Stochastic dynamic solution for off-design operation optimization of combined cooling, heating, and power systems with energy storage

Applied Thermal Engineering 163 (2019) 114356

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Stochastic dynamic solution for off-design operation optimization of combined cooling, heating, and power systems with energy storage Kuang Jiyuan, Chenghui Zhang, Bo Sun

T

⁎

School of Control Science and Engineering, Shandong University, Jinan, PR China

HIGHLIGHTS

nonlinearity, and storage are all considered in CCHP system operation. • Uncertainty, 24-hour CCHP system operation is modeled as a Gaussian random process. • The stochastic dynamic solution is proposed to solve the optimization problem. • AAdvantage of proposed method is significant, especially with strong randomness. • The optimality of the proposed method is guaranteed by the principle of optimality. • ARTICLE INFO

ABSTRACT

Keywords: CCHP system Energy storage Off-design performance Stochastic dynamic programming Operation optimization

Combined cooling, heating, and power (CCHP) systems are generally considered as a viable solution to meeting both the electric and thermal demands of buildings in metropolises. To ensure that CCHP systems operate at the optimal schedule, operation optimization is crucial. The nonlinearity of off-design performance, randomness of renewables and loads, and relationship between adjacent stages stemming from energy storage are three challenges that should be simultaneously addressed. In this paper, we propose a stochastic dynamic solution to perform the abovementioned operation optimization, which is resolved into a dynamic optimization problem consisting of many small static optimization problems. By applying the proposed method, the expected most economical operating schedule of a CCHP system with storage units can be obtained. As demonstrated through a case study, the proposed method reduces the operation cost by 13.4 RMB compared with a linear model used to describe the off-design performance. Moreover, a CCHP system with strong randomness shows operation cost reduction worth 10.4 RMB (1.66%) compared with that of the deterministic optimization. Hence, the stochastic dynamic solution can address off-design performance efficiently and is especially practical for use in CCHP systems with stochastic renewables.

1. Introduction Combined cooling, heating, and power (CCHP) systems can satisfy various kinds of demands with a high energy efficiency and low carbon emission [1,2]. In the operation of a CCHP system, energy storage is regarded as an indispensable component because it can significantly reduce the primary energy consumption by allowing energy generators to operate under more optimal conditions [3,4]. Therefore, the off-design performance of the main equipment should be considered [5]. Currently, several types of renewables are commonly applied to CCHP systems. Although energy storage can mitigate some of the adverse characteristics of these types of non-manageable energy sources, their stochastic performance can still have considerable effects [6]. Such a

⁎

complex CCHP system should be carefully managed to obtain an optimal performance [7–10]. Specifically, the off-design performance requires nonlinear models to describe the equipment of CCHP systems [11]. The uncertainties of renewables and loads make operation optimization a stochastic optimizing problem [12]. The presence of energy storage leads to correlations between adjacent operating stages in multistage optimization. On the other hand, optimization related to systems without storage is a quasi-steady single-point optimization and relatively easier to handle [13,14]. In this study, we manage to address these three factors in the operation optimization of CCHP systems, thereby providing a distinctive solution for future studies. Both off-design performance and randomness have already been simultaneously considered for the operation optimization of CCHP

Corresponding author. E-mail addresses: [email protected] (J. Kuang), [email protected] (C. Zhang), [email protected] (B. Sun).

https://doi.org/10.1016/j.applthermaleng.2019.114356 Received 31 May 2019; Received in revised form 7 September 2019; Accepted 7 September 2019 Available online 09 September 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

Applied Thermal Engineering 163 (2019) 114356

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Nomenclature ε σ π

Abbreviation CCHP COP FEL FTL MILP MINLP PGU PLR

combined cooling, heating, and power coefficient of performance following electric load following thermal load mixed-integer linear programming mixed-integer nonlinear programming power generation unit part-load ratio

Subscripts br c e exc exh g h i j l pgu pump pumph pumpc pv pwe r rexh rj s sh wp

Symbols C E F G H Q S V f k p sta u v var

efficiency prediction error standard deviations probability distribution

cooling electricity optimization objective (operation cost) natural gas heating heating value of natural gas state point capacity limit optimal operating cost stage proportion static decision stage operating cost variable

systems with no storage structure [15,16]. Hence, only the operation optimization of CCHP systems with storage is referred to in the following. Most previous studies on the operation optimization of CCHP systems have focused on off-design performance. Some researchers have used a nonlinear model to describe off-design performance. Deng et al. [17] developed a mixed integer nonlinear programming (MINLP) model for operation optimization of CCHP systems based on real data from an energy station in Tianjin. Bao et al. [18] developed an MINLP model based on predicted values and proposed an improved particle swarm optimization algorithm as a solution to day-ahead scheduling. Lorestani et al. [19] developed a simulation model for optimization of CCHP systems with renewables in which all the main components are nonlinearly modeled. Two operational strategies, namely following electric load (FEL) and following thermal load (FTL), were used for comparison, and a newly developed evolutionary particle swarm optimization (EPSO) was proposed as the optimization solution. In fact, the off-design performance data were usually measured under discrete operation points. Therefore, piecewise approximation (a special nonlinearity) is also regarded as a practical method in the modeling of a tri-generation. Bischi et al. [20] employed several mixed integer linear programming (MILP) models for piecewise approximation of a nonlinear CCHP system. The model was written in a modeling language for mathematical programming, and the resulting MILP was solved using the IBM ILOG CPLEX optimizer. Luo et al. [21] proposed a two-stage optimization and control structure for a CCHP system with a piecewise linearapproximated nonlinear-performance curve. MILP was applied to model the operating schedule problem. For CCHP systems with storage units, the relationship between adjacent time intervals significantly increases the amount of variables in operation optimization. Because a large-scale nonlinear optimization is difficult to solve, different solutions have been proposed. Deng et al. [17] set a certain operating rule for storage so that the complexity of

