Transshipment model-based linear programming formulation for targeting hybrid power systems with power loss considerations

Energy 75 (2014) 24e30

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Transshipment model-based linear programming formulation for targeting hybrid power systems with power loss considerations Cheng-Liang Chen*, Chieh-Ting Lai, Jui-Yuan Lee Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 December 2013 Received in revised form 12 May 2014 Accepted 16 May 2014 Available online 14 June 2014

This article aims at targeting the minimum electricity outsourced from the grid and the minimum battery capacity for temporal storage of surplus power in a HPS (hybrid power systems) with known power demands and RE (renewable energy) sources. The given continuous or intermittent power demands and RE supplies can be AC (alternating current) or DC (direct current) with known average power ratings without uncertainty. An ETM (expanded transshipment model) is proposed to determine the allocation of electricity generated from RE sources as well as the storage and outsourcing policies for electricity surplus and deficit. This model also considers possible power losses from the conversion between AC and DC, the charging (for storage) and discharging of electricity (to supplement RE supply), and self-discharge during storage. The problem of targeting the electricity outsourcing and storage for an HPS is formulated as a LP (linear program). A literature example is used to demonstrate the application of the proposed ETM. The promising results show that the proposed model can provide preliminary analysis for optimal power management in an HPS, and is an improved method compared to the PPA (power pinch analysis). © 2014 Elsevier Ltd. All rights reserved.

Keywords: RE (renewable energy) HPS (hybrid power system) Targeting MP (mathematical programming) ETM (expanded transshipment model) LP (linear programming)

1. Introduction The global-scale climate change is one of the most challenging crises of the modern world. The burning of fossil fuels by humansdwith the emission of GHGs (green-house gases)dsince the industrial era has been recognized as one of the main reasons resulting in the warming earth. Therein, the fossil fuel-fired power plants are the main sector for GHG emissions. One of the critical endeavors to deal with the global warming emergency is to increase the use of low-carbon electricity. Recently, several clean and renewable energy sources such as on land or offshore wind, photovoltaic solar, and various biomass materials, have grown into potential power alternatives due to their significant technical progress. In some industrialized countries, such as Germany, England, and Denmark, more than 10% of electricity is generated from wind turbine and/or photovoltaic devices, and the proportion will be expanded dramatically in the near future. However, the intermittent and unpredictable availability of some renewable energy sources creates a new challenge for matching electricity generation and consumption. The so-called HPS (hybrid power system)

* Corresponding author. Tel.: þ886 2 33663039; fax: þ886 2 23623040. E-mail address: [email protected] (C.-L. Chen). http://dx.doi.org/10.1016/j.energy.2014.05.059 0360-5442/© 2014 Elsevier Ltd. All rights reserved.

consists of two or more RE power sources and is complemented by the public grid. To overcome the variable and unpredictable supply of some renewables in an HPS, some kind of storage facilities is required to match the discrepancy of electricity supply and demand in both quantity and availability. The focus of previous research works of the HPS was mainly on the optimization and performance evaluation of integrated power generation systems with battery storage. Prasad and Natarajan [1] developed a software tool to optimize the capacity of an integrated power system including wind turbine, photovoltaic solar cell, and battery back up. Jakhrani et al. [2] presented a novel mathematical model to determine the optimal sizing of a standalone PV (photovoltaic) system for a remote area, where the required PV area and the associated battery storage capacity were optimized under various load demands. Bajpai and Dash [3] reviewed recent research works on the modeling, sizing, optimization and management of relevant components in a typical hybrid renewable energy system. Dai and Mesbahi [4] used the mixedinteger programming approach to study the load management of off-grid hybrid power systems. Further to these progresses on modeling and optimization of hybrid power systems, the PI (process integration) principles were extended recently from the traditional application areas such as heat [5], mass [6], water [7], heat and power [8], and energy

