A Bi-Level Peak Regulation Optimization Model for Power Systems Considering Ramping Capability and Demand Response

Abstract

In the context of constructing new power systems, the intermittency and volatility of high-penetration renewable generation pose new challenges to the stability and secure operation of power systems. Enhancing the ramping capability of power systems has become a crucial measure for addressing these challenges. Therefore, this paper proposes a bi-level peak regulation optimization model for power systems considering ramping capability and demand response, aiming to mitigate the challenges that the uncertainty and volatility of renewable energy generation impose on power system operations. Firstly, the upper-level model focuses on minimizing the ramping demand caused by the uncertainty, taking into account concerned constraints such as the constraint of price-guided demand response, the constraint of satisfaction with electricity usage patterns, and the constraint of cost satisfaction. By solving the upper-level model, the ramping demand of the power system can be reduced. Secondly, the lower-level model aims to minimize the overall cost of the power system, considering constraints such as power balance constraints, power flow constraints, ramping capability constraints of thermal power units, stepwise ramp rate calculation constraints, and constraints of carbon capture units. Based on the ramping demand obtained by solving the upper-level model, the outputs of the generation units are optimized to reduce operation cost of power systems. Finally, the proposed peak regulation optimization model is verified through simulation based on the IEEE 39-bus system. The results indicate that the proposed model, which incorporates ramping capability and demand response, effectively reduces the comprehensive operational cost of the power system.

1. Introduction

The integration of high proportions of renewable energy into power systems poses unprecedented challenges to their secure and stable operation. The variability and intermittency of generation outputs from renewable energy units significantly increase the demand for peak regulation (PR) [1]. However, with the gradual reduction in the share of traditional thermal power generation, the PR ability of the power system has become constrained, making it difficult to rapidly respond to changing load demands and the uncertainty of renewable energy outputs. This not only reduces the economic efficiency and reliability of the power system but also raises concerns about the power system stability [2,3]. Consequently, enhancing PR ability of the power system is essential to better accommodate the variability and intermittency of renewable energy, improve response speed and flexibility, and ensure the secure and stable operation of the power system.

Flexible ramping capacity of the power system refers to the capability of the power system to respond to sustained disturbances caused by net load fluctuations and uncertainties within a 5–15 min timescale. Numerous power markets, both domestically and internationally, are actively exploring and developing market-based mechanisms for procuring ramping flexibility of the power system. Zhang et al. [4] proposed a comprehensive bidding strategy for virtual power plants to participate in power markets, addressing the synergy between flexible ramping product (FRP) and energy markets and managing wind power uncertainty through a two-stage distributionally robust optimization model and the use of flexible resources. Chen et al. [5] proposed a stochastic economic dispatch model incorporating the trading of FRPs to enhance system flexibility and ensured fair compensation for flexible resource providers, validated through wind power scenario simulations generated using the copula function and quasi-Monte Carlo methods. To address the limitations of current FRP models, such as the neglect of spatiotemporal correlations among uncertainty sources and the omission of security constraints like transmission limits, Fang et al. [6] proposed a deliverable FRP approach based on a distributionally robust chance-constrained multi-interval optimal power flow model that considers the spatiotemporal correlation of wind power and demand uncertainties and introduced a novel uncertainty-contained locational marginal price derived from the proposed asymmetrical affine policy. Kwon et al. [7] proposed a flexible ramping capacity model that accounted for the practical ramping capability of generation resources and net load uncertainty, integrating ramping constraints into unit commitment and economic dispatch processes. Yang et al. [8] proposed a day-ahead optimal dispatch model for a coupled system of renewable energy sources and thermal power units, considering ladder-type ramping rate and flexible spinning reserve constraints.

To address the challenges posed by the increased load demand and the integration of renewable energy, extensive research has been conducted on the peak regulation ability of the power system. Wei et al. [9] proposed an optimization model for the deep PR, addressing the challenges posed by the large-scale integration of wind power. The proposed model minimized thermal power unit fuel injection cost by considering energy storage accident standby and unplanned disconnection risks. Yang et al. [10] proposed a two-layer optimization dispatching model to enhance the utilization of renewable energy by addressing the imbalance in PR resources among provincial power grids. Wu et al. [11] proposed a deep PR optimization model for wind and thermal power based on cooperative game theory, by introducing a modified Shapley value method for profit allocation that considers cooperative fairness and deep PR ability. Yue et al. [12] introduced a collaborative optimization method combining electrolytic aluminum load regulation with thermal-power deep peak shaving to reduce renewable energy curtailment. Yang et al. [13] proposed a unit commitment model, which aimed at minimizing total cost by employing the particle swarm optimization algorithm. In addition, the trade-off between deep PR of thermal units and wind power curtailment was explored.

Existing studies have made significant progresses in power system PR optimization and ramping capacity analysis, but there are still some research gaps. Firstly, the impact of customer-side demand response on the ramping demand and its positive effect on the reduction in the ramping demand are seldom considered in the existing ramping demand and ramping capacity assessment methods. Secondly, the existing models on PR optimization do not fully consider the uncertainty of the ramping demand due to load forecast errors and renewable energy forecast errors.

To address the above deficiencies, this paper proposes the calculation method for the ramping demand of the power system and a two-layer optimization model for the PR considering the ramping capacity and demand response. The proposed model not only considers the uncertainty in the calculation of the ramping demand and the impact of customer-side demand response on the ramping demand but also takes into account the positive effects of the energy storage and carbon capture units on reducing the comprehensive operation cost of the power system. The upper and lower models proposed in this paper are formulated as the linear programming and mixed-integer second-order conic programming problems, respectively, which are efficiently solved using the GUROBI 11.0.2 solver on the AMPL modeling platform.

2. Calculation Method for Ramping Demand

The ramping demand in power systems is typically determined by the predicted net load variation and forecast errors, as illustrated in Figure 1.

The x-axis of Figure 1 represents the time periods, and the y-axis represents the net load. The ramping demand in a power system primarily includes ramping-up and ramping-down demands. In Figure 1, the net load variation refers to the difference in the net load between two adjacent time periods. In the actual operation of power systems, to effectively take into account the volatility and uncertainty of power demands and generation outputs of renewable energy, it is necessary to determine the maximum ramping-up and ramping-down demands between two adjacent time periods, as indicated by the shaded areas in Figure 1.

