Multi-Objective Optimal Operation for Steam Power Scheduling Based on Economic and Exergetic Analysis

Abstract

Steam supply scheduling (SSS) plays an important role in providing uninterrupted reliable energy to meet the heat and electricity demand in both the industrial and residential sectors. However, the system complexity makes it challenging to operate efficiently. Besides, the operational objectives in terms of economic cost and thermodynamic efficiency are usually contradictory, making the online scheduling even more intractable. To this end, the thermodynamic efficiency is evaluated based on exergetic analysis in this paper, and an economic-exergetic optimal scheduling model is formulated into a mixed-integer linear programming (MILP) problem. Moreover, the ε-constraint method is used to obtain the Pareto front of the multi-objective optimization model, and fuzzy satisfying approach is introduced to decide the unique operation strategy of the SSS. In the single-period case results, compared with the optimal scheduling which only takes the economic index as the objective function, the operation cost of the multi-objective optimization is increased by 4.59%, and the exergy efficiency is increased by 9.3%. Compared with the optimal scheduling which only takes the exergetic index as the objective function, the operation cost of the multi-objective optimization is decreased by 19.83%, and the exergy efficiency is decreased by 2.39%. Furthermore, results of single-period and multi-period multi-objective optimal scheduling verify the effectiveness of the model and the solution proposed in this study.

1. Introduction

As the material basis of human survival and development, energy plays an increasingly important role in promoting social and economic development as well as in improving people’s living standards [1]. The energy situation and environmental problems have recently attracted worldwide attention. Steam supply scheduling (SSS) consumes primary energy to provide energy for an enterprise and simultaneously produces a substantial number of pollutants. This paper focuses on optimizing operation of the SSS to reduce the operation cost and improve the thermodynamic efficiency, thus the economic–exergetic operation of the system can be realized [2,3].

Scholars have conducted in-depth studies on the operation optimization of SSS and have made some achievements. Grossman proposed a mixed-integer linear programming (MILP) model framework for utility systems [4], and a mixed-integer nonlinear programming (MINLP) problem based on a successive MILP approach was solved [5]. Based on utility systems modeling, numerous scholars have focused on optimal operation strategy so as to achieve cost minimization and energy distribution. In order to reduce operation cost, a model was established which integrated the start and stop of utility operating units under different process requirements [6]. In [7], the multi-period with different electricity or steam demand was introduced into the operation strategy of the utility system, and the optimal choice of units for each period was determined by using the two-stage approach. Based on the linear single-device models in public utility systems, the influence of the change of external electricity price on the system operation scheduling was studied by taking into account the steam equilibrium, fuel supply and devices operation constraints [8]. A multi-period MILP model for byproduct gases, steam and power distribution optimization was proposed in steam power plants, and the experimental results showed that the proposed model could effectively reduce system cost [9]. Given the uncertainty of device efficiency and process demand, a data-driven method was proposed to achieve the tradeoff between optimality and robustness of operational decisions in utility system optimization, and the experiments demonstrated the effectiveness of the method [10], [11]. Besides, a method for the simultaneous synthesis of heat exchanger networks and utility systems was presented, and the two-stage algorithm was used to identify the best tradeoff between utility systems and heat exchanger networks costs [12,13]. Due to the escalating environmental crisis, several scholars have conducted extensive researches on the environmental issue. Central utility systems with adjoining waste-to-energy networks were integrated to form an ecological friendly energy management system, and the feasibility of the combination of the two networks was demonstrated from environmental and economic perspectives through experiments [14]. By taking into account the impact of pollutants on environment, the utility system consisting of boilers, gas turbines with heat recovery steam generators, ST and CT has been developed [15]. Considering the environmental performance of the entire site utility system, the structural design was optimized to minimize the total annual cost [16]. In addition, the multimodal genetic algorithm was used in the exergoenvironmental analysis of a combined heat and power plant [17]. In general, most studies have focused on the economic and environmental operation that consider energy quantity saving in the SSS, without considering the quality distinction between different energy resources.

