An inexact robust nonlinear optimization method for energy systems planning under uncertainty

Renewable Energy 47 (2012) 55e66

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An inexact robust nonlinear optimization method for energy systems planning under uncertainty C. Chen 1, Y.P. Li*, G.H. Huang 2, Y. Zhu 1 MOE Key Laboratory of Regional Energy Systems Optimization, Sino-Canada Energy and Environmental Research Academy, North China Electric Power University, Beijing 102206, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 January 2012 Accepted 2 April 2012 Available online 10 May 2012

In this study, an interval-robust nonlinear optimization (IRNO) method is developed for planning energy system and managing CO2 emissions with trading scheme, through incorporating interval-parameter programming (IPP) within a robust optimization (RO) framework. In the modeling formulation, two recourse actions were adopted to make the model robustness. One of recourse actions was launched to capture the notion of risk in stochastic programming. The other recourse action was seized the risk of shortage electricity amount during the energy system programming process, which successfully emphasizing the safety of energy system under high-variability. The IRNO method is applied to a case of planning energy system with considering the CO2 emissions management. A number of solutions under different robustness levels have been generated. The results obtained can help generate desired decision alternatives that will be able to enhance energy supply safety with a low system-failure risk level and particularly useful for risk-aversive decision makers in handling high-variability conditions.
2012 Elsevier Ltd. All rights reserved.

Keywords: Robust optimization Energy systems Stochastic Uncertainty CO2 trading

1. Introduction With rapid economic development, continual urban expansion and life standard improvement, energy demand and supply have been experiencing sharply increase [1e4]. According to recent IEA report [5], the global energy demand is growing at a rate of about 1.6% per year, and is expected to reach about 700
1018 Joule/Year by 2030, with more than 80% of worldwide primary energy production still coming from combustion of fossil fuels [6]. In China, total energy consumption reached 2457 million tons of coal equivalents (Mtce) in 2007, equivalent to 72 exajoules (EJ) with an increment of 9.3% from 2005 [7]. However, the trend of increased energy demand leads to the fossil fuel reserves continuously shrinking, while the global oil storage was 1258 billion barrels in 2008, which would be exhausted in future 42 years [8]. Meanwhile, energy system plays a primary role for climate change since consumptions of fossil fuels result in a large amount of greenhouse gas emissions [9]. For example, approximately 38% of current UK greenhouse gas emissions can be attributed to the energy supply

* Corresponding author. Tel.: þ86 1051971255; fax: þ86 1051971284. E-mail addresses: [email protected] (C. Chen), [email protected] (Y.P. Li), [email protected] (G.H. Huang), [email protected] (Y. Zhu). 1 Tel.: þ86 10 51971215; fax: þ86 10 51971284. 2 Tel.: þ86 1061772928; fax: þ86 1051971284. 0960-1481/$ e see front matter
2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2012.04.007

sector [10]. Such situations have forced decision makers to contemplated and proposed comprehensive and ambition plans for energy system managing and planning. However, such planning efforts are often complicated with many factors such as properties of resources and efficiencies of technologies, which are uncertain and cannot be expressed as deterministic values. For example, various parameters of energy systems (e.g., energy supply cost) can hardly be acquired as deterministic values but easily expressed as intervals; the interval values can be used for describing uncertain information of planning issues which fluctuates within a range without distribution information. The other system’s parameters (e.g., energy demand) may be expressed as probability distributions when the data obtained are enough to fit certain types of distributions based on historical records and literature reviews. Moreover, many processes are linked to the energy management systems, including supply of energy resources, energy production, import/ export, conversion, transmission, consumption and carbon dioxide (CO2) mitigation technologies options available [11]. Such processes are undergoing dramatic changes stemming from new regulations, re-urbanization, population growth, regional/ community sustainable development, transportation shifts and economic expansion, resulting in significant effects on energy activities, socio-economy and environment [12]. It is thus desired to develop effectively optimization methods for supporting

56

C. Chen et al. / Renewable Energy 47 (2012) 55e66

planning energy systems associated with such complexities and uncertainties [13e18]. Stochastic programming methods have received extensive attentions since they could directly integrate uncertain information expressed as probability distributions into the modeling formulations [9,10,19e22]. However, one potential limitation of the conventional stochastic programming methods is that they are incapable of considering the variability of the recourse values since they are based on an assumption that the decision maker is risk neutral [23,24]. As a result, it may become infeasible when the decision maker is risk averse under high-variability conditions. Energy systems are often associated with various system-failure risks (e.g., resource-availability risk, energy supply risk) due to multiple uncertainties and complexities, energy supply, the desired energy resources allocation patterns and greenhouse gas management may vary with time under high-variability conditions. For example, the desired energy resources allocation patterns may vary with time under high-variability conditions, which may result in a high risk of electricity shortage particularly when energy demand-level is high. Unfortunately, the conventional stochastic programming methods have difficulties in emphasizing safety of energy system under high-variability. Address the above issues, it is required that the related decisions be made with enhanced security for energy systems. To deal with such difficulties, robust optimization (RO) method was proposed for incorporating risk aversion into optimization models and finding robust solutions for system management problems, as an extension of recourse stochastic programming methods [25]. In RO models, uncertain parameters which derived from noisy, incomplete of erroneous data are handled as random variables with discrete distributions [26]. In general, RO method has the following advantages: (i) it integrates goal programming with a scenario-based description of problem data, and generates a series of solutions that are progressively less sensitive to realizations of the model data from a scenario set; (ii) it is useful for helping decision makers to quantitatively evaluate tradeoff between system economy and stability; and (iii) it is effective to penalize the costs that are above the expected values, as well as to capture the notion of risk in stochastic programming [23e29]. The conventional RO methods were effective in handling random variables when their probability distributions were available; however, the quality of available information about the uncertainties is often not satisfactory for establishing probability distributions. Moreover, even if the uncertainties expressed as probability distributions are available, it could be difficult to reflect them in large-scale stochastic models [10,30]. In energy systems, various uncertain components may exist and may not be available as probability distribution, such as cost parameters, total electricity demand, residual capacities and capacity expansion limitations. Interval-parameter programming (IPP) is effective for handling uncertainties express as intervals [31]. It improves upon the existing methods with the following natures: (i) it allows uncertainties to be directly communicated into the optimization process and resulting solutions, (ii) it does not lead to more complicated inter- mediate models, and thus has a relatively low computational requirement, and (iii) it can be easily integrated with other optimization methods [30,31]. This requires that IPP be introduced into the RO framework to reflect the uncertainties with varied quality levels and presentation formats in energy systems [22,30e34]. Therefore, this study aims to develop an interval-robust nonlinear optimization (IRNO) method for energy system planning, through coupling interval-parameter programming (IPP) with robust optimization (RO) to deal with uncertainties expressed as probability distributions and interval values. Robust recourse actions will be considered to improve the robustness of the IRNO model. A case