2.1. Deterministic Ramping Demand

The deterministic ramping-up and ramping-down demands are determined by the net load variation. The net load can be defined as the difference between the power demand and the generation outputs of the renewable energy. The calculation method of the net load considering the demand response is shown in (1).

$$P_t^{\mathrm{net},\mathrm{d}}=P_t^{\mathrm{d},\mathrm{pre}}+\Delta P_t^{\mathrm{L}}-P_t^{\mathrm{res},\mathrm{pre}}\forall t\in {\Omega }^{\mathrm{T}}$$

(1)

where ${\Omega }^{\mathrm{T}}$ represents the set of the time horizons, which is set to 24 h (one day) in this study. $P_t^{\mathrm{net},\mathrm{d}}$ represents the net load of the power system at time period t. $P_t^{\mathrm{d},\mathrm{pre}}$ represents the forecasted power demand of the power system at time period t. $\Delta P_t^{\mathrm{L}}$ represents the power changes due to demand response at time period t. $P_t^{\mathrm{res},\mathrm{pre}}$ represents the forecasted outputs of renewable energy in the power system at time period t.

The calculation methods for the deterministic ramping-up and ramping-down demands of the power system are shown in (2)–(4):

$$\Delta P_t^{\mathrm{net},\mathrm{d}}=P_{t+1}^{\mathrm{net},\mathrm{d}}-P_t^{\mathrm{net},\mathrm{d}}\forall t\in {\Omega }^{\mathrm{T}}$$

(2)

$$R_t^{\mathrm{up},\mathrm{d}}=\max (0,\Delta P_t^{\mathrm{net},\mathrm{d}})\forall t\in {\Omega }^{\mathrm{T}}$$

(3)

$$R_t^{\mathrm{down},\mathrm{d}}=\left|\min (0,\Delta P_t^{\mathrm{net},\mathrm{d}})\right|\forall t\in {\Omega }^{\mathrm{T}}$$

(4)

where $\Delta P_t^{\mathrm{net},\mathrm{d}}$ represents the net load variation in the power system at time period t. $R_t^{\mathrm{up},\mathrm{d}}$ and $R_t^{\mathrm{down},\mathrm{d}}$ represent the deterministic ramping-up demand and ramping-down demand of the power system at time period t, respectively.

Equation (2) represents the calculation method for the net load variation. Equations (3) and (4) represent the calculation method of the deterministic ramping-up demand and ramping-down demand.

2.2. Ramping Demand Considering Uncertainty

The uncertainty of the ramping demand primarily arises from load forecasting errors and renewable generation output forecasting errors in the power system. If the deterministic ramping demand belongs to ramping-up demand, the method for calculating the ramping-up demand considering the uncertainty is presented by (5):

$$R_t^{\mathrm{up},\mathrm{un}}=R_t^{\mathrm{up},\mathrm{d}}+P_t^{\mathrm{d},\mathrm{pre},\mathrm{up}}+P_t^{\mathrm{res},\mathrm{pre},\mathrm{down}}\forall t\in {\Omega }^{\mathrm{T}}$$

(5)

where $R_t^{\mathrm{up},\mathrm{un}}$ represents the ramping-up demand of the power system considering uncertainty at time period t. $P_t^{\mathrm{d},\mathrm{pre},\mathrm{up}}$ represents the upper bound of the load forecasting error for the power system at time period t. $P_t^{\mathrm{res},\mathrm{pre},\mathrm{down}}$ represents the lower bound of the renewable generation output forecasting error for the power system at time period t.

If the deterministic ramping demand of the power system is the ramping-down demand, the method for calculating the ramping-down demand considering the uncertainty is given by (6).

$$R_t^{\mathrm{down},\mathrm{un}}=R_t^{\mathrm{down},\mathrm{d}}+P_t^{\mathrm{d},\mathrm{pre},\mathrm{down}}+P_t^{\mathrm{res},\mathrm{pre},\mathrm{up}}\forall t\in {\Omega }^{\mathrm{T}}$$

(6)

where $R_t^{\mathrm{down},\mathrm{un}}$ represents the ramping-down demand of the power system considering uncertainty at time period t. $P_t^{\mathrm{d},\mathrm{pre},\mathrm{down}}$ represents the lower bound of the load forecasting error for the power system at time period t. $P_t^{\mathrm{res},\mathrm{pre},\mathrm{up}}$ represents the upper bound of the renewable generation output forecasting error for the power system at time period t.

In this work, extensive historical data on load demands and renewable energy generation outputs are employed to determine the lower bound and the upper bound of the load forecasting error and the renewable generation output forecasting error. The probability distribution function of forecasting errors at each time period can be derived from the historical data. At each time period, the lower and upper bounds of the forecasting errors are defined as the 5th and 95th percentiles of the corresponding probability distribution functions.

3. Bi-Level Peak Regulation Optimization Model

3.1. Upper-Level Model

The objective function of the upper-level model is to minimize the ramping demand of the power system, as shown in (7).

$$\min {\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}(R_t^{\mathrm{up},\mathrm{un}}+R_t^{\mathrm{down},\mathrm{un}})}$$

(7)

The following constraints should be satisfied:

• Constraints of the load adjustment amount

$${\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}\Delta P_t^{\mathrm{L}}=0}$$

(8)

Equation (8) indicates that the sum of load adjustment at the scheduling time periods equals zero, which means that the total load demand remains unchanged.

2.

Constraints of price-driven demand response

The price-driven demand response strategy uses price signals to guide users to adjust their electricity consumption patterns, thereby optimizing the load curve. By introducing the demand response, peak shaving and valley filling can be achieved, and peak regulation can be alleviated. The proposed price-driven demand response model can be described as follows:

$${\epsilon }_{t,\tau }={\displaystyle \frac{\Delta P_t^{\mathrm{L}}}{P_t^{\mathrm{d},\mathrm{pre}}}}{\displaystyle \frac{D_{\tau }^{\mathrm{L}0}}{\Delta D_{\tau }}}\forall \tau ,t\in {\Omega }^{\mathrm{T}}$$

(9)

$$D_t^{\min }\le D_t^{\mathrm{L}0}+\Delta D_t\le D_t^{\max }\forall t\in {\Omega }^{\mathrm{T}}$$

(10)

where ${\epsilon }_{t,\tau }$ represents the demand price elasticity coefficient between time periods t and τ. $D_{\tau }^{\mathrm{L}0}$ represents the electricity price at time period τ before customers participate in demand response. $\Delta D_{\tau }$ represents the change in electricity price at time period τ after customers participate in demand response. $D_t^{\min }$ and $D_t^{\max }$ represent the minimum and maximum electricity prices at time period t after customers participate in demand response, respectively.

