Optimal Scheduling of Combined Electric and Heating Considering the Control Process of CHP Unit and Electric Boiler

Abstract

In order to solve the problem of new energy consumption, a combined electric and heating system (CEHS) dynamic optimal scheduling method considering the optimal control of combined heat and power (CHP) unit and electric boiler is proposed from the perspective of unit technology transformation, to optimize the thermoelectric coupling relationship and improve the regulation capacity of the CEHS. Firstly, the electric and heat output models of CHP units considering the optimal control process, were constructed and used to analyze the electric–thermal characteristics and the impact of unit pressure safety under variable load input. On this basis, CHP units, electric boilers, wind power units, and thermal power units are optimally scheduled to minimize system operating costs. Finally, a simultaneous method of “discrete first, optimize later” is proposed to solve the dynamic optimal scheduling problem. The simulation results verify that the optimal scheduling considering the optimal control of CHP units and the retrofitting of electric boilers can promote the consumption of wind power and improve the overall operating economy of the system while ensuring the feasibility of the CEHS scheduling scheme.

1. Introduction

Combined heat and power technology has been widely used for its efficient utilization of fossil fuels and plays an active role in alleviating energy shortage and air pollution [1,2]. However, as the installed capacity of CHP units increases, the inherent thermoelectric coupling characteristics lead to a subsequent decrease in the regulation ability of electric power systems [3]. In the CEHS with a high proportion of CHP units, the rigid power output of the units restricted by “determining power generation by heat” seriously limits the space for new energy consumption [4], leading to serious electricity restriction problems. Therefore, the technical transformation of CHP units to enhance their electric output flexibility and promote the adaptability of renewable energy has attracted much attention in recent years.

In order to improve the electrical output flexibility of CHP units, researchers have extensively studied CHP units in CEHS and explored their dispatch flexibility potential. On the one hand, the CHP unit is retrofitted with a direct decoupling scheme by adding equipment through physical modifications. In [5,6], the optimal scheduling strategy of CEHS containing electric boilers was studied to verify the impact of coordinated heat supply from electric boilers on wind and coal abandonment. Literature reports [7,8] used heat storage devices to break the thermoelectric coupling relationship of “determined power by heat” and analyzed the effects of different installation positions of heat storage devices. These transformations reduce the minimum feasible power load of the unit, thus expanding the feasible domain to improve the operational flexibility of the unit. On the other hand, existing studies have used optimal control methods to improve the short-term rapid active regulation of the unit based on the intrinsic characteristics of CHP units. This scheme is an indirect decoupling technical scheme, which can significantly change the unit’s electrical output in a short time without affecting the heat load supply. In [9,10], an optimal control strategy for CHP units is investigated to increase the maximum allowable ramping rate to more than 4% of the rated power. The rapid variable load capacity of CHP units can complement the rapid fluctuation of wind power and electric load, and make full use of the scheduling flexibility of the units, which will be beneficial to wind power consumption [11]. However, most of the current CEHS optimization scenario studies consider the effect of the direct decoupling or indirect decoupling scheme transformation on the system scheduling results independently, and less in-depth analysis of the complementary effect of the optimization scheduling flexibility when the two transformation schemes coordinate heating.

In the traditional CEHS optimal scheduling research, a convex polygon combination [12,13] is used to describe the electrical and thermal characteristics of CHP units, which often ignores or simplifies the operation process after its transformation. The internal mechanism of the CHP unit is very complex, and it is not easy to describe its nonlinear characteristics with simplified algebraic equations [14,15]. Especially for the CHP unit that has been optimized and modified, its operation state regulation is a dynamic process system, that includes regulating the temperature, pressure, and mass flow of the fluid in the boiler, steam turbine, and heating network [16]. Therefore, considering the coupling relationship of variables in CHP units and the characteristics of the optimization control process, the refined modeling of the physical processes of these components is the basis for exploring the flexibility of CHP units. The control process constraints of fine modeling of CHP units [9,10] are incorporated into the CEHS optimal scheduling, making the traditional scheduling problem a dynamic optimization problem with differential algebraic equation (DAE) constraints. The traditional CEHS optimal scheduling solution method, considering the constraints of a unit algebraic equation, is mainly divided into two categories: traditional mathematical optimization algorithms [17,18,19] and intelligent heuristic algorithms [20,21,22,23]. In engineering applications, traditional mathematical optimization has a high solution speed, but it has specific restrictions on the linearity and convexity of the model. Generally, certain assumptions or simplifications need to be introduced into the model, making it difficult to guarantee the solution’s accuracy. However, the intelligent algorithm has fewer restrictions on the complexity of the model, but it has the problems of poor convergence and low speed of solution. The above algorithms are difficult to solve the dynamic optimization problem of economic dispatch with unit differential algebraic equation constraints. They cannot meet the calculation requirements of the system’s economic dispatch and unit control process at the same time. Therefore, the need to study the CEHS optimal scheduling model considering the constraints of the CHP unit control process will become more urgent. When considering the solution’s efficiency and accuracy, the optimization model’s solution method under such dynamic constraints also needs to be further studied.

In this paper, a dynamic optimal scheduling method of CEHS considering the optimal control process of the CHP unit and electric boiler are proposed. The contributions of this paper are as follows: (1) A sophisticated system of differential algebraic equations (DAEs) is proposed to inscribe the CHP unit model, considering the CHP unit’s internal structure and control process, and describing the physical processes experienced by the steam of the boiler, turbine, and heating network. (2) A CEHS dynamic optimization scheduling problem is proposed based on consideration of the above CHP unit model. In order to solve the resulting large-scale nonlinear optimization problem, a simultaneous method of “discrete first, optimize later” is proposed.

The remainder of this paper is organized as follows: Section 2 establishes the CHP unit model. Section 3 constructs the CEHS dynamic optimal scheduling problem. Section 4 proposes a simultaneous method solution strategy. Section 5 provides case studies and discussions. Finally, conclusions are given in Section 6.

2. CHP Uuit Optimal Control Process Modeling

Figure 1 illustrates the internal composition and structure of an extracted steam CHP unit with an optimized control module. From the energy flow conversion point of view, the coal enters the boiler for the combustion reaction, producing high temperature steam that enters the high-pressure (HP) cylinder. Then, the steam is extracted to the intermediate-pressure (IP) cylinder, and part of it enters the low-pressure (LP) cylinder to produce electricity, while the other part is supplied directly to the heat source. In terms of the optimal control process, the amount of fuel is controlled by regulating $V_{\mathrm{B}}$, and the amount of steam entering the HP cylinder is controlled by regulating butterfly valve $V_{\mathrm{T}}$. The steam extraction regulating butterfly valve $V_{\mathrm{H}}$ controls the amount of steam entering the LP cylinder to do work and the amount of steam for heating. When the unit’s output power changes, the controller acts to cause the steam flow into the LP cylinder to change, and the control variables $V_{\mathrm{B}}$, $V_{\mathrm{T}}$, and $V_{\mathrm{H}}$ respond quickly to complete the regulation. During the regulation process, these variables affect the main steam pressure ${\psi }_{\mathrm{t}}$, the electric output $P_{\mathrm{H}}$, and the heat output $m_{\mathrm{H}}$ to different degrees. The coordinated relationships among electric output, main steam pressure, heat output, coal-feed rate, regulating valve opening, and heat source valve reflect the dynamic characteristics of CHP units.