In 1953, Rant put forward the concept of exergy, which is a physical quantity that synthesizes the first and the second law of thermodynamics to measure working ability. This concept can focus on the quality and quantity of energy [18,19,20], which provides a highly efficient method to evaluate the energy efficiency of the system [21,22,23]. Certain scholars have recently studied the energy system by means of the exergetic analysis. A kind of solid oxide fuel cell integrated with gas and steam trigeneration systems was optimized, and the energy, exergy and economy of the system were analysed in [24]. In addition, the influence of parameter changes on system performance was further studied. An existing CHP system was analysed in terms of energy, exergy and environmental (3E) aspects [25]. In order to analyse the performance and optimize parameter of the geothermal power plant, a system optimization model was formulated to maximize the exergy efficiency, which was solved through the gravitational search algorithm [26]. A study was conducted to examine the energy and exergy performance as well as multi-objective optimization of an exhaust air heat recovery system, which could provide reference for system planning [27]. In [28], considering the total cost, carbon dioxide emission and exergetic destruction, a multi-objective optimization of district heating system was carried out and the Pareto front was obtained with the weighting method. By means of exergy, exergoeconomic and exergoenvironmental analysis, the optimal integration of steam and power system with a steam power plant as the source and a utility system as the sink was investigated, and the experimental results reflected that the integration of steam power plant and utility system is a favorable option [29]. However, most studies have applied exergetic analysis to the performance evaluation and parameter optimization of the energy system, while there are relatively few quality studies on the energy such as heat and electricity in the SSS.

In this paper, exergy is introduced into the operation optimization of SSS. Firstly, the multi-objective mixed-integer linear programming (MOMILP) model of SSS is established by using the exergetic analysis method to reduce the operation cost and exergy input for SSS. At the same time, the Pareto front of the multi-objective optimization model is obtained with the ε-constraint approach, and the compromise solution on the Pareto front was acquired with the fuzzy satisfying approach. Finally, the effectiveness of the proposed model and solution method was verified by the results of single-period and multi-period multi-objective optimal scheduling.

This paper is organized as follows: the MOMILP model of SSS is developed in Section 2. In Section 3, the multi-objective operation strategy of SSS is presented to obtain the Pareto front, and a tradeoff is conducted between these different objectives. Case studies are analysed in Section 4, and the conclusion is summarised in Section 5.

2. Problem Formulation of Multi-objective Optimal Operation of SSS

The SSS converts primary energy (fuel coal, fuel gas, fuel oil) into secondary energy (electricity, steam and hot water) to provide enterprises with the required process steam, thermal energy and electricity, and its typical structure is illustrated in Figure 1. To realize the economic and efficient operation of the SSS, the mathematical model of each equipment is built and the concept of exergy is adopted to evaluate all types of energy. Subsequently, the MOMILP optimal model of the SSS is formulated.

2.1. Objectives

2.1.1. Economic Objective

Generally, the economic objective of SSS operation is to minimize the total cost of the entire period, including fuel consumption cost, electricity or steam purchase cost, equipment operation and maintenance cost, depreciation cost and equipment start/stop cost. The specific expression is as follows:

$$\begin{array}{ll}{f}_1=& {\displaystyle \sum _{Bn}{\displaystyle \sum _i{\displaystyle \sum _t{F}_{Bn,t,i}}}}{p}_i+{\displaystyle \sum _r{\displaystyle \sum _{t}P{S}_{r,t}}{p}_r}+{\displaystyle \sum _{t}P{E}_t}{p}_e\\ & +{\displaystyle \sum _n{\displaystyle \sum _{t}C{M}_n}}{Y}_{n,t}+{\displaystyle \sum _n{\displaystyle \sum _{t}CS{T}_n{Z}_{n,t}}}+{\displaystyle \sum _n{\displaystyle \sum _{t}CS{P}_{n}Z{S}_{n,t}}}\end{array}$$

(1)

2.1.2. Exergetic Objective

Exergy represents the maximum amount of useful work that can be obtained from a given form of energy, that is, the quality of energy. For different kinds of energy, the quality of the same quantity of energy is not necessarily the same. Hence, the quality of different energy of the SSS is evaluated with the exergetic analysis method, and then the maximum exergy efficiency can be achieved. Among them, exergy efficiency can be defined as the ratio of the total output exergy to the total input exergy [21], and the formula is as follows:

$$\psi =\frac{{\displaystyle \sum _t\dot{E}{x}_{out,t}}}{{\displaystyle \sum _t\dot{E}{x}_{in,t}}}$$

