Planning regional-scale electric power systems under uncertainty: A case study of Jing-Jin-Ji region, China

Applied Energy 212 (2018) 834–849

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Planning regional-scale electric power systems under uncertainty: A case study of Jing-Jin-Ji region, China

T

⁎

L. Yua,b, Y.P. Lia,c, , G.H. Huangd, Y.R. Fand, S. Yine a

Department of Environmental Engineering, Xiamen University of Technology, Xiamen 361024, China Sino-Canada Energy and Environmental Research Center, North China Electric Power University, Beijing 102206, China c School of Environment, Beijing Normal University, Beijing 100875, China d Institute for Energy, Environment and Sustainable Communities, University of Regina, Regina, Sask. S4S 7H9, Canada e State Grid Henan Economic Research Institute, Zhengzhou 450052, China b

H I G H L I G H T S

G RA P H I C A L AB S T R A C T

copula-based stochastic fuzzy-cred• Aibility programming method is proposed.

and interactions • Multi-uncertainty among multiple random variables are examined.

Its applicability can be verified in • planning the EPS of Jing-Jin-Ji region. of various scenarios and • Solutions different credibility levels are analyzed.

can provide multiple joint • CSFP-REPS planning strategies for the REPS.

A R T I C L E I N F O

A B S T R A C T

Keywords: Copula Electric power systems Interactions Joint planning Multiple uncertainties Regional-scale

In this study, a copula-based stochastic fuzzy-credibility programming (CSFP) method is developed for planning regional-scale electric power systems (REPS). CSFP cannot only deal with multiple uncertainties presented as random variables, fuzzy sets, interval values as well as their combinations, but also reflect uncertain interactions among multiple random variables owning different probability distributions and having previously unknown correlations. Then, a CSFP-REPS model is formulated for planning the electric power systems (EPS) of the JingJin-Ji region, where multiple scenarios with different joint and individual probabilities as well as different credibility levels are examined. Results reveal that electricity shortage would offset [4.8, 5.2]% and system cost would reduce [3.2, 3.3]% under synergistic effect scheme. Results also disclose that the study region’s future electricity-supply pattern would tend to the transition to renewable energies and the share of renewable energies would increase approximately 10% over the planning horizon. Compared to the conventional stochastic programming, the developed CSFP method can more effectively analyze individual and interactive effects of multiple random variables, so that the loss of uncertain information can be mitigated and the robustness of solution can be enhanced. Moreover, based on the main effect analysis and regression analysis, CSFP-REPS can provide multiple joint planning strategies in a cost- and computation-effective way. Findings are useful for reflecting interactions among multiple random variables and disclosing their joint effects on modeling outputs of REPS planning problems.

⁎

Corresponding author at: Department of Environmental Engineering, Xiamen University of Technology, Xiamen 361024, China. E-mail address: [email protected] (Y.P. Li).

https://doi.org/10.1016/j.apenergy.2017.12.089 Received 18 August 2017; Received in revised form 18 December 2017; Accepted 25 December 2017 0306-2619/ © 2017 Elsevier Ltd. All rights reserved.

Applied Energy 212 (2018) 834–849

L. Yu et al.

1. Introduction

REPS. Besides, joint-probabilistic programming (JPP) for reflecting interactive relationships among a set of probabilistic constraints are based on assumptions that all of random variables employed to probabilistic constraints are normally and independently distributed [30]. In practical REPS planning problems, multiple random variables may present different probability distributions and the associated correlation may be previously unknown [31]. In detail, one region may contain several individual EPS with each at an urban-scale; each individual EPS has its unique characteristics in population, resources, economy, environment and geographic location. The previous studies are limited to these assumptions, and may be not suitable for the cases where random variables present different probability distributions with unknown correlations. Copula approach for modeling multivariate joint distributions was proposed by Sklar in 1959, which can characterize multivariate joint distributions by its respective marginal distributions and bind them together independent of the types of individual marginal distributions through using copula functions [32,33]. Therefore, this study aims to develop a copula-based stochastic fuzzy-credibility programming (CSFP) method for the joint planning of regional-scale electric power systems (REPS). CSFP will integrate copula-based stochastic programming (CSP), fuzzy-credibility constrained programming (FCP), and interval-parameter programming (IPP) within a mixed-integer linear programming (MILP) framework. Then, a CSFPREPS model is formulated to planning the REPS of Beijing, Tianjin and Hebei province (abbreviated as “Jing-Jin-Ji” region). In CSFP-EPS, a series of scenarios with different joint and individual probabilities as well as different credibility levels will be considered. Results will provide decision supports for: (a) reflecting uncertain interactions among multiple random variables and disclosing their impacts on system outputs; (b) achieving tradeoffs among system violation risk, environmental requirement and system cost; (c) identifying joint planning strategies of Jing-Jin-Ji region in a cost- and computation-effective way.