Equation (9) represents the calculation method for the demand price elasticity coefficient. Equation (10) indicates that the electricity price after customers participate in demand response remains within a certain range.

3.

Constraints of electricity usage satisfaction and electricity cost satisfaction

$$1-{\displaystyle \frac{{\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}\left|\Delta P_t^{\mathrm{L}}\right|\Delta t}}{{\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}{P}_t^{\mathrm{d},\mathrm{pre}}\Delta t}}}\ge S^{w,\min }$$

(11)

$$1-{\displaystyle \frac{{\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}(P_t^{\mathrm{d},\mathrm{pre}}+\Delta P_t^{\mathrm{L}})(D_t^{\mathrm{L}0}+\Delta D_t)-P_t^{\mathrm{d},\mathrm{pre}}{D}_t^{\mathrm{L}0}}}{{\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}{P}_t^{\mathrm{d},\mathrm{pre}}{D}_t^{\mathrm{L}0}}}}\ge S^{p,\min }$$

(12)

where $S^{w,\min }$ and $S^{p,\min }$ represent the minimum values of electricity usage satisfaction and electricity cost satisfaction, respectively.

Equation (11) represents the constraint of electricity usage satisfaction. Equation (12) represents the constraint of electricity cost satisfaction.

3.2. Lower-Level Model

The objective function of the lower-level model is to minimize the total cost of the power system, as shown in (13).

$$\min F=F^P+F^O+F^V+F^Y$$

(13)

$$F^P={\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}({\displaystyle \sum _{g\in G_i}(a_g{(P_{g,t}^{\mathrm{G}})}^2+b_g{P}_{g,t}^{\mathrm{G}}}}+c_g))\forall i\in N$$

(14)

$$F^O={\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}{\displaystyle \sum _{i\in N}{\displaystyle \sum _{g\in G_i}(x_{g,t}(1-x_{g,t-1})S_g)}}}$$

(15)

$$F^V={\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}{K}^s\phi E_{c,t}^J}\forall c\in C_i,i\in N$$

(16)

$$F^Y=c_t^p{\displaystyle \sum _{t\in {\Omega }^{\mathrm{T}}}{\displaystyle \sum _{g\in G_i}(m_g{P}_{g,t}^{\mathrm{G}}-}\sigma P_{g,t}^{\mathrm{G}}-E_{c,t})}\forall i\in N,c\in C_i$$

(17)

where G_i represents the set of generators connected to bus i. N denotes the set of buses. C_i represents the set of carbon capture units connected to bus i. F is the total cost of the power system. F^P is the operation cost of thermal power units. F^O is the start-up cost of thermal power units. F^V is the cost of ethanolamine solution. F^Y is the carbon trading cost. a_g, b_g, and c_g represent the cost coefficients for power generation by generator g at time period t. $P_{g,t}^{\mathrm{G}}$ represents the active power output of generator g at time period t. $x_{g,t}$ is a binary variable representing the operational status of generator g at time period t, which equals 1 if the generator is in operation during time period t. S_g denotes the startup cost of generator g. K^S represents the cost coefficient for ethanolamine solution. $\phi$ represents the operation coefficient of ethanolamine solution. $E_{c,t}^J$ represents the net CO₂ capture amount of carbon capture unit c at time period t. $c_t^p$ represents the carbon trading price at time period t. m_g represents the carbon emission intensity coefficient. $\sigma$ represents the carbon trading coefficient. $E_{c,t}$ represents the total CO₂ capture amount of carbon capture unit c at time period t.

The following constraints should be satisfied:

• Power balance constraints

$${\displaystyle \sum _{g\in G_i}{P}_{g,t}^{\mathrm{G}}}+{\displaystyle \sum _{c\in C_i}{P}_{c,t}^{\mathrm{J}}}+{\displaystyle \sum _{w\in W_i}{P}_{w,t}^{\mathrm{res}}}+{\displaystyle \sum _{ji\in {\Omega }^{\mathrm{L}}}{P}_{ji,t}^{\mathrm{L}}}={\displaystyle \sum _{ij\in {\Omega }^{\mathrm{L}}}{P}_{ij,t}^{\mathrm{L}}}+P_{i,t}^{\mathrm{D}}+P_{i,t}^{\mathrm{ch}}-P_{i,t}^{\mathrm{dch}}\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(18)

$${\displaystyle \sum _{g\in G_i}{Q}_{g,t}^{\mathrm{G}}}+{\displaystyle \sum _{ji\in {\Omega }^{\mathrm{L}}}{Q}_{ji,t}^{\mathrm{L}}}={\displaystyle \sum _{ij\in {\Omega }^{\mathrm{L}}}{Q}_{ij,t}^{\mathrm{L}}}+Q_{i,t}^{\mathrm{D}}\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(19)

where W_i represents the set of renewable energy units connected to bus i. ${\Omega }^{\mathrm{L}}$ represents the set of transmission lines. $P_{c,t}^{\mathrm{J}}$ represents the net output of carbon capture unit c at time period t. $P_{w,t}^{\mathrm{res}}$ represents the predicted active power output of renewable energy unit w at time period t. $P_{ij,t}^{\mathrm{L}}$ represents the actual active power flow through the line ij at time period t. $P_{i,t}^{\mathrm{D}}$ represents the active power demand at bus i at time period t. $P_{i,t}^{\mathrm{ch}}$ and $P_{i,t}^{\mathrm{dch}}$ represent the charging and discharging power of energy storage at bus i at time period t, respectively. $Q_{g,t}^{\mathrm{G}}$ represents the reactive power output of generator g at time period t. $Q_{ij,t}^{\mathrm{L}}$ represents the reactive power flow through the line ij at time period t. $Q_{i,t}^{\mathrm{D}}$ represents the reactive power demand at bus i at time period t.

