29	2521.74	6142.75	0.011	1.18
Method 2 (GWh)	1.15	0.89	2594.59	680.45	18.64	3002.4	333.29	2521.74	6142.75	0.011	1.18
Method 3 (GWh)	4.90	5.45	2594.59	655.93	46.98	2929.28	374.4	2521.65	6142.74	0.0158	20.56

.

According to the results in Table 3, when the peak shaving depth is set to 0% and internal constraints such as ramp rates are not considered, the algorithm proposed in this paper yields simulation results similar to the traditional equivalent load duration curve method, and the computational efficiency of the proposed method is improved by about two times, which verifies the effectiveness of the proposed algorithm. Further analysis shows that when the peak shaving depth is set to 60%, the ramp rate is 2%, and internal unit constraints are considered, the power generation of the base-load unit is similar to that of the traditional model. However, the power generation of the flexible peak shaving units shows greater fluctuations compared to the traditional simulation methods, while the power generation of the start–stop peak shaving units is generally higher than that of the traditional methods. This is because the flexible peak shaving units, when responding to load demands, take into account their dynamic adjustment characteristics, adjusting the output flexibly according to actual needs. However, due to constraints on the unit parameters, state transitions are limited, leading to larger fluctuations in power generation. Additionally, due to the limitations of ramping capacity, peak shaving units cannot respond quickly to load changes, forcing start–stop peak shaving units to respond more frequently, which results in higher power generation compared to the units in the traditional method.

Furthermore, from the perspectives of LOLP and EENS, the values obtained using the proposed method are higher than those calculated by traditional simulation methods. This can be attributed to the fact that our method incorporates the impact of losses occurring during flexible deep load-following operations, thereby better reflecting the operational constraints faced by generating units in real-world scenarios. Unlike traditional models, which assume that units consistently operate at their rated output, the proposed approach fully captures the dynamic adjustments of units during load-following operations. As a result, the obtained reliability indices more accurately align with actual system behavior.

6.3. Impact of Different Peak Shaving Schemes

This section analyzes the impact of unit peak shaving operation on the system under different peak shaving schemes. The range of the unit’s load change rate is set from 1% to 5%, and the peak shaving depth is set to 90%, 80%, 70%, 60%, 50%, and 40%. Based on the integration of a 300 MW wind turbine, different peak shaving schemes are obtained by pairing the various peak shaving parameters. PPS is then conducted to study the system’s reliability, economic performance, and renewable energy absorption capacity.

Analysis of Simulation Results Based on LOLP, Expected Wind Curtailment, Start–Stop Counts, and Generation Costs: As shown in Figure 6, under varying ramp rates, the LOLP increases with deeper peak shaving depths. This is due to the fact that as the peak shaving depth increases, the units operate outside of their design conditions, significantly affecting their reliability. Additionally, it can be observed that when the ramp rate is between 2% and 3%, the LOLP decreases. This is because an increase in ramp rate allows the units to respond to load demand more promptly while ensuring reliable operation. However, as the ramp rate rises from 4% to 5%, the LOLP increases again. The larger fluctuations in output at higher ramp rates can severely compromise the unit’s operational reliability, reduce its lifespan, and consequently affect the overall system’s reliability.

As shown in Figure 7, the expected wind curtailment decreases with an increase in peak shaving depth. This is because a deeper peak shaving depth creates more space for wind power absorption, effectively facilitating the integration of renewable energy. It is also observed that when the ramp rate is between 2% and 3%, compared to 1%, the wind curtailment is relatively alleviated. The flexibility in the ramping capability of the units is an important factor in ensuring wind power absorption. However, when the ramp rate is set to 4% to 5%, the expected wind curtailment increases again. The severe fluctuations in load demand at higher ramp rates significantly impact the unit’s thermal operation, thereby hindering the absorption of wind power.

With an increase in peak shaving depth, the start–stop frequency of the units also rises. The larger the peak shaving depth, the greater the load variation range of the units, making it more difficult for flexible peak shaving units to complete peak shaving tasks within the required time. As a result, start–stop peak shaving units must respond to load demands by starting and stopping promptly. It can be observed from Figure 8 that the start–stop frequency is minimized when the ramp rate is between 2% and 3%, as the units can maintain both flexibility and reliability at this ramp rate. Moreover, before reaching a 50% peak shaving depth, the start–stop frequency is the lowest at a 4% ramp rate. At this point, the load adjustment range of the units is relatively small, and the 4% ramp rate enables the units to adjust flexibly within a certain range, thus reducing the frequency of starts and stops.

As illustrated in Figure 9, the generation cost of the system decreases with an increase in the peak shaving depth of the random units. This is because, on one hand, the units can operate at a reduced output to meet load demands, and on the other hand, the higher peak shaving depth leads to an increased failure rate, forcing the units to operate at a lower output. Additionally, before reaching a 65% peak shaving depth, the generation cost increases with the ramp rate, as the units operate in a high-load adjustment state. A higher ramp rate results in increased fuel consumption, thereby raising the generation cost. However, when the peak shaving depth exceeds 65%, an excessively high ramp rate reduces the unit’s reliability, decreases generation output, and increases dynamic costs, ultimately leading to a reduction in the generation cost.

Based on the above analysis, the system operates most optimally when the ramp rate is between 2% and 3% and the peak shaving depth ranges from 55% to 65%. This configuration closely aligns with practical scenarios, thus validating the reasonableness of the proposed algorithm.

7. Conclusions

In this study, we proposed an improved PPS method that integrates coal-fired unit degradation dynamics and wind power fluctuations within a multi-state reliability framework. Unlike traditional methods, our approach dynamically updates state transition probabilities based on operating conditions, ramping constraints, unit degradation, and load demand, leading to a more accurate and computationally efficient reliability assessment. The UGF framework was employed to optimize computational efficiency and overcome state-space explosion issues.

The simulation results of the IEEE-RTS79 test system demonstrate that the proposed method more accurately reflects real-world operational conditions. For conventional generating units, system reliability is improved when the ramp rate is set between 2 and 3% and the load-following depth is maintained at 55–65%. However, if the ramp rate is increased from 1% to 5%, the LOLP increases by 48%, confirming that forcibly enhancing a unit’s load-following capability leads to a higher failure rate, thereby exposing the entire system to greater risk.

While this study focuses on wind power integration, the method can be extended to solar power and hybrid renewable systems. Future research will explore economic factors, electricity market pricing, and further real-world validations. This approach provides a more practical and efficient tool for reliability assessment in modern power systems.