lithium bromide absorption chiller cooling electric heat exchanger exhaust generator heating internal combustion engine jacket water loss power generation unit heat pump heating energy provided by heat pump cooling energy provided by heat pump photovoltaic combination of PV, wind power, and electric load recovered thermal energy recovered thermal energy of exhaust recovered jacket water energy storage solar hot water wind power

optimization was reduced, while Bao and Lorestani et al. [18,19] applied heuristic algorithms to solve the complex problem. Although both of these types of studies reported good results, randomness was neglected in their operation optimization. Randomness is becoming a research hotspot in CCHP system operation optimization. In previous studies, deterministic point prediction (DPP) has been applied to handle uncertainty. Moreover, rolling optimization has also been utilized to fix operation plans after the error has already been generated. Gu et al. [22] proposed an online optimal operation approach for CCHP systems based on model predictive control with feedback correction to adjust the prediction error. In addition, a hybrid algorithm based on integrating time-series analysis and Kalman filters was employed in the DPPs of renewable energy resources. Further, the rolling optimization model used in the online operation was expressed as a general linear programming problem. Technically, the probability distribution function is the most accurate way to describe a stochastic phenomenon. In contrast, the DPP omits a considerable amount of information. Hence, studies on prediction mainly focus on prediction intervals [23]. The probabilistic forecasting of renewable energy power and loads has gained considerable attention in recent years [24,25]. Some studies have reported that a better operating schedule can be obtained when probabilistic forecasting results are employed in optimization [26,27]. Di Somma et al. [28] employed stochastic linear programing to model a CCHP system with uncertainties, which were simulated using the roulette wheel mechanism and the Monte Carlo simulation method. The forecast error was modeled using probability distribution, while the stochastic optimal operating schedule was solved using the branch-and-cut method. However, the off-design performance was not considered as it tends to limit the robustness of the solution method. Viviani et al. [29] considered the correlation between uncertain energy prices and energy demands by applying the Monte Carlo sampling technique to generate loading and pricing scenarios. The resulting stochastic multi-scenario 2

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MINLP model was solved by the branch-and-bound solver ANTIGONE (Algorithms for coNTinuous/Integer Global Optimization of Nonlinear Equations). Both randomness and nonlinearity were considered in this study, and a thermal storage tank was involved in the super structure. However, the operation optimization was solved independently in each stage. In other words, the relationship between adjacent time steps derived from storage was not considered. Hence, the three factors mentioned before have still not been considered simultaneously. For CCHP systems without storage, the operation optimization can be solved independently at each stage [30]. There are less than ten optimization variables to solve the stochastic nonlinear problem. For optimization of CCHP systems with storage units, the outputs of different pieces of equipment at each stage are taken as equivalent optimization variables in most existing studies [17–21]. In other words, hundreds of correlative variables should be considered. Undoubtedly, simultaneously considering both randomness and nonlinearity in optimization is difficult. To make it easier, fewer optimization variables are preferred. Facci et al. [31] considered storage units and off-design performance of a CCHP system with a simple structure. Dynamic programming and a meta-heuristic optimization were introduced to reduce the difficulty of operation optimization. The optimization complexity was significantly reduced by referring to the principle of optimality, and it was declared that the computation would have been impossible using other methodologies, even on a large computer cluster. For a more complex CCHP system, we also applied dynamic programming to deal with storage units. Furthermore, the concept of variable cost was introduced to simplify the component dispatch optimization with offdesign performance [32]. Both of these studies managed to reduce the complexity of operation optimization. However, they did not consider the randomness of the renewables and loads, which is highly important because renewables are widely applied. Based on our previous research, a host of renewables and their accompanying uncertainty are involved. In this study, we consider both the randomness of renewables and loads as well as off-design performance in the operation optimization of a CCHP system with energy storage units. The proposed problem is modeled as a dynamic optimization problem consisting of many static optimization problems. The dynamic problem is actually a distinctive Gaussian random process and reflects the dynamic and stochastic processes in the multistage operation of CCHP systems. (Dynamic refers to the relationship between adjacent stages with respect to the storage units. This relationship is

described by a difference equation.) The arrangement of equipment and the corresponding operating cost for each stage were determined by a static problem, which is repeatedly called for by a dynamic problem. Specifically, the dynamic and static optimization problems were linked by the total heating and cooling production. Finally, a stochastic dynamic solution is proposed to solve the aforementioned optimization model. The advantages of the proposed solution are compared with our previously proposed method [32]. The main contributions of this paper can be summarized as follows: 1. Both the randomness and off-design performance of a CCHP system with storage are considered in the operation optimization. 2. This optimization problem is modeled as a Gaussian random process involving stochastic optimization problems and is solved efficiently by using the proposed stochastic dynamic solution. 3. The optimality of the solution obtained using the proposed method is guaranteed by the principle of optimality. The remainder of this paper is organized as follows. In Section 2, we briefly introduce the superstructure and characteristic performance of the CCHP system. In Section 3, the optimization of CCHP system operation is modeled as a dynamic problem involving many static ones. In Section 4, the proposed stochastic dynamic solution is represented according to the optimization model. In Section 5, two control groups are designed and compared with the proposed method. Finally, in the last section, we summarize the main conclusions from this study and provide some directions for future work. 2. CCHP system characteristic modeling Without loss of generality, the structure of a representative scheme of the CCHP system is shown in Fig. 1 [11] introducing wind power, photovoltaic (PV) cell, and solar heated water. Specifically, the power generation unit (PGU) consumes natural gas and generates electricity and thermal energy simultaneously. The thermal recovery exchanger recovers the waste thermal energy in the jacket water and exhaust gas. Furthermore, the absorption chiller uses the recovered energy to produce cooling water. Similarly, the domestic hot water heat exchanger uses the recovered energy to produce domestic hot water. The chiller and exchanger are assisted by separate heat pumps. The thermal storage tanks operate at the calculated optimal schedule. Note that the renewable outputs and energy demand are stochastic. Moreover, the