Equations (18) and (19) represent the active power balance equation and the reactive power balance equation, respectively.

2.

Power flow constraints

$$P_{ij,t}^{\mathrm{L}}=-G_{ij}{e}_{i,t}+G_{ij}{c}_{ij,t}+B_{ij}{s}_{ij,t}\forall ij\in {\Omega }^{\mathrm{L}},t\in {\Omega }^{\mathrm{T}}$$

(20)

$$Q_{ij,t}^{\mathrm{L}}=(B_{ij}-b_{ij}/2)e_{i,t}-B_{ij}{c}_{ij,t}+G_{ij}{s}_{ij,t}\forall ij\in {\Omega }^{\mathrm{L}},t\in {\Omega }^{\mathrm{T}}$$

(21)

$$c_{ij,t}^2+s_{ij,t}^2\le e_{i,t}{e}_{j,t}\forall ij\in {\Omega }^{\mathrm{L}},t\in {\Omega }^{\mathrm{T}}$$

(22)

$${(P_{ij,t}^{\mathrm{L}})}^2+{(Q_{ij,t}^{\mathrm{L}})}^2\le {(S_{ij}^{\max })}^2\forall ij\in {\mathsf{\Omega }}^{\mathrm{L}},t\in {\Omega }^{\mathrm{T}}$$

(23)

where $G_{ij}$ and $B_{ij}$ represent the conductance and susceptance, respectively, which corresponds to the real part and imaginary part of the element in the i-th row and j-th column of the bus admittance matrix. $e_{i,t}$ is an auxiliary variable introduced to satisfy $e_{i,t}$ = $V_{i,t}^2$. $c_{ij,t}$ and $s_{ij,t}$ are auxiliary variables introduced to satisfy $c_{ij,t}$ = $V_{i,t}{V}_{j,t}\cos {\theta }_{ij,t}$ and $s_{ij,t}$ = $V_{i,t}{V}_{j,t}\sin {\theta }_{ij,t}$. b_ij represents the shunt susceptance of line ij. $S_{ij}^{\max }$ represents the maximum apparent power of line ij.

Equations (20) and (21) provide the calculation for power flow though the lines. Equation (22) represents the second-order cone relaxation constraint associated with auxiliary variables. Equation (23) specifies that the power flow though the lines remains within the threshold range.

3.

Voltage constraints

$${(V^{\min })}^2⩽e_{i,t}⩽{(V^{\max })}^2\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(24)

where $V^{\min }$ and $V^{\max }$ represent the minimum and maximum voltage values of bus i at time period t, respectively.

4.

Thermal power unit ramping capability constraints

$${\displaystyle \sum _{i\in N}{\displaystyle \sum _{g\in G_i}{R}_{g,t}^{\mathrm{cap},\mathrm{up}}}}\ge R_t^{\mathrm{up},\mathrm{un}}\forall t\in {\Omega }^{\mathrm{T}}$$

(25)

$${\displaystyle \sum _{i\in N}{\displaystyle \sum _{g\in G_i}{R}_{g,t}^{\mathrm{cap},\mathrm{down}}}}\ge R_t^{\mathrm{down},\mathrm{un}}\forall t\in {\Omega }^{\mathrm{T}}$$

(26)

$$R_{g,t}^{\mathrm{cap},\mathrm{up}}\le R_{g,t}^{\mathrm{up}}\forall i\in N,g\in G_i,t\in {\Omega }^{\mathrm{T}}$$

(27)

$$R_{g,t}^{\mathrm{cap},\mathrm{down}}\le R_{g,t}^{\mathrm{down}}\forall i\in N,g\in G_i,t\in {\Omega }^{\mathrm{T}}$$

(28)

$$R_{g,t}^{\mathrm{cap},\mathrm{up}}\le P_g^{\max }-P_{g,t}^{\mathrm{G}}\forall i\in N,g\in G_i,t\in {\Omega }^{\mathrm{T}}$$

(29)

$$R_{g,t}^{\mathrm{cap},\mathrm{down}}\le P_{g,t}^{\mathrm{G}}-P_g^{\min }\forall i\in N,g\in G_i,t\in {\Omega }^{\mathrm{T}}$$

(30)

where $R_{g,t}^{\mathrm{cap},\mathrm{up}}$ and $R_{g,t}^{\mathrm{cap},\mathrm{down}}$ represent the ramping-up and ramping-down capabilities of generator g at time period t, respectively. $R_{g,t}^{\mathrm{up}}$ and $R_{g,t}^{\mathrm{down}}$ represent the ramping-up and ramping-down rates of generator g at time period t, respectively. $P_g^{\mathrm{G},\max }$ and $P_g^{\mathrm{G},\min }$ represent the maximum and minimum values of the active power output of generator g, respectively.

Equation (25) indicates that the total ramping-up capability provided by all units should not be smaller than the ramping-up demand. Equation (26) specifies that ramping-up capability provided by all units should not be smaller than the ramping-down demand. Equation (27) indicates that the ramping-up capability of a unit does not exceed its ramping-up rate. Equation (28) indicates that the ramping-down capability of a unit does not exceed its ramping-down ramping rate. Equation (29) stipulates that the ramping-up capability of a unit does not exceed the difference between the unit’s maximum generation output and its current generation output. Equation (30) stipulates that the ramping-down capability of a unit does not exceed the difference between its current generation output and its minimum generation output.

5.