(2)

Figure 1 shows that the energy types on the load side include electric energy and heat energy, and the input energy includes fossil energy (fuel coal, fuel oil, fuel gas), purchased steam and electricity. Therefore, the load demand exergy and the input exergy of the SSS are expressed as follows:

$$\{\begin{array}{l}\dot{E}{x}_{out,t}=\dot{E}{x}_{out,t}^e+\dot{E}{x}_{out,t}^h\\ \dot{E}{x}_{in,t}=\dot{E}{x}_{in,t}^{coal}+\dot{E}{x}_{in,t}^{oil}+\dot{E}{x}_{in,t}^{gas}+\dot{E}{x}_{in,t}^e+\dot{E}{x}_{in,t}^h\end{array}$$

(3)

Fuel coal, fuel oil and fuel gas are all chemical fuels, and their specific exergy is generally expressed by a lower heating value (LHV) and an exergy factor [30,31]. The exergy factors of different types of fuels are slightly different, but basically the same [32]. The exergy in chemical fuels can be expressed by the following formula:

$$\{\begin{array}{l}{\alpha }_i={\gamma }_{i,LHV}{\zeta }_i\\ \dot{E}{x}_{in,t}^i={\alpha }_i{F}_{Bn,t,i}/3.6\end{array}$$

(4)

The exergy of electricity is equal to electricity because it can be completely converted into work. However, the work done by heat energy is limited by the Carnot factor, and its heat exergy is equal to the work done by the Carnot cycle. Therefore, the heat and electricity exergy of SSS are described as follows:

$$\dot{E}{x}_{out,t}^h={\displaystyle \sum _r(1-\frac{T_a}{T_r^h})D{S}_{r,t}{h}_r/3.6}$$

(5)

$$\dot{E}{x}_{in,t}^h={\displaystyle \sum _r(1-\frac{T_a}{T_r^h})\mathrm{P}{S}_{r,t}{h}_r/3.6}$$

(6)

$$\dot{E}{x}_{out,t}^e=D{E}_t$$

(7)

$$\dot{E}{x}_{in,t}^e=P{E}_t$$

(8)

The energy demand of SSS is predictable, that is, the total output exergy can be calculated. Therefore, the exergetic objective could be converted from maximum exergy efficiency to minimum exergy input of the SSS. The formula is as follows:

$$f_2={\displaystyle \sum _t\dot{E}{x}_{in,t}}$$

(9)

2.2. Constraints

2.2.1. Device Constraints

Industrial boilers generally convert the chemical energy in fuel into heat energy, and then transfer this heat energy to water through different heating surfaces, and finally produce the high-pressure (HP) or medium-pressure (MP) steam required by the system. The model is represented by Equation (10). Equation (11) indicates that the boiler load should be placed within a certain safety range:

$$F_{Bn,t,i}{\gamma }_{i,LHV}=a_B{}_n{Y}_{Bn,t}+b_{Bn}{G}_B{}_{n,t}$$

(10)

$$G_{Bn}^L{Y}_{Bn,t}\le G_{Bn,t}\le G_{Bn}^U{Y}_{Bn,t}$$

(11)

The operation characteristics of a simple ST, including backpressure and CT, can be expressed by the linear relationship between steam intake and power output, and the steam intake and output power of ST should be placed within a certain safety range. The general constraints are detailed as follows:

$$P_{Tn,t}=d_{Tn}{Y}_{Tn,t}+e_{Tn}{G}_{Tn,in,t}$$

(12)

$$P_{Tn}^L{Y}_{Tn,t}\le P_{Tn,t}\le P_{Tn}^U{Y}_{Tn,t}$$

(13)

$$G_{Tn,in}^L{Y}_{Tn,t}\le G_{T\mathrm{n},in,t}\le G_{Tn,in}^U{Y}_{Tn,t}$$

(14)

The double extraction CT generates steam with different pressure, and at the same time it can generate the power required to meet the external electricity load. Its output power is related to steam intake, industrial sectors or heating steam extractions. The model generally uses a linear function to represent the relationship among steam intake, adjustable extraction and power of the double extraction CT. It can be described as follows:

$$P_{cn,t}=g_{cn}{G}_{cn,in,t}-{\displaystyle \sum _{oj}{h}_{cn,oj}}{G}_{cn,oj,t}^{out}-f_{cn}{Y}_{cn,t}$$

