Risk-based scheduling of smart apartment building under market price uncertainty using robust optimization approach

Sustainable Cities and Society 48 (2019) 101549

Contents lists available at ScienceDirect

Sustainable Cities and Society journal homepage: www.elsevier.com/locate/scs

Risk-based scheduling of smart apartment building under market price uncertainty using robust optimization approach

T

Afshin Najafi-Ghaleloua, Kazem Zarea, , Sayyad Nojavanb ⁎

a b

Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Department of Electrical Engineering, University of Bonab, Bonab, Iran

ARTICLE INFO

ABSTRACT

Keywords: Smart apartment building Market price uncertainty Robust optimization approach Robust management Robust mixed-integer linear programming

Nowadays market price uncertainty is one of the important challenging issues in the optimal scheduling of smart apartment building (SAB). So, in this paper, a robust optimization approach (ROA) is proposed for robust scheduling of SAB in the presence of price uncertainty. For modeling the market price uncertainty, upper and lower limits of the market price are considered instead of the forecasted market prices. The proposed sample system includes a SAB that contains ten smart homes (SHs) with different living habits and equipped with different equipment, i.e. combined heat and power (CHP), boiler, battery storage system (BSS), thermal storage system (TSS) and smart appliances. To assess the effectiveness of a home energy management system (HEMS) on the performance of the proposed problem, two different controlling scenarios are studied, namely normal and smart scenarios. It should be mentioned that the proposed model is formulated as mixed-integer linear programming (MILP) which guarantees global optimal solution and carried out with general algebraic modeling system (GAMS) software.

1. Introduction The HEMS is a technology platform comprised of both software and hardware that allows the users of the SHs to monitor energy usage and production and to manually or automatically control the use of energy within the household (Rastegar, Fotuhi-Firuzabad, & Zareipour, 2016). The HEMS may allow a user's to do a range of things such as turn equipment on and off remotely, set appliances to operate on schedules, set up conditional rules for appliances operation, manage the flow of energy from CHP, boiler and other generators through the home or in and out of the storages (Marzband, Yousefnejad, Sumper, & DomínguezGarcía, 2016). One of the important issues in the optimal scheduling of SAB is seeking strategies to meet the heat and electricity demands under uncertainty situations. Notably, the ROA is one of the powerful tools to find the robust scheduling strategies in the presence of market price uncertainty (Nojavan, Najafi-Ghalelou, Majidi, & Zare, 2017). 1.1. Literature review Optimal energy management of SH without considering uncertainties have been reviewed as follows: To minimize the consumption cost of the SH and practical management of renewable sources, a new SH community architecture has been provided in Anees and Chen ⁎

(2016). An evolutionary accretive comfort algorithm has been utilized in (Khan, Javaid, & Khan, 2018) to control and manage the appliances’ priorities within the SH. In order to increase the energy efficiency and reduce the peak demand in the SH, a novel home energy management algorithm has been provided in Elma and Selamogullari (2015). Optimal scheduling of SH in the presence of solar thermal storage system has been studied in Najafi-Ghalelou, Nojavan, Majidi, Jabari, and Zare (2018a) A novel multi-objective home energy management method has been provided in Sattarpour, Nazarpour, and Golshannavaz (2018) to simultaneous reduction of SH's bill cost and load profile deviation. In order to minimize the total operation cost of the SH and maximize the comfort level of residents, a multi-objective model of the SH has been provided in Rahmani-Andebili (2017). In order to control the energy management of multi- households, a multi-objective cost-load optimization planning problem has been provided in Shakouri and Kazemi (2017). In order to determine the optimum timing of exchanging power among the SHs, a novel optimization algorithm based on the harmony search algorithm, bat algorithm, particle swarm optimization algorithm and the differential evolution algorithm has been provided in Marzband, Fouladfar, Akorede, Lightbody, and Pouresmaeil (2018). An interoperable internet-of-things platform has been provided in Iqbal et al. (2018) to optimal management of SH under the cloud and web-ofobjects architecture. Finally, in order to minimize the bill costs of the

Corresponding author. E-mail address: [email protected] (K. Zare).

https://doi.org/10.1016/j.scs.2019.101549 Received 27 July 2018; Received in revised form 31 March 2019; Accepted 10 April 2019 Available online 18 April 2019 2210-6707/ © 2019 Elsevier Ltd. All rights reserved.