Fig. 1. Structure and energy flux of the CCHP system. 3

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parameters of the equipment and concrete load of the system are presented in a case study.

is provided below. The presence of storage results in a close relationship between adjacent stages in the operation and thus must be focused upon. According to the right part of Fig. 1, the state-transition equation based on the stored energy between the k th hour and (k + 1) th hour can be expressed as

2.1. Off-design performance The PGU is a small gas-fired internal combustion generating set. The efficiency of the generator is and the efficiency of the internal combustion engine is i . With the exception of energy converting into electric power, a large portion of fossil energy is converted to waste thermal energy, whose proportion in jacket water, exhaust, and dissipation are denoted as pj , pexh , and pd , respectively. The values of these parameters under different part-load ratios (PLRs) are listed in Table 1. The PGU is connected to a thermal recovery exchanger to recover the thermal energy. The transfer efficiency from the consumed natural gas, G , to the recovered thermal energy, Hr , is defined as r , and it can be calculated as follows: r

= (1

pgu )·[(1

lrj )· pj +

rexh ·pexh ]

Hs (k + 1) = Cs (k + 1)

(10 01)· H

(1)

(2)

Hsh (k ) + Cload (k )

h (k )

+ sh (k ) , (k )

(4)

c

(5)

C (k ) = Cbr (k ) + Cpumpc (k )

(6)

Hexc (k )

+

exc

Cbr (k ) (7)

br

In the case of no storage units, the heating and cooling loads are directly supplied by the left part of Fig. 1. A k-stage operation-optimization problem only needs k independent calculations of the static problem. As for a CCHP system with storage, supposing the quantity of storage status is m, the complexity of operation optimization is mk in most existing studies [17–21]. This complexity can be reduced to k· m2 with respect to dynamic programming. Similar to this method, the objective of optimization of the operating cost is given in recursive form as

3. CCHP system operation optimization analysis and statement

Table 1 Performance factors of a small naturally aspirated internal combustion engine generator [11].

For the CCHP system given in Fig. 1, both thermal and electric demands should be met. The N hour operation cost F (optimization objective) can be expressed as

PLR

N k=1

( )

Hs (k ) H (k ) + 1 0 · 0 1 Cs (k ) C (k )

H (k ) = Hexc (k ) + Hpumph (k ) and

Hr (k ) =

where is the prediction error. In particular, wp is the error of wind power, pv is the error of PV, sh is the error of solar hot water, e is the error of the electric load, h is the error of the heating load, and c is the error of the cooling load, with the standard deviations of each error component denoted as wp , pv , sh , e , h , and c , respectively. The distributions of renewables and loads are assumed to be independent. Continuous variables cannot be processed by a computer. In our method, the error distribution was discretized by referring to Refs. [27,28] (see Fig. 2). It should be noted that other distributions are also suitable for our proposed method.

(Fgrid (k ) + E gas (k ))

·

where Hpumph and Cpumpc denote the heating and cooling contributions of the heat pump, respectively; and Cbr and Hexc denote the chiller and exchanger outputs, respectively. Different ratios of Cbr , Hexc , Hpumph , and Cpumpc result in different costs but can obtain the same H and C . The left part of Fig. 1 shows the combination of concrete equipment, based on which the optimal dispatch (Epgu , Cbr , Hexc , Hpumph , and Cpumpc ) to obtain certain H and C is calculated. This calculation is regarded as a static optimization problem. Epgu is relevant to the recovered thermal energy Hr . Considering the following equation, Epgu is totally determined by Cbr and Hexc :

Except for certain factors, the renewables and loads are stochastic. Therefore, prediction error is unavoidable. Without loss of generality, the forecast error of loads and renewables are assumed to comply with a Gaussian distribution [25,27], i.e.,

F=

c

where Hs and Cs represent the amount of stored heating and cooling energies, respectively; Vh and Vc are the capacities of the heating and cooling tanks, respectively; h and c are the heating and cooling storage efficiencies, respectively; H and C denote the total heating and cooling production, respectively; and Hload and Cload are the heating and cooling energy demands, respectively. The control input variables in system (4) are H and C . Hence, each multistage operation corresponds to a certain energy storage process. The optimization of the multistage storage plan is regarded as a dynamic optimization problem. H and C link the left and right parts of Fig. 1 and can be expressed according to the structure of the left part as follows:

2.2. Stochastic performance

2)

0

load (k )

where lrj is the thermal energy loss ratio of jacket water in the thermal recovery process, and rexh is the efficiency of recovering thermal energy from the exhaust. The PGU model can be fitted using the aforementioned off-design operating data. Furthermore, PLRs of less than 20% are not allowed in the system operation owing to the low energy utilization rate. According to the fitted model, there is a clear correlation between the generator output (Epgu ), consumed natural gas (G ), and recovered thermal energy (Hr ). Further, the efficiencies of the absorption chiller and domestic hot water heat exchanger are br and exc , respectively, while the heating and cooling efficiencies of the heat pumps are denoted as COPh and COPc , respectively.