(15)

$$P_{cn}^L{Y}_{cn,t}\le P_{cn,t}\le P_{cn}^U{Y}_{cn,t}$$

(16)

$$G_{cn,in}^L{Y}_{cn,t}\le G_{cn,in,t}\le G_{cn,in}^U{Y}_{cn,t}$$

(17)

$$G_{cn,oj}^{out,L}{Y}_{cn,t}\le G_{cn,oj,t}^{out}\le G_{cn,oj}^{out,U}{Y}_{cn,t}$$

(18)

The main function of the pressure reducer and attemperator is to adjust the steam from high temperature and high pressure to the relatively low temperature and low pressure required by the system. The general form of the model is shown in Equation (19) [10,33]. For a given steam system, once the steam pressure and temperature of each level steam are specified, the enthalpy h_Ln,_out, h_Ln,in, and h_Ln,w are constant. By defining a parameter η_Ln., the expression is shown in Equation (20):

$$G_{Ln,in,t}=\frac{h_{Ln,out,t}-h_{Ln,w,t}}{h_{Ln,in,t}-h_{Ln,w,t}}{G}_{Ln,out,t}$$

(19)

$$G_{Ln,in,t}={\eta }_{Ln}{G}_{Ln,out,t}$$

(20)

2.2.2. Balance Constraints

Electricity balance and steam balance should be considered to meet the energy demand and energy distribution among various energy devices in the SSS should also be considered. Therefore, the steam and electricity balance model are expressed as follows:

$$D{S}_{r,t}=P{S}_{\mathrm{r},\mathrm{t}}+{\displaystyle \sum _n^{}{G}_{Bn,t}}{Y}_{Bn,t}+{\displaystyle \sum _n^{}{G}_{cn,oj,t}^{out}{Y}_{cn,t}+{\displaystyle \sum _n^{}{G}_{\mathrm{Ln},\mathrm{out},\mathrm{t}}}{Y}_{Ln,t}-}{\displaystyle \sum _n^{}{G}_{Tn,in}{}_{,t}{Y}_{T\mathrm{n},t}}-{\displaystyle \sum _n^{}{G}_{\mathrm{Ln},\mathrm{in},\mathrm{t}}}{Y}_{Ln,t}-Los{s}_r$$

(21)

$$D{E}_t=P{E}_t+{\displaystyle \sum _n^{}{P}_{Tn,t}}{Y}_{T\mathrm{n},t}+{\displaystyle \sum _n^{}{P}_{\mathrm{cn},t}}{Y}_{\mathrm{cn},t}$$

(22)

2.2.3. Logical Constraints on Device Start and Stop

Due to the different demand for steam and power in different periods, the equipment is easy to start and stop in the adjacent two periods, thus resulting in the start and stop cost of the equipment. In this study, Equations (23) and (24) are used to represent the start and stop logic of the equipment:

$$Z_{n,t}=Y_{n,t}-Y_{n,t-1},Y_{n,0}=0$$

(23)

$$Z{S}_{n,t}=Y_{n,t}-Y_{n,t+1},Y_{n,t+1}=0$$

(24)

In summary, the MOMILP optimal model of SSS can be represented as follows:

$$\begin{array}{l}{\min \mathrm{obj}}_1=f_1(x_t,y_t)\\ {\min \mathrm{obj}}_2=f_2(x_t,y_t) \\ \mathrm{subject} \mathrm{to} \{\begin{array}{l} \mathrm{g}(x_t,y_t)\le 0\\ \mathrm{h}(x_t,y_t)=0\\ x_t\in (x^L,x^U)\\ y_t\in (0,1)\end{array}\end{array}$$

(25)

3. Solution and Decision of the Optimal Condition

In order to achieve optimal operation of the economy and exergetics of the SSS, this section uses the ε-constraint method to solve the proposed multi-objective problem, so as to obtain the Pareto front. The Pareto curve is used to determine the optimal solution with the fuzzy satisfying approach.