Sustainable Cities and Society 48 (2019) 101549

A. Najafi-Ghalelou, et al.

SjBattery , SThermal Stored energy in the BSS/TSS at time t (kWh) j, t ,t Battery Cj , t , DjBattery , CThermal , DThermal Charge/discharge rate of the BSS/ j, t j, t ,t TSS at time t (kW) FBattery, FThermal The initial amount of the stored energy in the BSS/ TSS (kWh) P jImport , P jExport Imported/exported power from/to the upstream grid ,t ,t at time t (kW) QjDemand Heat demand of the jth SH at time t ,t γCHP Heat to power ratio of the CHP generator λt Market price (£/MWh)

Nomenclature Sets t j i θ

index index index index

of of of of

time (h) SH smart appliances smart appliances’ operation time

Parameter

Binary variable

LCHP, LBoiler, LBattery, LThermal the capacity of the CHP, boiler, BSS and TSS (kWh) δ time interval of simulation (h) ηCHP, ηBoiler CHP/boiler efficiency (%) ηCh,Battery, ηDCh,Thermal, ηCh,Thermal, ηDch,Thermal BSS/TSS charge/discharge efficiency (%) CLBattery, DLBattery, CLThermal, DLThermal charge/discharge limit of the BSS/TSS (kW) MBattery, MThermal, MGrid maximum capacity of the BSS, TSS and purchased power from the upstream grid (kW) Finish T jStart earliest starting/latest finishing time of the ith appli, i , T j, i ance related to the jth SH (h) Pi,App Consumption power of the ith smart appliances (kW) λGas Natural gas price (£/kWh) λSell Cost of selling power to the upstream grid (£/kWh) BCBattery, TCThermal Maintenance cost of the BSS/TSS (£/kWh) Pi Length of the operation time of each smart appliance (h) min , tmax Lower/upper bound of the market price (£/MWh) t

BjBattery ,t BThermal j, t App j, i , t

BGrid j, t

Binary variable; equal to 1 if the BSS is charged at time t; equal to 0 if the BSS is discharged at time t Binary variable; equal to 1 if the TSS is charged by the jth SH at time t; equal to 0 if the TSS is discharged at time t Binary variable; equal to 1 if the ith smart appliances related to the jth SH be active at time t; otherwise 0 Binary variable; equal to 1 if electricity is imported from the upstream grid by the jth SH at time t; equal to 0 if the electricity is exported to the upstream grid by the jth SH at time t

Abbreviation SAB SH CHP BSS TSS HEMS

Variable

smart apartment building smart home combined heat and power battery storage system thermal storage system home energy management system

Boiler P CHP The output power of the CHP/boiler at time t (kW) j, t , Q j, t

various papers. For instance, fuzzy mathematical programming, stochastic programming (Ghalelou, Fakhri, Nojavan, Majidi, & Hatami, 2016), sensitivity analysis, information gap decision theory and ROA (Akbari, Nasiri, Jolai, & Ghaderi, 2014). The ROA is one of the robust approaches that relies on the lower and upper limits of the estimated market price which leads to guarantee a robust decision. In this paper, the proposed model of the SAB can be formulated as mixed-integer linear programming (MILP) which guarantees the global optimal solution. Also, with considering the lower and upper limits for the estimated market price, the proposed model of the SAB can be reformulated as robust mixed-integer linear programming (RMILP) which leads to a robust decision within the defined market price bounds (Bertsimas & Sim, 2003).

multi-smart apartment buildings, a novel HEMS model has been provided in Najafi-Ghalelou, Zare, and Nojavan (2018b). Risk-based models of SHs have been investigated as follows: Robust scheduling of smart apartment building utilizing information gap decision theory technique has been provided in Najafi-Ghalelou, Nojavan, and Zare (2018c). A novel home energy management controller based on genetic, wind-driven optimization, bacterial foraging optimization and binary particle swarm optimization algorithms has been provided in (Javaid et al., 2017). Information gap decision theory approach has been utilized in Najafi-Ghalelou, Nojavan, and Zare (2017) to robust scheduling of SH under market price uncertainty. To tackle the uncertainty of photovoltaic panel, a novel robust optimization model for load scheduling of a SH has been provided in (Wang et al., 2015a). Riskbased scheduling of smart apartment building considering solar thermal storage system and market price uncertainty has been provided in (Najafi-Ghalelou, Nojavan, and Zare (2018d). In order to minimize the total cost and carbon emissions of the microgrid under uncertainty, a robust multi-objective model has been presented in Wang, Li, Ding, Sun, and Wang (2017). A scenario-robust optimization approach has been provided in Craparo, Karatas, and Singham (2017) to analyze the performance of a hybrid microgrid under uncertain situation. Finally, it is noteworthy that detailed study of further SH-based papers are provided in Table 1.