2.1. Dynamic Relational Mechanism Modeling

As shown in Figure 1, the CHP unit has a direct-blow pulverizing system with a medium-speed mill, one intermediate reheat, a condensing steam turbine, and a condensing feed water system with three HP preheaters, four LP preheaters, and one deaerator [24]. This CHP unit model is derived from a mechanistic analysis including mass conservation and energy conservation and is divided into four independent parts to analyze the dynamic relationships between the building variables.

(1) The coal-pulverizing system model describes the relationship between the coal-feed rate command $V_{\mathrm{B}}$ and the coal-feed rate in furnace $V_{\mathrm{f}}$ [25].

$$V_{\mathrm{m}}\left(t\right)=V_{\mathrm{B}}\left(t-t_{\mathrm{B}}\right)$$

(1)

$$\frac{d{V}_{\mathrm{f}}\left(t\right)}{dt}=-\frac{V_{\mathrm{f}}\left(t\right)}{T_{\mathrm{f}}}+\frac{V_{\mathrm{B}}\left(t-t_{\mathrm{B}}\right)}{T_{\mathrm{f}}}$$

(2)

where $V_{\mathrm{B}}\left(t\right)$ is the mass flow rate of coal-feed to the unit; $V_{\mathrm{m}}\left(t\right)$ is the mass flow rate of coal feed to the mill in the pulverizing system; $V_{\mathrm{f}}\left(t\right)$ is the coal-feed rate in the boiler; $t_{\mathrm{B}}$ is the delay time constant of pulverizing process; $T_{\mathrm{f}}$ is the inertia time constant of pulverizing.

(2) The drum-boiler system model reflects the relationship between the input and output energy of the boiler, and the drum pressure is the state sign of the boiler energy balance [26]. The dynamic relationship can be described as follows:

$$\frac{d{\psi }_{\mathrm{d}}\left(t\right)}{dt}=-\frac{K_3{\psi }_{\mathrm{t}}\left(t\right)V_{\mathrm{T}}\left(t\right)}{C_{\mathrm{d}}}+\frac{K_1{V}_{\mathrm{f}}\left(t\right)}{C_{\mathrm{d}}}$$

(3)

$${\psi }_{\mathrm{d}}\left(t\right)-{\psi }_{\mathrm{t}}\left(t\right)=K_2{\left(K_1{V}_{\mathrm{f}}\left(t\right)\right)}^{1.5}$$

(4)

where ${\psi }_{\mathrm{d}}\left(t\right)$ is the drum pressure; ${\psi }_{\mathrm{t}}\left(t\right)$ is the main steam pressure; $V_{\mathrm{T}}\left(t\right)$ is the turbine regulator valve opening; $C_{\mathrm{d}}$ is the drum storage coefficient.

(3) The turbine system model depicts the relationship between the input work and output power of the steam turbine. The input work of CHP units is partly transferred to the electric power output and partly to the heat supply. The equation is as follows:

$$\frac{d{P}_{\mathrm{H}}\left(t\right)}{dt}=\frac{K_3{K}_4{\psi }_{\mathrm{t}}\left(t\right)V_{\mathrm{T}}\left(t\right)}{T_{\mathrm{t}}}+\frac{K_5{\psi }_{\mathrm{z}}\left(t\right)V_{\mathrm{H}}\left(t\right)}{T_{\mathrm{t}}}-\frac{P_{\mathrm{H}}\left(t\right)}{T_{\mathrm{t}}}$$

(5)

$${\psi }_1\left(t\right)=0.01{\psi }_{\mathrm{t}}\left(t\right)V_{\mathrm{T}}\left(t\right)$$

(6)

$$\frac{d{\psi }_{\mathrm{z}}\left(t\right)}{dt}=-\frac{K_6{m}_{\mathrm{r}}\left(t\right)\left(96{\psi }_{\mathrm{z}}\left(t\right)-{\epsilon }_{\mathrm{r}}\left(t\right)+103\right)}{C_{\mathrm{z}}}+\frac{K_3\left(1-K_4\right){\psi }_{\mathrm{t}}\left(t\right)V_{\mathrm{T}}\left(t\right)}{C_{\mathrm{z}}}-\frac{K_5{\psi }_{\mathrm{z}}\left(t\right)V_{\mathrm{H}}\left(t\right)}{C_{\mathrm{z}}}$$

(7)

where $P_{\mathrm{H}}\left(t\right)$ is the electric power output; $T_{\mathrm{t}}$ is the turbine inertia time constant; ${\psi }_{\mathrm{z}}\left(t\right)$ is the IP cylinder discharge pressure; $V_{\mathrm{H}}\left(t\right)$ is the extraction regulating butterfly valve opening; ${\psi }_1\left(t\right)$ is the turbine first stage pressure; $m_{\mathrm{r}}\left(t\right)$ is the circulating water mass flow rate; ${\epsilon }_{\mathrm{r}}\left(t\right)$ is the circulating water return temperature; $C_{\mathrm{z}}$ is the heat storage coefficient of the heat network heater.

(4) The heating system model reflects the relationship between input energy and output energy at the heat source and can be described as:

$$m_{\mathrm{H}}\left(t\right)=K_7{K}_6{m}_{\mathrm{r}}\left(t\right)\left(96{\psi }_{\mathrm{z}}\left(t\right)-{\epsilon }_{\mathrm{r}}\left(t\right)+103\right)$$

(8)

where $m_{\mathrm{H}}\left(t\right)$ is the heat source mass flow rate.

2.2. Optimized Control Methods

Based on the above dynamic relationship model, an optimal control method for the change direction and step size of control variables ($V_{\mathrm{T}}\left(t\right)$, $V_{\mathrm{B}}\left(t\right)$, and $V_{\mathrm{H}}\left(t\right)$) is designed, and the heat-electric power coordination and distribution control strategy is adopted [10,24], as shown in Equations (9)–(11).