N(0,

h

0

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

(3)

where Fgrid represents the power cost of the grid and Egas represents the cost of consumed natural gas. To operate the CCHP system in the most economic manner, the optimal operation schedule of each device (PGU, absorption chiller, heat exchanger, and heat pumps) should be calculated. We have determined a new way to formulate this optimization problem. The operation optimization modeling, which clarifies the solving framework, 4

i

0.0000 0.1020 0.1809 0.2250 0.2637 0.2871 0.3085 0.3184 0.3184 0.3039 0.2886

g

0.0000 0.7700 0.7800 0.8200 0.8400 0.8600 0.8750 0.8850 0.9000 0.9100 0.9200

pj

pexh

pd

0.5628 0.5227 0.5031 0.4903 0.4865 0.4861 0.4892 0.4818 0.4745 0.4507 0.4336

0.2764 0.2955 0.3006 0.3097 0.3108 0.3125 0.3237 0.3285 0.3285 0.3169 0.3147

0.1608 0.1818 0.1963 0.2000 0.2027 0.2014 0.1870 0.1898 0.1971 0.2324 0.2517

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Fig. 3. Arrangement of Sk .

Fig. 2. Error distribution discretization.

min f (k ) = f (k k=N

1) + v (k )

adjusting the heating and cooling production. However, the randomness of energy demands and solar hot water results in deviations that follow the normal distribution. The objective of the dynamic problem is to search the optimal multistage energy storage plan corresponding to the most economically expected operating cost. For visualization, Sk (Hs , Cs ) is used to denote the heating and cooling energy storages of stage k, where 0 Hs Vh and 0 Cs Vc . Vh and Vc denote the capacities of the heating and cooling storage tanks, respectively. After setting m and n, Sk can be discretized into V V (m + 1)·(n + 1) state points. State point Sk p mh , q nc can be expressed

(8)

where N is the number of stages of total optimization, f is the multistage operating cost, and v is the stage cost. Stage operating cost v is calculated as follows:

v (k ) = Eprice (k )·E grid (k ) + Gprice·G (k )

(9)

where Eprice and Gprice denote the prices of electric and natural gas, respectively. Egrid is the amount of electricity transmitted from/to the grid and can be positive or negative. The corresponding Eprice of buying and selling electricity are Epricebuy and Epricesell , respectively. Based on Table 1, consumed natural gas G is determined by Epgu , and Egrid is calculated as follows:

Egrid (k ) = Eload (k ) +

Hpumph (k ) COPh

Ewp (k ) +

pv (k )

+ +

Cpumpc (k ) COPc wp (k )

+

Epgu (k ) e (k ).

(

)

simply as Skp, q , where 0 p m and 0 q n . Therefore, Sk can be denoted by Sk {Skp, q, p , q} , which is arrayed as depicted in Fig. 3. In a 24-h operation, tracking the stored heating and cooling energies, Skp, q , in each stage can provide the multistage energy storage plan of the CCHP system. Regardless of the uncertainties of cooling and heating loads, the contribution of the most economical equipment arrangement from Skp, q to Ski,+j 1 is denoted as uk (Skp, q, Ski,+j 1) . The corresponding operating cost is denoted as vk (Skp, q, Ski,+j 1) . The method to solve uk and vk is given in the static optimization problem. Taking the decision uk (Skp, q, Ski,+j 1) at Skp, q , there is a possibility of deviating from the plan and achieving Ski++ 1 i, j + j when considering randomness. By considering the randomness of the heating and cooling storage, the 5 × 5 square expression of joint probability distribution k can be obtained according to Fig. 2. The dynamic problem is thus visualized as a discrete dynamic random process, which can be simply determined by controlling H and C . The detailed states of each stage follow a Gaussian random process during the multistage operation, and stochastic dynamic programing was employed. The function fk (Skp, q, SN + 1) denotes the most economically expected operating cost from Skp, q to the final state SN + 1, and vk (Skp, q, Ski,+j 1) is the

Epv (k ) (10)

In summary, the operation of CCHP systems can be modeled as a state-transition equation (dynamic problem) whose control variables are H and C . If the stage operation cost v could be directly denoted by H and C , the dynamic optimization problem could be directly addressed by stochastic dynamic programming. Unfortunately, v has a complex relationship with heating and cooling productions and cannot be solved directly. Hence, the static optimization problem is defined to calculate the optimal equipment dispatch and the expected operating cost in different heating/cooling production stages. In other words, the dynamic problem focuses on the whole multistage operation optimization, and some detailed information is required in this calculation. This information is obtained by repeatedly solving the static problems. 4. Stochastic dynamic solution

(

V

V

)

(

V

V

)

most economical operating cost from Sk p mh , q nc to Sk + 1 i mh , j nc . Hence, the backward stochastic dynamic programming equation can be expressed as follows:

The difficulties of a dynamic problem are its dynamic process constraints due to the presence of energy storage and the randomness of solar hot water as well as heating and cooling demands. In contrast, the difficulties of a static problem involve the off-design performance and randomness of renewables and electricity demands. The dynamic problem is solved through stochastic dynamic programming, while the static problem is solved by extracting the core element, namely the variable cost. Finally, the operating schedule corresponding to the expected minimum CCHP system operation cost is determined.

fk (Skp, q, SN + 1) = min i, j

{

i, j

[

k(

i, j)· fk + 1 (Ski++ 1 i, j + j, SN + 1)]

}

+ vk (Skp, q, Ski,+j 1) = fk + 1 (Sk + 1, SN + 1) + vk (Skp, q, Sk + 1 ) fN + 1 (SN + 1, SN + 1) = 0, (11) where Sk + 1 represents the optimal state point selected from Sk + 1. The complexity at the boundary of the energy storage requires explanation. Once the maximum storage capacity is reached, the excess energy would be wasted. When the thermal tanks are exhausted, the extra energy demands due to the prediction error are fulfilled by heat pumps. The additional costs of heating and cooling are denoted as fpumph and fpumpc , respectively. Therefore, Sk was expanded out of the natural boundary. The stochastic dynamic programming flow diagram is shown