1.3. Contributions

1.2. Procedure

In this paper, a robust optimization approach is proposed for robust scheduling of the SAB under market price uncertainty. One of the advantages of ROA in comparison with other stochastic programming methods is that obtained results by ROA approach is effective and needs low computational burden. Also, to assess the effects of the HEMS on the performance of the proposed problem and total operation cost of the SAB, two different controlling scenarios are studied. According to the mentioned explanation, the novelty and contributions of this paper are presented as follows:

Due to uncertain nature of the market price, uncertainty modeling is important to determine the strategies for optimal operation of the system. In order to deal with uncertainties in different power system models, different techniques and methods have been addressed in

1. Presenting a method for robust scheduling of the SAB considering lower and upper limits for market prices, not estimated market prices. 2. Considering the lower and upper limits of the market prices instead 2

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(continued on next page)

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8

– (–)

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Zhang, Shah, and Papageorgiou (2013) Zhang et al. (2014) Ghazvini, Soares, Abrishambaf, Castro, and Vale (2017) AnvariMoghaddam, Monsef, and Rahimi-Kian (2015b) Danandeh, Zhao, and Zeng (2014) Brahman, Honarmand, and Jadid (2015) Golpîra and Khan (2019) Wang et al. (2015b) Gazafroudi et al. (2019) Xiao et al. (2018) Wakui et al. (2017) Kampelis et al. (2018) Talha et al. (2019) Tascikaraoglu, Boynuegri, and Uzunoglu (2014) Sanjari, Karami, and Gooi (2016) Mosaddegh, Canizares, and Bhattacharya (2017) Luo, Ranzi, Kong, Dong, and Wang (2018)

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Minimizing coS2 emission

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Table 1 (continued)

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–(–)

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Abushnaf and Rassau (2018) Samuel et al. (2018) Najafi-Ghalelou et al. (2018c,a) Najafi-Ghalelou et al. (2018c,b) Najafi-Ghalelou et al. (2017) Wang et al. (2015a) Bassamzadeh, Ghanem, Lu, and Kazemitabar (2014) Good, Karangelos, NavarroEspinosa, and Mancarella (2015) Rahmani-Andebili (2017) Yao, Samadi, Wong, and Schober (2016) Vasilj, Gros, Jakus, and Zanon (2017) Thomas, Deblecker, and Ioakimidis (2018) Zachar and Daoutidis (2018) This paper

Wind speed (Uncertainty)

Minimizing coS2 emission

Ref.

Table 1 (continued)

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Fig. 1. Schematic of the proposed smart buildings model.

of the forecasted prices which lead to guarantee a robust decision. 3. Risk-constrained scheduling of smart appliances in the presence of distributed energy sources with the aim of minimizing the total operation cost of the SAB. 4. Appropriate operational strategies are obtained utilizing ROA.

1. Optimal scheduling of the SAB‘s energy consumption problem is solved by considering the lower bound of the market price. 2. Then, by stepping up the market price bound, the related RMILP problem is solved for each interval within the predefined market price bound to achieve different scheduling strategies. 3. A steady increase in the market prices will continue until the upper bound of the market price is reached.

1.4. Structure of proposed paper

3. Problem formulation

The rest of this paper is categorized as follows: The general description of the ROA is described in Section 2. The deterministic mathematical model of the SAB is presented in Section 3. Also, the ROA-based mathematical formulation is presented in Section 3, too. The flowchart of the ROA is provided in Section 4. Input data, case study, results, and assessments in details are provided in Section 5. Finally, the conclusion is provided in Section 6.

In this section, the mathematical formulation of the studied problem is presented through various sections. 3.1. Deterministic formulation of the smart apartment building The overall architecture of the proposed MILP-based model of the smart apartment building that contains 10 SHs is demonstrated in Fig. 1, which chiefly includes advanced metering infrastructure, HEMS, smart meter, in-home display equipment, CHP, boiler, BSS, TSS, and smart appliances. The advanced metering infrastructure is responsible for two-way data flow between upstream grid and smart meter. The smart meter measures and processes the records then transmits them to the upstream grid. It is typically installed between the HEMS and the advanced metering infrastructure. There are different means for communication links between the smart meter and HEMS, for instance ZigBee, Z-Wave, Wi-Fi, or a wired protocol. The HEMS organizes, controls and also schedules smart appliances alongside the controllable distributed energy resources such as CHP and etc. All smart appliances’ operation along with controllable distributed energy resources would

2. Model In this section, the developed ROA-based model of the SAB is explained. 2.1. Method description ROA is one of the powerful tools to find the robust optimal scheduling strategy in the presence of market price uncertainty. In this method, some RMILP problems are solved consecutively as described as follows (Bertsimas & Sim, 2003; Nojavan, Mohammadi-Ivatloo, & Zare, 2015a): 10

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Fig. 2. Flowchart of the proposed algorithm.

have been scheduled by HEMS while the procedure supervised through in-home display. In order to obtain an optimal schedule over a time horizon, the HEMS needs a variety of input data such as thermal demands, the characteristics of appliances such as earliest starting time, latest finishing time, market price, gas tariff and etc.