The optimal control achieves the three essential control requirements of the CHP unit’s variable load process of thermoelectric cooperation, rapid thermal state recovery, and accurate energy balance. Considering the safety and stability of the unit, the output variable fluctuations during regulation should satisfy the path constraints of Equations (12)–(14). At the same time, the steady-state values of each output variable should not exceed the allowed error range and should also satisfy the final value constraints of Equations (15)–(17). In the process of variable load regulation, $V_{\mathrm{T}}\left(t\right)$, $V_{\mathrm{B}}\left(t\right)$, and $V_{\mathrm{H}}\left(t\right)$ need to satisfy the boundary constraints of Equations (18)–(20) due to the limited value constraints of the CHP unit’s component properties. In summary, the mathematical models of the CHP unit optimal control methods are described as:

$$V_{\mathrm{T}}\left(t\right)=K_{\mathrm{PT}}\left({\psi }_{\mathrm{t}}^{\mathrm{sp}}-{\psi }_{\mathrm{t}}\left(t\right)\right)+K_{\mathrm{IT}}{{\displaystyle \sum }}_{t=t_0}^{t_e}\left({\psi }_{\mathrm{t}}^{\mathrm{sp}}-{\psi }_{\mathrm{t}}\left(t\right)\right)$$

(9)

$$V_{\mathrm{B}}\left(t\right)=K_{\mathrm{PB}}\left[P_{\mathrm{H}}^{\mathrm{sp}}-\left(m_{\mathrm{H}}\left(t\right)-\frac{K{e}^{-t/T_{\mathrm{c}}}}{T_{\mathrm{c}}}{P}_{\mathrm{H}}\left(t\right)\right)\right]+K_{\mathrm{IB}}{{\displaystyle \sum }}_{t=t_0}^{t_e}\left[P_{\mathrm{H}}^{\mathrm{sp}}-\left(m_{\mathrm{H}}\left(t\right)-\frac{K{e}^{-t/T_{\mathrm{c}}}}{T_{\mathrm{c}}}{P}_{\mathrm{H}}\left(t\right)\right)\right]$$

(10)

$$V_{\mathrm{H}}\left(t\right)=K_{\mathrm{PH}}\left(m_{\mathrm{H}}^{\mathrm{sp}}-m_{\mathrm{H}}\left(t\right)\right)+K_{\mathrm{IH}}{{\displaystyle \sum }}_{t=t_0}^{t_e}\left(m_{\mathrm{H}}^{\mathrm{sp}}-m_{\mathrm{H}}\left(t\right)\right)$$

(11)

$$\left|{\psi }_{\mathrm{t}}^{\mathrm{sp}}-{\psi }_{\mathrm{t}}\left(t\right)\right|\le {\mathsf{\Gamma }}_{{\psi }_{\mathrm{t}}}$$

(12)

$$\left|P_{\mathrm{H}}^{\mathrm{sp}}-P_{\mathrm{H}}\left(t\right)\right|\le {\mathsf{\Gamma }}_{P_{\mathrm{H}}}$$

(13)

$$\left|m_{\mathrm{H}}^{\mathrm{sp}}-m_{\mathrm{H}}\left(t\right)\right|\le {\mathsf{\Gamma }}_{m_{\mathrm{H}}}$$

(14)

$$\left|{\psi }_{\mathrm{t}}\left(t_e\right)-{\psi }_{\mathrm{t}}^{\mathrm{sp}}\right|\le {\mathsf{\delta }}_{{\psi }_{\mathrm{t}}}$$

(15)

$$\left|P_{\mathrm{H}}\left(t_e\right)-P_{\mathrm{H}}^{\mathrm{sp}}\right|\le {\mathsf{\delta }}_{P_{\mathrm{H}}}$$

(16)

$$\left|m_{\mathrm{H}}\left(t_e\right)-m_{\mathrm{H}}^{\mathrm{sp}}\right|\le {\mathsf{\delta }}_{m_{\mathrm{H}}}$$

(17)

$$V_{\mathrm{T}}^{\min }\le V_{\mathrm{T}}\left(t\right)\le V_{\mathrm{T}}^{\max }$$

(18)

$$V_{\mathrm{B}}^{\min }\le V_{\mathrm{B}}\left(t\right)\le V_{\mathrm{B}}^{\max }$$

(19)

$$V_{\mathrm{H}}^{\min }\le V_{\mathrm{H}}\left(t\right)\le V_{\mathrm{H}}^{\max }$$

(20)

where $K_{\mathrm{PT}}$ and $K_{\mathrm{IT}}$ are the control parameters of $V_{\mathrm{T}}\left(t\right)$; $K_{\mathrm{PB}}$ and $K_{\mathrm{IB}}$ are the control parameters of $V_{\mathrm{B}}\left(t\right)$; $K_{\mathrm{PH}}$ and $K_{\mathrm{IH}}$ are the control parameters of $V_{\mathrm{H}}\left(t\right)$; $K$ and $T_{\mathrm{c}}$ are the control parameters of heat–electric power coordination and distribution. In the control optimization cycle, ${\psi }_{\mathrm{t}}^{\mathrm{sp}}$ is the main steam setting value; $P_{\mathrm{H}}^{\mathrm{sp}}$ is the unit power generation power setting value; $m_{\mathrm{H}}^{\mathrm{sp}}$ is the heat source mass flow setting value; ${\mathsf{\Gamma }}_{{\psi }_{\mathrm{t}}}$, ${\mathsf{\Gamma }}_{P_{\mathrm{H}}}$, and ${\mathsf{\Gamma }}_{m_{\mathrm{H}}}$ are the fluctuation ranges of the main steam pressure, electric power, and extracted steam flow, respectively; ${\mathsf{\delta }}_{{\psi }_{\mathrm{t}}}$, ${\mathsf{\delta }}_{P_{\mathrm{H}}}$, and ${\mathsf{\delta }}_{m_{\mathrm{H}}}$ are the error ranges of the main steam pressure, electric power, and steam flow rate of steam extraction; $V_{\mathrm{T}}^{\min }$ and $V_{\mathrm{T}}^{\max }$ are the minimum and maximum values of the variation of the turbine regulator opening, respectively. $V_{\mathrm{B}}^{\min }$ and $V_{\mathrm{B}}^{\max }$ are the minimum and maximum values of the variation of coal-feed mass flow rate of the unit, respectively. $V_{\mathrm{H}}^{\min }$ and $V_{\mathrm{H}}^{\max }$ are the minimum and maximum values of the variation of the opening of the extraction regulating butterfly valve, respectively.

In the traditional scheduling process, the modeling approach of an algebraic system of equations often needs to be revised to portray the electro–thermal coupling characteristics of the optimal control process of CHP units. When the constraints on climbing and standby capacity of CHP units are too loose, the standby scheduling plan cannot be realized precisely, which brings about hidden system safety problems; while when the constraints are too strict, the depth of unit variable load is reduced, and it is not easy to give full play to the remaining generation capacity of the unit, which affects the economy of system operation. At the same time, the adjustment process of CHP units executing the scheduling plan will reduce the feasibility of the scheduling plan because the variables related to the safe and stable operation of the units are ignored or simplified, and there may be safety hazards in the of operation of the units. Therefore, to ensure CEHS fine scheduling, it is necessary to quantify and analyze the characteristics of CHP units after optimal control modification and establish precise model constraints for CHP unit operation.