3.1. ε-Constraint Based Solution

In the optimization problem of SSS, the two objectives of reducing the operation cost and exergy input are considered to affect each other, that is, it is difficult for both sides to reach the optimal simultaneously. Therefore, the ε-constraint method is used to solve this multi-objective optimal problem. The ε-constraint method preserves one of the objectives in the objective function and transforms the rest of the objective functions into constraints. Thereby the multi-objective optimization problem is transformed into a series of single-objective optimization problems, which can be solved by modifying the value range of the constraints condition step by step. The details are presented as follows:

$$\begin{array}{l}\min f_1\\ \{\begin{array}{l}\mathrm{subject} \mathrm{to} f_2\le \epsilon \\ \hspace{1em} \mathrm{Equations} (1)-(24)\end{array}\end{array}$$

(26)

where the value of ε is considered to be expressed by the following equation:

$$\epsilon =f_{2,\max }-(f_{2,\max }-f_{2,\min })(a-1)/(a_{\max }-1)$$

(27)

where a = 1,2, …, a_max, a_max is the maximum number of cycles; f_2,min and f_2,max are the maximum and minimum values of f₂ obtained when f₂ and f₁ are considered as a single-objective function, respectively. Furthermore, the economic objective is taken as f₁ as shown in Equation (1) and the exergetic objective as f₂ as shown in Equation (9) in this paper.

3.2. Decision based on Fuzzy Satisfying Approach

To achieve the coordination and unification of the multi-objective of SSS, the fuzzy satisfying approach [34] is introduced to help the operator establish a trade-off between the economic objective and the exergetic objective. The target values of each operation strategy are normalized according to (28). Subsequently, the membership function value of each operation strategy is calculated according to (29), and the best operation strategy is selected according to Equation (30):

$${\mu }_k^a=\{\begin{array}{ll}1& f_k^a\le f_{k,\min }\\ \frac{f_{k,\max }-f_k^a}{f_{k,\max }-f_{k,\min }}& f_{k,\min }\le f_k^a\le f_{k,\max }\\ 0& f_k^a\ge f_{k,\max }\end{array}$$

(28)

$${\mu }^a=\min ({\mu }_1^a,{\mu }_2^a)$$

(29)

$${\mu }^{\max }=\max ({\mu }^1,{\mu }^2,\dots {\mu }^{a_{\max }})$$

(30)

To make the fuzzy satisfying approach clearer, a small example is given in

Table 1. An example of the fuzzy satisfying approach.

a	f1	f2	µ1	µ2	µmax
1	57883.09281	575814.554	1	0	0
2	58242.90797	560822.484	0.956601	0.105264	0.105264
3	58697.95742	545830.368	0.901716	0.210528	0.210528
4	59153.00687	530838.253	0.846831	0.315792	0.315792
5	59608.05632	515846.137	0.791946	0.421056	0.421056
6	60063.10577	500854.021	0.73706	0.52632	0.52632
7	60566.37947	485861.905	0.676359	0.631584	0.631584
8	61116.28338	470869.789	0.610033	0.736848	0.610033
9	61666.18728	455877.674	0.543707	0.842112	0.543707
10	66174.01862	433390.6	0	1	0

, and the multi-objective optimization strategy selected is scheme 7, which is the bold part of Table 1. In addition, Figure 2 summarizes the specific process of the multi-objective operation optimization.

4. Case Study

To verify the effectiveness of the MILP model of SSS with economic and exergetic objectives, the optimal model of single-period and multi-period of SSS are solved and results are analysed in this section. Moreover, it is necessary to declare the case studied in this paper does not consider optimization situation of neighbouring enterprises, which is a partial optimization.