Minimizing P CHP j, t ,

maintenance cost of the BSS and TSS are provided in the third and the fourth terms. Finally, the fifth and the sixth terms are shown the cost and profit of exchanged power with upstream grid. 3.1.1. CHP generator constraints The capacity limitation of the CHP generator can be expressed as Eq. (2) (Najafi-Ghalelou et al., 2018c; Zhang, Liu, & Papageorgiou, 2014).

, j, t ; D jBattery , j, t ; DThermal , j, t; P jImport , j, t ; P jExport , j, t j, t; QjBoiler ,t j, t ,t ,t ,t

J

P CHP j, t

LCHP ,

j,

t

(2)

j =1

J

T

Gas

CHP

j=1 t =1 J

× P CHP j, t

J

T

Gas

+

3.1.2. Boiler constraints The capacity limitation of the boiler can be expressed as (NajafiGhalelou et al., 2018c; Zhang et al., 2014):

× QjBoiler ,t Boiler

j=1 t =1

T

J

BCBattery × DjBattery ,t

+ j=1 t=1 J

×

T

j =1

LBoiler ,

j,

t

(3)

TC Thermal × DThermal j, t

+ j=1 t =1 J

QjBoiler ,t

T

+

t j=1 t=1

× P jImport ,t

J

T Sell

3.1.3. Battery storage system constraints The presented BSS in this paper plays a central BSS role in the SAB model. In others words, one BSS is shared among SHs of the same SAB. So, each SH (j) of the SAB can discharge the BSS as much as it has charged it by itself at previous periods (Zhang et al., 2014). The capacity limitation of the BSS and the stored energy in the BSS at each period t are expressed as follows:

× P jExport ,t

j=1 t=1

(1) According to the provided objective function, the first and second terms indicate the cost of the purchased gas by the CHP and boiler. The 11

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SjBattery ,t

LBattery ,

j,

DjBattery × ,t

t

Dch, Battery

(4)

j =1

Battery SjBattery = SjBattery × ,t , t 1 + ( × C j, t

Ch, Battery )

× DjBattery ,t Dch, Battery

,

j,

t

CjBattery ,t DjBattery ,t

To avoid the net accumulation at the end of the sample day, stored energy in the BSS at the end of the sample day should be equal to the initial amount. In this regard, Eq. (6) is provided.

" "

=

SjBattery , 24 " "

=

F Battery ,

j,

t

DjBattery ,t

DLBattery ,

j,

t

j =1 J j =1

CjBattery ,t

CLBattery ,

j,

t

t

(9)

MBattery × BjBattery , ,t MBattery × (1

j,

t

BjBattery ), ,t

(10)

j,

t

(11)

3.1.4. Thermal storage system Alike the BSS, TSS plays the role of a central storage system, and its energy is equal to the sum energies of the sub-TSS in each SH. Hence, each SH is allowed to discharge the TSS as much as it has charged it by itself at the previous periods (Zhang et al., 2014). The capacity limitation of the TSS and stored energy in the TSS at each period t are provided as follows:

(6)

The charge/discharge power of the BSS at time t must be limited by its designed charge/discharge limitation. In this regard, Eqs. (7) and (8) are provided. Also, to limit the discharge rate of the BSS at time t with the amount of the stored energy in the BSS at time t − 1, Eq. (9) is provided. J

j,

BSS is not allowed to charge and discharge at the same time. In this regard, Eqs. (10) and (11) are provided. (5)

SjBattery , 1

SjBattery ,t 1 ,

J

SThermal j, t

LThermal ,

j,

t

(12)

j =1

(7)

SThermal = SThermal + ( × CThermal × j, t j, t 1 j, t × DThermal j, t Dch, Thermal

(8)

Fig. 3. Heat demand of smart homes 1–6. 12

,

Ch, Thermal )

j,

t (13)

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Fig. 4. Heat demand of smart homes 7–10.

Fig. 5. Lower/expected/upper values of market prices.

To avoid the net accumulation, stored energy at the beginning and the end of the sample day should be equal together. In this regard, to find the best state of charge of the TSS, stored energy at the beginning and the end of the sample day is set to be equal.

SThermal j, 1 " "

=

SThermal j, 24 " "

=

F Thermal,

j,

t

J

j =1

DThermal j, t

DLThermal ,

j,

t

CLThermal ,

j,

t

SThermal j, t 1 ,

j,

t

(16)

j =1

DThermal × j, t

(14)

Dch, Thermal

In order to limit the charge/discharge power of the TSS, Eqs. (15) and (16) are provided. Furthermore, to limit the discharge rate of the TSS at time t with the stored energy at time t − 1, Eq. (17) is provided. J

CThermal j, t

(17)

TSS is not allowed to charge and discharge at the same time. In this regard, Eqs. (18) and (19) are expressed.