study will then be provided for demonstrating the applicability of the developed method. The results obtained can help generate desired decision alternatives that will be able to not only enhance energy supply safety with a low system-failure risk level but also managing CO2 emissions with an effective trading scheme. 2. Methodology Robust optimization (RO) method could not only penalize the costs that are above the expected values, but capture the notion of risk under uncertainty [23]. In fact, the RO method is a hybrid of stochastic and goal programs, to balance tradeoff between the expected recourse costs and the variability of these random values [25]. A general RO model can be formulated as follows:

Min f ¼ CT1 X þ

s X

ph DT2 Y þ r

h¼1

s X

ph DT2 Y  ph

s X

h¼1

!2 DT2 Y

(1a)

h

subject to

Ar X  Br ;

r˛M; M ¼ 1; 2; .; m1

~ ih ; Ai X þ A0i Y  w xj  0; xj ˛X;

(1b)

i˛M; i ¼ 1; 2; .; m2 ; h ¼ 1; 2; .; s

j ¼ 1; 2; .; n1

yjh  0; yjh ˛Y;

(1c) (1d)

j ¼ 1; 2; .; n2 ; h ¼ 1; 2; .; s

(1e)

In the above modeling formulation, the random variables take ~ ih with probability levels Ph, where h ¼ 1, 2, ., s discrete values w P  and ph ¼ 1. x j and yjh represent the first- and second-stage 2  s P decision variables, respectively. The term of DT2 Y  ph DT2 Y h

is a variability measure on the second-stage penalty costs; the nonnegative factor r represents a weight coefficient. Depending on the value of r, the optimization may favor solutions with a higher s P expected second-stage cost ph DT2 Y in exchanging for a lower h

variability in the second-stage penalty costs as measured by  2 s P DT2 Y  ph DT2 Y [29]. When r ¼ 0, the RO model becomes h

a conventional two-stage stochastic program (TSP); the objective is only to minimize the first- and second-stage costs. This also implies that the decision makers possess a risk neutral attitude and would not consider the variability of the uncertain recourse costs. However, when r ¼ 1, the decision makers can consider the variability of the second-stage cost based on a risk-aversive attitude [35]. Uncertainties may vary with time under high-variability conditions during the programming process, resulting in the results becoming more uncertain with a higher risk decision making. Then, the above RO model can be re-formulated as follows [25,26]:

Min f ¼CT1 X þ

s X

ph DT2 Y þ r

h¼1

þu

s X

ph dh

s X h¼1

ph DT2 Y  ph

s X

!2 DT2 Y

h

ð2aÞ

h¼1

subject to

DT2 Y  ph

s X h

DT2 Y  0

(2b)

C. Chen et al. / Renewable Energy 47 (2012) 55e66

Ar X  Br ;

r˛M; M ¼ 1; 2; .; m1

~ ih ; Ai X þ A0i Y  w xj  0;

(2c)

i˛M; i ¼ 1; 2; .; m2 ; h ¼ 1; 2; .; s

xj ˛X; j ¼ 1; 2; .; n1

yjh  0;

(2d)

(2f)

h

(iii) recourse cost that seized the risk of shortage electricity amount (dh) during the energy system programming process, weighted by parameter u [25]. Model (2) can not only effectively reflect random uncertainty but also guarantee solutions to be more stable and reliable. In real world problems, the quality of available information about the uncertainties is often not satisfactory for establishing probability distributions, and some random events can only be quantified as discrete intervals, leading to dual uncertainties [35]. However, RO is infeasible to treat the uncertainties expressed as interval values. Fortunately, interval-parameter programming (IPP) is effective for handling uncertainties in objective function and constraints as well as those that cannot be quantified as probability distributions, since interval numbers are acceptable as its uncertain inputs [34,36e38]. Therefore, one potential approach for accounting for random and interval uncertainties is to introduce the IPP into the RO framework; this leads to an interval-robust nonlinear optimization (IRNO) model as follows:

s X

þr

h¼1 s X

h¼1

 ph D T2 Y

 ph D T2 Y  ph

þu

s X

h¼1 s X

h¼1

 ph D T2 Y

 ph D T2 Y  ph

s X

!2  D T2 Y

(4a)

h 

þu

ph dh

h¼1

subject to

 D T2 Y  ph

s X

 D T2 Y  0

(4b)

h

  A i X  Br ;

r˛M; M ¼ 1; 2; .; m1

(4c)

 0  ~ A i X þ Ai Y  wih ; r˛M; M ¼ 1;2;.;m1 ; h ¼ 1;2;.;s

(4d)

x j  0;

 x j ˛X ; j ¼ 1; 2; .; n1

(4e)

y jh  0;

 y jh ˛Y ; j ¼ 1; 2; .; n2 ; h ¼ 1; 2; .; s

(4f)

 Solutions of x jopt ðj ¼ 1; 2; .; n1 Þ and yjhopt ðj ¼ 1; 2; .; n2 Þ can be obtained through submodel (4). Based on the above solutions, the second submodel corresponding to fþ can be formulated as follows:

s X

Min f þ ¼ CTþ1 X þ þ

!2  D T2 Y

s X

þr

(3a)

s X

þr

h

h¼1 s X



ph dh

h¼1

þ ph Dþ T2 Y

þ ph Dþ T2 Y  ph

þu

h¼1

s X

!2 þ Dþ T2 Y

(5a)

h þ

ph dh

h¼1

subject to

 D T2 Y  ph

s X

Min f  ¼ CT1 X  þ

The objective function value of model (2) includes: (i) expected cost (energy resource supply cost, variable cost for power conversion, capital cost for capacity expansions of power conversion technologies and cost for treating excess carbon dioxide); (ii) recourse cost that is above the expected values, as well as to capture the notion of risk in stochastic programming, s P P which was judged by the value of r sh ¼ 1 ph ðDT2 Y  ph DT2 YÞ2 ;

s X

The IRNO method can be transformed into two deterministic submodels, which correspond to the lower and upper bounds of the objective function value. When the objective is to be minimized, the submodel corresponding to f  can be firstly formulated (assume that B  0; f  > 0):