4.1. Case Description

This study takes the SSS of petrochemical enterprises as an example (Figure 1), which includes four different levels of steam, namely, high-pressure steam (HP, 9.5 MPa and 535 °C), medium pressure steam (MP, 3.5 MPa and 425 °C), low-pressure steam (LP, 1 MPa and 300 °C) and low and low-pressure steam (LLP, 0.3 MPa and 200 °C). B1–B2 are coal-fired boilers that produce HP steam with a blowdown rate of 8%; B3–B5 are dual fuel boilers, which burn oil and gas to produce MP steam. The amount of gas is determined by the processing unit, and the maximum available gas import capacity is 12 t/h. Double extraction CT (CC1, CC2) produces MP and LP steam as well as electric energy. T1 and T2 steam turbines generate power, and T3 produces LLP steam and electric energy. As can be seen from Figure 1, the condensate is recycled and converted into boiler feed water. Furthermore, the minimum value of condensing steam amount of steam turbine is 63 t/h, the maximum value of condensing steam amount is 142 t/h, and the condensing pressure is 5.9 kPa. Pressure reducer and attemperator (L1, L2, L3 and L4) can convert high-temperature and high-pressure steam into relatively low-level steam. The study allows the maximum electricity import capacity from neighbouring enterprises of 50,000 kW. The maximum MP steam, LP steam and LLP steam import capacity from neighbouring enterprises of 100, 50 and 50 t/h. The effects of device 1, device 2 and other devices on the system are not considered in this study, and the loss of the system is neither considered, that is, loss_r = 0.

Table 2. Model parameters of boiler and steam turbine (ST).

Parameter	Value	Parameter	Value
aB1	95.8	fc1	−179
aB2	95.8	fc2	−179
aB3	19.47	gT3	73.37
aB4	5.818	gc1	252
aB5	5.159	gc2	252
bB1	0.8488	hc1,o1	−235
bB2	0.8488	hc1,o2	−102
bB3	0.931	hc2,o1	−235
bB4	1.021	hc2o2	−102
bB5	1.032	hT3	−23.3
dT1	−1459	ηL1	0.933
dT2	−1650	ηL2	0.933
eT1	86.2	ηL3	0.913
eT2	88.6	ηL4	0.923
fT3	−116.7

indicates the model parameters of boiler and steam turbine.

Table 3. Boiler equipment information.

Boiler	B1	B2	B3	B4	B5
Rated evaporation (t/h)	320	320	140	75	65
Minimum evaporation (t/h)	160	160	80	35	50
Start and stop cost (￥)	10,000	10,000	13,000	13,000	13,000
Equipment operation cost (￥/h)	100	100	200	185	170
Steam temperature (°C)	535	535	425	425	425
Steam pressure (MPa)	9.5	9.5	3.5	3.5	3.5
Feedwater temperature (°C)	211.23	211.23	163	163	163
Feedwater pressure (MPa)	15.08	15.08	3.5	3.5	3.5

and

Table 4. Steam turbine equipment information.

Equipment	Start and Stop Cost (￥)	Equipment Operation Cost (￥/h)	Power Rating (kW)		Maximum Steam Intake (t/h)	Maximum First Extraction Steam (t/h)	Maximum Second Extraction Steam (t/h)
			Maximum	Minimum
CC1	12,000	200	50,000	20,000	380	190	180
CC2	12,000	200	50,000	20,000	380	190	180
T1	6000	160	6500	1500	85	—	—
T2	6000	140	6500	1500	85	—	—
T3	6000	110	6000	3500	100	—	—

list the equipment parameters, start/stop costs and equipment operation costs of boilers and ST.

Table 5. The unit price of resources.

Fuel Coal (￥/t)	Fuel Gas (￥/t)	Fuel Oil (￥/t)	Electricity (￥/kWh)	MP Steam (￥/t)	LP Steam (￥/t)	LP Steam (￥/t)
750	1200	3500	0.45	180	135	80

indicates the unit price of resources.

Table 6. Parameters of the resource.

Fuel	Lower Heating Value (kJ/kg)	Exergy Factor	/	Enthalpy (kJ/kg)	Industrial Steam Production	Value (t/h)
Coal	24,440	1.08	MP	3280.7	IMPSP	57
Gas	32,503	1	LP	3051.7	ILPSP	76.8
Oil	40,245	1.06	LLP	2865.9	ILLPSP	34.9
			Water of Ln	632.2

shows the parameters of the resource, and

Table 7. Start and stop time of equipment.

Equipment	B1	B2	B3	B4	B5	CC1	CC2	T1	T2	T3
Start time (h)	10	10	7.5	7.5	7.5	3.3	3.3	3.3	3.3	3.3
Stop time (h)	10	10	7.5	7.5	7.5	3.3	3.3	3.3	3.3	3.3

lists the start and stop time of the equipment.