(15) 13

CThermal j, t

M Thermal × BThermal , j, t

DThermal j, t

M Thermal × (1

j,

BThermal ), j, t

t

(18)

j,

t

(19)

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Fig. 6. Robust cost of the smart apartment building. Table 2 Comparison of operation cost of the smart apartment building. Scenario 1 Normal scenario

Scenario 2 Smart scenario

Reduction cost

Pessimistic case of ROA

Total cost The cost of produced power by the CHP generator The cost of produced heat by the boiler The cost of discharged power by the battery storage system The cost of discharged heat by the thermal storage system The cost of purchasing power from the upstream grid Revenue from selling power to the upstream grid

19.635 2.518 3.324 0.250 0.026 7.968 0.808

12.618 2.996 3.074 0.224 0.030 4.508 0.636

−35.74 +18.98 +7.52 −10.4 +15.38 −43.42 −21.29

Deterministic case

Total cost The cost of produced power by the CHP generator The cost of produced heat by the boiler The cost of discharged power by the battery storage system The cost of discharged heat by the thermal storage system The cost of purchasing power from the upstream grid Revenue from selling power to the upstream grid

16.453 2.453 3.359 0.254 0.026 8.007 0.833

11.348 2.719 3.221 0.224 0.029 4.513 0.69

−31.03 +10.84 −4.11 −11.81 +11.54 −43.64 −17.17

Optimistic case of ROA

Total cost The cost of produced power by the CHP generator The cost of produced heat by the boiler The cost of discharged power by the battery storage system The cost of discharged heat by the thermal storage system The cost of purchasing power from the upstream grid Revenue from selling power to the upstream grid

13.139 2.129 3.521 0.231 0.017 8.109 0.866

9.856 2.149 3.513 0.208 0.019 4.746 0.779

−24.99 +0.94 −0.23 −9.96 +11.76 −41.47 −10.05

3.1.5. Smart appliances A set of the most common smart appliances, namely washing machine, dish washer, microwave, and fridge have been considered to be controlled. The occupant's comfort level is one of the major issues in the SAB problems. Thus, to schedule and control the smart appliances there is need to take into account all possible challenges such as user's preference and market price uncertainty. In this regard, Eq. (20) is provided to consider the user's preferences. So, each smart appliance Finish should be active within the specific period (T jStart ) which is set , t , T j, t and determined by the users of each SH user (Zhang et al., 2014). T jFinish Pi ,i t = T Start j, i

App j , i, t

= 1,

j,

i,

3.1.6. Imported/exported power from/to the upstream grid To avoid the simultaneous import/export power from/to the upstream grid, Eqs. (21) and (22) are provided.

P jImport ,t

M Grid × BGrid j, t ,

P jExport ,t

M Grid × (1

j,

BGrid j, t ),

t

(21)

j,

t

(22)

3.1.7. Energy balance constraints The electricity demand is fulfilled by the electricity generated by the CHP, the electricity received from the BSS (discharge mode), imported power from the upstream grid minus the send electricity to the BSS (charge mode) and exported power to the upstream grid.

t (20) 14

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Fig. 7. Imported/exported power from/to the upstream grid.

Table 3 Total obtained results. Case 1 Normal scenario (kWh)

Case 2 Smart scenario (kWh)

Decreased value (%)

Pessimistic case of ROA

The output power of the CHP The output power of the boiler The charge power of the BSS The Discharge power of the BSS The stored energy in the BSS The charge power of the TSS The discharge power of the TSS Stored energy in the TSS The imported power from the upstream grid The exported power from the upstream grid

65.29 209.37 97.86 99.92 124.24 34.92 51.31 169.30 537.95 53.84

77.67 193.61 86.29 89.48 133.03 43.53 59.57 174.73 512.97 42.36

+18.97 −7.53 −11.82 −10.45 +7.07 +24.65 +16.11 +3.21 −4.64 −21.31

Deterministic case

The output power of the CHP The output power of the boiler The charge power of the BSS The discharge power of the BSS Stored energy in the BSS The charge power of the TSS The discharge power of the TSS The stored energy in the TSS The imported power from the upstream grid The exported power from the upstream grid

63.61 211.55 99.70 101.57 125.49 34.86 51.25 167.13 541.48 55.51

70.49 202.85 86.19 89.39 125.88 40.88 57.02 167.46 523.77 46.00

+10.82 −4.11 −13.55 −12 +0.31 +17.24 +11.27 +0.2 −3.27 −17.13

Optimistic case of ROA

The The The The The The The The The The

55.20 221.74 89.15 92.06 124.41 15.98 33.11 97.43 551.09 57.74

55.70 221.26 79.42 83.28 118.51 20.21 37.17 143.33 543.83 51.93

+0.9 −0.22 −10.91 −9.54 −4.74 +20.47 +12.29 +47.11 −1.32 −10.06

output power of the CHP output power of the boiler charge power of the BSS discharge power of the BSS stored energy in the BSS charge power of the TSS discharge power of the TSS stored energy in the TSS imported power from the upstream grid exported power from the upstream grid

15

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Fig. 8. The output power of the CHP generator.