3. CEHS Scheduling Model

In this paper, the economic optimization of the scheduling system is the goal, and the output of thermal power units, the output of wind power units, the output of electric boiler equipment, the flow rate and valve opening of CHP units are the scheduling optimization variables to regulate the electricity–heat supply. The feasibility of the scheduling scheme is ensured by constraints. Each unit is considered to carry out its intra-day scheduling, while the CHP units are simulated with the optimal control process in real-time to match the best unit electric–heat output scheme. In this section, a CEHS scheduling model is developed, decision variables for the scheduling time period, control variables for the control time domain, and output variables are defined, and the CEHS scheduling model is represented in the abstract form of a dynamic optimization problem containing DAEs constraints.

The constraints are as follows: Equations (1)–(20) are dynamic model constraints for CHP unit operation, which can characterize the electric–heat output constraints, ramping capability constraints and reserve capacity constraints, and dynamic characteristic relationships of the optimal control process of the unit, and the parameters are detailed in Appendix A

Table A1. Working point parameters of CHP unit.

Static Paramete		Dynamic Parameter		Work Point
$K_1$	2.37	$t_{\mathrm{B}}$	15	$V_{\mathrm{B}}$	126.58
$K_2$	0.00035	$T_{\mathrm{f}}$	120	$V_{\mathrm{T}}$	66.895
$K_3$	0.269	$C_{\mathrm{d}}$	3300	$V_{\mathrm{H}}$	54.526
$K_4$	0.651	$T_{\mathrm{t}}$	12	$P_{\mathrm{H}}$	235
$K_5$	2.096	$C_{\mathrm{z}}$	160	$m_{\mathrm{H}}$	400
$K_6$	0.00039	$m_{\mathrm{r}}$	2500	${\psi }_{\mathrm{z}}$	0.35
$K_7$	6.1538	${\epsilon }_{\mathrm{r}}$	70	${\psi }_{\mathrm{t}}$	16.67

and

Table A2. CHP unit control system and thermal system parameters.

Control system Parameter		Thermal System Parameter
$K_{\mathrm{PT}}$	−0.9	$C_{\mathrm{sys}}$	$3.1896\times {10}^8$
$K_{\mathrm{PB}}$	0.1	$\Delta h$	$2.3637\times {10}^3$
$K_{\mathrm{PH}}$	0.01
$K_{\mathrm{IT}}$	−1
$K_{\mathrm{IB}}$	0.01
$K_{\mathrm{IH}}$	10

. Equations (22)–(24) are the electric boiler (EB) operation constraints. Equations (25)–(29) are the thermal system operation constraints. In the optimal control cycle, the heat variation of the extracted steam flow fluctuation is calculated by Equations (25)–(27), and the heat variation of supplying production and domestic heating is constrained by Equations (28) and (29). Equations (30)–(33) are the thermal power (TP) unit operation constraints. Equation (34) is the wind power (WP) operation constraint. Equations (35) and (36) are the system electric power and heat power balance constraints. Equation (37) is the power system network security constraint.

$$\begin{array}{c}min{{\displaystyle \sum }}_{T\in {\mathsf{\Omega }}^{\mathrm{T}}}\left\{{{\displaystyle \sum }}_{l\in {\mathsf{\Omega }}^{\mathrm{G}}}\left[a_{2,l}{\left(P_{G,l}^T\left(t_e\right)\right)}^2+a_{1,l}{P}_{G,l}^T\left(t_e\right)+a_{0,l}\right]+{{\displaystyle \sum }}_{m\in {\mathsf{\Omega }}^{\mathrm{H}}}\left[b_{2,m}{\left(P_{H,m}^T\left(t_e\right)\right)}^2+b_{1,m}{P}_{H,m}^T\left(t_e\right)+b_{0,m}+\right.\right.\hfill \\ \left.c_{2,m}{\left(Q_{H,m}^T\left(t_e\right)\right)}^2+c_{1,m}{Q}_{H,m}^T\left(t_e\right)+c_{0,m}\left(P_{H,m}^T\left(t_e\right)·Q_{H,m}^T\left(t_e\right)\right)\right]+{{\displaystyle \sum }}_{n\in {\mathsf{\Omega }}^{\mathrm{w}}}\left[\theta \left(P_{W,n}^{\mathit{real},T}\left(t_e\right)-P_{W,n}^T\left(t_e\right)\right)\right]+\hfill \\ \left.{{\displaystyle \sum }}_{u\in {\mathsf{\Omega }}^{\mathrm{EB}}}\left[c_{EB}{Q}_{EB,u}^T\left(t_e\right)/{\eta }_{EB}\right]\right\}\hfill \end{array}$$

(21)

s.t. (1)–(8), (9)–(20)

$$Q_{EB,u}^T\left(t\right)={\eta }_{EB}{P}_{EB,u}^T\left(t\right)$$

(22)

$$P_{EB,u}^{\min }\le P_{EB,u}^T\left(t\right)\le P_{EB,u}^{\max }$$

(23)

$$-R_{EB,u}^{\mathrm{dn},T}\le P_{EB,u}^T\left(t_e\right)-P_{EB,u}^{T-1}\left(t_e\right)\le R_{EB,u}^{\mathrm{up},T}$$

(24)

$$Q_{H,m,equ}^T=m_{H,m,equ}^T\cdot \Delta h$$

(25)

$$m_{H,m,equ}^T={{\displaystyle \sum }}_{t=t_0}^{t_e}{m}_{H,m}\left(t\right)/\left(t_e-t_0\right)$$

(26)

$$C_{\mathrm{sys}}\frac{d{\tau }_{\mathrm{indoor}}}{dt}={{\displaystyle \sum }}_{m\in {\mathsf{\Omega }}^{\mathrm{H}}}{Q}_{H,m,equ}^T-\left(Q_{\mathrm{Load}}^T-{{\displaystyle \sum }}_{u\in {\mathsf{\Omega }}^{\mathrm{EB}}}{Q}_{EB,u}^T\right)$$

(27)

$$\left|{{\displaystyle \sum }}_{m\in {\mathsf{\Omega }}^{\mathrm{H}}}{Q}_{H,m,equ}^T-\left(Q_{\mathrm{Load}}^T-{{\displaystyle \sum }}_{u\in {\mathsf{\Omega }}^{\mathrm{EB}}}{Q}_{EB,u}^T\right)\right|\le \Delta q$$

(28)

$$\left|{{\displaystyle \sum }}_{T\in {\mathsf{\Omega }}^{\mathrm{T}}}\left[{{\displaystyle \sum }}_{m\in {\mathsf{\Omega }}^{\mathrm{H}}}{Q}_{H,m,equ}^T-\left(Q_{\mathrm{Load}}^T-{{\displaystyle \sum }}_{u\in {\mathsf{\Omega }}^{\mathrm{EB}}}{Q}_{EB,u}^T\right)\right]\right|\le \Delta Q$$

(29)

$$P_{G,l}^{\min }\le P_{G,l}^T\left(t\right)\le P_{G,l}^{\max }$$

(30)

$$R_{G,l}^{\mathrm{up},T}=min\left\{\left(P_{G,l}^{\max }-P_{G,l}^T\left(t_e\right)\right),\Delta T\cdot R_{G,l}\right\}$$