(2e)

yjh ˛Y; j ¼ 1; 2; .; n2 ; h ¼ 1; 2; .; s

Min f  ¼ CT1 X  þ

57

subject to s X

 D T2 Y  0

(3b)

h

  A r X  Br ;

þ Dþ T2 Y  ph

s X

þ Dþ T2 Y  0

(5b)

h

r˛M; M ¼ 1; 2; .; m1

(3c)

þ þ Aþ i X  Br ;

 0  ~ A i X þ Ai Y  wih ; i˛M; i ¼ 1;2;.;m2 ; h ¼ 1;2;.;s

(3d)

þ 0 þ ~þ Aþ i X þ Ai Y  wih ; r˛M; M ¼ 1;2;.;m1 ; h ¼ 1;2;.;s

(5d)

  x j  0; xj ˛X ;

j ¼ 1; 2; .; n1

(3e)

xþ j  0;

(5e)

  y jh  0; yjh ˛Y ;

j ¼ 1; 2; .; n2 ; h ¼ 1; 2; .; s

(3f)

yþ  0; yþ ˛Y þ ; jh jh

r˛M; M ¼ 1; 2; .; m1

þ xþ j ˛X ; j ¼ 1; 2; .; n1

j ¼ 1; 2; .; n2 ; h ¼ 1; 2; .; s

(5c)

(5f)

58

C. Chen et al. / Renewable Energy 47 (2012) 55e66

Solutions of xþ ðj ¼ 1; 2; .; n1 Þ and yþ ðj ¼ 1; 2; .; n2 Þ can jopt jhopt be obtained through submodel (5). Through integrating the solutions of submodels (4) and (5), the solutions for IRNO model can be obtained.

Period 1 25

20

3. Case study In this study, the system consists of multiple energy sources (domestic production and external importation), multiple technologies (those for energy transmission, storage and conversion) and multiple facilities, which form a complex network in the regional context [38]. Conventional and renewable energy resources with limited availabilities are employed for meeting local electric-power demands. The data of electricity demand within the planning three periods (with each period having five years) are acquired through survey and forecast measures. Fifty specimens of electricity demand were selected for each period. Thus, 150 specimens would be needed in the study. The electricity demand distribution curve for each period can thus be obtained. Through KeS test, the electricity demands for the three periods fit the normal distributions. Then, electricity demand for each period was assumed to be discrete with six probability levels (i.e., low, lowmedium, medium, medium-high, high, and very high). Fig. 1 shows the electricity demand distribution, which fit normal distribution through KeS test. Table 1 presents the electricity demand and the corresponding probabilities under each demandlevel. Due to uncertainties existing in electricity demands, decisions have to be made over a multistage framework. Assume that each power conversion technology has a pre-regulated generation target. If the target is not exceeded, the system will be encountered the regular cost; otherwise, the system will be subject to penalties resulted from the extra labor, management, operation and maintenance costs, or capacity expansion and higher costs for imported energy. From a long-term planning view, decision makers have to face a dilemma of either investing more funds on capacity expansion of existing facilities or turning to other energy production options or putting extra funds into energy imports at raised prices [9]. In the study system (Fig. 2), four main components are considered. They are (i) energy resources supply sector (coal, gas, petroleum); (ii) the energy conversion sector (coal-fired, gas-fired, petroleum-fired electricity conversion technology and CO2-emission mitigation technology); (iii) electricity demand sector (i.e., industrial, commercial, agricultural, residential sectors); (iv) CO2emission management sector. Three conventional energy resources (e.g., coal, gas and petroleum) and one renewable resource (wind) are employed with limited availabilities for meeting energy demands in the study system. It covers of three periods, with each period having five years. Two measures are used to reduce the amount of CO2-emission: (i) capture and storage, and (ii) chemical absorption [39]. Trading scheme is also proposed to mitigate the CO2 emissions, such that some power plants can sell their credit to the other power plants with higher electric-power profitability. CO2 emissions can thus be reallocated to the most efficient power plants instead proportionally allocated to each power plant. Under trading mechanism, each power plant is no longer constrained by its own emission permissions but theoretically by the aggregate number of CO2-emission limit from the power system, which can minimum the system cost at a certain level of CO2-emission permissions. Increasingly, with consideration of energy security and market risks associated with energy system, the IRNO model is useful for managing a variety of energy resources and technologies under high-variability conditions. It can help facilitating a systematic analysis of the proposed problem as well as bringing risk aversion into optimization models in order to find robust solutions to energy

15

10

5

0 -2

0

2

4

6

8

10

12

14 5

x 10

Period 2 25

20

15

10

5

0

0

2

4

6

8

10

12

14 5

x 10 Period 3 25

20

15

10

5

0

2

4

6

8

10

12

14

16 5

x 10

Fig. 1. Probability distributions for electricity demand in three planning periods.

C. Chen et al. / Renewable Energy 47 (2012) 55e66 Table 1 Electricity demands associated with different probabilities.

I X T  X

Periods

Demand-level

Probability (%)

Electricity demand (103 GWh)

Period 1

Low Low-medium Medium Medium-high High Very high Low Low-medium Medium Medium-high High Very high Low Low-medium Medium Medium-high High Very high

3 10 37 37 11 2 3 15 13 42 22 5 5 7 17 42 15 14

[0, 200] [200, 400] [400, 600] [600, 800] [800, 1000] [1000, 1200] [200, 300] [300, 500] [500, 600] [600, 800] [800, 1000] [1000, 1400] [400, 700] [400, 800] [800, 900] [900, 1100] [1100, 1200] [1200, 1600]

Period 2

Period 3

i¼1 t¼1

I X T  X i¼1 t¼1

Min f

¼

þ




X1 t

þ

I X T X H X i¼1 t¼1 h¼1



I X

T X

H X

NS2 t




þr

t¼1 h¼1

þ

NS3 t

h

 Qith



pth PP it




2  pth 4 CC

ijc th

 Yith



#2 þr

  RCit þ Zith  US t ;