Fig. 9. The output power of the boiler.

16

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Fig. 10. Charge/discharge rate of the battery storage system.

Fig. 11. Stored energy in the battery storage system.

17

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Fig. 12. Charge/discharge rate of the thermal storage system.

Fig. 13. Stored energy in the thermal storage system.

18

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Fig. 14. The operation time of different appliances in different smart homes of the smart apartment building.

Fig. 15. The operation time of different appliances in different smart homes of the smart apartment building.

19

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Fig. 16. The operation time of different appliances in different smart homes of the smart apartment building.

I i=1

Pi 1 =1

Pi,App ×

App j , i, t

= P CHP + DjBattery j, t ,t

CjBattery + P jImport ,t ,t

j,

In the RMILP formulation, to control the level of the robustness, it is essential to define a new parameter Γ0 as an integer variable which has value in [0, |J0|], where J0 = {t|dt > 0}. It is noteworthy that to consider the price deviations in the proposed objective function, Γ0 should be set equal to |J0| and to ignore the effect of the price deviations in the proposed objective function, Γ0 should set equal to zero. Finally, the form of the RMIP optimization model related to the (25)–(28) is reformulated as follows:

t

P jExport , ,t (23) The heat demand is fulfilled by the heat energy generated by the CHP, produced heat by the boiler, received heat from the TSS (discharge mode) minus the sent heat to the TSS (charge mode).

QjDemand = ,t

CHP

× P CHP + QjBoiler + DThermal j, t ,t j, t

CThermal , j, t

j,

t

T

(24)

Minimize

xt , qot , yt , t; z 0

The objective function of the SAB's model (1) with considering the relevant constraints (2)–(24) is the MILP formulation, which can be expressed as standard form as follows (Bertsimas & Sim, 2003; NajafiGhalelou, Nojavan, Zare, & Mohammadi-Ivatloo, 2019; Nojavan, Mohammadi-Ivatloo, & Zare, 2015b): T

et xt

(25)

t=1

Subject to

amt xt

bm ,

m = 1,..., M

xt

0,

t = 1,..., T

xt

{0,1} for some t = 1,..., T

+

qot t=1

(29)

z 0 + qot

dt yt ,

qot

t = 1,..., T

0,

yt

0,

z0

0

xt

yt ,

t

t = 1,..., T

J0

(30) (31) (32) (33)

t = 1,..., T

(34)

The objective function (29) and the related constraints (30)–(34) are obtained through the linearizing technique and duality theorem (Bertsimas & Sim, 2003; Nojavan et al., 2015b). The auxiliary variable yt is used to get the corresponding linear declarations. Finally, variables qot and z0 are dual variables of the optimization model (25)–(28) which are utilized to determine the upper/lower bound of the known parameters et. It is noteworthy that according to Eq. (29), if Γ0 be equal to 0, z0 would be free to be any positive number. Hence, Eq. (30) will be satisfied without any difficulties. Also, the value of z0 will not have any effect on the surmised objective function (29). Notably, it is better to increase the value of the z0 instead of increasing the value of the qot in which any value of the z0 does not have any effect on the value of the objective function. Consequently, it can be concluded that it is better to increase the value of the z0 as well as decreasing the value of the qot with the aim of satisfying Eq. (30) and minimizing the objective

T t=1

t=1

0

Subject to

3.2. Robust formulation based on proposed ROA

Minimize

T

et xt + z 0

(26) (27) (28)

In Eq. (25), et is assumed as the coefficient of the objective function. This parameter is assumed to be unknown within the known bounds. So, we can reformulate the RMILP optimization problem (25)–(28). Also, the new parameter dt illustrates the deviance from the nominal coefficient et. Therefore, all parameters of the et will have a different value in each interval [et, et + dt]. 20

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function (29). In contrast, with moving the value of Γ0 from 0 to 1, any value of z0 will have impact on the objective function (29). Therefore, it is better to select the lowest value of the z0. On the other hand, in order to satisfy Eq. (30), the value of the qot should be increased in which these changes are considered as a cost in the presumed objective T function (29) with use of t = 1 qot term. it should be noted that if the value of qot did not consider in the objective function, the amount of z0 would be decreased to control the increase of the Γ0. Therefore, the effects of these changes would not be considered in the provided objective functions (29).