C.-L. Chen et al. / Energy 75 (2014) 24e30

planning [9], etc., to the analysis of the HPS [10]. Sreeraj et al. [11] and Bandyopadhyay [12] studied the design of isolated energy systems through pinch analysis, where the uncertainties of power sources and the cost of the generated power were taken into account to determine the optimum battery capacity. Ho et al. [13] also developed an electric cascade analysis method for the design of a solar-based HPS. Given the RE power supply and power demand data, one critical problem is to target the minimum electricity required to import from the grid for shortage, and the battery capacity needed for storage of surplus power. Wan Alwi et al. [14] offered a power pinch graphical tool for targeting the outsourcing electricity of an HPS. Wan Alwi et al. [15] and Mohammad Rozali et al. [16] proposed the power cascade table and the storage cascade table to give precise numerical calculations for the electricity targets of the HPS including the required power storage capacity. Mohammad Rozali et al. [17] extended the numerical targeting calculations by considering the possible losses during the conversion, transfer and storage of surplus power. However, the numerical calculations for power targeting could be complicated when the number of time intervals increases. Instead of using the battery for storing the excess power, Mohammad Rozali et al. [18] extended the power pinch analysis to the pumped-hydro storage system, where the storage unit can deliver electricity in AC. Recently, Chen et al. [19] established the transshipment model-based LP (linear programming) formulations for implementing the targeting calculations of the HPS. Chen et al. [20] expanded the transshipment model for considering power losses. The targeting problem, when considering the power losses from electricity transfer and storage, was formulated as a MILP (mixed-integer linear programming) problem. However, the conversion losses were not included. This article aims to modify the ETM based mathematical formulations for targeting the HPS with the consideration of power losses during the conversion, transfer and storage of surplus power. Given the available power supply and demand data, the problem of targeting the power shortage, which will be complemented by grid, and the battery capacity for storage of excess power will be formulated as a LP problem. The proposed model considers all possible power allocation options, and the LP formulation guarantees the global optimal solution to the HPS. In the following sections, the problem statement is given first. The modified ETM, which considers all possible power dispatching options and the possible losses during electricity conversion, transfer, and storage, is next presented. One numerical example adapted from literature will be used to demonstrate the application of the proposed model for targeting the electricity import and the battery storage capacity in an HPS. Note that the presented model will be used mainly for preliminary analysis and design to give an ideal of optimal/efficient power allocation in an HPS. The developed model can be expanded when more detailed design is required, in which case some of the parameters would have to be treated as optimization variables.

25

intermittent. Without loss of generality, the RE power sources are concerning with AC or DC, thus the set of RE power sources I can be divided into I A and I D for AC and DC, I ¼ I A ∪I D . Similarly, the power demands, the power sellers and the power buyers in grid are all concerning with AC and DC. The corresponding sets are divided for AC and DC, J ¼ J A ∪J D , M ¼ MA ∪MD , and N ¼ N A ∪N D . It is further assumed that the available times and their average power ratings for RE power sources and consumer demands are known without uncertainty. The generated RE power is used first to satisfy the consumers demand. The surplus RE power supply can be temporally stored in a battery for later use, and the deficit electricity is complemented by the discharged electricity from the battery and the power imported from the public grid. Meanwhile, the excess electricity generated by RE sources can be exported to potential buyers. The electricity from the grid, the power produced from the renewable sources, the consumer demanding loads, and the exported power can be AC or DC. However, a battery is used for storing excess power in DC form. A power conditioning system is involved in the typical HPS for rectification of AC (alternating current) to DC (direct current) and for inverse of DC to AC, if necessary. The efficiency parameters for rectification and inversion of power as well as the charging/ discharging of power are usually determined by the hardware performance. Constant efficient parameters were previously used by Wan Alwi et al. [15] and Mohammad Rozali et al. [16]. In this rec article, fixed average recovery ratios are given for the rectifier (rAD inv for ACeDC) and the inverter (rDA for DCeAC) for conversion between AC and DC. For a selected battery type, fixed power recha from AC supply or r cha covery ratios are also given for charge (rAD DD dch to AC demand or r dch to DC from DC supply) and discharge (rDA DD demand). The objective is to determine the minimum quantity of electricity imported from the grid, and the minimum battery capacity required for storing surplus electricity from RE power sources. 3. Model formulation For allocating the available power from various RE power sources to customer demands, and also the management of storage and discharge of excess RE powers, the ETM that was used for optimal synthesis of heat exchanger networks [21] is modified for targeting an HPS, as shown in Fig. 1. In this simple ETM for targeting the HPS, the studied period is divided into a set K of time intervals with elapsed time durations Dk ; k2K. The division of the time horizon is based on the given average power rating of RE sources i2I and the power consumption rate for customers j2J at various time intervals. The

2. Problem statement The problem addressed in this paper can be formally stated as follows. A typical HPS consists of a set I of RE power sources, a set J of power consumers, a set M of power sellers in the public grid supported by different power plants, a set N of power buyers in the grid which can purchase surplus RE power, and a battery for storing surplus power in DC (direct current) for later discharge to supplement the temporal shortage of RE power. The RE power sources may include wind, solar, ocean, biomass, etc., which can generate electricity continuously or intermittently, and the power demanded by the consumers can also be continuous or

Fig. 1. The expanded transshipment model (ETM) for modeling the hybrid power system considering power losses in electricity transformation.