(6e)

ct; h

(6f)

ci; t; h

(6g)

T X H X

i

ns I X T X H X X

T X

¼ 1; ¼ 0;

if capacity expansion of is undertaken if otherwise

#

þ






pth


NE t




X4 th

þ

t¼1 h¼1

t ¼1 h¼1




 Qith

þ

B it




 Zith





T X

T X

H X

t ¼ 1 jc ¼ 1 t ¼ 1 h ¼ 1

  Nit  Zith  Mit Qith ;

i¼1 t¼1 h¼1

T X H X

"

pth



t¼1 h¼1



pth A it

t ¼1 t ¼1 h¼1 32 I   X

nc X T T X H  X  X  5 þu
DEC pth CC ijc t
DECijc th ijc th 

subject to

I X T X H X

 PV it
Wit þ

 pth
CC ijc t
DECijc th þ r

"

pth A it

I X i¼1

i ¼ 1 t ¼ 1 h ¼ 1 jc ¼ 1

H X

ci; t; h (6h)

X3 t

   pth A it
Qith þ Bit
Zith þ

i¼1 t¼1 h¼1 T X H X

X2 t

ct; h

(4) Constraints for capacity expansion of electricity generation facilities:

" NS1 t

  RCit þ Zith  Ut ;

   0  Yith  Wit þ lDWit ;

system. Therefore, based on the proposed IRNO method, the study problem can be formulated as follows:



59

i¼1




 PP it
Yith

 Qith

AWit
s it

 pth
PP it
Yith

þ

B it






 Zith



2

#2

ð6aÞ

ci; t; h

(6i)

(1) Constraints for mass balance of fossil fuels: (5) Regional carbon trading constraint: I X

T  X

i¼1 t¼1

     Wit þ lit DWit þ Yith FE it  X1t þ X2t þ X3t ;

ci; t (6b)

0

nc X T X jc ¼ 1 t ¼ 1

 hjc DEC itjc h  Eith ;

ci; t; h

(6j)

(2) Constraints for electricity supply and demand balance:   Eith I X T X H  X i¼1 t ¼1 h¼1

I  X   s Wit þ lit DWit þYith þX4 th þ i  Dth ; ci;t;h i¼1

(6c)

nc X T X jc ¼ 1 t ¼ 1

I X T X i¼1 t¼1

(3) Constraints for electricity peak load demand:



     Wit þ lit DWit þ Yith  CF it RCit þ Zith ;

ci; t; h

 hjc t DEC itjc h  Sit ;

 S it  ð1  mÞKt ;

ci; jc ; t; h

(6k)

ci; t

(6l)

Eit ¼ lit Wit

(6d)

(6) Carbon mitigation constraints:

(6m)

facilities

for

the

disposal

capacity

60

C. Chen et al. / Renewable Energy 47 (2012) 55e66

Industry

Fossil fuels

Renewable resources

Commerce

Coal

Coal-fired pwer plant

Natural gas

Gas-fired power plant

Petroleum

Petroleum-fired power plant

Wind-power

Wind-power plant

Agriculture

Residents

Regional power system

Greenhouse effect

Carbon dioxide

Emission permit trading

Extra-regional power grid

Treat measure

Captured and storage

Chemical absorption

Fig. 2. Schematic of energy systems.

DEC itjc h 

nc X jc ¼ 1

MC ijc t ;

ci; t; h

(6n)

(7) Non-negative constraints:

 PPit
Yith 

T X H X t¼1 h¼1

 Pth PPit
Yith  0;

   A it
Qith þ Bit
Zith 

H X

ci; t; h

(6o)

     Pth A it
Qith þ Bit
Zith

h¼1

 0;

ci; t; h

 CC ijc t
DECijc th 

(6p) H X h¼1

 Pth CC ijc t
DECijc th  0;

    X1 t ; X2t ; X3t ; Yith ; DECijc th  0;

ci; jc; t; h

ci; jc ; t; h

(6q)

(6r)

where: f, expected system cost f over the planning horizon ($); i, type of power conversion technology, i ¼ 1, 2, 3, 4; i ¼ 1 for coal-fired power conversion technology, i ¼ 2 for natural gasfired power conversion technology, i ¼ 3 for biomass generation; j, type of energy resource, j ¼ 1, 2, 3; j ¼ 1 for coal, j ¼ 2 for natural gas, j ¼ 3 for biomass, j ¼ 4 for imported electricity; t, time period, t ¼ 1, 2, 3.

h, electricity demand-level, h ¼ 1, 2, . , H; jc, type of CO2 control measure, jc ¼ 1, 2, . , np; jc ¼ 1 for (CS); jc ¼ 2 for (CA); 3 NS1 t , cost for coal supply in period t ($10 /TJ); 3 , cost for gas supply in period t ($10 /TJ); NS2 t 3 NS3 t , cost for petroleum supply in period t ($10 /TJ); , cost for imported electricity supply in period t ($103/GWh); NE t , operating cost of power conversion technology i for prePV it regulated electricity generation in period t ($103/GWh); PP it , operating cost of power conversion technology i for excess electricity generation in period t ($103/GWh); A it , fixed-charge cost for capacity expansion of power conversion technology i during period t ($/tonne); B it , variable cost for capacity expansion of power conversion technology i in period t ($106/GW); CC ijc t , operating and penalty cost of control measure jc for excess CO2 emissions ($/tonne); Ut , the lower load demand in period t (GW); US t , peak load demand in period t (GW); Pth, probability of demand-level h occurrence in period t (%); , random variable of total electricity demand of level h in D th period t (GWh); RCit, residual capacity of conversion technology i in period t (GW); FE it , units of energy carrier per units of electricity production for power conversion technology i (TJ/GWh); CFit, units of electricity production per units of capacity for power conversion technology i in period t (GWH/GW); Mit , variable upper bounds for capacity expansion of power conversion technology i in period (GW); Nit , variable lower bounds for capacity expansion of power conversion technology in period t, and Nit  0 (GW); h jc , average efficiency of CO2 control measure js1 (%); Kt , discharge limit of total CO2 emissions for the whole power system (tonne);