4.1. Dynamic optimization problem solution The dynamic optimization problem is a framework of the stochastic dynamic solving method and corresponds to the right part of Fig. 1. In terms of energy storage with stochastic performance, the operation of a CCHP system can be represented as a special random process. Specifically, the stored energy in the next stage can be roughly controlled by 5

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in Fig. 4. The prediction error is relatively small compared with the capacity of thermal storage tanks. Hence, the multistage storage optimization is divided into two steps. First, the stored energy is discretized with low accuracy, and general dynamic programming is used to search for the shortest path. Second, the stored energy is discretized with high accuracy around the path obtained in the first optimization, and stochastic dynamic programming is employed to handle the uncertainty of heating and cooling loads.

v = Eprice·

E+

H Hexc C Cbr + COPh COPc

Epgu +

pwe

+ Gprice· G (Epgu ) (12)

where E is the difference between electrical load Eload and renewable output, and pwe denotes the combination of pv , wp , and e . By disregarding the uncertainty of the CCHP system and referring to the economic concept of variable cost and constant cost, cost function v is split into constant cost vsta and variable cost vvar : (13)

v = vsta + vvar

When the PGU is turned off, the operating cost can be expressed as follows:

4.2. Static optimization problem solution Static optimization problems are the elementary units of the total operation optimization and correspond to the dispatch of the components in the left part of Fig. 1. Specifically, a static problem can be described as the calculation method of the detailed device dispatch and the corresponding expected operation cost, given the total amount of heating and cooling production. In addition, this single stage optimization is known as a static problem because it is considered quasi steady [30]. According to optimization modeling, stage operating cost v is calculated as

vsta = Eprice·

E+

H C + COPh COPc

(14)

where vsta is the constant cost determined by an uncontrollable factor in the static problem. Turning the generator on results in an additional gas cost, while the generated power offsets the power bought from the grid. vvar represents the change in cost resulting from the generator being operated at various power levels. The operating cost continues to decrease as the PLR increases (provided that the derivative of vvar with respect to Epgu is negative). Hence, the key to solving the static

Fig. 4. Stochastic dynamic programming flow diagram. 6

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Fig. 5. Hourly energy demands and renewable outputs in day-ahead prediction.

Fig. 6. Real values of energy demands and renewable outputs.

Table 2 Uncertainty parameters. Standard deviations of predicted errors pv ,

e,

wp ,

h,

and

and

Values (%) 30

sh

5

c

problems is to obtain the minimum value of function vvar , which is calculated as

vvar = Gprice· G (Epgu ) Eprice· Epgu +

Hexc C + br COPh COPc

(15) Fig. 7. Peak and valley time prices [11].

The method to solve this deterministic problem has been described in our previous research [32]. The data required for this study are listed in the Appendix. However, the prediction error of wind power, PV cell, and electric load would make the operation cost uncertain. The discrete probability distribution of pwe is denoted as pwe , which is shown in Fig. 2. The expected operational cost is calculated as

Table 3 Parameters of CCHP system [11,32,35].

2

vexpect =

[ i= 2

pwe (i · pe )· v (i )]

(16)

where pwe (i · pe ) is the probability of different conditions, and v (i ) is the operation cost under different conditions. In this section, the static optimization problem, which is the elementary unit of the stochastic dynamic solving method, was solved. By referring to the concept of variable cost, all static problems can be simplified to the same model. The expected operation costs obtained in static problems are calculated and fed back to the dynamic problem. Therefore, the dynamic problem can obtain the multistage operation schedule of a CCHP system.

Parameter

Value

Rated COP for electrically driven heat pump COPh , COPc Cold storage coefficient c Heat storage coefficient h Capacity of heat storage unit Vh Capacity of cold storage unit Vc Rated power of generator Erated Capacity of absorption chiller Vbr Efficiency of domestic hot water heat exchanger exc

3 0.99 0.98

Efficiency of absorption chiller Recovery efficiency of exhaust Heating value of natural gas Q Price of natural gas Gprice

0.8 0.8 35,500 kJ/m3 2.7 RMB/m3

Energy loss ratio of jacket water in heat exchanger lrj br re

Efficiency of power plant Efficiency of power transmission

7

180 kW·h 180 kW·h 90 kW 150 kW 0.8 0.2

0.35 0.92

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Fig. 9. Simulation without probabilistic randomness.

5. Case study

The price of electricity (in Chinese Yuan, RMB) per hour is shown in Fig. 7. The parameters of the CCHP system are listed in Table 3.

5.1. Load description and basic data

5.2. Results and discussion

In the conventional simulation of the daily operation of CCHP systems, the value of the stored energy is considered to be zero at the beginning and end of each day, 0:00 AM. Owing to the low electricity price, the CCHP system does not operate late at night, when it consumes the stored energy from the tanks. Hence, the beginning and end of a day in this study was set as early morning; in all the following figures, the first stage corresponds to 6:00–7:00 AM. For our test case, the energy demands were obtained from Refs. [31,33]. Electric, cooling, and heating loads of a typical summer day (average values of energy demands from a whole summer) were selected as the day-ahead forecast values. The outputs of the renewables were obtained from Refs. [25,34], as shown in Fig. 5. The forecast errors were set as listed in Table 2 [24,25]. The loads and renewables are assumed to obey normal distributions [25–29]. The simulation renewable outputs and energy demands are shown in Fig. 6.