Stepwise ramping rate constraints of thermal power units

The different ramping rates of thermal power units under different peak-shaving conditions are considered in this paper, and the ramping-up and ramping-down rates are calculated by (31) and (32), respectively.

$$R_{g,t}^{\mathrm{up}}=\left\{\begin{array}{l}{R}_g^{1,\mathrm{up}},{\alpha }_1{P}_g^{\mathrm{G},\max }\le P_{g,t}^{\mathrm{G}}\le {\alpha }_2{P}_g^{\mathrm{G},\max }\\ R_g^{2,\mathrm{up}},{\alpha }_2{P}_g^{\mathrm{G},\max }\le P_{g,t}^{\mathrm{G}}\le {\alpha }_3{P}_g^{\mathrm{G},\max }\\ R_g^{3,\mathrm{up}},{\alpha }_3{P}_g^{\mathrm{G},\max }\le P_{g,t}^{\mathrm{G}}\le {\alpha }_4{P}_g^{\mathrm{G},\max }\end{array}\right.\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(31)

$$R_{g,t}^{\mathrm{down}}=\left\{\begin{array}{l}{R}_g^{1,\mathrm{down}},{\alpha }_1{P}_g^{\mathrm{G},\max }\le P_{g,t}^{\mathrm{G}}\le {\alpha }_2{P}_g^{\mathrm{G},\max }\\ R_g^{2,\mathrm{down}},{\alpha }_2{P}_g^{\mathrm{G},\max }\le P_{g,t}^{\mathrm{G}}\le {\alpha }_3{P}_g^{\mathrm{G},\max }\\ R_g^{3,\mathrm{down}},{\alpha }_3{P}_g^{\mathrm{G},\max }\le P_{g,t}^{\mathrm{G}}\le {\alpha }_4{P}_g^{\mathrm{G},\max }\end{array}\right.\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(32)

where ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, and ${\alpha }_4$ represent the power interval range parameters for different peak-shaving states of the thermal power unit. $R_g^{1,\mathrm{down}}$, $R_g^{2,\mathrm{down}}$, and $R_g^{3,\mathrm{down}}$ represent the ramping-up rates of the thermal power unit under different peak-shaving states.

Equations (31) and (32) are conditional constraints. To improve the solving efficiency of the proposed model, (31) and (32) can be linearized, as shown in (33)–(40), respectively:

$${\xi }_{g,t}^1+{\xi }_{g,t}^2+{\xi }_{g,t}^3=1\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(33)

$$R_{g,t}^{\mathrm{up}}=R_g^{1,\mathrm{up}}{\xi }_{g,t}^1+R_g^{2,\mathrm{up}}{\xi }_{g,t}^2+R_g^{3,\mathrm{up}}{\xi }_{g,t}^3\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(34)

$$P_{g,t}^{\mathrm{G}}\le {\alpha }_2{P}_g^{\mathrm{G},\max }{\xi }_{g,t}^1+{\alpha }_3{P}_g^{\mathrm{G},\max }{\xi }_{g,t}^2+{\alpha }_4{P}_g^{\mathrm{G},\max }{\xi }_{g,t}^3\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(35)

$$P_{g,t}^{\mathrm{G}}\ge {\alpha }_1{P}_g^{\mathrm{G},\max }{\xi }_{g,t}^1+{\alpha }_2{P}_g^{\mathrm{G},\max }{\xi }_{g,t}^2+{\alpha }_3{P}_g^{\mathrm{G},\max }{\xi }_{g,t}^3\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(36)

$${\zeta }_{g,t}^1+{\zeta }_{g,t}^2+{\zeta }_{g,t}^3=1\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(37)

$$R_{g,t}^{\mathrm{down}}=R_g^{1,\mathrm{down}}{\zeta }_{g,t}^1+R_g^{2,\mathrm{down}}{\zeta }_{g,t}^2+R_g^{3,\mathrm{down}}{\zeta }_{g,t}^3\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(38)

$$P_{g,t}^{\mathrm{G}}\le {\alpha }_2{P}_g^{\mathrm{G},\max }{\zeta }_{g,t}^1+{\alpha }_3{P}_g^{\mathrm{G},\max }{\zeta }_{g,t}^2+{\alpha }_4{P}_g^{\mathrm{G},\max }{\zeta }_{g,t}^3\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(39)

$$P_{g,t}^{\mathrm{G}}\ge {\alpha }_1{P}_g^{\mathrm{G},\max }{\zeta }_{g,t}^1+{\alpha }_2{P}_g^{\mathrm{G},\max }{\zeta }_{g,t}^2+{\alpha }_3{P}_g^{\mathrm{G},\max }{\zeta }_{g,t}^3\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(40)

where ${\xi }_{g,t}^1$, ${\xi }_{g,t}^2$, ${\xi }_{g,t}^3$, ${\zeta }_{g,t}^1$, ${\zeta }_{g,t}^2$, and ${\zeta }_{g,t}^3$ are binary variables introduced for linearizing (31) and (32).

6.

System reserve constraints

$${\displaystyle \sum _{i\in N}{\displaystyle \sum _{g\in G_i}{x}_{g,t}{P}_g^{\mathrm{G},\max }}}\ge {\displaystyle \sum _{i\in N}{P}_{i,t}^{\mathrm{D}}}+R_t^{\mathrm{D}}\forall t\in {\Omega }^{\mathrm{T}}$$

(41)

where $R_t^{\mathrm{D}}$ represents the required reserve capacity at time period t.

7.

Generation output constraints of thermal power units

$$x_{g,t}{P}_g^{\mathrm{G},\min }\le P_{g,t}^{\mathrm{G}}\le x_{g,t}{P}_g^{\mathrm{G},\max }\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(42)

Equation (42) indicates that the power generation outputs of the thermal power unit are not less than its minimum value and not greater than its maximum value.

8.

Logical constraints

$$y_{g,t}-z_{g,t}=x_{g,t}-x_{g,t-1}\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(43)

$$y_{g,t}+z_{g,t}\le 1\forall i\in N,g\in G_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(44)

$$x_{g,t}-x_{g,t-1}\le x_{g,t^{\prime }}\forall i\in N,g\in G_i,t^{\prime }\in [t+1,\min \{t+U{T}_x,T\}],t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(45)

$$x_{g,t-1}-x_{g,t}\le 1-x_{g,t^{\prime }}\forall i\in N,g\in G_i,t^{\prime }\in [t+1,\min \{t+D{T}_x,T\}],t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(46)

where $y_{g,t}$ is a binary variable indicating whether unit g starts up at time period t. $z_{g,t}$ is a binary variable indicating whether unit g shuts down at time period t. $U{T}_x$ represents the minimum up time duration for thermal power units. $D{T}_x$ represents the minimum down time duration for thermal power units. T is the number of time periods in the scheduling horizon.

Equation (43) represents the logical relationships among the three binary variables. Equation (44) indicates that the sum of the start-up state variable and the shut-down state variable for unit g does not exceed 1. Equations (45) and (46) enforces the minimum up time constraint and minimum down-time constraint, respectively.