C. Chen et al. / Renewable Energy 47 (2012) 55e66

Conversion technology

Period 1

Period 2

Period 3

Regular and surplus costs for power generation by each power conversion technology (103 $/GWh) Coal-fired power Regular cost [5.0, 7.0] [5.5, 7.5] [6.0, 8.0] Operating cost [3.5, 5.0] [4.0, 5.5] [4.5, 6.0] Gas-fired power Regular cost [4.5, 6.5] [5.0, 7.0] [5.5, 7.5] Operating cost [2.5, 4.5] [3.0, 5.0] [3.5, 5.5] Petroleum-fired Regular cost [4.3, 6.0] [4.6, 6.5] [5.0, 7.0] power Operating cost [2.0, 4.0] [2.5, 4.5] [3.0, 5.0] Wind power plant Regular cost [2.5, 3.5] [3.0, 4.0] [3.5, 4.5] Operating cost [1.5, 2.5] [2.0, 3.0] [2.5, 3.5] Fixed ($106) and variable ($106/GW) costs for capacity expansion Coal-fired power Fixed cost [325, 395] [385, 455] Variable cost [700, 850] [750, 900] Gas-fired power Fixed cost [300, 375] [350, 425] Variable cost [650, 800] [700, 850] Petroleum-fired Fixed cost [295, 335] [340, 395] power Variable cost [600, 750] [650, 800] Wind power plant Regular cost [800, 900] [880, 970] Operating cost [1900,2450] [1950,2500]

[445, 515] [800, 950] [400, 455] [750, 900] [380, 435] [700, 850] [960, 1040] [2000, 2550]

Units of electricity production per units of capacity ($103 GWh/GW) Coal-fired power 90 95 100 Gas-fired power 75 80 85 Petroleum-fired power 82 88 94 Wind power 65 68 75 Units of energy carrier per units of electricity production (TJ/GWh) Coal-fired power [10.0, 12.5] [9.9, 12.4] [9.8, 12.3] Gas-fired power [9.0, 11.5] [8.9, 11.4] [8.8, 11.3] Petroleum-fired power [8.5, 10.5] [8.4, 10.4] [8.3, 10.3] Wind power [0.11, 0.13] [0.1, 0.12] [0.09, 0.11] Variable upper bounds for capacity expansion of each conversion (103 GW) Coal-fired power 0.7 0.5 Gas-fired power 0.5 0.6 Petroleum-fired power 0.45 0.5 Wind power 0.1 0.2

technology 0.3 0.7 0.6 0.3

be raised over the planning horizon, being [287.9, 359.9]
103 TJ in period 1, [316.7, 396.7]
103 TJ in period 2, and [344.9, 432.8]
103 TJ in period 3. For the petroleum, there would be [172.5, 215.6]
103, [189.8, 237.7]
103 and [206.6, 259.3]
103 TJ during periods 1 to 3. The results indicated that coal would play the most important role in the energy supply activities and the resources supply would slightly increase as the electricity demand rise. Fig. 4 shows the optimized electricity generation plans of different power conversion technologies over the planning horizon.

4. Result analysis

1400 1200 1000

3

Amount (10 TJ)

800 600 400 200

Period 1

Period 2 Lower bound

Period 3 Upper bound

Fig. 3. Energy supply over the planning horizon.

Petroleum

Gas

Coal

Petroleum

Gas

Coal

0 Petroleum

In the IRNO modeling formulation, a number of robustness weighting levels were examined. The r and u are goal programming weight, which correspond to the robustness level of risk control. The results as follows presents the solutions obtained through the IRNO model under the scenario of r ¼ 1 and u ¼ 260. Generally, higher values of r and u mean higher robustness levels. The optimized solutions with a low system-failure risk level can be obtained when r ¼ 1 and u ¼ 260. Fig. 3 shows the energy resources supply schemes over the planning horizon through IRNO model. In this study, coal, gas and petroleum and fossil fuels are employed for meeting electricity demands. As shown in Fig. 3, the amount of coal supply would increase over the planning horizon, being [777.9, 972.4]
103 TJ in period 1, [855.7, 1071.8]
103 TJ in period 2, and [931.8, 1169.5]
103 TJ in period 3. The amount of natural gas supply would

Planning periods

Gas

In the above model, decision variables are sorted into two categories: continuous and binary. The continuous variables represent energy flows (from supply side to demand one) and capacity expansion levels, while the binary ones indicate whether or not individual convention/processing technology development or expansion actions will be undertaken. The representative costs and technical data were investigated based on governmental reports and other related references [10,16,33,34,40,41]. The data of economic and technological datum of each power conversion technology are presented in Table 2.

Table 2 Economic and technological data.

Coal

l i , CO2-emission coefficient for coal, gas and petroleum; AW it , Cost for electricity generation ($/GWh); V1, positive deviations of given scenarios’ recourse costs from the expected ones for electricity generation process. V2, positive deviations of given scenarios’ recourse costs from the expected ones for capability expansion activity. V3, positive deviations of given scenarios’ recourse costs from the expected ones for treating excess CO2 process. ADV1, the weighted values of the expected deviations for electricity generation process. ADV2, the weighted values of the expected deviations for capability expansion activity. ADV3, the weighted values of the expected deviations for treating excess CO2 process. X1 t , coal supply (TJ); X2 t , gas supply (TJ); X3 t , petroleum supply (TJ); X4 t , the amount of imported electricity (TJ); Wit , pre-regulated electricity generation target of power conversion technology i which is promised to end-users in period t (103 GWh);  , excess electricity generation of power conversion techYith nology i by which electricity generation target is exceeded when electricity demand-level is h (103 GWh);  , continuous variables about the amount of capacity expanZith sion of power conversion technology i when electricity demandlevel is h (GW);  , binary variables for identifying whether or not a capacity Qith expansion action of power conversion technology i needs to be undertaken when electricity demand-level is h; DEC ijc th , the amount of excess CO2 treated by control measure jc under electricity demand-level h;  , the amount of CO -emissions from power plant i during Eith 2 period t under level h (tonnes); S it , the reallocated CO2-emission permit to power plant i with trading scheme (tonnes);

61

62

C. Chen et al. / Renewable Energy 47 (2012) 55e66

a

80 60

350

Lower bound

300

50

Amount (10 GWh)