In rolling optimization simulations, the actual value is known at the end of each stage, thereby generating new operation discrepancy each hour. The operating deviations of wind power, PV, and electric load are absorbed by the power grid. Similarly, heating and cooling deviations as well as deviations of solar hot water are absorbed by the thermal storage tanks. In the most extreme case, the additional energy would be wasted if the tank is already full, whereas the heat pump would activate if the tanks are unable to provide energy. The heat pumps will work if operated by the controller even if the tanks are full. Previous studies have neglected either off-design performance or randomness. In comparison, we reduced our nonlinear model to a general linear model to simulate the first case. Meanwhile, the proposed stochastic solving method is degraded into a deterministic solving method (proposed in Ref. [32]) to simulate the second case. Note that the optimality of the degraded method is also guaranteed by the

Fig. 8. Simulation without off-design performance.

8

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Fig. 10. Simulation results of stochastic dynamic solving method.

Fig. 11. Simulation with renewables considering deterministic optimization.

Table 4 Target comparison of a whole day of operation without renewables. Scenario

FEL

Linear model optimization

Deterministic optimization

Proposed method

Operating cost (RMB)

1420.9

1340.7

1332.4

1327.3

energy is precisely determined, deviations clearly exist in the electricity. Hence, the actual PGU output is different from that optimized using a linear model. Fortunately, this error would not be significant when the PGU operates at approximately 0.5–0.9 PLR, but it would undoubtedly harm the performance of the CCHP system. The simulation result of the optimization method without probabilistic randomness (solution proposed in Ref. [32]) is shown in Fig. 9, and that of the proposed stochastic dynamic solving method is shown in Fig. 10. The proposed method considers both the off-design performance and probabilistic randomness of the CCHP system. Hence, its performance should be better than those of other methods. In reality, CCHP systems usually operate at the following electric load (FEL). Specifically, the electricity produced by a generator is equal to the electric load. The recovered thermal energy is applied to drive the chiller and heat exchanger, while the additional energy is stored in tanks. The heat pumps supply energy when the stored energy is exhausted. While the FEL is a weak method, it provides practical operation. The difference in cost between FEL and the optimal operating schedule is minimal when the system capacity is optimized. A comparison of operating costs under different operation modes is

principle of optimality. Only MATLAB 2014a was utilized to realize our proposed method, and all the solving procedures were typed as scripts. For a 24-stage operation optimization, the proposed stochastic dynamic solution would spend approximately 6 s to address the calculation on an average desktop (i3, 3.4 GHz). For 48-stage operation optimization, the calculation time is approximately 12 s. To show the influence of randomness, first, the CCHP system without renewables was used in the simulation. In other words, only the randomness of loads was considered (CCHP system with weak uncertainty). The simulation results without considering the off-design performance are shown in Fig. 8. The nonlinear relationship between power generation and thermal recovery is simplified as a linear model. When the recovered thermal 9

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Table 6 Statistical result.

Table 5 Target comparison of a whole day operation with renewables. Deterministic optimization

Proposed method

Operating cost (RMB)

626.8

616.4

Standard deviation

1.63%

1.30%

presented in Table 4. According to Table 4, all the optimization operating modes performed better than the FEL mode. However, the linear-model optimization results in deviation in optimization modeling and underperforms. Hence, the proposed method results in a considerable improvement (13.4 RMB) by considering the off-design performance. Although the proposed method considers the uncertainty of loads, slight advantages (5.1 RMB) over deterministic optimization are observed. The reason is that the uncertainty factors are relatively weak, and no significant difference is observed between predicted and real values. It is essential to consider the off-design performance in operation optimization. However, the influence of randomness should be continually investigated. To research the performance of the proposed method, strong randomness is introduced to the CCHP system. Considering a CCHP system with wind power, PV, and photovoltaicthermal, the deterministic optimization and stochastic dynamic solving methods were applied under the same system, loads, and renewable outputs. The only difference is that the proposed method utilizes the probability distribution of the prediction error, which is neglected in deterministic optimization (solution proposed in Ref. [32]). Theoretically, more information would benefit the decision. The simulation results are shown in Figs. 11 and 12. The simulation results are compared in Table 5. The operating costs of the proposed stochastic solving method were reduced by 1.66% (10.4 RMB) compared with that of the deterministic optimization. Compared with the simulation results without renewables, it can be concluded that the stochastic dynamic solving method can obtain a better result, especially in a CCHP system with strong randomness. For the relationship between adjacent stages, each single decision has a chaotic impact on the whole multistage operation performance. Hence, the advantage of a single stage is difficult to explain. Fortunately, a significant difference is observed from stages 10 to 11. During deterministic optimization, which neglects the probability distribution, the PGU is turned on, whereas the proposed method results in the PGY being turned off. The reasons for the different decisions can be analyzed by referring to Fig. 13 and Appendix Table 2. Considering the electric load, wind power, and PV, the expected value of the power demand is approximately 19 kW·h during this period (Fig. 13). In addition, two heat pumps operate at 80 kW during this period (Fig. 11 (a)), consuming approximately 26 kW·h of the electric demand. The total power demand is approximately 45 kW·h, and the corresponding variable cost is −4.8 (Table A2). Thus, it seems reasonable to turn on the PGU. However, the proposed method considered the prediction error and realized the potential risk of renewables’ fluctuation. The operation cost expectation is uneconomical according to the probability calculation. Hence, the proposed method recognized a worthwhile gamble, leading to a better result at this stage. The stochastic dynamic solving method considers more information to make more reasonable probabilistic decisions. The advantage of one single decision in real operation can never be guaranteed. However, this advantage would accumulate and be observed after multistage operation. To validate our proposed method, additional simulations were carried out 120 times, and the statistical result of the advantage of our proposed method compared with the deterministic method (solution proposed in Ref. [32]) is given in Table 6. In general, the conclusion can be drawn that the proposed solution performs better. In summary, the stochastic dynamic solution considered both the off-design performance and randomness of a CCHP system with storage. The method has considerable advantages over a simplified linear model

Fig. 12. Simulation with renewables considering the stochastic dynamic solving method.