9.

Operation constraints of carbon capture units

Due to the CO₂ capture function of carbon capture units, the power output of the carbon capture unit comprises the net power output of the carbon capture unit and the power due to the carbon capture consumption, as shown in (47).

$$P_{\mathrm{c},t}=P_{c,t}^{\mathrm{J}}+P_{c,t}^{\mathrm{B}}\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(47)

where $P_{\mathrm{c},t}$ represents the power output of carbon capture unit c at time period t. $P_{c,t}^{\mathrm{B}}$ represents power demand due to the carbon capture consumption.

The net power output of the carbon capture unit should meet the following constraints:

$$P_{\mathrm{c}}^{\min }\le P_{c,t}^{\mathrm{J}}\le P_{\mathrm{c}}^{\max }\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(48)

$$P_{\mathrm{c}}^{\min }-\lambda \eta {\delta }^{\max }{m}_c^{\mathrm{C}}{P}_{\mathrm{c}}^{\max }-P^{\mathrm{H}}\le P_{c,t}^{\mathrm{J}}\le P_{\mathrm{c}}^{\max }-P^{\mathrm{H}}\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(49)

where $P_{\mathrm{c}}^{\min }$ and $P_{\mathrm{c}}^{\max }$ represent the upper and lower limits of net power output of the carbon capture unit c, respectively. λ is the energy consumed per unit of CO₂ captured. $m_c^{\mathrm{C}}$ is the carbon emission intensity of the carbon capture unit. η is the carbon capture efficiency. ${\delta }^{\max }$ is the maximum flue gas diversion ratio at time t. $P^{\mathrm{H}}$ is the fixed power demand of the carbon capture unit.

The power demand due to the carbon capture consumption for the carbon capture unit should meet the following constraints:

$$P_{c,t}^{\mathrm{B}}=P_{c,t}^{\mathrm{r}}+P^{\mathrm{H}}\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(50)

$$P_{c,t}^{\mathrm{r}}=\lambda E_{c,t}^{\mathrm{J}}\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(51)

$$E_{\mathrm{c},t}=m_c^{\mathrm{C}}{P}_{c,t}^{\mathrm{r}}\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(52)

$$E_{c,t}^{\mathrm{J}}=\eta {\delta }_t{E}_{\mathrm{c},t}+E_t^v\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(53)

$$0\le E_{c,t}^{\mathrm{J}}\le \gamma \eta m_c^{\mathrm{C}}{P}_{\mathrm{c},t}\forall i\in N,c\in C_i,t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(54)

$$V_t^{\mathrm{ca}}=E_t^v{M}^v/(M^{{\mathrm{CO}}_2}\theta {\rho }^v{\delta }^v)\forall t\in {\mathsf{\Omega }}^{\mathrm{T}}$$

(55)

where $P_{c,t}^{\mathrm{r}}$ represents the operating power demand of the carbon capture unit at time period t. $E_t^v$ represents the amount of CO₂ provided by the storage tank at time period t. ${\delta }_t$ represents the split ratio of the carbon capture plant at time period t. $V_t^{ca}$ represents the volume of solution required to capture the CO₂ provided by the storage tank at time period t. $M^v$ and $M^{{\mathrm{CO}}_2}$ represent the molar masses of ethanolamine solution and CO₂, respectively. $\theta$ represents the desorption amount in the regenerator. ${\rho }^v$ represents the concentration of the ethanolamine solution. ${\delta }^v$ represents the density of the ethanolamine solution.

Equation (50) indicates that the power demand due to the carbon capture consumption of the carbon capture unit is equal to the sum of operating power demand and fixed power demand of the carbon capture unit. Equation (51) calculates the operating power demand of the carbon capture unit. Equation (52) shows that the total CO₂ capture amount of carbon capture unit is derived from the product of the carbon emission intensity and operating power demand of the carbon capture unit. Equation (53) indicates that the net CO₂ capture amount of carbon capture unit c at time period t is equal to the sum of CO₂ capture amount of carbon capture unit c itself and the amount of CO₂ supplied by the storage tank at time period t. Equation (54) specifies the minimum and maximum limits of the net CO₂ capture amount of carbon capture unit c at time period t. Equation (55) calculates the volume of solution required to capture the CO₂ provided by the storage tank at time period t.

10.

Operation constraints of energy storage

$$E_{i,t}^{\mathrm{ES}}=E_{i,t-1}^{\mathrm{ES}}+\left(P_{i,t}^{\mathrm{ch}}{\eta }_i^{\mathrm{ch}}-{\displaystyle \frac{P_{i,t}^{\mathrm{dch}}}{{\eta }_i^{\mathrm{dch}}}}\right)\Delta t\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(56)

$$x_i^{\mathrm{ES}}{E}_i^{\mathrm{ES},\min }\le E_{i,t}^{\mathrm{ES}}\le x_i^{\mathrm{ES}}{E}_i^{\mathrm{ES},\max }\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(57)

where $E_{i,t}^{\mathrm{ES}}$ denotes the energy of the energy storage station at bus i at time period t. ${\eta }_i^{\mathrm{ch}}$ and ${\eta }_i^{\mathrm{dch}}$ represent the charging and discharging efficiencies of the energy storage station, respectively. $\Delta t$ is the length of each time period. $E_i^{\mathrm{ES},\max }$ and $E_i^{\mathrm{ES},\min }$ are the upper and lower limits of the energy capacity of the energy storage station, respectively.

Equation (56) calculates the energy of the energy storage station at bus i at time period t. Equation (57) shows the minimum and maximum energy levels of the energy storage station.