4.2. Single-Period Case

Table 8. Single-period steam and electricity demands.

Steam (t/h)			Electricity (kW)
MP	LP	LLP
180	125	150	64,400

reports the demand for steam and electricity over a single period time, without considering the start and stop costs of the equipment in the economic objective. During the solution process, the maximum number of cycles n in the ε-constraint method is set to 20.

Figure 3 shows the single-period Pareto front for the SSS, the points on it are all optimal values, which can provide different operation strategies for operators. Furthermore, the multi-objective optimal operation strategy can be obtained with the fuzzy satisfying approach, which is the point marked on the Pareto curve in Figure 3.

Table 9. Optimal scheduling results of boilers load for SSS.

Boilers Load (t/h)/Situation	Multi-Objective Operation (Standard)	Economic Operation	Exergetic Operation
B1 steam production	320	0	0
B1 coal consumption	41.33	0	0
B2 steam production	0	320	247.03
B2 coal consumption	0	41.33	34.36
B3 steam production	0	103.00	0
B3 oil consumption	0	0	0
B3 gas consumption	0	8.55	0
B4 steam production	0	0	0
B4 oil consumption	0	0	0
B4 gas consumption	0	0	0
B5 steam production	0	0	0
B5 oil consumption	0	0	0
B5 gas consumption	0	0	0

and

Table 10. Optimal scheduling results of purchased resources for SSS.

Purchased Resources/Situation	Multi-Objective Operation (Standard)	Economic Operation	Exergetic Operation
PMP steam (t/h)	21.74	0	100
PLP steam (t/h)	0	0	0
PLLP steam (t/h)	0	0	0
PE (kW)	19,838.62	743.04	19,838.62

show the optimal scheduling results of boilers load and purchased resources for SSS.

Evidently, compared with the multi-objective operation, the energy conversion equipment such as boilers and ST meets the demand for steam and most electricity in the economic operation. Consequently, less steam and electricity are purchased. By contrast, exergetic operation purchases more steam from the neighbouring enterprises. The multi-objective operation establishes a tradeoff between the economic objective and exergetic objective to satisfy the multi-objective optimal operation by coordinating the consumption of different types of energy (fossil energy, heat energy and electric energy).

Table 11. Comparison of multi-objective optimization results with single-objective optimization results.

Situation	Indicator	Value	Growth Rates
Multi-objective operation (standard)	Operation cost/￥	44,667.83	0
	Input exergy/kW	334,216	0
	Exergy efficiency/%	75.83%	0
Economic operation	Operation cost/￥	42,615.71	−4.59%
	Input exergy/kW	380,921.40	13.97%
	Exergy efficiency/%	66.53%	−9.3%
Exergetic operation	Operation cost/￥	53,526.94	19.83%
	Input exergy/kW	323,989.8	−3.06%
	Exergy efficiency/%	78.22%	2.39%

indicates the operation cost, input exergy and exergy efficiency in multi-objective optimization and single-objective optimization. Based on the results of multi-objective optimization, the growth rate of operation cost, input exergy and exergy efficiency in economic operation and exergetic operation are calculated. Results reveal that compared with the multi-objective optimal operation, the operation cost of the economic operation is decreased by 4.59%, while the input exergy is increased by 13.97%. On the contrary, for the exergetic operation, its input exergy is decreased by 3.06%, while its operation cost is increased by 19.83%.

Compared with SSS optimal scheduling which only takes economic or exergetic as the objective function, from the above calculated data, it can see that the multi-objective operation can comprehensively consider energy efficiency from the point of view of economic and exergetic, make a tradeoff between the economic index and exergetic index, and pay attention to the quality and quantity of energy simultaneously, so as to achieve the purpose of reducing cost and increasing efficiency. Furthermore, this paper is in line with the sustainable energy development strategy of the world today.

4.3. Multi-Period Case

A multi-period case is established in this section to further verify the effectiveness of the proposed multi-objective model and solution method. The multi-period model includes six periods, each with a duration of 720 h, which is consistent with the solution method and the ε-constraint parameter setting in Section 4.2.

Table 12. Multi-period steam and electricity demands.