(31)

$$R_{G,l}^{\mathrm{dn},T}=min\left\{\left(P_{G,l}^T\left(t_e\right)-P_{G,l}^{\min }\right),\Delta T\cdot R_{G,l}\right\}$$

(32)

$$-R_{G,l}^{\mathrm{dn},T}\le P_{G,l}^T\left(t_e\right)-P_{G,l}^{T-1}\left(t_e\right)\le R_{G,l}^{\mathrm{up},T}$$

(33)

$$0\le P_{W,n}^T\left(t_e\right)\le P_{W,n}^{real,T}$$

(34)

$${{\displaystyle \sum }}_{l\in {\mathsf{\Omega }}^{\mathrm{G}}}{P}_{G,l}^T\left(t_e\right)+{{\displaystyle \sum }}_{m\in {\mathsf{\Omega }}^{\mathrm{H}}}{P}_{H,m}^T\left(t_e\right)+{{\displaystyle \sum }}_{n\in {\mathsf{\Omega }}^{\mathrm{W}}}{P}_{W,n}^T\left(t_e\right)=P_{\mathrm{Load}}^T\left(t_e\right)+{{\displaystyle \sum }}_{u\in {\mathsf{\Omega }}^{\mathrm{EB}}}{P}_{EB,u}^T\left(t_e\right)$$

(35)

$${{\displaystyle \sum }}_{m\in {\mathsf{\Omega }}^{\mathrm{H}}}{Q}_{H,m}^T\left(t_e\right)+{{\displaystyle \sum }}_{u\in {\mathsf{\Omega }}^{\mathrm{EB}}}{Q}_{EB,u}^T\left(t_e\right)=Q_{\mathrm{Load}}^T\left(t_e\right)$$

(36)

$$P_{Line,s}^{\min }\le P_{Line,s}^T\left(t\right)\le P_{Line,s}^{\max } ,s\in {\mathsf{\Omega }}^{\mathrm{B}}$$

(37)

where the operating cost coefficients of TP units are $a_2$, $a_1$, and $a_0$; $P_G^T$ is the TP unit output; the operating cost coefficients of CHP units are $b_2$, $b_1$, $b_0$, $c_2$, $c_1$, and $c_0$; $P_H^T$ and $Q_H^T$ are the electric output and heat output of CHP units; the WP penalty cost factor is $\theta$; $P_W^T$ is the actual WP output, and $P_W^{real}$ is the forecasting WP output; $c_{EB}$ is the EB operating cost factor, and $Q_{EB}^T$ is the EB heat output; the scheduling period is $T\in {\mathsf{\Omega }}^{\mathrm{T}}$, and the time interval is $\Delta T$; ${\mathsf{\Omega }}^{\mathrm{G}}$, ${\mathsf{\Omega }}^{\mathrm{H}}$, ${\mathsf{\Omega }}^{\mathrm{W}}$, and ${\mathsf{\Omega }}^{\mathrm{EB}}$ are the index set of the TP units, CHP units, WP units, and EBs, respectively; $l$, $m$, $n$, $u$ indicate the number of TP units, CHP units, WP units, and EBs, respectively; $P_{EB}^T$, $P_{EB}^{\min }$, and $P_{EB}^{\max }$ are the EB power consumption and its regulation upper and lower limits, respectively; ${\eta }_{EB}$ is the electric heat conversion efficiency of EB; $R_{EB}^{\mathrm{up}}$ and $R_{EB}^{\mathrm{dn}}$ are the upper and lower reserve capacity of EB; $Q_{\mathrm{Load}}^T$ is the heat load demand; $Q_{H,equ}^T$ is the average equivalent thermal power of the regulation process of the CHP unit; $\Delta h$ is the enthalpy drop of heat extraction; $m_{H,equ}^T$ is the average equivalent flow of the regulation process; $t_0$ and $t_e$ are the start and end time points of the optimal control of the unit; $C_{\mathrm{sys}}$ is the heat capacity of the heating area, ${\tau }_{\mathrm{indoor}}$ is the production process demand temperature, and $d{\tau }_{\mathrm{indoor}}/dt$ is the rate of temperature change; $\Delta q$ is the allowable heat variation value within the scheduling period (5 min), and $\Delta Q$ is the allowable heat variation value within the total scheduling time (2 h); $R_G$ is the ramping capability of TP units; $P_G^{\min }$ and $P_G^{\max }$ are the minimum and maximum values of TP unit output, respectively; $R_G^{\mathrm{up}}$ and $R_G^{\mathrm{dn}}$ are the upper and lower reserve capacity of TP units; $P_{\mathrm{Load}}^T$ is the electrical load demand. $P_{Line,s}^T$ is the transmission power of branch s in time period $T$; $P_{Line,s}^{\min }$ and $P_{Line,s}^{\max }$ are the lower and upper limits of the maximum transmission power of branch $s$; ${\mathsf{\Omega }}^{\mathrm{B}}$ is all the transmission lines in the power system.

To facilitate the use of matrices and vectors to represent constraints and variables, we set $P_{\mathrm{EB}}\left(t\right)=V_{\mathrm{EB}}\left(t\right)$, $P_{\mathrm{G}}\left(t\right)=V_{\mathrm{G}}\left(t\right)$, and $P_{\mathrm{W}}\left(t\right)=V_{\mathrm{W}}\left(t\right)$. Since only the DAEs model of the optimal control process of the CHP unit is considered in this paper, the outputs of other units or devices are equal everywhere in the control time domain $\left[t_0,t_e\right]$, we have:

$$P_{G,l}^T\left(t_0\right)=P_{G,l}^T\left(t_1\right)=P_{G,l}^T\left(t_2\right)=\cdots =P_{G,l}^T\left(t_e\right)$$

(38)

$$P_{W,u}^T\left(t_0\right)=P_{W,u}^T\left(t_1\right)=P_{W,u}^T\left(t_2\right)=\cdots =P_{W,u}^T\left(t_e\right)$$

(39)

$$P_{EB,u}^T\left(t_0\right)=P_{EB,u}^T\left(t_1\right)=P_{EB,u}^T\left(t_2\right)=\cdots =P_{EB,u}^T\left(t_e\right)$$

(40)

In addition, it should be noted that the ${\psi }_{\mathrm{t}}^{\mathrm{sp}}$ is given by the $P_H^T$ and the $Q_H^T$ during the optimal control cycle of the T-th scheduling time period [27], $P_{\mathrm{H}}^{\mathrm{sp}}=P_H^T$, and the $m_{\mathrm{H}}^{\mathrm{sp}}$ is calculated by the $Q_H^T$ using Equation (25).