to the step 5, which in this step the hourly optimal scheduling strategies are constructed based on the achieved results. 5. Numerical simulation As shown in Fig. 1, the proposed SAB model contains ten SHs with different living habits and equipped with different equipment, i.e. CHP, boiler, BSS, TSS and smart appliances. The starting time of the case study is 8:00 AM and the ending time is 8:00 AM of the next morning with the time interval of 30 min. The main purpose of the mentioned paper is to minimize the operation cost of the SAB in the presence of market price uncertainty. To model the market price uncertainty, ROA is employed which relies on the lower and upper limits of the forecasted market prices. Also, the performance of the proposed algorithm assessed through two different controlling scenarios: normal and smart scenarios. The normal scenario describes a situation in which the users of the SAB do not possess or use a HEMS; therefore there is no ability to manage the smart appliances in a cost-effective way under market price uncertainties. Also, the priorities and preferences of residents are not considered in this scenario. Unlike the previous scenario, the smart scenario benefits from a fully-featured of the HEMS as well as the priorities and preferences of the residents. Obtained strategies from ROA can be utilized by the HEMS to take appropriate decisions against the price fluctuations.

3.2.1. Robust-based scheduling of the smart apartment building The robust based scheduling of the SAB has been formulated as follows:

Minimizin g PCHP j, t ,

j, t ; QjBoiler , j, t; DjBattery , j, t; DThermal , j, t ; P jImport , j, t; P jExport , j, t ; z 0; qj, t ,t j, t ,t ,t ,t

J

T

× P CHP j, t

Gas

CHP

j=1 t =1 J

J

T

Gas

+

× QjBoiler ,t Boiler

j=1 t =1

T

BCBattery × DjBattery ,t

+ j=1 t=1 J

×

T

TC Thermal

+

×

5.1. Input data

DThermal j, t

j=1 t =1 J

T t

+

×

J

P jImport ,t

Sell

j=1 t=1

0

×

The technical information of the CHP, boiler, BSS, TSS and smart appliances are adopted from (Najafi-Ghalelou et al., 2018d; Zhang et al., 2014). The heat demands of each SH are presented in Figs. 3 and 4 (Najafi-Ghalelou et al., 2018d; Zhang et al., 2014). The lower/expected/upper value of the market prices are provided in Fig. 5 (Zhang et al., 2014). Also, the cost of selling power to the upstream grid is set equal to 3 p/kWh (Najafi-Ghalelou et al., 2018d). Finally, the natural gas price is set equal to 2.7 p/kWh (Najafi-Ghalelou et al., 2018d).

P jExport ,t

j=1 t=1 J

+ z0

T

T

+ j=1 t=1

qj, t (35)

Constraints (2)

z 0 + qj, t

(36)

(24)

dt × yj, t

t = 1,..., T

j = 1,..., J

(37)

qj, t

0 t = 1,..., T

j = 1,..., J

(38)

yj, t

0 t = 1,..., T

j = 1,..., J

(39)

z0

The proposed ROA-based model of the SAB is formulated as the RMILP problem and solved using CPLEX solver (The GAMS Software, 2012) in GAMS (Brooke, Kendrick, Meeraus, & Raman, 2008) optimization platform. In this paper, 11 iterations are considered to construct the optimal scheduling strategies. Eq. (1) is minimized subject to constraints (2)–(24) with considering the deterministic market prices as input data. In this condition, the total operation cost of the SAB is 16.45 £ and 11.35 £ in normal and smart scenarios, respectively. In other word, in the smart scenario, the operation cost of the SAB is decreased by 31.03% in comparison with the normal scenario. The RMILP optimization problem (35)–(41) is simulated for ten iterations (Gz replaces in steps of amount δ = 0.1) to build the optimal scheduling strategies. In other words, the proposed robust objective function (35) subject to constraints (36)–(41) is minimized taking into account iteration value of Gz with a fixed step which is increased with adding a constant positive value to the lower bound of the market price. Robustness operation cost of the SAB in each interval, in normal and smart scenarios are presented in Fig. 6. By comparing the obtained results, it can be seen that the total operation cost of the SAB in the smart scenario of the optimistic case is decreased approximately 25% more in comparison with the normal scenario. On the other hand, the total operation cost of the SAB in the smart scenario of the pessimistic case is decreased nearly 35.74% more in comparison with the normal scenario. It can be concluded that with increasing the level of market prices, total operation cost of the SAB is increased and gradually become more resistant against the fluctuation of the market prices. Also,

(40)

0

P jImport ,t

5.2. Simulation results

yj, t

t = 1,..., T

j = 1,..., J

(41)