40 20 10

200

Period 1

150

Period 2

100

Wind

Period 3

50

Target (Lower)

Target (Upper)

Optimum target

Fig. 4. Optimized electricity generation plans for each power plant.

b

450

hi gh

M

Ve ry

Hi g

ed iu m -h i

gh

ed iu m M

Period 3

-m ed

Period 2

Lo w

Period 1

h

0

Lo w

Petroleum

Gas

Coal

Wind

Petroleum

Gas

Coal

Wind

Petroleum

Gas

Coal

0

250

iu m

30

3

Amount (10 3 GWh)

70

Upper bound

400

3

Amouont (10 GWh)

350 300 250

Period 1

200

Period 2

150

Period 3

100 50

M

Ve ry

hi gh

h Hi g

ed iu m -h i

gh

ed iu m M

Lo w

-m ed

Lo w

iu m

0

Fig. 5. (a) Lower and (b) upper bounds of imported electricity over the planning horizon.

(e.g., coal-fired) would be [2.88, 4.84]
106, [2.11, 2.53]
106 and [2.33, 2.79]
106 tonnes during periods 1e3. For the petroleumfired power plant, the trading amount of CO2-emission permit would be [1.07, 1.11]
106, [0.21, 0.27]
106 and [0.24, 0.25]
106 tonnes during periods 1 to 3. And it would be [1.27, 1.57]
106 tonnes in period 1, [2.61, 2.82]
106 tonnes in period 2, and [3.57, 4.00]
106 tonnes in period 3 for the wind power plant. 5. Discussions If the variability of the uncertain recourse cost is ignored, the study problem can be formulated as an interval two-stage stochastic programming (ITSP) model. In ITSP model, the

8 6

CO2 trading quantity (10 6 t)

Optimized electricity generation targets for each conversion tech l D s nology can be obtained based on WD it ¼ Wit þ itopt W þ it and lit ˛ [0, 1]. Thus, when Wit approach their lower bounds (i.e., when lit ¼ 0), a relatively low cost would be obtained; however, a higher penalty may have to be paid when the electricity demand is not satisfied. Conversely, when Wit approach their upper bounds (i.e., when lit ¼ 1), a higher cost would be generated but, at the same time, a lower risk of violating the promised targets (and thus lower penalty). For the coal-fired conversion technology, the results (i.e., l11opt ¼ l12opt ¼ l13opt ¼ 0) indicate that optimized coal-fired electricity generation targets would be 22.5
103, 25
103 and 27.5
103 GWh during periods 1e3, which are their lower-bound  ; W  ; and W  Þ. For the gas-fired conversion techtargets ðW11 12 13 nology, the results (i.e., l21opt ¼ l22opt ¼ l23opt ¼ 0) indicate that the optimized electricity generation targets would be 6.5
103 GWh in period 1, 7.2
103 GWh in period 2 and 8.0
103 GWh in period 3, respectively. For the petroleum-fired conversion technology, the results (i.e., l31opt ¼ l32opt ¼ l33opt ¼ 0) indicate that the optimized electricity generation targets would be 5.5
103, 9.6
103 and 14.8
103 GWh during periods 1e3. This is mainly because the manager would have to promise lower-bound targets in order to satisfy the CO2-mitigation requirement. In comparison, for the wind power the results (i.e., l41opt ¼ l42opt ¼ l43opt ¼ 1) indicate that the optimized electricity generation target would be 2.0
103, 3.5
103 and 5.0
103 GWh. This is indicated that renewable recourses would be encouraged over the planning horizon. Fig. 5 shows the imported electricity over the planning horizon. It would be necessary to import electricity from other regions over the planning horizon, which mainly because both the high electricity demand of this region and the total amount of carbon dioxide emitted would be confined with a certain level during the planning periods. Imported electricity would increase from [15.9, 40.9]
103 TJ in period 1 to [88.3, 163.3]
103 TJ in period 3 under low electricity demand-level. Besides, the results also indicated that import electricity would be raised with the increasing electricity demands. Imported electricity would play an important role, particularly when the demand-level is high. Fig. 6 shows the trading amount of CO2-emission permit from IRNO model. Coal-fired power plant would purchase partial CO2emission permit from other power plants (e.g., gas-fired, petroleumfired and wind power) over the planning horizon. For example, the amounts of CO2 emissions purchased from other power plants would be [4.84, 4.91]
106 tonnes in period 1, [4.92, 4.95]
106 tonnes in period 2, and [6.14, 7.04]
106 tonnes in period 3. Seller would be the role for gas-fired power, petroleum-fired power and wind power plants. For example, for the gas-fired power plant, the amount of CO2-emission permit sold to other power plants

4 2 0 -2 -4 -6 -8 Period 1

Period 2

Period 3

Period 1

Lower bound

Coal

Period 2

Period 3

Upper bound

Gas

Petroleum

Wind

Fig. 6. Solutions of trading amount of CO2-emission permit from IRNO model.

C. Chen et al. / Renewable Energy 47 (2012) 55e66

objective is to minimize the sum of the first-stage and expected second-stage costs, based on an assumption that the decision maker is risk neutral [17]. Thus, we have:

" Min f



¼

þ

pth


NE t




X4 th

þ

t¼1 h¼1

þ

i¼1 t ¼1 h¼1

þ

T X

H X

i¼1 t¼1 h¼1

þ

PV it




i¼1

I X T X H X

I X

#

I X

0

Wit

nc X T X jc ¼ 1 t ¼ 1

 pth
PP it
Yith

h

pth A it

ns I X T X H X X i ¼ 1 t ¼ 1 h ¼ 1 jc ¼ 1




 Qith

þ

B it




 Zith

  Eith

i

 pth
CC ijc t
DECijc t

ci; t; h

(7i)

(5) Regional carbon trading constraint:

    NS1
X1 t þ NS2
X2t þ NS3
X3t T X H X

  Nit  Zith  Mit Qith ;

63

ð7aÞ

 hjc DEC itjc h  Eith ;

nc X T X jc ¼ 1 t ¼ 1

I X T X i¼1 t¼1

ci; t; h

 hjc t DEC ijc th  Sit ;

 S it  ð1  mÞKt ;