Scenario

Average value

Fig. 13. Expected and real power shortfalls. 10

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that does not consider the off-design performance. Moreover, a significant advantage of the proposed method over the deterministic method was verified by repeated simulations.

1. The operation optimization of a CCHP system with storage is innovatively modeled as a random process involving many static optimization problems, thereby providing a new angle for future research. 2. The optimality of the solution obtained using the proposed method is guaranteed by the principle of optimality. 3. The proposed operation optimization solution can be applied to the optimal design of CCHP systems since the calculation time is short enough (6 s for 24-hour operation optimization).

6. Conclusions Operation optimization of a CCHP system with storage considering off-design performance and probabilistic randomness is a dynamic nonlinear stochastic problem. In this study, the originally complex operation optimization was resolved to a dynamic optimization problem consisting of many static ones, and a stochastic dynamic solution was proposed. The dynamic problem was solved by employing stochastic dynamic programming, while the static problem was further simplified and solved directly. In the case study, the simulation results indicated that the proposed method achieved a considerable advantage over that which neglects off-design performance and randomness. Hence, the conclusion can be drawn that the proposed method can simultaneously consider the three factors mentioned before—nonlinearity of off-design performance, randomness of renewables and loads, and relationship between adjacent stages stemming from energy storage—and achieve a reasonable advantage. To the best of our knowledge, this is the first solution for the operation optimization of CCHP systems with storage units that considers both off design performance and randomness of loads and renewables. Furthermore, it can be employed to solve most of distributed energy systems. In terms of a system without uncertainty or nonlinearity, the proposed method can be reduced to a deterministic or linear method. The major contributions of this work can be summarized as follows:

It should be noted that some factors have still not been considered in this study. For example, source-load collaborative optimization is a research hotspot. Future research will focus on combining the proposed solution with a heuristic algorithm to involve load management in optimization. Moreover, the model extension may introduce multiple objective decisions to supply valid supports for CCHP system designing. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgement This work was supported by the National Natural Science Foundation of China (grant numbers 61821004, 61733010, 61573224), the Young Scholars Program of Shandong University (grant number 2016WLJH29), and the Fundamental Research Funds of Shandong University (grant number 2018JC060).

Appendix This Appendix presents the data on specific static problems of a concrete CCHP system under different electricity prices. As discussed in Section 3.2, the stage operating cost consists of constant cost vsta and variable cost vvar , where vvar is the main variable of a static problem. Tables A1 and A2 are derived by combining the plant modeling discussed in Section 2 with the system parameters in Tables 1 and 4. Table A1 PGU model. PLR

Epgu (kW·h)

Hr (kW·h)

G (m3)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

9 18 27 36 45 54 63 72 81 90

69.12 70.45 76.38 80.70 87.69 94.98 104.09 115.15 130.12 149.04

11.62 12.94 14.84 16.48 18.48 20.29 22.67 25.48 29.70 34.37

Table A2 Variation trend of variable cost vvar (RMB). PLR

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Electricity price/RMB 0.4

0.6

0.8

0.9

1.1

20.60372 20.40306 20.2139 21.12166 21.49097 22.54851 23.04238 24.91251 27.71156 33.91609 40.9125

16.99092 14.91688 12.8567 11.64814 9.986914 8.871719 7.176949 6.760969 7.17 10.77631 14.96359

13.37812 9.430707 5.499508 2.174617 −1.51714 −4.80507 −8.68848 −11.3906 −13.3716 −12.3635 −10.9853

11.57172 6.687618 1.820911 −2.56214 −7.26917 −11.6435 −16.6212 −20.4663 −23.6423 −23.9334 −23.9598

7.958918 1.20144 −5.53628 −12.0357 −18.7732 −25.3203 −32.4866 −38.6179 −44.1839 −47.0731 −49.9087