In summary, the CEHS dynamic optimal scheduling problem considering the optimal control process of CHP units can be expressed in the following form:

$$\min J\left(\mathit{x}\left(t\right)\right)$$

$$\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}\dot{\mathit{x}}\left(t\right)=F(\mathit{x}\left(t\right),\mathit{y}\left(t\right),\mathit{u}\left(t\right))\\ G_{\mathrm{E}}(t,\mathit{x}\left(t\right),\mathit{y}\left(t\right),\mathit{u}\left(t\right))=0\\ G_{\mathrm{IE}}(t,\mathit{x}\left(t\right),\mathit{y}\left(t\right),\mathit{u}\left(t\right))<0\\ H_{\mathrm{E}}\left(t_e,\mathit{x}\left(t_e\right),\mathit{y}\left(t_e\right),\mathit{u}\left(t_e\right)\right)=0\\ H_{\mathrm{IE}}\left(t_e,\mathit{x}\left(t_e\right),\mathit{y}\left(t_e\right),\mathit{u}\left(t_e\right)\right)<0\\ \mathit{x}\left(t_0\right)={\mathit{x}}_0\\ {\mathit{u}}_{\mathit{L}}\le \mathit{u}\left(t\right)\le {\mathit{u}}_U\end{array}\right.$$

(41)

where the state variable is $\mathit{x}={\left[{\psi }_{\mathrm{d}},P_{\mathrm{H}},{\psi }_{\mathrm{z}},V_{\mathrm{f}}\right]}^{\mathrm{T}}$; the control variable is $\mathit{u}={\left[V_{\mathrm{T}},V_{\mathrm{B}},V_{\mathrm{H}},V_{\mathrm{G}},V_{\mathrm{W}},V_{\mathrm{EB}}\right]}^{\mathrm{T}}$; the output variable is $\mathit{y}={\left[{\psi }_{\mathrm{t}},P_{\mathrm{H}},m_{\mathrm{H}},P_{\mathrm{G}},P_{\mathrm{W}},P_{\mathrm{EB}}\right]}^{\mathrm{T}}$. $F$ is the DAEs dynamic model of CEHS optimal scheduling; $G_{\mathrm{E}}$ and $G_{\mathrm{IE}}$ are equation and inequality path constraints, $H_{\mathrm{E}}$ and $H_{\mathrm{IE}}$ are the final value constraints at $t_e$ time.

4. Solution Strategy

For the above CEHS dynamic optimization scheduling problem with DAE constraints, the simultaneous method [28,29,30], widely used in relevant fields, can solve the optimization proposition. The simultaneous algorithm is an idea or scheme to solve problems, and is a combination of multiple methods. It can also be called a “strategy”. The simultaneous solution strategy usually follows the fixed problem-solving mode of “all discretization and then optimization”. The solution framework of “first discrete, then optimized” is shown in Figure 2. Firstly, the control, differential, and algebraic variables are discretized simultaneously using Lagrange interpolation polynomials based on orthogonal collocation points, and a nonlinear programming (NLP) proposition is obtained. Then the proposition is solved by the interior point method and other optimization algorithms. The simultaneous method has significant advantages in dealing with the optimal problem with complex constraints. This method combines the solution of the DAE model with the optimization problem. It only needs to solve the DAE once to avoid many calculations in the middle process [29].

4.1. Discrete Format

In this paper, the Lagrange interpolation function based on the Radau orthogonal configuration points approximates the original functions of the state and control variables. The whole time domain $\left[t_0,t_e\right]$ is divided into N-segment finite elements, and on each segment finite element $\left[t_{i-1},t_i\right]$ ($i=1,2,\cdots ,N$), the interpolation equation of the state variables is as follows:

$$x\left(t\right)={{\displaystyle \sum }}_{j=0}^K{l}_j\left(\tau \right)x_{ij},\mathrm{which}{l}_j\left(\tau \right)={{\displaystyle \prod }}_{k=0,\ne j}^K\frac{\tau -{\tau }_k}{{\tau }_j-{\tau }_k}$$

(42)

where $K$ is the interpolation order, and K = 3 is chosen so that the discretization solution has 5th order accuracy; $x_{ij}$ is the value of the state variable at the $j$-th configuration point of the $i$-th finite element.

The initial and final value conditions of the state variables are

$$x_{1,0}=x_0, x_e=x_{N,K}$$

(43)

Since the state variables are derivable, the values of the state variables on the nodes at adjacent finite element connections should be continuous, so there is a continuity condition as follows:

$$x_{i+1,0}={{\displaystyle \sum }}_{j=0}^K{l}_j\left(1\right)x_{ij}, i=1,2,\cdots ,N-1.$$

(44)

The interpolating polynomial for the control variables is as follows:

$$u\left(t\right)={{\displaystyle \sum }}_{j=0}^K{\overline{l}}_j\left(\tau \right)u_{ij},\mathrm{which}{\overline{l}}_j\left(\tau \right)={{\displaystyle \prod }}_{k=1,\ne j}^K\frac{\tau -{\tau }_k}{{\tau }_j-{\tau }_k}$$

(45)

The advantage of the Lagrange interpolation polynomial is that the values of the variables at each configuration point are exactly equal to their coefficients, and we have

$$t_{ij}=t_{i-1}+\left(t_i-t_{i-1}\right){\tau }_j$$

(46)

$$x\left(t_{ij}\right)=x_{ij},u\left(t_{ij}\right)=u_{ij}$$

(47)

Substituting Equations (42) and (45)–(47) into the differential equations of proposition (41), the configuration equations are obtained as follows:

$$R_i\left({\tau }_j\right)={{\displaystyle \sum }}_{k=0}^K{\overline{l}}_k\left({\tau }_j\right)x_{ik}-h_{i}F\left(x_{ij},u_{ij}\right)=0, i=1,2,\cdots ,N; j=1,2,\cdots ,K.$$

(48)

$$t=t_{i-1}+h_i\tau ,h_i=t_i-t_{i-1}$$

(49)

So far, the NLP proposition after the discretization of the original proposition (41) is

$$\min J\left(\mathit{x}\left(t\right)\right)$$

$$\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}\sum _{k=0}^K{\overline{l}}_k\left({\tau }_j\right)x_{ik}-h_{i}F\left(x_{ij},y_{ij},u_{ij}\right)=0\\ G_{\mathrm{E}}\left(x_{ij},y_{ij},u_{ij}\right)=0\\ H_{\mathrm{E}}\left(x_{NK},y_{NK},u_{NK}\right)=0\\ G_{\mathrm{IE}}\left(x_{ij},y_{ij},u_{ij}\right)<0\\ H_{\mathrm{IE}}\left(x_{NK},y_{NK},u_{NK}\right)<0\\ u_L\le u_{ij}\le u_U\\ i=1,2,\cdots ,N;j=1,2,\cdots ,K.\\ x\left(t_0\right)=x_0,x\left(t_e\right)=x_e\\ x_{i+1,0}=\sum _{j=0}^K{l}_j\left(1\right)x_{ij},i=1,2,\cdots ,N-1.\end{array}\right.$$

(50)

4.2. Optimal Solution

The interior point method has the characteristics of fast convergence and strong stability and has obvious advantages in solving optimization problems with many inequality constraints [31]. In the field of dynamic optimization research related to chemical variable load process [32], autonomous parking [29], and satellite formation reconstruction [28], this method is often used to solve large-scale NLP problems after dispersion. Its basic idea is to transform the original optimization problem into an equivalent problem containing only equality and simple inequality constraints and obtain the search direction by perturbing the first-order optimality conditions.