(7j)

ci; jc ; t; h

(7k)

ci; t

(7l)

subject to

Eit ¼ lit Wit

(1) Constraints for mass balance of fossil fuels:

I X T  X i¼1 t¼1

     Wit þ lit DWit þ Yith FE it  X1t þ X2t þ X3t ;

ci; t

(7m)

(6) Carbon mitigation constraints:

facilities

for

the

disposal

capacity

(7b) DEC ijc th 

(2) Constraints for electricity supply and demand balance:

I X T X H  X i¼1 t ¼1 h¼1

I  X   s Wit þ lit DWit þ Yith þ X4 th þ i  Dth ; ci; t; h i¼1

(7c) (3) Constraints for electricity peak load demand:



     Wit þ lit DWit þ Yith  CF it RCit þ Zith ;

I X T  X i¼1 t¼1

I X T  X i¼1 t¼1

  RCit þ Zith  Ut ;

  RCit þ Zith  US t ;

    Wit þ lDWit ; 0  Yith

ci; t; h

ct; h

(7d)

(7e)

ct; h

(7f)

ci; t; h

(7g)

(4) Constraints for capacity expansion of electricity generation facilities:

 Qith



¼ 1; ¼ 0;

if capacity expansion of is undertaken if otherwise

ci; t; h (7h)

nc X jc ¼ 1

MC ijc t ;

ci; t; h

(7n)

The detailed nomenclatures for the variables and parameters are same with model (6). Tables 3 and 4 provide solutions from the ITSP and IRNO models. Energy supply amounts and optimized electricity targets from the ITSP model were both higher than those from the IRNO model, while the import electricity through ITSP was lower than it through IRNO. Solutions of the ITSP also provide two extremes  ¼ $½3:18; 4:39
1012 ), which of the expected system cost (i.e., fopt are both lower than those obtained through IRNO model. This is mainly because the objective of ITSP model is to minimize the sum of the first-stage cost and the second-stage random penalty, without considering the variability of the uncertain recourse cost. The results also reveal that low system cost is also linked with high risk of system instability. Meanwhile, the solution intervals of ITSP model were wider than those of IRNO model, implying that IRNO can generate more stable solutions. In comparison to ITSP, two recourse actions were conducted in the IRNO model to improve the robustness of this model. One recourse action was to penalize the second-stage costs that were above the expected value, which was weighted by parameter r. Another action was adequate considered the high variability and uncertainties of power supply and demand, and capture the risk of shortage electricity amount during the energy system programming process, which was weighted by parameter u. A number of solutions would be obtained from IRNO model through changing parameters r and u. The IRNO encourages managers to make decisions with considering system stability and economy, as well as ensures management policies be made with reasonable consideration of both system cost and risk. Through solving the IRNO model under various r values, different expected deviations can be obtained. Fig. 7 shows the weighted values of the expected deviations for electricity generation process, capability expansion activity and treating excess

C. Chen et al. / Renewable Energy 47 (2012) 55e66

Table 3 Electricity generation target, energy supply amount and optimized target. Electricity generation target (103 GWh) Conversion technology

Period 1

Period 2

Period 3

Coal-fired power Gas-fired power Petroleum-fired power Wind power

[22.5, 50.0] [6.5, 20.0] [5.5, 10.5]

[25.0, 60.0] [7.2, 25.0] [6.0, 15.5]

[27.5, 70.0] [8.0, 30.0] [6.5, 20.5]

[0, 5.0]

[1.0, 5.0]

[2.0, 5.0]

Solutions of energy supply amount and optimized target technology

Coal Gas Petroleum Wind Coal Gas Petroleum Wind Coal Gas Petroleum Wind

Periods

Period Period Period Period Period Period Period Period Period Period Period Period

1 1 1 1 2 2 2 2 3 3 3 3

ITSP (103)

IRNO (103)

Energy supply (TJ)

optimization target (GWh)

Energy supply (TJ)

optimization target (GWh)

[744,1034] [384,491] [209,267] d [849,1182] [438,561] [238,305] d [951,1328] [491,630] [267,343] d

23.92 26 14 4 27.58 30 16 7 31.24 34 21 10

[778,972] [288,360] [173,216] d [856,1071] [317,397] [190,238] d [914,1170] [345,433] [207,259] d

22.5 6.5 5.5 2 25 7.2 6 3.5 27.5 8 6.5 5

CO2 process (i.e., ADV1, ADV2 and ADV3) under different P P P  robustness levels (r). The ADV1 ¼ r Ii ¼ 1 Tt¼ 1 H h ¼ 1 pth V1ith , PI PT PH and ADV3 can be ADV2 ¼ r i ¼ 1 t ¼ 1 h ¼ 1 pth V2 ith P P P   obtained based on r Ii ¼ 1 Tt¼ 1 H h ¼ 1 pth V3ith . Here, V1ith ;

V2 ith ;

50

40 30

20 10

p = 0.5

500

ITSP

IRNO

Period 1 Low Low-medium Medium Medium-high High Very high

3 10 37 37 11 2

[8.1, 33.2] [33.2, 83.0] [83.2, 133.0] [133.2, 183.0] [183.2, 233.0] [233.2, 283.0]

[15.9, 40.9] [40.9, 90.9] [90.9, 140.9] [140.9, 190.9] [190.9, 240.9] [240.9, 290.9]

Period 2 Low Low-medium Medium Medium-high High Very high

3 15 13 42 22 5

[30.0, 54.8] [80.0, 104.9] 130 180 230 [230.0, 329.8]

[39.6, 64.6] [89.65, 114.6] 139.6 189.6 239.6 [239.6, 339.6]

Period 3 Low Low-medium Medium Medium-high High Very high System cost

5 [76.1, 150.9] 7 [151.1, 175.9] 17 [176.1, 200.9] 42 [201.9, 250.9] 15 [251.1, 275.9] 14 [276.1, 375.9] $ [3.18, 4.39]
1012

[88.3, 163.3] [163.3, 188.3] [188.2, 213.3] [213.2, 263.2] [263.3, 288.3] [288.2, 388.3] $ [31.23, 41.47]
1012

1.0

Upper bound

400

300 200 100

0 p = 0.4

p = 0.5

p = 0.7

Lower bound 300 Expected variance ($10 6)

Demand-level Probability (%) Imported electricity (103 TJ)

p = 0.8

ADV2

V3 ith

Table 4 Solutions for imported electricity through ITSP and IRNO models.

p = 0.7

Lower bound

and equal lower than the expected ones, then zero. Thus, they would merely measure the recourse costs that are higher than the expected ones. The results indicate that the ADV1, ADV2 and ADV3 would be raised with r level enforcement. For example, for ADV1, the weighted values would be $[9.93,

Periods

ADV1

p = 0.4

 V2 ith ; and V3ith represent positive deviations of given scenarios’ recourse costs from the expected ones. When recourse costs are

V1 ith ;

60

0

Expected variance ($10 6)

Conversion technology

Expected variance ($10 3 )

64

p = 0.8

1.0

Upper bound

ADV3

250

200 150

100 50

0 0.4

0.5 Lower bound

0.7

0.8

1.0

Upper bound

Fig. 7. Weighted values of the expected deviations.