26

C.-L. Chen et al. / Energy 75 (2014) 24e30

determination of these time intervals is according to the timing of power generation and consumption, not the power ratings. As shown in Fig. 1, the average power rating of RE sources i2I and the power consumption rate for customers j2J within time interval k2K are given constant values. With the elapsed time durations

de Ejk

¼

8 > > > > <

P ci2I A

> > > > :

P ci2I D

0

P

Eijk þ

cm2MA

Eijk þ

P cm2MD

Emjk þ

inv @ rDA

0 Emjk þ

rec @ rAD

X ci2I D

X ci2I A

Eijk þ Eijk þ

X cm2MD

X cm2MA

Dk ; k2K, the total power generated and consumed within the sp de in AC or DC, can be found whole kth time interval, Eik and Ejk accordingly. Further to the RE sources, the customer deficit power can be complemented by available suppliers m2M≡MA ∪MD in AC or DC from grid. The surplus RE power can be temporarily charged into a battery in DC or sold to potential buyers gd ex denote the total power n2N ≡N A ∪N D in AC or DC. Let Emk and Enk imported from supplier m in grid and the power exported to buyer n at time interval k. Ri;k1 is the residual power from RE source i which is stored in the battery and can be discharged into AC or DC to supplement the need from customers at time interval k. For describing the electricity transformation in the ETM model, Eijk is the power supply from RE source i to customer j at kth time interval. Eink is the power exported from source i to electricity buyer n. Emjk is the complementary electricity imported from power seller m in grid and is used by customer j at time interval k. The proposed ETM is used for modeling when and how much electricity is transformed in an HPS. Details of the ETM and the resulting LP formulation will be elucidated in the following. Eqs. (1) and (2) show that the power generated by RE source i at time insp sp;use terval k (Eik ) is used by customers (Eik ) or charged into a battery sp;sav for later use (Eik ). Similarly, the current residual power stayed in the battery (Rsp ) can be discharged for subsequent use (Rsp;use ) or i;k1 ik sp;sav keeps stored (Rik ). sp sp;use sp;sav Eik ¼ Eik þ Eik

ci2I ; k2K

¼ Rsp;use þ Rsp;sav Rsp i;k1 ik ik

(1)

ci2I ; k2K

(2)

Eq. (3) reveals the power from RE source i remained in the sp battery, Rik , which has taken into account the self-discharge effect and the recovery ratio for charging the electricity into the battery.

! sp Rik

¼

sp;sav Rik

( 

sp;sav

1  s  Dk

þ Eik

cha rAD

if i2I A

cha rDD

if i2I D

(3) ci2I ; k2K

sp;use

sp;use

Rik

¼

X

Eijk þ

cj2J

¼

X cj2J

X

Eink

cn2N

Rijk

ci2I ; k2K

ci2I ; k2K

1

(4)

(5)

X

dch Emjk A þ rDA

Rijk

if j2J A

ci2I

1

X

dch Emjk A þ rDD

c j2J ; k2K

(6)

if j2J D :

Rijk

ci2I

Additional constraints denoted by USa and USb are raised to emphasize the periodic operation of an HPS. Therein, scenario a (Sa) assumes an empty battery for start-up operation of an HPS. Export of electricity is prohibited in Sa to evaluate the total surplus power over the whole operating period. Scenario b (Sb) is used for normal periodic operation, where the power remained in the battery at the end of an operation period is used as the initial storage for the next period.

( USa ¼ n

 

 sp R Rsp i0  i0

¼ 0; ci2I ;

X

X

X

) Eink ¼ 0

ci2I o cn2N ck2K

 sp R ¼ Rsp ; ci2I USb ¼ Rsp ; Rsp i0 iK i0 iK

The objective of the HPS design problem is to minimize the power imported from the public grid, as shown in the following LP, P1, where x1 and U denote the unknown variables to be determined and the feasible searching space.

P1 :

X X X min JyP1 ¼ Emjk x1 2U1 ∩Uy cm2M cj2J ck2K (

y2fSa; Sbg

sp;use sp;sav sp Eik ; Eik ; R 0 ; Rsp;use ; Rsp;sav ; Rijk ; Eijk ; Eink ; Emjk ; ik ik ik x1 ¼ 0 0 ci2I ; j2J ; k2K; k 2K ; m2M; n2N U1 ¼ fx1 jEqs: ð1Þ  ð6Þg

)

The solution of P1 gives the minimum imported power from grid, ðJyP1 Þ ; y2fSa; Sbg. However, the power stored in the battery over these time intervals, Rsp , may have multiple solutions. Addiik' tional constraint is needed to restrict the maximum power imported from the grid during the minimization of the battery capacity. The problem of minimizing the total amount of power stored over the time horizon can be formulated as the following LP, P2, where Rk' is the total amount of power stored in the battery at time interval k'2K0 .