With the growing blackouts frequency and rapid socioeconomic development, electricity demand-supply security is becoming particularly urgent for planning regional-scale electric power systems (REPS). Energy supply, resource conservation, and environmental protection continue to be the major concerns for many countries throughout the world [1,2]. Although China has made great efforts to improve energy utilization efficiencies, upgrade power transmission infrastructures, expand power conversion facilities, as well as develop renewable energies, two major hurdles still restrict effectively planning electric power systems (EPS) [3–5]. The electricity-demand target of REPS is regulated by each individual EPS instead of being addressed as a group. When a new target electricity-demand is added, a magnitude of extra effort (e.g., investment, human labor, and land use) is required. However, the possible adverse impacts (meeting one electricity-demand while sacrificing the other’s energy resources) among various individual EPS schemes can lead to a low efficiency of EPS planning resource input [6]. Besides, for each individual EPS, the main solutions to satisfy the tremendous electricity shortage are capital-investment based measures (e.g., capacity expansion and power import). Although the measures can bring positive effects on one individual EPS, they may impose negative effects on the other EPS simultaneously [7,8]. Therefore, to facilitate the achievement of multiple electricity-demand targets in a cost-effective way, joint planning for REPS is desired [9]. Joint planning for REPS can effectively integrate electricity demands of multiple EPS into one general framework, which not only brings conjunct satisfaction effects of REPS but also produces individual satisfaction effects for each EPS simultaneously [10]. Previously, a number of systems analysis approaches for planning REPS were advanced [11–16]. For example, Devlin and Talbot [11] used dynamic programming for planning the renewable energy resources in the EPS of Ireland, where the changed transport costs in each time horizon were obtained. Sun et al. [15] utilized a static deterministic linear model for planning the China’s REPS development, in which real energy use patterns among interregional energy spillover effects were examined. Generally, the above studies for planning REPS are mainly based on deterministic analysis methods through converting into a linear programming (LP) model; they also narrow themselves in planning the REPS with conjunct targets for each individual EPS, which are incapable of reflecting the complex interactions among REPS [17]. For a realworld REPS, the unique energy, environmental and economic features of individual EPS could influence each other, which enforce the REPS become more complicated [18,19]. Besides, some system parameters are not available as deterministic values owing to the incompleteness or impreciseness of observed information. Previously, a number of inexact optimization methods were developed for planning the REPS in response to such complexities and uncertainties [20–24]. For example, Narayan and Ponnambalam [23] developed a risk-averse stochastic programming (SP) model for planning the EPS, where random parameters of uncertain nature resources, imprecise renewable energy generation and cost as well as dynamic demand needs were tackled. Lotfi and Ghaderi [25] proposed a novel possibilistic price-based mixed integer linear programming approach for planning the mid-term electric power systems, in which uncertainties in the objective function and constraints were solved based on possibilistic distributions. Wang et al. [26] proposed a multistage joint-probabilistic chance-constrained fractional programming approach for planning the provincial EPS of Saskatchewan (Canada), where joint probabilities existed in carbon emissions were reflected. Generally, stochastic programming (SP) has advantage in dealing with random variables with known probability distributions; fuzzy programming (FP) is effective for representing the possible degree of event occurrence for imprecise data described by fuzzy possibility distributions [27–29]. However, few of them are focused on analyzing interactive relationships among multiple random parameters in the