The charging and discharging power of the energy storage system should satisfy the following constraints:

$$0\le P_{i,t}^{\mathrm{ch}}\le y_{i,t}^{\mathrm{E},\mathrm{ch}}{P}_i^{\mathrm{ch},\max }\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(58)

$$0\le P_{i,t}^{\mathrm{dch}}\le y_{i,t}^{\mathrm{E},\mathrm{dch}}{P}_i^{\mathrm{dch},\max }\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(59)

In the equations, the notation is as follows: $P_i^{\mathrm{ch},\max }$ and $P_i^{\mathrm{dch},\max }$ represent the upper limits for charging and discharging rates of the energy storage station, respectively. $y_{i,t}^{\mathrm{E},\mathrm{ch}}$ is a binary variable indicating the charging state of the energy storage station at time period t, which equals 1 if the device is charging. $y_{i,t}^{\mathrm{E},\mathrm{dch}}$ is a binary variable indicating the discharging state of the energy storage station at time period t, which equals 1 if the device is discharging. Variables $y_{i,t}^{\mathrm{E},\mathrm{ch}}$ and $y_{i,t}^{\mathrm{E},\mathrm{dch}}$ should satisfy the following constraint.

$$y_{i,t}^{\mathrm{E},\mathrm{ch}}+y_{i,t}^{\mathrm{E},\mathrm{dch}}\le 1\forall i\in N,t\in {\Omega }^{\mathrm{T}}$$

(60)

Equation (60) represents the logical constraint for charging and discharging states, which means that the energy storage station cannot be simultaneously in charging and discharging states.

$$E_{i,T}^{\mathrm{ES}}=E_{i,1}^{\mathrm{ES}}\forall i\in N$$

(61)

where $E_{i,1}^{\mathrm{ES}}$ and $E_{i,T}^{\mathrm{ES}}$ represent the energy of the energy storage station at the first and last time period in the scheduling horizon, respectively.

Equation (61) ensures that the energy of the energy storage station remains unchanged at the first and last time period in the scheduling horizon.

4. Case Study

In this section, an improved IEEE 39-bus test system is employed for simulation analysis to demonstrate the effectiveness of the proposed method. The program is performed on the AMPL platform, and the commercial solver GUROBI 11.0.2 is employed to solve the model. The simulation is conducted on a desktop computer with an Intel(R) Core(TM) i7-10700 processor and 16 GB of memory. The total computational time of the proposed method for the 39-bus power system is 75.95 s.

4.1. Case Study Parameters

The time resolution considered in the model in this paper is 15 min, resulting in a total of 96 periods. The modified IEEE 39-bus system includes nine thermal power units, one carbon capture unit, one wind farm, and three energy storage stations. The topology of the modified IEEE 39-bus system is illustrated in Figure 2. The relevant parameters of the thermal power units are detailed in

Table 1. Parameters of thermal power units.

Units	Maximum Output/MW	Minimum Output/MW	Fuel Cost			Carbon Emission Intensity/(t/(MW·h))
			a/($/MW2)	b/($/MW)	c/$
1	646	60	0.00198	16.7	900	0.9
2	725	70	0.00198	16.7	900	0.9
3	652	60	0.00712	22.3	370	0.98
4	508	50	0.0022	25.5	600	1.05
5	687	60	0.00198	16.7	900	1.05
6	580	50	0.00712	22.3	370	0.95
7	564	50	0.00712	22.3	370	0.98
8	865	80	0.00198	16.7	900	1.05
9	1100	100	0.00048	16.2	1000	1.05

, and the parameters of the carbon capture unit are provided in

Table 2. Parameters of carbon capture units.

Parameters	Value
λ/(MW·h/t)	0.269
φ/(kg/t)	1.5
Ks/($/kg)	8.2
$c_t^p$/($/t)	100
$M^{{\mathrm{CO}}_2}$/(g/mol)	44
$M^v$/(g/mol)	61.07
η	0.9
γ	1.05
θ/(mol)	0.24
ρv/%	30
δv/(g/mol)	1.01

.

4.2. Optimization Results of Price-Driven Demand Response

This section presents the optimization results of price-driven demand response. The net load curves before and after demand response are shown in Figure 3. In Figure 3, the black line represents the net load before demand response, while the red line represents the net load after demand response. It should be pointed out that the black lines and red lines overlap at some time periods; this is because the net loads before demand response and after demand response are identical at these time periods. The net load change and electricity price change before and after demand response are shown in Figure 4. It is seen from Figure 3 and Figure 4 that the demand response can effectively smooth the net load of the power system. During peak load periods, the increase in electricity prices leads to a reduction in peak load through demand-side response. During low load periods, the decrease in electricity prices encourages the load demand to increase. Therefore, the total ramping demand can be reduced, and the pressure on peak regulation can be alleviated.

4.3. Ramping Demand and Peak Regulation Optimization Results

The ramping-up and ramping-down demand curves obtained by using the method proposed in this paper are shown in Figure 5.

It is seen from Figure 5 that there are ramping-up demands at the time periods [11, 46], [50, 58], and [62, 80] and there are ramping-down demands at the time periods [0, 11], [46, 50], [58, 62], and [80, 96]. The ramping-up and ramping-down demands are determined by the combination of the power load curve, the wind power output curve, and concerned predicted errors. For example, at the time period [11, 14], the power demand increases while the wind power generation output decreases. This results in the ramping-up demand in the power system, which shows a gradually increasing trend. At the time period [15, 18], both the power load and wind power generation output increase. Since the increase in power load is greater than the increase in wind power generation output, the power system still has the ramping-up demand, which shows a gradually decreasing trend. The generation output curve of selected thermal power units obtained by using the proposed method is shown in Figure 6. It can be seen from Figure 6 that at the time period [1, 20], the overall generation outputs of thermal power units show a declining trend due to the decrease in the load demand in the power system in this period. At the time period [30, 46], the generation outputs of most thermal power units show an increasing trend, which are mainly attributed to the increase in the load demand in the power system in this period.

The charging and discharging power curves of energy storage stations are shown in Figure 7. It is seen from Figure 7 that the energy storage stations primarily engage in charging at the nighttime periods and discharging at the peak time periods. In the nighttime, the residential and commercial power demand is relatively low, the energy storage stations charge, and the stored energy is used to discharge in peak periods in the daytime. At the peak time periods, the power demand increases, and the energy storage stations discharge to meet the power demand and reduce the impact of the peak loads on the secure operation of power systems. Note that at the time period 60, most energy storage stations begin to charge and then discharge at the time period 72. The charging action at the time periods [60, 72] helps to ensure that sufficient energy is available for discharging in the peak load period following the time period 72 to alleviate the sharp increase in power demands.

The effectiveness of the proposed model is validated by comparing the solutions of different scenarios. The following scenarios are introduced and analyzed.