P2 :

Eq. (4) is the distribution of electricity from the RE source i used by all customers j2J and exported to all potential buyers in grid n2N . Eq. (5) shows the supplementary power discharged from the battery.

Eik

Eq. (6) show that the power requested by customer j at time interval k come from RE sources i2I and is supplemented by the power discharged from the battery and all possible suppliers m2M in grid. Note that the requirement of electricity rectification and inversion between AC and DC is considered.

X X sp min JyP2 ¼ Rik' x2 2U2 ∩Uy 0 ci2I ck'2K X sp ¼ Rk' y 2fSa; Sbg 0

o n ck'2K sp x2 ¼ x1 ∪ Rk' ; ck0 2K0 8  9 <  X =   X X Emjk ¼ JyP1 ; y2fSa; Sbg U2 ¼ U1 ∩ x2  :  cm2M ; cj2J ck2K Based on the solution of P2, the minimum battery capacity, ðRÞy , for both scenarios, Sa and Sb, can be found straightforwardly,

C.-L. Chen et al. / Energy 75 (2014) 24e30

 sp sp sp  ðRÞy ¼ max R0 ; R1 ; …; RK y

y2fSa; Sbg

(7)

Further to the LP formulation for targeting the outsourced electricity and the battery capacity, some follow-up calculations can help elucidate the electricity transformation in an HPS. For example, the insight-based power cascade table is sometimes desirable to provide simple visualization on network targeting and design. Some useful follow-up calculations based on the optimized data from P2, will be depicted in the numerical example.

27

Table 2 The given interval-dependent electricity supplies and demands (kWh). k

Time (h)

sp

AC 1 2 3 4 5 6 P

0e2 2e8 8e10 10e18 18e20 20e24 0e24

de Ejk

Eik

300 100

400

AC 140 420 140 560 140 280 1680

DC

120 480

600

DC 60 180 60 240 60 120 720

AC

100 400

500

DC 40 120 40 160 40 80 480

AC

AC

100 400

80 320 80

500

480

4. Numerical example One numerical example studied by Mohammad Rozali et al. [16] is adopted to show the adequacy of the proposed ETM for modeling a HPS with power loss consideration. The available RE power supplies (wind, biomass, and solar) and power demands (appliances 1e5) and the elapsed operating time, the average power ratings within the assigned time slots, and the total amount of electricity supplies/demands are given in Table 1. Note that the solar power is DC, and appliances 1 and 3 require DC. Other sources and supplies are AC. The public grid consists of suppliers in both AC and DC, while there is only one buyer for electricity in AC. According to the given elapsed time for each power sources and demands, the 24 h is divided into six time intervals with specified power loads, as shown in Table 2. In this study, the recovery ratios for the rectifier and the inverter rec ¼ r inv ¼ 0:95. Several batteries with different are both given as rAD DA self-discharge rates and recovery ratios for charging surplus power into the battery and discharging stored power from the battery are listed in Table 5 [23]. Therein, the lead acid battery is studied in more detail as follows, where the self-discharge rate is dch ¼ r cha ¼ 0:9 for diss ¼ 0:004%=h, the recovery ratios are rDD DD charging DC from the battery or charging surplus DC to the battery, dch ¼ r cha ¼ 0:8551 for discharging the elecand recovery ratios rDA AD tricity in battery to AC and charging surplus AC to the battery. Similar to the analysis done by Chen et al. [20], two scenarios for applying the ETM are investigated, including the start up operation (Sa) and the normal operation (Sb) with electricity import when it is demanded. The two scenarios, Sa and Sb, in the LP model of P1 entail 127/126 constraints, and both scenarios contain 351 continuous variables. The numbers of constraints and variables are increased by 8 and 7 in P2 for both scenarios. The LP optimization models are implemented in the GAMS environment [22] on an Intel Core i5 CPU760, 2.80 GHz, 2.0 GB RAM processor, with CPLEX as the LP solver. The solutions can be found within 0.02 CPU second for both scenarios. The results of solving P1 and P2 in sequence for the two scenarios when using the lead acid battery are illustrated in Table 3, Table 1 The available power supplies and demands and their working time intervals and ratings for the example, taken from Ref. [17]. Power sources

Time (h)

Power rating

Electricity

AC

DC

From

To

Interval

(kW)

Gen (kWh)

Solar

2 0 8

10 24 18

8 24 10

50 70 60

400 1680 600

Wind Biomass Power demands

Time (h)

Power rating

Electricity

AC

DC

From

To

Interval

(kW)

Use (kWh)