The iterative steps are: (1) Select the starting point within the constrained manifold and find the direction of descent using the gradient method; (2) Return to the search center through nonlinear projection transformation so that each gradient descent operation is optimal; (3) Gradual convergence to optimality in the feasible domain along the fastest descent direction; (4) Strict restriction by the penalty function method keeps the iteration point in the feasible domain, and the objective function value will be increased to prevent the iteration points from crossing the boundary when the iteration points are near the boundary.

The optimization problem, as in Equation (50), can be abbreviated as

$$\min J\left(x\right)$$

$$\mathrm{s}.\mathrm{t}.\{\begin{array}{c}e\left(x\right)=0\\ c\left(x\right)\le 0\end{array}$$

(51)

where $e\left(x\right)=0$ is the equation constraint and $c\left(x\right)\le 0$ is the inequality constraint.

Equation (51) can be converted by the interior point method to

$$\min J_{\mu }\left(x,s\right)=\min J_{\mu }\left(x\right)-\mu {\displaystyle \sum }_i\ln \left(s_i\right)$$

$$\mathrm{s}.\mathrm{t}.\{\begin{array}{c}e\left(x\right)=0\\ c\left(x\right)+s=0\end{array}$$

(52)

where the relaxation variable $s_i$ corresponds to the inequality constraint and $s_i$ is positive to ensure that the barrier function $\ln \left(s_i\right)$ is bounded; the minimum value of the $J_{\mu }\left(x,s\right)$ function is the minimum value of $J\left(x\right)$ when $\mu$ is reduced to 0.

The evaluation function that terminates the internal iterations is

$$J_{\mu }\left(x,s\right)+\upsilon \Vert e\left(x\right),c\left(x\right)+s\Vert$$

(53)

where the parameter $\upsilon$ can increase the number of iterations to obtain a more optimal solution.

5. Case Study

The system structure is shown in Figure 3, containing three thermal power units, one wind farm, one CHP plant, and one electric boiler for CEHS energy supply. The abandoned wind penalty cost is 850 RMB/MWh, and the parameters related to CHP units, TP units, and EBs are shown in Appendix A Table A1, Table A2,

Table A3. Cost coefficient of CHP unit.

Coefficient	Value	Coefficient	Value
$b_0$ (RMB/h)	2740	$c_0$ (RMB/MW2h)	0.00254
$b_2$ (RMB/MW2h)	0.00698	$c_2$ (RMB/MW2h)	0.000233
$b_1$ (RMB/MWh)	112	$c_1$ (RMB/MWh)	38.4

,

Table A4. Operation parameters of TP unit.

Parameter		G1	G2	G3
Upper limit of power/MW		20	30	50
Lower limit of power/MW		10	12	25
Ramping rate/(MW/5 min)		3	4.5	7.5
Cost coefficient	$a_2$ (RMB/MW2h)	0.0004	0.0009	0.0015
	$a_1$ (RMB/MWh)	100	150	225
	$a_0$ (RMB/h)	200	500	167

and

Table A5. Operation parameters of EB.

Parameter	Value
Electrical power upper and lower limits/MW	[40, 10]
Ramping rate/(MW/5 min)	30
Electrical–heat conversion coefficient	0.98
Maintenance costs/(MW/RMB)	21

. Electric and heat load demand and wind power forecast power, as shown in Appendix A Figure A1. The total scheduling time is 2 h, and each period is 5 min, a total of 24 scheduling time periods.

The simulations are carried out in the following four modes, respectively.

Mode 1: CHP static model with the feasible region modeled by algebraic equations [33], whose extremal points are shown in Appendix A Figure A2 (CPLEX Solution).

Mode 2: CHP static model + electric boiler (CPLEX Solution).

Mode 3: CHP dynamic model, modeled by DAEs considering the optimal control process of CHP units as proposed (Simultaneous Method Solution).

Mode 4: CHP dynamic model + electric boiler (Simultaneous Method Solution).

5.1. CEHS Scheduling Results Analysis

5.1.1. Analysis of the Power Output of Each Unit

Comparing the output of each unit in the four modes, we analyze the effect of the CHP unit dynamic modeling method and the addition of the electric boiler method on the CEHS operation results. As can be seen in Figure 4a, after considering the optimal control process of the CHP unit, modes 3 and 4 have smaller TP units electric output in several periods, with an overall reduction of 11.95% and 6.66% compared with modes 1 and 2, respectively. In Figure 4b, the CHP units of modes 3 and 4 have a greater range of variability and possess greater or lesser electric output at multiple periods compared to modes 1 and 2. Adding an electric boiler achieves the electric and thermal decoupling of the CHP unit, which improves the system regulation capability. As shown in Figure 4c, the EB meets the heat load demand and reduces the heat and electric output of the CHP unit during severe wind abandonment periods, thus consuming more wind power. From Figure 4d, the heat output of the CHP unit is the same in each time period of modes 1 and 3, while the heat output of modes 2 and 4 with EB is lower than that of modes 1 and 3, and the reduced heat output is compensated by EB to balance.

Overall, the mode with the CHP unit dynamic model has a smaller TP unit and CHP unit electric output in multiple time periods, which provides more room for wind power feed-in. The mode with the addition of EB gives the system more flexibility to effectively coordinate the supply of lower cost WP unit.

5.1.2. Comparative Analysis of Scheduling Solutions

Table 1. Comparison of scheduling schemes with different modes.

Scheme Indicators	Mode 1	Mode 2	Mode 3	Mode 4
Calculation time/s	4.48	4.75	109.42	111.38
Total cost/RMB	154,827	126,310	128,817	101,490
TP output/MWh	124.68	128.27	109.78	119.73
TP cost/RMB	23,404	23,757	21,102	22,161
CHP electric output/MWh	393.03	410.27	381.58	384.44
CHP heat output/MWh	529.09	474.20	529.09	484.33
CHP cost/RMB	70,854	70,702	69,541	68,130
AWP/MWh	71.26	36.09	44.91	12.05
Penalty cost/RMB	60,570	30,675	38,174	10,240
EB heat output/MWh	-	54.89	-	44.76
EB cost/RMB	-	1176	-	959

Note: The total scheduling time is 2 h.

compares the scheduling schemes of the four modes in terms of each unit’s CEHS operating cost and output. It can be seen that the physical retrofit scheme of adding an EB (mode 2) and the optimized control retrofit scheme of the CHP unit (mode 3) can achieve the same effect in terms of total cost and wind power dissipation. In the actual engineering transformation, the optimized control transformation is significantly lower than the cost of adding an EB, which is more practical for engineering promotion. Compared with mode 1, mode 4 results in a significant reduction in the electric output of coal-consuming units, a significant reduction of 83.1% in abandoned wind power (AWP), and a reduction of 53,337 RMB in CEHS operating costs.