22.66]
103 when r ¼ 0.4, $[17.77, 47. 16]
103 when r ¼ 0.7 and $[18.84, 49.51]
103 when r ¼ 1. For ADV2, the weighted values would be $[198.26, 236.73]
106 when r ¼ 0.4, $[379.18, 473. 14]
106 when r ¼ 0.7 and $[421.39, 473.47]
106 when r ¼ 1. For ADV3, the weighted values would be $[91.61, 128.80]
106 when r ¼ 0.4, $[175.21, 204.23]
106 when r ¼ 0.7 and $[194.68, 257.57]
106 when r ¼ 1.

C. Chen et al. / Renewable Energy 47 (2012) 55e66

65

become higher under higher u levels. As shown in Fig. 9, higher system cost could guarantee high system reliability. Conversely, lower system cost will lead to higher system-failure risk. In fact, the decision maker is more willing to either choose a more stable and conservative solution with higher cost, or a more variable and advantageous solution with a lower cost. 6. Conclusions

Fig. 8. System costs under different robustness levels.

Fig. 8 presents the net system costs under various r values from the IRNO model (u ¼ 260). The results indicate that the system cost would increase as the robustness levels are enforced. For example, the system cost would be $[12.96, 16.59]
1012 when r ¼ 0.4, $[13.16, 16.63]
1012 when r ¼ 0.5, and $[31.23, 41.47]
1012 when r ¼ 1. A plan with lower robust levels would result in lower system cost, implying that the manager has an optimistic attitude; however, it might be associated with a higher risk levels (e.g., shortage of electricity demand). Conversely, a plan with a higher robust level would better resist from shortage of electricity demand. Thus, decision with a higher robust level would correspond to a lower risk of system failure (i.e., higher system reliability). There is a tradeoff between system cost and system reliability. Fig. 9 shows the tradeoff between the expected system Recourse cost cost and the recourse cost (r ¼ 1). P  2 ½i:e:; value of u Ii ¼ 1 AW it
ðsit Þ  seized the risk of electricity shortage, weighted by parameter u. Here, s it presents the electricity shortage. As shown in Fig. 9, the system cost would slightly increase as the u value raise (e.g., $[31.23, 39.04]
1012 when u ¼ 50, $[31.24, 41.47]
1012 when u ¼ 100, and $[31.23, 41.47]
1012 when u ¼ 260). At the same time, the value of P  2 would decrease considerably with the u Ii ¼ 1 AW it
ðsit Þ enforcement of u (e.g., $[14.1, 20.1]
106 when u ¼ 50, $[9.7, 15.5]
106 when u ¼ 100, and $[1.8, 2.9]
106 when u ¼ 260). This illustrated the risk of power shortage would reduce with the improvement of robustness levels, and the model robustness (closeness to a feasible solution) and system stability would

50

20 16 14

40

12 10 8 30

6

System cost ($1012 )

Recourse cost ($10 6 )

18

An interval-robust nonlinear optimization (IRNO) method has been developed for supporting energy systems planning under uncertainty. This method incorporates interval-parameter programming and robust optimization within a stochastic programming framework. It can deal with uncertainties contain a combination of deterministic, interval and distributional information, and can thus facilitate the reflection for different forms of uncertainties. The developed model has been applied to a case of energy system management under uncertainty. Interval solutions under different robust levels have been generated. In its solution process, based on an interactive algorithm, the IRNO model is transformed into two deterministic submodels, which correspond to the lower and upper bounds of the desired objective function values. In summary, the results obtained are valuable for supporting: (i) adjustment or justification of allocation patterns of regional energy resources and services; (ii) formulation of local policies regarding energy supply, conversion, consumption and CO2 management; (iii) analysis of the effect of robustness level in this study case and indepth analysis of tradeoffs between system cost and decision maker’s satisfaction degree. ITSP can effectively handle uncertainties presented as probabilistic distributions and intervals. However, a potential limitation of the conventional ITSP is that it can only account for the expected second-stage cost without any consideration on the variability of the recourse values [18]. As a result, the ITSP may become infeasible when the decision maker is risk averse under high-variability conditions. Compared with the ITSP model, the IRNO model would be more robustness through two recourse actions. One of recourse actions was launched to capture the notion of risk in stochastic programming. The other recourse action was seized the risk of shortage electricity amount during the energy system programming process, which successfully emphasizing the safety of energy system under high-variability. The modeling results can help generate desired decision alternatives that will be particularly useful for risk-aversive decision makers in handling high-variability conditions. Moreover, compared with the existing RO methods, the IRNO can incorporate more uncertain information within its modeling framework. In conclusion, the IRNO (i) uses discrete random variables and discrete intervals to reflect uncertainty properties, (ii) obtains higher system cost than ITSP model, but achieve a more stable generation schemes, (iii) provides opportunities to managers to make decisions based on their own preferences on system stability and economy; this ensures that the management policies and plans be made with reasonable consideration of both system cost and risk. Acknowledgments

4 2 20

0

Recourse cost (Lower)

Recourse cost (Upper)

System (Lower)

System (Upper)

Fig. 9. Recourse and system costs under different u values.

This research was supported by the MOE key Project program (311013), the Program for Innovative Research Team in University (IRT1127), the Major Project Program for the Natural Sciences Foundation (51190095), and National High-tech R&D (863) Program (2012AA091103). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.

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C. Chen et al. / Renewable Energy 47 (2012) 55e66

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