App.1

0 8 0 8 8

24 18 24 18 20

24 10 24 10 12

30 50 20 50 40

720 500 480 500 480

App.2 App.3 App.4 App.5

including the distribution of electricity from all RE sources and discharged from the battery, the outsourced electricity from grid, and the total amount of electricity stored over the time horizon. Note the targeted electricity imported from grid is 225 kWh in AC for start-up mode and 174.2 kWh in AC for normal operation, both are smaller than the reported targets in Ref. [16]. As shown in Table 3(a), in the first time interval of the start-up operation mode (no power stored initially), the power generated from the biomass (140 kW) is used by appliances 1 (63.2 kWh) and 2 (42.1 kWh) and the rest 34.7 kWh is sent to storage in the battery. Therein 10% of the electricity is lost during the charging and the effective electricity stored in the battery is 29.7 kWh. In the second time interval, 315.8 kWh out of the 420 kWh power from biomass plant is used by appliances 1 (189.5 kWh) and 3 (126.3 kWh). The surplus power from the biomass (104.2 kWh) and all electricity from the wind turbine (300 kWh) is sent for storage. The resulting accumulation in the battery becomes 375.3 kWh, where 256.5 kWh is contributed by the wind and 118.8 kWh from biomass. In the third time interval, electricity supplied by the three renewable sources (100, 140, and 120 kWh) are all used up by appliances. For example, the 100 kWh from wind turbine is totally allocated to appliance 2. The 140 kWh from biomass is used by appliances 4 (81 kWh) and 5 (59 kWh). The 120 kWh from solar in DC is distributed to appliances 1 (60 kWh), 3 (40 kWh) and 4 (20 kWh). In this time interval, 24.6 kWh in the battery marked with the wind turbine is discharged to supplement the need by appliance 5, and 231.9 kWh remains. In the fourth interval, the electricity from solar (480 kWh in DC) is all used by appliances 1e3 (240, 80, and 160 kWh). The 560 kWh from biomass plant is distributed to appliances 2, 4, and 5 (145.5, 175.1 and 239.4 kWh). The battery provides 208.7 kWh out of 256.5 kWh from the storage of wind power and the whole 94.2 kWh from the storage of biomass for discharging to complement the need of appliances 2 and 5. Note that 47.8 kWh is remained in the battery for later use by appliance 3 in DC. The deficit power in appliance 4 is complemented by 225 kWh import from grid in AC. In the fifth interval, 140 kWh from biomass is used by appliances 1 (17.9 kWh), 3 (42.1 kWh), and 5 (80 kWh). The residue in the battery (47.8 kWh) is discharged into 43 kWh in DC to support the need of appliance 1. In the last interval, the biomass generates 280 kWh, where 210.5 kWh is used by appliances 1 (126.3 kWh) and 3 (84.2 kWh). The remaining 69.5 kWh is charged into the battery, and the effective storage becomes 59.4 kWh. Note the targeting electricity imported from grid is 225 kWh in AC. The DC supply in grid is not used in this case. The total surplus electricity remained in the battery at all time intervals are also shown in Table 3(a). The battery capacity, 375.3 kWh, is the maximum of these surplus values which has been marked in the table. Note also that the surplus power stored in the battery at the last interval can be used for the next operating period. This results in the normal operation mode where the surplus electricity at the last interval is exactly used as the initial storage for the next operating

28

C.-L. Chen et al. / Energy 75 (2014) 24e30

Table 3 Optimized data from proposed LP formulation for the example with one AC/one DC power sellers and one AC power buyer in grid. (a) Start-up operation sp