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References

[18] Z. Bao, Q. Zhou, Z. Yang, Q. Yang, L. Xu, T. Wu, A multi time-scale and multi energy-type coordinated microgrid scheduling solution, IEEE Trans. Power Syst. 30 (5) (2015) 2257–2276. [19] A. Lorestani, M.M. Ardehali, Optimal integration of renewable energy sources for autonomous tri-generation combined cooling, heating and power system based on evolutionary particle swarm optimization algorithm, Energy 145 (2018) 839–855. [20] A. Bischi, L. Taccari, E. Martelli, E. Amaldi, G. Manzolini, P. Silva, et al., A detailed MILP optimization model for combined cooling, heat and power system operation planning, Energy 74 (2014) 12–26. [21] Z. Luo, Z. Wu, Z. Li, H. Cai, B. Li, W. Gu, A two-stage optimization and control for CCHP microgrid energy management, Appl. Therm. Eng. 125 (2017) 513–522. [22] W. Gu, Z. Wang, Z. Wu, Z. Luo, Y. Tang, J. Wang, An online optimal dispatch schedule for CCHP microgrids based on model predictive control, IEEE Trans. Smart Grid 8 (5) (2017) 2332–2342. [23] L. Liu, Y. Zhao, D. Chang, J. Xie, Z. Ma, Q. Sun, et al., Prediction of short-term PV power output and uncertainty analysis, Appl. Energy 228 (2018) 700–711. [24] A. Vaghefi, M.A. Jafari, E. Bisse, Y. Lu, J. Brouwer, Modeling and forecasting of cooling and electricity load demand, Appl. Energy 136 (2014) 186–196. [25] Z. Ziadi, M. Oshiro, T. Senjyu, A. Yona, N. Urasaki, T. Funabashi, et al., Optimal voltage control using inverters interfaced with PV systems considering forecast error in a distribution system, IEEE Trans. Sustain. Energy 5 (no2) (2014) 682–690. [26] J. Xie, T. Hong, GEFCom2014 probabilistic electric load forecasting: an integrated solution with forecast combination and residual simulation, Int. J. Forecast. 32 (2016) 1012–1016. [27] H.A. Aalami, S. Nojavan, Energy storage system and demand response program effects on stochastic energy procurement of large consumers considering renewable generation, IET Gener. Transm. Distrib. 10 (2016) 107–114. [28] M. Di Somma, G. Graditi, E. Heydarian-Forushani, M. Shafie-khah, P. Siano, Stochastic optimal scheduling of distributed energy resources with renewables considering economic and environmental aspects, Renew. Energy 116 (2018) 272–287. [29] V.C. Onishi, C.H. Antunes, E.S. Fraga, H. Cabezas, Stochastic optimization of trigeneration systems for decision-making under long-term uncertainty in energy demands and prices, Energy 175 (2019) 781–797. [30] H. Cho, A.D. Smith, P. Mago, Combined cooling, heating and power: a review of performance improvement and optimization, Appl. Energy 136 (2014) 168–185. [31] A.L. Facci, S. Ubertini, Meta-heuristic optimization for a high-detail smart management of complex energy systems, Energy Convers. Manage. 160 (2018) 341–353. [32] J. Kuang, C. Zhang, F. Li, B. Sun, Dynamic optimization of combined cooling, heating, and power systems with energy storage units, Energies 11 (2018) 2288. [33] Office of Energy Efficiency and Renewable Energy. Commercial Reference Buildings. Available online: < http://energy.gov/eere/buildings/commercialreference-buildings > (accessed on 13 November 2012). [34] Z. Ning, K. Chongqing, X. Qing, L. Ji, Modeling conditional forecast error for wind power in generation scheduling, IEEE Trans. Power Syst. 3 (2014) 1316–1324. [35] Z. Yang, L. Yuehong, Y. Chengchu, W. Shengwei, MPC-based optimal scheduling of grid-connected low energy buildings with thermal energy storages, Energy Build. 86 (2015) 415–426.

[1] L. Ju, Z. Tan, H. Li, Q. Tan, X. Yu, X. Song, Multi-objective operation optimization and evaluation model for CCHP and renewable energy based hybrid energy system driven by distributed energy resources in China, Energy 111 (2016) 322–340. [2] M. Liu, Y. Shi, F. Fang, Combined cooling, heating and power systems: a survey, Renew. Sustain. Energy Rev. 35 (2014) 1–22. [3] M.A. Lozano, J.C. Ramos, M. Carvalho, Structure optimization of energy supply systems in tertiary sector buildings, Energy Build. 41 (2009) 1063–1075. [4] A.D. Smith, P.J. Mago, N. Fumo, Benefits of thermal energy storage option combined with CHP system for different commercial building types, Sustain. Energy Technol. Assess. 1 (2013) 3–12. [5] V. Palomba, M. Ferraro, A. Frazzica, S. Vasta, F. Sergi, V. Antonucci, Experimental and numerical analysis of a SOFC-CHP system with adsorption and hybrid chillers for telecommunication applications, Appl. Energy 216 (2018) 620–633. [6] M. Liu, Y. Shi, F. Fang, Load forecasting and operation strategy design for CCHP systems using forecasted loads, IEEE Trans. Control Syst. Technol. 23 (5) (2015) 1672–1684. [7] R. Zeng, H. Li, L. Liu, X. Zhang, G. Zhang, A novel method based on multi-population genetic algorithm for CCHP–GSHP coupling system optimization, Energy Convers. Manage. 105 (2015) 1138–1148. [8] L. Guo, W. Liu, J. Cai, B. Hong, C. Wang, A two-stage optimal planning and design method for combined cooling, heat and power microgrid system, Energy Convers. Manage. 74 (2013) 433–445. [9] X. Song, L. Liu, T. Zhu, T. Zhang, Z. Wu, Comparative analysis on operation strategies of CCHP system with cool thermal storage for a data center, Appl. Therm. Eng. 108 (2016) 680–688. [10] H. Al Moussawi, F. Fardoun, H. Louahlia-Gualous, Review of tri-generation technologies: design evaluation, optimization, decision-making, and selection approach, Energy Convers. Manage. 120 (2016) 157–196. [11] D. Wei, A. Chen, B. Sun, C. Zhang, Multi-objective optimal operation and energy coupling analysis of combined cooling and heating system, Energy 98 (2016) 296–307. [12] M. Hu, H. Cho, A probability constrained multi-objective optimization model for CCHP system operation decision support, Appl. Energy 116 (2014) 230–242. [13] F. D’Ettorre, P. Conti, E. Schito, D. Testi, Model predictive control of a hybrid heat pump system and impact of the prediction horizon on cost-saving potential and optimal storage capacity, Appl. Therm. Eng. 148 (2019) 524–535. [14] L. Urbanucci, D. Testi, J.C. Bruno, An operational optimization method for a complex polygeneration plant based on real-time measurements, Energy Convers. Manage. 170 (2018) 50–61. [15] L. Ji, D. Niu, G. Huang, An inexact two-stage stochastic robust programming for residential micro-grid management-based on random demand, Energy 67 (2014) 186–199. [16] L. Urbanucci, D. Testi, Optimal integrated sizing and operation of a CHP system with Monte Carlo risk analysis for long-term uncertainty in energy demands, Energy Convers. Manage. 157 (2018) 307–316. [17] N. Deng, R. Cai, Y. Gao, Z. Zhou, G. He, D. Liu, et al., A MINLP model of optimal scheduling for a district heating and cooling system: A case study of an energy station in Tianjin, Energy 141 (2017) 1750–1763.

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