2. Development of CSFP-REPS model 2.1. Copula-based stochastic fuzzy-credibility programming A decision maker is responsible for allocating electricity-supply patterns, capacity expansions and pollutant mitigation with a minimum system cost over a long-term planning horizon [34]. In the REPS, the total electricity demand may vary from each individual EPS in each period, and the relationship of electricity demands among individual EPS may be previously unknown. These can be presented as random variables, and the interactive relationship among electricity demands in individual EPS can be reflected through copulas [35–37]. Copula functions connect univariate marginal distribution functions with the multivariate probability distribution:

F (x1,x2,…,x n ) = C (FX1 (x1),FX2 (x2),…,FXn (x n ))

(1)

where FX1 (x1),FX2 (x2),…,FXn (x n ) are marginal distributions of electricity demands (X1,X2 ,…,Xn ) . If these marginal distributions are continuous, a single copula function C exists, which can be written as [38]:

C (u1,u2,…,un ) = F (F X−11 (u1),F X−21 (u2),…,F X−n1 (un ))

(2)

Based on Chen et al. [33], Simic and Dabic-Ostojic, [39], a general copula-based stochastic programming (CSP) model can be formulated as: I

Min E =

K

T

∑∑∑

di,k,t x i,k,t

i=1 k=1 t=1

(3a)

subject to: K

Pr

p ⎧ ∑ ai,k,t xi,k,t ⩽ (bi,ti,t )⎫⎬ ⩾ 1−pi,t , ∀ i,t ⎨ k=1 ⎩ ⎭

[constraints of individual electricity demands] 835

(3b)

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C (1−p1,t ,1−p2,t ,…,1−pi,t ) = 1−pt , ∀ t

K

(3c)

Cr

[constraint of total electricity demand]

ai,k,t x i,k,t ⩽ ei,t , ∀ i,t

k=1

(3d)

(5c)

IFCP can deal with uncertainties existed in the credibility constraints associated with pollutant mitigation, and it can also handle economic data through using interval values even having unknown probability distributions and unknown possibility distributions. However, IFCP may encounter difficulties in handling multiple uncertainties existed in REPS owing to spatial and temporal variations of obtained data and incompleteness or impreciseness of observed information [52]. One effective approach for dealing with such complex uncertainties can be developed through integrating IFCP into CSP. Then a copula-based stochastic fuzzy-credibility programming (CSFP) model can be formulated as follows:

[constraints of pollutant emissions]

x i,k,t ⩾ 0 , ∀ i,k ,t

(5b)

x i±,k,t ⩾ 0 , ∀ i,k ,t

K

∑

⎧ ∑ a ± x ± ⩽ ei,̃ t ⎫⎬ ⩾ λ± , ∀ i,t ⎨ k = 1 i,k,t i,k,t ⎩ ⎭

(3e)

[nonnegative constraints] where E is the expected system cost over the planning horizon ($); ai,k,t are sets of parameters in the constraints (i.e. energy consumption rate, electricity-conversion efficiency, pollutant-emission coefficient); di,k,t denote the sets of parameters in the objective function (i.e. purchasing electricity resource cost, importing electricity cost, service time of electricity-conversion technology, fixed and variable costs for electricity-generation and capacity-expansion), Pr is a probability measure first proposed by Charnes et al. [40] and has been widely used in many p research areas [34,37,41–43]; (bi,ti,t ) are electricity demands with unknown distribution information; ei,t are allowed pollutant-emission amounts; x i,k,t are decision variables (i.e. electricity-generation amount, imported electricity amount, exported electricity amount); i is the name of individual EPS (i = 1, 2 …, I); k is the name of electricity-conversion technology (k = 1, 2, … K); t is the time period (t = 1, 2, …, T); C is the best copula determined previously; pi,t are probabilistic violation levels of electricity demands for each individual EPS i in period t; 1−pt is a prescribed joint probability level of the total electricity demand in REPS. Generally, the CSP model can deal with uncertainties expressed as random variables (i.e. electricity demands) in the right-hand sides of constraints (i.e. constraints of electricity demands) with unknown probability distributions and correlations. Nevertheless, in practical REPS planning problems, series of environmental objectives related to pollutant mitigation can be existed in constraints’ right-hand sides with possibility distributions. Such uncertainties could result in complexities in REPS planning management associated with the satisfaction degree of system performance. Fuzzy-credibility constrained programming (FCP) is effective for dealing with independent uncertainties in the constraints’ right-hand sides (i.e. constraints of pollutant emissions) [44]. In FCP, relationships of satisfaction degree and system-failure risk employed to the credibility constraints (i.e. constraints of pollutant emissions) can be effectively reflected based on the fuzzy sets. According to Zhang and Huang [45], constraint (3d) can be transformed as:

I

Min E ± =

K

T

∑∑∑

di±,k,t x i±,k,t

(6a)

i=1 k=1 t=1

subject to: K

Pr

p ⎧ ∑ a ± x ± ⩽ bi,ti,t ⎨ k = 1 i,k,t i,k,t ⎩

( )±⎫⎬ ⩾ 1−pi,t ,

∀ i,t

⎭

C (1−p1,t ,1−p2,t ,…,1−pi,t ) = 1−pt , ∀ t

(6b) (6c)

K

Cr

⎧ ∑ a ± x ± ⩽ ei,̃ t ⎫⎬ ⩾ λ± , ∀ i,t ⎨ k = 1 i,k,t i,k,t ⎩ ⎭

x i±,k,t ⩾ 0 , ∀ i,k ,t

(6d) (6e)

where ‘−’ and ‘+’ represent lower and upper bounds of the interval values, respectively. The detailed solution algorithm for solving the CSFP model is depicted in Appendix A to this paper. CSFP has advantages of: (a) dealing with multiple uncertainties presented as random variables, fuzzy sets and interval values as well as their combinations; (b) reflecting uncertain interactions among multiple random variables through using copula functions even having different probability distributions and previously unknown correlations; (c) facilitating the achievement of conjunct targets in a cost-effective way. However, in CSFP, marginal and joint probability distributions of random variables as well as fuzzy membership functions of system parameters need to be determined firstly, which may lead to increased complexities, improved data requirement and reduced computation efficiency.

K

Cr

⎧ ∑ ai,k,t xi,k,t ⩽ ei,̃ t ⎫⎬ ⩾ λ , ∀ i,t ⎨ k=1 ⎩ ⎭

2.2. CSFP-REPS modeling formulation

(4)

In this study, the incomplete economic parameters expressed as discrete intervals can be tackled by IPP. Electricity demands varied by a number of factors such as population growth, technology innovation, economic development, weather change as well as the general randomness in individual usage can be handled by CSP. The imprecise allowed pollutant emissions due to subjective experience of experts and stakeholders can be solved by FCP. Based on the proposed CSFP method, a CSFP-REPS model is formulated for planning the REPS. The objective of CSFP-REPS aims at obtaining power-generation pattern and facility-capacity expansion scheme with a minimized system cost within the pollutant discharge requirement and the given joint constraintviolation level. The system cost includes cost for energy resource purchase, cost for electricity import, cost for electricity generation, capital investment for capacity expansion, cost for electricity transmission, and cost for pollutant treatment. The detailed objective function is:

where ‘∼’ is the fuzzy sets; ei,̃ t are allowed pollutant-emission amounts; Cr is the credibility measure first proposed by Liu [46] and has been widely used in many research area [47,48]; λ is a significant credibility level which should be greater than 0.5 [49]. In REPS management problems, economic data are often expressed as discrete intervals with known lower and upper bounds owing to many factors such as socio-economic, technical, legislational, institutional and political aspects. The IPP technique is capable of handling the uncertainties expressed as intervals without probability distributions and fuzzy membership functions [50,51]. Through introducing IPP into FCP, an interval-fuzzy credibility-constrained programming (IFCP) model can be formulated as follows: I

Min E ± =

K

T

∑∑∑ i=1 k=1 t=1

di±,k,t x i±,k,t

(5a)

Min E ± = (1) + (2) + (3) + (4) + (5) + (6)

subject to: 836

(7a)

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EGAi±,k = 2,t × FEi±,k = 2,t ⩽ NYLi±,j = 3,t

(1) Purchase cost for energy resources: In real-world REPS planning problems, energy purchase cost in EPS consists of purchase cost of local and imported energy in response to the conversion activities, including the purchase price of energy resources, workers’ wage, truck rental fees and the cost for energy transport losses. 3

3

i=1

(2) Capacity limitation constraint of electricity-conversion facilities: The study system involves several electricity-conversion technologies to meet the overall energy demand from end-users. This constraint guarantees that each capacity of electricity-conversion technology is greater than the amount of output.