k 0 1 2 3 4 5 6

Eik

k

Eijk j¼1 0 0 0 0 0 0

1 2 3 4 5 6 k 1 2 3 4 5 6

0 300 100 0 0 0

Rijk j¼1 0 0 0 0 47.8 0

sp;use

sp;sav

Eik 140 420 140 560 140 280

63.2 189.5 0 0 17.9 126.3

0 0 0 0 0 0

0 0 120 480 0 0

0 0 100 0 0 0

0 0 60 240 0 0

j¼2 0 0 100 0 0 0

0 0 0 0 0 0

j¼2 0 0 0 208.7 0 0

sp

Eik 105.3 315.8 140 560 140 210.5

0 0 0 145.5 0 0

0 0 0 0 0 0

0 0 120 480 0 0

0 300 0 0 0 0

0 0 0 80 0 0

j¼3 0 0 0 0 0 0

0 0 0 0 0 0

j¼3 0 0 0 0 0 0

34.7 104.2 0 0 0 69.5

42.1 126.3 0 0 42.1 84.2

0 0 0 0 0 0

Rik 0 0 256.5 256.5 47.8 0 0

0 0 0 0 0 0

0 0 40 160 0 0

j¼4 0 0 0 0 0 0

0 0 0 0 0 0

j¼4 0 0 0 0 0 0

sp;sav

0 29.7 118.8 94.2 0 0 59.4

0 0 0 0 0 0

0 0 256.5 47.8 0 0

0 0 20 0 0 0

j¼5 0 0 0 0 0 0

0 0 24.6 0 0 0

0 0 0 0 0 0

j¼5 0 0 0 0 0 0

59.4 89.1 340.2 340.2 47.8 0 59.4

0

0 0 81 175.1 0 0

sp;use

Rik

0

sp

Rik 0 29.7 94.2 0 0 0

0 0 59 239.4 80 0

0 0 0 94.2 0 0

0 0 0 0 0 0

0 0 0 208.7 47.8 0

0 0 24.6 94.2 0 0

0 0 0 0 0 0

Rk 0 29.7 375.3 350.7 47.8 0 59.4

0 0 0 0 0 0

Em¼1;j;k j¼1 0 0 0 0 0 0

j¼2 0 0 0 0 0 0

j¼3 0 0 0 0 0 0

j¼4 0 0 0 225 0 0

j¼5 0 0 0 0 0 0

0 0 0 0 0 0

Em¼2;j;k j¼1 0 0 0 0 0 0

j¼2 0 0 0 0 0 0

j¼3 0 0 0 0 0 0

j¼4 0 0 0 0 0 0

j¼5 0 0 0 0 0 0

(b) Normal operation k 0 1 2 3 4 5 6

sp Eik

k

Eijk j¼1 0 189.5 0 0 0 0

1 2 3 4 5 6 k 1 2 3 4 5 6

0 300 100 0 0 0

Rijk j¼1 0 0 0 0 0 0

sp;use Eik

140 420 140 560 140 280

63.2 0 0 0 17.9 126.3

0 0 0 0 47.8 0

0 0 120 480 0 0

0 189.5 100 0 0 0

0 0 60 240 0 0

j¼2 0 0 0 0 0 0

0 0 0 0 0 0

j¼2 0 0 0 69.9 0 0

sp;sav Eik

105.3 126.3 140 560 140 210.5

0 0 100 0 0 0

0 0 0 292.4 0 0

0 0 120 480 0 0

0 110.5 0 0 0 0

0 0 0 0 0 0

j¼3 0 0 0 0 0 0

0 0 0 0 0 0

j¼3 0 0 0 0 0 0

34.7 293.7 0 0 0 69.5

42.1 126.3 0 0 42.1 84.2

0 0 0 0 0 0

0 0 0 0 0 0

Rsp ik 0 0 94.5 69.9 0 0 0

0 0 40 160 0 0

j¼4 0 0 20 0 0 0

0 0 0 0 0 0

j¼4 0 0 24.6 0 0 0

period, as shown in Table 3(b). The detailed transformation of electricity in the normal operation mode is skipped due to space limitation. The target of the power imported from grid is reduced from 225 kWh for start-up operation to 174.2 kWh for the normal operation. However, the required maximum battery capacity is increased from 375.3 kWh to 434.7 kWh. Based on the optimized power transformation data shown in Table 3, some follow-up calculations, such as shown in Eqs. 8e11, can help constructing the power cascade table for comparing the proposed LP formulation with the insight-based power pinch analysis method. Eqs. (8) and (9) are the imported/exported electricity in AC or DC. Eq. (10) is the power in AC or DC to be charged into the battery in DC type. Eq. (11) shows the power in AC or DC discharged from the battery for supplementing the customer demands.

Rsp;sav ik

0 0 40 316 0 0

0 0 0 0 0 0

gd;A

Ek

0 0 69.9 0 0 0

0 0 20 0 0 0

j¼5 0 0 0 0 0 0

0 0 0 0 0 0

j¼5 0 0 0 0 0 0

X

¼

59.4 89.1 340.2 47.8 0 0

0 0 0 0 0 0

0 0 0 244 80 0

0 0 0 0 0 0

X

cm2MA cj2J

gd;D

Ek

X

¼

X

cm2MD cj2J

Ekex;A ¼

Ekex;D

0 0 0 0 0 0

Rsp;use ik

¼

0

X

@

cn2N A

X cn2N D

0 @

X ci2I A

X ci2I D

0 0 24.6 69.9 0 0

0 0 0 292.4 47.8 0

Rsp k 59.4 89.1 434.7 410.1 47.8 0 59.4

0 0 0 0 0 0

0 0 80 80 0 0

Em¼1;j;k j¼1 j¼2 0 0 0 0 0 0 0 158.2 0 0 0 0

j¼3 0 0 0 0 0 0

j¼4 0 0 0 0 0 0

j¼5 0 0 0 16 0 0

0 0 0 0 0 0

Em¼2;j;k j¼1 j¼2 0 0 0 0 0 0 0 0 0 0 0 0

j¼3 0 0 0 0 0 0

j¼4 0 0 0 0 0 0

j¼5 0 0 0 0 0 0

Emjk Emjk

ck2K

inv Eink þ rDA

Eink þ

rec rAD

X ci2I D

X ci2I A

(8)