Period	Steam (t/h)			Electricity (kW)
	MP	LP	LLP
1	180	125	150	64,400
2	350	165	75	73,700
3	320	185	60	85,700
4	250	200	95	58,700
5	200	175	105	67,600
6	335	300	120	78,300

indicates the steam and electricity demands of the six periods. The optimal scheduling results are detailed as follows.

Figure 4 depicts the steam distribution among the equipment. It can be seen that under the premise of fully considering the steam purchase, the boiler and ST jointly produce steam, and the system supplements the regulation of pressure reducer and attemperator, thus the integrated operation of steam production and supply at all levels can be realized.

Table 13. Optimal scheduling results of start/stop of multi-period operation equipment.

Equipment/Period	1	2	3	4	5	6
B1	1	1	1	0	0	1
B2	0	1	1	1	1	1
B3	0	0	0	0	0	0
B4	0	0	0	1	1	0
B5	0	0	0	0	0	0
CC1	1	1	1	1	1	1
CC2	1	1	1	1	1	1
T1	0	0	0	0	0	0
T2	0	0	0	0	0	0
T3	0	1	0	0	1	0

shows the optimal scheduling results of the start and stop of multi-period operation equipment. Figure 5 describes the optimal scheduling results of fuel consumed in the system. Evidently, changes in steam and electricity demand lead to the inevitably start and stop of equipment, thus changing fuel consumption. Due to the relative high steam and electricity demand compared with other periods, the B1 and B2 are in operation in periods 2, 3 and 6. Furthermore, since the MP steam demand of period 1 is lower than that of periods 4 and 5, boilers producing MP steam are closed in period 1. Moreover, considering that the energy of the system is converted from the HP steam generated by B1 and B2, a large amount of coal is consumed.

Figure 6 shows the optimal scheduling results of electricity. Under the premise of purchasing electricity (Figure 6 PE), it can be observed that the coordinated operation of T1, T2, T3, CC1 and CC2 can meet the electricity demand. Besides, T1 and T2 are stopped in each period and double extraction CT is used more frequently during the operation process. On the one hand, this is because CC1 and CC2 can satisfy most of the electricity demand. Moreover, CC1 and CC2 can generate electricity and produce both MP and LP steam to meet the steam demand by consuming HP steam. On the other, T1 and T2 only generate electricity. In order to save operation costs, it is not necessary to maintain the operation of all units. In addition, T3 is used more frequently than T1 and T2, partly because T3 can generate electricity and LLP steam simultaneously. On the other hand, it can be seen from Table 4 that the operation cost of T3 is lower than that of T1 and T2. Accordingly, this operation strategy can save economic costs. Furthermore, in order to balance both economic and exergetic objectives, the system neither over purchases energy, nor blindly consumes chemical fuel to meet the electricity demand, thus realizing the primary energy saving and improving the thermodynamic efficiency of the system in multi-period operation.

Therefore, the multi-period case study in this section can provide guiding significance for the actual operating system, and the corresponding unit output plan can be made from the two aspects of system economy and thermodynamic efficiency. Furthermore, this study can help reduce greenhouse gas emissions and improve the thermodynamic efficiency under the premise of meeting the power and thermal demand of enterprises.

5. Conclusions

In order to achieve a good balance between enhancing energy efficiency and reducing system cost, this paper adopts the exergetic analysis method in thermodynamics to evaluate the effective energy contained in different kinds of energy. At the same time, the exergetic objective function is built. Considering the cost of electricity and steam, combined with the mathematical model of each equipment, an SSS optimal model based on economic index and exergetic index is further built. Utilizing the ε-constraint method to obtain the Pareto front of multi-objective optimization problems, the fuzzy satisfying approach is introduced to determine the optimal operation strategy. Taking the single-period operation as an example, it can be seen that the multi-objective optimization operation strategy can consider the economic and exergetic of the system by comparing with the single- objective optimization results. Meanwhile, the single-objective optimization only takes the economic or exergetic index as the objective function. Moreover, it can be verified by the results of multi-period scheduling that the multi-objective model and solution is effective. In addition, to deal with the multi-objective problem, the fuzzy satisfying approach is introduced to obtain the optimal results. However, the optimal results may rely on the fuzzy satisfying approach. Therefore, to get better multi-objective optimal results, our future work will focus on the effectiveness of various multi-objective optimal methods.