5

∑∑∑

PECi±,j,t × NYLi±,j,t (7b)

j=1 t=1

EGAi±,k,t ⩽ RCi±,k,t × STi±,k,t (2) Cost for electricity import: When the existing and expected expansion capacities of electricity-conversion technologies are insufficient, electricity of importing from adjacent power grids become inevitable and indispensable. The cost for electricity import can be determined by the amounts of imported electricity and unit price. 3

5

∑∑

3

PEJi±,t × PEi±,t − ∑

i=1 t=1

OEJi±,t × OEi±,t

C (1−p1 ,1−p2 ,1−p3 ) = 1−p

(7c)

i=1 t=1

(3) Fixed and variable costs for electricity generation: The electricity-generation cost always covers a wide range which including unit start-up cost, workers’ wage, equipment-maintenance cost and taxation expense. 3

8

5

∑∑∑

3

FGCi±,k,t

×

(RCi±,k,t = 0

+

ECi±,k,t

×

YCi±,k,t )

i=1 k=1 t=1

+

8

Pr

VGCi±,k,t (7d)

Pr

(4) Fixed and variable costs for capacity expansion: When the existing capacities of electricity-conversion technologies are insufficient, expansions of them become necessary. The investment for expanding electricity-conversion technologies contains finance investment, labor fee, equipment maintenance and operation cost, as well as taxation expense. 3

8

Pr

(FECi±,k,t + VECi±,k,t ,) × ECi±,k,t × YCi±,k,t

(EGAi±,k,t + ECi±,k,t × YCi±,k,t × STi±,k,t ) × CUi±,k,t

i=1 k=1 t=1

(10c)

⎫ ⩾ 1−p3 ⎬ ⎭

(10d)

(4) Constraints of pollutant emissions: These constraints are used for ensuring that the pollutant-emission amounts should be satisfied by the pollutant-emission permits. 8

5

∑∑∑

⎫ ⩾ 1−p2 ⎬ ⎭

8 EGAi±= 3,k,t × (1−ZLi±= 3,k,t ) × TEi±= 3,k,t ⎞ ⎧ ± ± ∑ ⎛⎜ ⎟ + PEi = 3,t −OEi = 3,t ⩾ ⎨ k = 1 + ECi±= 3,k,t × YCi±= 3,k,t × STi±= 3,k,t ⎝ ⎠ ⎩

Demandi±= 3,t

(7e)

(5) Cost for electric power transmission losses: High-voltage power transmission is adopted to reduce resistance losses in long distance electric power transmission. However, electric power transmission distances often reach to thousands of kilometers, which will lead to a large amount of electricity losses; these losses cannot be ignored [53]. 8

(10b)

8 EGAi±= 2,k,t × (1−ZLi±= 2,k,t ) × TEi±= 2,k,t ⎞ ⎧ ± ± ∑ ⎛⎜ ⎟ + PEi = 2,t −OEi = 2,t ⩾ ⎨ k = 1 + ECi±= 2,k,t × YCi±= 2,k,t × STi±= 2,k,t ⎝ ⎠ ⎩

Demandi±= 2,t

5

i=1 k=1 t=1

3

⎫ ⩾ 1−p1 ⎬ ⎭

i=1 k=1 t=1

× (EGAi±,k,t + ECi±,k,t × YCi±,k,t × STi±,k,t )

∑∑∑

(10a)

8 EGAi±= 1,k,t × (1−ZLi±= 1,k,t ) × TEi±= 1,k,t ⎞ ⎧ ± ± ∑ ⎛⎜ ⎟ + PEi = 1,t −OEi = 1,t ⩾ ⎨ k = 1 + ECi±= 1,k,t × YCi±= 1,k,t × STi±= 1,k,t ⎝ ⎠ ⎩

Demandi±= 1,t

5

∑∑∑

(9)

(3) Constraints of electricity demands: These constraints are established to ensure that the total electricity generated from the existing and future expanding capacities, and purchased from other power grids should not be less than the amount of electricity demands. The security of electricity supply is a critical issue for the decision makers.