Compared with the CHP static model scheduling, the optimization scheduling with dynamic model constraints takes longer to calculate. However, it can meet the real-time requirements of the 2 h scheduling system, and the system economy has apparent advantages.

As shown in Figure 5, the overall view of the CHP unit retrofitted modes reduces the wind abandonment in several time periods. Specifically, within the scheduling time of 2 h, the abandoned wind power by the four modes are 71.26 MWh, 36.09 MWh, 44.91 MWh, and 12.05 MWh, respectively. The results show that the model combining the two retrofit solutions can provide greater flexibility for the system, improve wind power consumption capacity, and mitigate wind abandonment the best.

5.2. Dynamic Process Analysis Considering Optimal Control of the CHP Unit

5.2.1. CHP Unit Electric Power Output Dynamic Process Analysis

Figure 6 displays the distribution of operating points of the CHP units for 24 time periods in different modes. It can be seen that the operating points of modes 2 and 4 with the addition of EB have a wider distribution in the horizontal direction. The operating points of modes 3 and 4 with optimal control considered have more operating points near the upper and lower boundaries in the longitudinal direction, and the distribution is also wider. This phenomenon is because the CHP unit’s optimal control strengthens its electrical power ramping ability, while the EB relieves the “determining power generation by heat” restriction so that the unit can operate with less heat power. Therefore, the CHP unit has more room for power optimization, thus verifying better operational flexibility.

In modes 3 and 4, when the CHP dynamic model constraints are considered to participate in the optimal scheduling calculation, the scheduling values of the unit electric output corresponding to their scheduling schemes and the unit optimal control process values are shown in Figure 7. In mode 3, the unit electric output in the T11 period is regulated from 245.32 MW of variable load at 3000 s to 185.32 MW at 3300 s. The regulation time of this process is about 150 s, and the average shortage during the period is 0.11 MW/s. The T12 period changes the load from 185.32 MW at 3300 s to 183.05 MW at 3600 s. The regulation time of this process is about 100 s, and the average shortage is 0.0055 MW/s. Similarly, the dynamic process of the electric output of the CHP unit in mode 4 can be analyzed. The results show that the optimal control of the unit regulates faster and can quickly track the electric output of the scheduling demand. The average shortage during regulation is slight, which can be compensated for by fine-tuning the output of other units and has less impact on the feasibility of the scheduling scheme.

5.2.2. CHP Unit Main Steam Pressure Dynamic Process Analysis

During the execution of the scheduling scheme, the main steam pressure variation of the CHP unit is related to the safe and stable operation. As shown in Figure 8, the maximum peaks of upward fluctuations were 17.37 MPa and 15.73 MPa, and the maximum peaks of downward fluctuations were 16.07 MPa and 14.36 MPa, respectively, when the main steam pressure was set at 16.67 MPa and 15 MPa in mode 3. During the regulation process, the pressure did not exceed the fluctuation range of $\pm 5\%$ of the set value, and all reached the set value at a steady state. In mode 4, the set value of the main steam pressure of the CHP unit changes with the change in working conditions. According to the change of working conditions, set the pressure steady-state value to 16.67 MPa, 15 MPa, 14.5 MPa, 14 MPa, 13.5 MPa, and 13 MPa, respectively. The maximum upward and downward peak fluctuations of the main steam pressure at different setting values did not exceed the fluctuation range of $\pm 5\%$, and all reached the setting value at a steady state. The simulation proves that the pressure fluctuations caused by the optimized control process of the CHP unit do not affect the safe and stable operation of the unit.

5.2.3. CHP Unit Heat Output Dynamic Process Analysis

The set value of the heat source mass flow rate varies with the heat demand value during the fast regulation of the CHP unit optimal control. As shown in Figure 9, the steady-state values for each time period of modes 3 and 4 do not exceed the corresponding set values, and the peak fluctuations of the regulation process do not exceed the set values, achieving a fast recovery of the heat power of the regulation process with fewer fluctuation effects.

The upward and downward fluctuations of the heat source mass flow in Figure 9 can be calculated as the average equivalent value ($m_{H,equ}$), as shown by the dashed line. Overall, the CHP units in modes 3 and 4 generate 638.64 kJ and 25,214.97 kJ less heat in the scheduling time (2 h) when meeting the heat load demand. This heat leads to an average reduction of 0.1784 °C and 7.0446 °C in the circulating water temperature of the units, causing an average reduction of $0.6007\times {10}^{-3}$ °C and $23.7161\times {10}^{-3}$ °C in the production and living temperatures. The data show that the heat fluctuation generated by the CHP unit’s dynamic regulation process under optimal control does not affect the heating temperature demand.

6. Conclusions

This paper proposes a CEHS dynamic optimal scheduling model considering the CHP unit control process and electric boiler. Then the simultaneous method is used to solve the dynamic optimization problem.

The economic dispatching results of the system and the operation status of CHP units under four different modes are compared through numerical examples, and the following conclusions are obtained:

(1) The CEHS scheduling model with optimized control and retrofit of CHP units and electric boilers can absorb more wind power while minimizing the total scheduling cost of the system. Under the static modeling of CHP units, the dispatching cost of combined transformation mode 2 is reduced by 18.42%. Under the dynamic modeling of CHP units, the dispatching cost of combined transformation mode 4 is reduced by 21.21%.

(2) In the CEHS scheduling model, when the CHP unit is modeled according to the method described in Chapter 1 (dynamic model constraints), the system can obtain the best economy. Under the single transformation mode, the system scheduling cost constrained by the unit dynamic model is reduced by 16.80%. Under the combined transformation mode, the system scheduling cost constrained by the unit dynamic model is reduced by 19.65%.

(3) When CHP units adopt dynamic model constraints into CEHS scheduling, the feasibility of scheduling scheme implementation can be effectively guaranteed. During the execution of the dispatching value of the CHP unit, the electrical output matching, pressure safety, heat supply stability, and other variable load indicators all meet the unit’s normal operation requirements.

(4) Compared with the optimal scheduling method constrained by the static model of CHP units, the proposed simultaneous method can effectively solve the dynamic optimization problem. Although the calculation time is increased, it can ensure the demand for scheduling optimization scenarios. The dynamic operation process of the CHP unit in Section 5.2 can reach the steady state quickly and accurately, and the solution algorithm can meet the convergence requirements.

To sum up, the dispatching model and solution method proposed in this paper can further expand the space for wind abandonment and dissipation in the power grid, save the dispatching costs and provide the basis for the power grid dispatching department to formulate the dispatching plan. This paper compares the short-term dispatching economy of the four modes under specific conditions. It does not consider such factors as the initial investment of the project, operation and maintenance costs, and the annual time scale. Further discussion and research is needed in practical application.