1 Eink A

1 Eink A

ck2K

(9)

C.-L. Chen et al. / Energy 75 (2014) 24e30

29

Table 4 The power cascade table for the example with one AC/one DC power sellers and one AC power buyer in grid. Interval-based power supply/demand t

Dt

(h)

(h)

Rating (kW)

Power (kWh)

Surp.(þ)

Charge

(a) Start-up operation

Into

Disch.

Supply

Demand

Supply

Demand

Defi. ()

Battery

AC

AC

AC

AC

AC

AC

DC

DC

DC

DC

DC

Battery

(After) DC

AC

DC

(kWh)

(kWh)

DC

AC

0 2

70

0

0

50

140

0

0

100

140

100

34.7

6

120

0

0

50

720

0

0

300

720

300

404.2

2

120

60

140

50

240

120

280

100

40

20

21

8

70

60

140

50

560

480

1120

400

560

80

259

2 8

(b) Normal operation Import

Disch.

Battery

(After) DC

AC

DC

59.4

29.7

89.1

410.1 225

18

309.8

174.2

47.8 0

40

50

140

0

80

100

60

100

70

0

0

50

280

0

0

200

280

200

47.8

43

20 4

DC

434.7

350.8

70

(kWh) AC

21

10

2

(kWh) DC

0

375.3

Import

43 0

0

59.4

59.4

69.5

24

Table 5 Targets for the imported electricity from the grid and the required storage capacity for some typical batteries. rec ¼ r inv ¼ 0:95; r cha ¼ r dch rAD DD DD DA

(a) Start-up

(b) Normal

cha ¼ r dch ¼ 0:9501 r cha rAD DD DA

Operation

Operation

Recovery ratios cha rDD

Battery type

1. 2. 3. 4. 5.

Lead acid [16] Nickel Cadmium ðNiCdÞ Lithium ion Sodium Sulphur ðNaSÞ; Vanadium redox ðVRBÞ; Zinc Bromine; or Regenerative fuel cell Metal air

Ekcha;A ¼

X

sp;sav

ci2I A

dch Ekdch;A ¼ rDA dch Ekdch;D ¼ rDD

Eik

X

;

Ekcha;D ¼

X

ci2I cj2J A

X

X

ci2I cj2J D

X ci2I D

sp;sav

Eik

ck2K

(10)

Rijk ; Rijk

ck2K

(11)

These follow-up calculations can provide the same computational details of the insight-based techniques such as the problem table and the power cascade table [16], as shown in Table 4. From this calculated power cascade table, there are surplus AC power at the first (34.7 kWh), the second (404.2 kWh), and the last (69.5 kWh) time intervals to be charged into the battery for the start-up operation mode. The battery discharges 21 kWh in AC at the third interval to complement the deficit. At the fourth interval, instead of using all 350.8 kWh power stored in the battery for supporting the need of the appliances, 259 kWh in AC is discharged and 47.8 kWh stays in the battery. The 47.8 kWh is discharged into 43 kWh in DC at the fifth interval to supplement the deficit of DC appliances. Thus there is no need to import electricity from the DC supplier in this case. Also note that there is no surplus power to export in this case study. In the original power cascade table presented by Mohammad Rozali et al. [16], the 350.8 kWh in the battery at the fourth interval is all discharged to 300 kWh in AC, and 43 kWh in DC is imported from grid at the fifth interval. Though the imported electricity at the fourth interval is reduced from 225 kWh to 184 kWh in AC, additional 43 kWh in DC is imported at the fifth

s

Imported

dch rDD

(%/h)