Scenario 1: The demand response and energy storage are not considered.

Scenario 2: The demand response is considered; however, the energy storage is not considered.

Scenario 3: The energy storage is considered; however, the demand response is not considered.

Scenario 4: The demand response and energy storage are considered, which represents the method proposed in this paper.

Comparisons of peak regulation solutions in different scenarios are shown in

Table 3. Comparisons of peak regulation solutions in different scenarios.

Cost	Results in Scenario 1	Results in Scenario 2	Results in Scenario 3	Results in Scenario 4
Operation cost of thermal power units/$	770,515.1	754,421.8	769,391.3	752,702.7
Carbon trading cost/$	160,608	99,801	156,379	96,535.9
Total cost/$	939,791.4	862,888.7	934,416.9	857,963.4

. Scenario 1 is defined to evaluate the peak regulation solution of the power system without any flexible resources involved. This scenario helps identify the limitations of traditional peak regulation methods. Demand response is an important flexibility resource that optimizes power system scheduling by adjusting users’ consumption behavior. Scenario 2 is used to assess the contribution of demand response to peak regulation, particularly to validate the impact of demand response on power system operation in the absence of energy storage. Scenario 3 is used to evaluate the effect of energy storage on peak regulation and examine the impact of energy storage on power system operation without the demand response. Scenario 4 incorporates both demand response and energy storage for peak regulation. Scenario 4 integrates both flexibility resources to verify the proposed bi-level optimization model. It helps to analyze how the synergy between demand response and energy storage achieves optimal operation of the power system during peak periods. It is seen from Table 3 that in Scenario 1, the operation cost of thermal power units, the carbon trading cost, and the total cost are the highest among those in all scenarios, indicating insufficient economic efficiency in the power system operation. Compared to Scenario 1, the demand response is considered in Scenario 2, and the operation cost of thermal power units, the carbon trading cost, and the total cost are lower than those in Scenario 1. This indicates that incorporating demand response into the peak regulation process helps to optimize generation scheduling and improve power system operation efficiency. In Scenario 3, the energy storage is introduced for peak regulation optimization, and the associated costs are lower than those in Scenario 1. This demonstrates that energy storage plays a positive role in peak shaving and reducing both generation and carbon trading cost. In Scenario 4, both the demand response and energy storage are considered, and the operation cost of thermal power units, the carbon trading cost, and the total cost reach their minimal values. Therefore, the proposed method in this paper is beneficial for significantly reducing the total cost and enhancing the economic operation efficiency of the power system.

4.4. Sensitivity Analysis

4.4.1. Impact of Renewable Energy Generation Forecasts on Scheduling Results

The impact of variations in renewable energy generation forecasts on power system scheduling results is illustrated in

Table 4. Impact of renewable energy generation forecasts on scheduling results.

Variations in Renewable Energy Generation Output Forecasts/%	Ramping-Up Capability/MW	Ramping-Down Capability/MW	Total Cost of the Power System/$
−20	11,500.3	7952.2	871,928.2
−10	11,784.3	8015.5	864,755.4
0	12,053	8076.2	857,963.4
10	12,349.8	8132.9	851,623.6
20	12,677.4	8188.3	845,717.3

. It is seen from Table 4 that both the ramping-up and ramping-down capacities of the power system increase with the increase in renewable energy generation forecasts. Specifically, when the variation in renewable energy generation forecasts increases from −20% to 20%, the ramping-up capacity increases from 11,500.3 MW to 12,677.4 MW, and the ramping-down capacity increases from 7952.2 MW to 8188.3 MW. This indicates that as renewable energy generation forecasts increase, the power system requires more ramping capacity to deal with the fluctuations of renewable energy generation output. In addition, it is seen from Table 4 that as renewable energy generation forecasts increase, the total cost of the power system gradually decreases. When the variation in renewable energy generation forecasts increases from −20% to 20%, the total cost of the power system decreases from $871,928.2 to $845,717.3. Therefore, the increase in renewable energy generation can enhance the economic efficiency of power system operation due to the reduction in fossil fuel power generation output.

4.4.2. Impact of Carbon Prices on Associated Costs

The impact of carbon prices on associated costs is shown in

Table 5. Impact of carbon prices on associated costs.

Carbon Trading Price/($/t)	Operation cost of Thermal Power Units/$	Carbon Trading Cost/$	Total Cost of the Power System/$
50	752,619.8	48,058.7	809,343.5
75	752,691.4	72,042.8	833,383.1
100	752,702.7	96,535.9	857,963.4
125	752,704.3	120,059.0	881,417.8
150	752,721.7	144,034.0	905,428.9

. It is seen from Table 5 that as the carbon trading price increases, the operation cost of thermal power units, carbon trading cost, and total cost of the power system increase. Therefore, the carbon trading price is a crucial factor influencing the operation cost of power systems. This underscores the importance of establishing reasonable carbon trading prices in the carbon trading market to ensure the economic operation of the power system. The long-term implications of changes in carbon pricing on power system operational strategies mainly lie in adjustment of generation mix. As carbon prices increase, the cost of traditional fossil fuel-based generation increases substantially, driving the power system towards adopting low-carbon generation technologies. The resulting pricing signal encourages the rapid development and deployment of renewable energy and energy storage technologies, gradually replacing high-carbon fossil fuel units and reducing the operational costs of the power system.

5. Conclusions

This paper presents a bi-level optimization model for power system peak regulation taking the ramping capability and demand response into account. The proposed upper- and lower-level models aim to minimize ramping demands considering uncertainty factors and the overall operation cost of the power system, respectively. A modified IEEE 39-bus system is employed to verify the effectiveness of the proposed peak regulation optimization model. It is concluded from the simulation results that (1) the demand response strategy based on price incentives can flatten the net load curve of the power system, effectively mitigating peak regulation burden and improving operation efficiency of the power system, and (2) the proposed peak regulation optimization model, which considers the ramping capability and demand response, effectively reduces the total operation cost of the power system and enhances economic performance.

It should be pointed out that the optimization of cutting parameters and accuracy improvement in forecasting errors are not considered in this paper. Studying how to optimize the cutting parameters and improve the accuracy of the forecasting errors are our research directions for the future.