4. The proposed algorithm to make robust based scheduling strategies The following algorithm is used to make robust scheduling strategies. Flowchart of the proposed algorithm is illustrated in Fig. 2 and described as follows: Step 1: Set prices t = tmin (t = 1,..., T ) , and Γ0 = 24 (The entire time of the simulation) to consider all possible deviation of the market prices. min ), (t = 1,..., T ) , where Gz is a factor Step 2: Set dtz = G z ( tmax t which increases with steady steps from 0 to 1 and the index z indicates the counter of iteration Step 3: The RMILP optimization formulation (35)–(41) is simulated to obtain the number of decision variables at each period t and each Export Battery CHP Thermal Boiler iteration z, P jImport , t , z , P j, t , z , P j, t , z , Q j, t , z , D j, t , z , Dj, t , z Step 4: For covering all ranges of factors of Gz, steps 2–4 will be repeated iteratively as long as the condition of Gz+1 > 1 is satisfied. Step 5: After reaching the parameter Gz to 1, exit the loop and enters 21

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by comparing the results from other perspective, it can be concluded that in the smart scenario, the slope changes are less and it means that the total operating cost is more resistant against the market price uncertainties in comparison with the normal scenario. For more comparison, total operation cost of the SAB in optimistic, deterministic and pessimistic cases of ROA without and with considering the features of the HEMS, numerically in details are presented in Table 2. Imported/exported power from/to the upstream grid is presented in Fig. 7. By comparing the obtained results, it can be seen that in normal scenario imported power from the upstream grid in pessimistic, deterministic and optimistic cases during the 24-h study case are 537.95 kWh, 541.48 kWh and 551.09 kWh, respectively. In the smart scenario, imported power from the upstream grid in pessimistic, deterministic and optimistic cases are 512.97 kWh, 523.77 kWh, and 543.83 kWh, respectively. So, with comparing the obtained results, it can be seen that with increasing the market price level, the HEMS tends to buy less power from the upstream grid and rely on the equipment within the SAB and consume gas instead of the electricity. Also, in the smart scenario, the amount of imported power from the upstream grid is lower than the normal scenario. The obtained results are reported numerically in Table 3 for better comparison. Also, obtained results are presented in the figure form at the rest of the paper. The output power of the CHP and boiler are presented in Figs. 8 and 9, respectively. By comparing the obtained results in optimistic, deterministic and pessimistic cases, it can be seen that in the pessimistic case, the HEMS tends to use CHP more than the boiler. One of the reasons for this trend is that CHP consumed gas and produced heat and power simultaneously. Therefore, with increasing the market price uncertainty, the HEMS tends to rely more on CHP rather than the boiler. The charge/discharge rate of the BSS and stored energy in the BSS are provided in Figs. 10 and 11, respectively. By comparing the obtained results, it can be realized that the use of the BSS does not depend on the fluctuation of the market price. Also, by analyzing the mentioned figures from other perspective, it can be seen that in the smart scenario, charge and discharge rates of the BSS are lower in comparison with the normal scenario. Charge/discharge rate of the TSS and stored energy in the TSS are presented in Figs. 12 and 13, respectively. In can be seen that in the pessimistic case, stored energy in the TSS and charge and discharge rates of the TSS are higher in comparison with other cases. One of the reasons of this state is that with increasing the market price, the HEMS tends to buy less power from the upstream grid and conversely consumes more gas to meet the heat and power demands. So, with increasing the market price uncertainty, the HEMS tends to use the TSS more and more. Also, in the smart scenario, stored energy in the TSS and charge and discharge rates of the TSS is higher in comparison with the normal scenario. The operation time of different appliances in each SH of the SAB in pessimistic, deterministic and optimistic cases of ROA are chosen randomly and illustrated in Figs. 14–16, respectively. For example, the cooker hub belonging to the 1st SH in pessimistic/deterministic/optimistic cases of ROA is set to be operated between the hours 15:00–15:30 in the normal scenario and the hours 15:30–16:00, 15:30–16:00 and 15:00–15:30 in the smart scenario, respectively.

studied in two controlling scenarios. It can be realized from the obtained results that in pessimistic/deterministic/optimistic case of ROA with considering the features of the HEMS, total operation cost of the SAB is decreased nearly 35.74%, 31.03%, and 24.99%, respectively. Also, by analyzing the obtained results from other perspectives, it can be seen that without considering the effects of the HEMS, total operation cost of the SAB in pessimistic case is increased 16.21% and 33.08% in comparison with deterministic and optimistic cases of ROA. While, by considering the effects of the HEMS, total operation cost of the SAB in pessimistic case is increased 10.06% and 21.89% in comparison with the deterministic and optimistic cases of ROA. With comparing the obtained results, it can be concluded that with considering the HEMS, the total operation cost of SAB will be more resistance against the market price fluctuation in caparison with the case that the features of the HEMS are ignored. 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