Electricity

0.9

0.004 0.016 0

225.0 225.1 149.7 225.0 351.7

1.0 0.9 0.7

Battery

Imported

Battery

Capacity

Electricity

Capacity

375.3 375.3 417.0 375.3 291.9

174.2 174.4 87.0 174.2 321.0

434.7 434.6 483.0 434.7 338.1

interval. This example shows that the proposed LP formulation can easily find the actual targets of the outsourced electricity and the battery capacity for storage of excess power. Further to the illustrative case study, the imported electricity from the public grid and the required storage capacity for several typical batteries are summarized in Table 5. The effects of the selfdischarge rate and the recovery ratios for electricity charging and discharging on power targets and battery size can be examined. For example, one can compare the imported electricity and the required battery capacity for the lead acid and the NiCd battery, where both batteries have the same recovery ratios chg dcg rDD ¼ rDD ¼ 0:9 but different self-discharge rates (0:004%=h vs. 0:016%=h). It is found that the self-discharge rate does not have much effect on the required electricity import for both operational scenarios since the time of electricity storage in the studied case is short, less than a day. The effect of the recovery ratios can be seen from the comparison of the results for the last three battery sets in Table 5. The imported electricity increases dramatically (149.7, 225, 351.7 kWh for start-up operation and 87, 174.2, 321 kWh for normal operation) with the decrease of the recovery ratios (1, 0.9, 0.7). Furthermore, the lower the recovery ratio, the smaller the required storage capacity. This information is useful for selection of electricity storage device in an HPS.

5. Conclusion This work deals with the targeting of minimum outsourced electricity from the grid and the minimum electricity storage capacity for an HPS. The ETM developed by Chen et al. [20] is

30

C.-L. Chen et al. / Energy 75 (2014) 24e30

extended for power allocation and for handling electricity storage and outsourcing. The major power losses that typically occur in an HPS, such as those from the conversion between AC and DC, electricity charging and discharging, and self-discharge during storage, are taken into account in the proposed ETM. Based on the ETM model, with the assumptions made for the HPS problem, including average power ratings used, known power generation and demands etc., the problem of targeting the outsourced electricity and the storage capacity is formulated as two sequential LP problems. A literature example was used to illustrate the proposed approach. Results of the example show that the presented model is promising to provide preliminary analysis for optimal power allocation in an HPS, and is an improved method compared to the power pinch analysis. Note that the uncertainties in the availability of renewable energy and even in the power demand might be not easy to predict in practice. Handling these uncertainties such as the availability of RE over various time intervals, require a more comprehensive model to take further details into account. This will be handled in an on-going research work.

Emjk

electricity imported from seller m and used by appliance j at interval k sp;sav sp;use Eik ; Eik power from source i for storage or use at interval k gd ex power import from seller m or export to buyer n at Emk ; Enk interval k Ekcha;A ; Ekcha;D power (in AC or DC) for charge at interval k Ekdch;A ; Ekdch;D discharged power (in AC or DC) from battery at interval k Ekde;A ; Ekde;D demanded power (in AC or DC) at interval k Ekex;A ; Ekex;D excess power, in AC or DC, exported to grid at interval k gd;A gd;D Ek ; Ek power (in AC or DC) imported from grid at interval k Eksd;A ; Eksd;D surplus/deficit power (in AC or DC) at interval k sp;A sp;D Ek ; Ek power (in AC or DC) from renewable sources at interval k Jy objective function, 2fP1; P2g; y2Sa; Sb Rijk power in battery related to source i released to appliance j a interval k Rsp power originated from source i at interval k ik sp;use Rsp;sav ; R stored power from source i for save or use at interval ik ik k

Acknowledgments References The authors thank the National Science Council of ROC for supporting this research under Grants NSC101-3113-E-002-004 and NSC102-2221-E-002-216-MY3. Nomenclature

Indices i j k m n

2I , index for power supply 2J , index for power demand 2K, index for time interval 2M, index for electricity seller in grid 2N , index for electricity buyer in grid

Sets I ¼ I A ∪I D {1, 2, …, I}, set of power suppliers, I A for AC and I D for DC J ¼ J A ∪J D {1, 2, …, J}, set of power demands, J A for AC and J D for DC K f1; 2; …; Kg, set of time intervals; K0 ¼ f0g∪K M ¼ MA ∪MD f1; 2; …; Mg, set of electricity sellers in grid, MA for AC and MD for DC N ¼ N A ∪N D f1; 2; …; Ng, set of electricity buyers in grid, N A for AC and N D for DC x set of all adjustable variables, 2f1; 2g U feasible searching space, 2f1; 2; Sa; Sbg Parameters sp Eik electricity given by renewable source i at interval k de Ejk electricity consumed by appliance j at interval k rec ; r inv recovery ratio for converting AC to DC and vice versa rAD DA cha , r cha recovery ratio for charging electricity in AC/DC to battery rAD DD dch dch recovery ratio for discharging from battery to electricity rDA , rDD in AC or DC s hourly self-discharge rate (z0:004%=h) Dk elapsed time (hours) at interval k Positive variables power transfer from renewable source i to appliance j at Eijk interval k Eink excess power from renewable source i exported to buyer n at interval k

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