5

∑

(8c)

Cr (7f)

∼ ⎧ ∑ (EGAi±,k,t + ECi±,k,t × YCi±,k,t × STi±,k,t ) × AMRi±,k,t,q ⩽ ESi,t,q⎫⎬ ⩾ λ± ⎨ k=1 ⎩ ⎭ (11a)

8

(6) Cost for pollutant mitigation: The cost of pollutant mitigation is calculated based on the various process activities, the associated emission rates and the unit cost of environmental facilities, as well as relevant financial subsidies. 3

8

5

∑

(EGAi±,k,t + ECi±,k,t × YCi±,k,t × STi±,k,t ) × AMRi±,k,t ,q ⩽

k=1

ESi,t ,q + (1−2λ±)(ESi,t ,q−ES i,t ,q) ,∀ i,t ,q

(11b)

3

∑∑∑∑

(EGAi±,k,t + ECi±,k,t × YCi±,k,t × STi±,k,t ) × (CPi±,t ,q

(5) Constraints for capacity expansion: These constraints are established to ensure that the capacity will satisfy the demand of electricity from a long-term planning point of view. The related optimization analysis will require the use of integer variables to indicate whether a particular facility development or expansion option needs to be undertaken.

i=1 k=1 t=1 q=1

+ CEi±,t ,q/STi±,k,t −SUi±,k,t )

(7g)

Constraints mainly consist of resources availability, electricity demand-supply balance (i.e. individual and joint violation risk), capacity expansion, and pollutant treatment. They can be depicted as follows:

= 1; if capacity expansion is undertaken ⎫ YCi±,k,t ⎧ ⎨ ⎬ ⎩= 0; if otherwise ⎭

(12a)

(1) Resource availability constraints: These constraints are established to ensure that the amount of energy utilization must be not less than the total available energy amounts.

RCi±,k,t = 0

(12b)

NYLi±,j,t ⩽ ARi±,j,t

(8a)

∑∑∑

EGAi±,k = 1,t × FEi±,k = 1,t ⩽ NYLi±,j = 1,t

(8b)

CapT ±

2

+

8

ECi±,k,t

×

YCi±,k,t

⩽

CapUk±,t

5

(RCi±,k,t = 0 + ECi±,k,t × YCi±,k,t ) ⩽

i=1 k=1 t=1

837

(12c)

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Fig. 1. The study area.

Chengde

Zhangjiakou Qinhuangdao BJ 10.7 GW

Tangshan

Langfang HB 60.3 GW

Baoding

TJ 15.0 GW

Cangzhou Shijiazhuang Hengshui

Xingtai Legend Handan

BJ: Beijing TJ: Tianjin HB: Hebei

Coal Gas Hydro Wind Solar Biomass Waste Pumped storage

Step 5: Repeat Steps (3) and (4) at every (p,p1 ,p2 ,p3 ) and λ level; then we can obtain all the solution results; Step 6: Analyze the calculation results and provide useful bases for decision makers.

(6) Nonnegative constraints: This constraint assures that only positive electricity-conversion activities are considered in the solution, eliminating infeasibilities while calculating the solution.

EGAi±,k,t ,NYLi±,j,t ,OEi±,t ,PEi±,t ⩾ 0 ; ∀ i,k ,j,t

(13)

The specific nomenclatures for variables and parameters are provided in Appendix B. All of the decision variables in the CSFP-REPS model are considered as interval values (e.g., NYLi±,j,t , PEi±,t , OEi±,t and EGAi±,k,t ), and the binary variable (YCi±,k,t ) is introduced to indicate whether a particular facility development or expansion option needs to be undertaken. The detailed solution process of handling the CSFPREPS model can be summarized as follows:

3. Application 3.1. Overview of the study system The CSFP-REPS model is applied to the REPS of Jing-Jin-Ji region. Jing-Jin-Ji region is located in the North China Plain, and it covers an administrative area of 216,000 km2 and a population of 111.4 million in 2015 (Fig. 1). The region’s gross domestic product (GDP) has increased to 6.9 × 1012 RMB¥ in 2016, occupying 10.1% of total national GDP. Accompanying with rapid urbanization processes, a series of energy crises and environmental issues have been emerged. For example, the gap of electricity between local demand and domestic supply reached 41.0 × 103 GW h of Beijing, 19.8 × 103 GW h of Tianjin and 87.4 × 103 GW h of Hebei, respectively. Besides, according to the National Ambient Air Quality Standard (NAAQS), most of the air

Step 1: Formulate the CSFP-REPS model; Step 2: Convert the CSFP-REPS model into two deterministic submodels, as detailed in the “Appendix A”; Step 3: Solve the each CSFP-REPS submodel to obtain a global optimal solution under a λ level; Step 4: Solve the each CSFP-REPS submodel (under the same λ level) to obtain a global optimal solution under a (p,p1 ,p2 ,p3 ) level; 838

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Technical aspects

Environmental factors

Economic elements

Social factors

Energy supply

Power conversion

Coal

Coal-fired power

Gas

Gas-fired power

Beijing power grid

Wind

Wind power

Tianjin power grid

Hydro

Hydro power

Hebei power grid

Biomass

Biomass power

Shaanxi power grid

Waste

Waste power

Shandong power grid

Photovoltaic

Photovoltaic power

Pumped storage

Pu-storage power

Power transimission

Policy fluctuations

Power utilization Primary industry Industry Residential Construction

Inner Mongolia power grid

Tertiary industry

Uncertainties

Uncertainties Imprecise economic data Incomplete technical data Various pollutant types Various scenario paterns Different individual EPS

Uncertain electricity demands Dynamic capacity variations Fluctuating forecast values Interactive random variables Changed air-pollutant emissions

EPS of Jing-Jin-Ji region

Related stochastic variables

Fuzzy-credibility constraints

Interval-parameters

Copula- based stochastic programming (CSP)

Fuzzy-credibility programming (FCP)

Interval-parameter programming (IPP)

Lower-bound submodel

Electricity supply

Copula-based stochastic fuzzy-credibility programming (CSFP) method

Capacity expansion

Power import

Power export

Pollutant reduction

Upper-bound submodel

System cost

Generation of decision alternatives Fig. 2. Framework of study system.

The REPS is considered as a complex interrelated network connected by several sectors, as shown in Fig. 2. For example, eight power-conversion technologies were considered for generating electricity. The powerconversion technologies involve non-renewable energy (i.e. coal and gas) and renewable energy (i.e. wind, hydro, biomass, waste, photovoltaic and pumped-storage). Six domestic or adjacent power grids (i.e. Beijing, Tianjin, Hebei, Shaanxi, Shandong and Inner Mongolia) were used for exporting electricity to satisfy the electricity demand of REPS. Besides, various economic and technical parameters were considered by the decision makers corresponding to the complex processes (i.e. energy supply, power conversion, transmission and utilization as well as pollutant reduction). These uncertainties couldnot only intensify the related optimization processes, but also affect the generated decision schemes.

pollutants (e.g., SO2, NOx and PM10) in Jing-Jin-Ji region were not satisfied with the requirement of NAAQS. Therefore, effective planning of Jing-Jin-Ji region is of great importance to deal with these problems. In the Jing-Jin-Ji region, millions of dollars has been invested to upgrade electricity distribution and transmission infrastructures. According to the 13th Five-year-plan (i.e. 2016–2020) about Jing-Jin-Ji region development, by the year of 2020, Beijing will construct six ultra-high voltage (UHV) transmission lines (i.e. Beijing-Shunyi, Beijing-Tongzhou, Beijing-New Aerotropolis, Fangshan-Nancai, Zhangnan-Changping and Mentougou-Weixian). More than 2.7 GW of power-supply capacity will be introduced to Beijing, and 3.0 GW of power-supply capacity will be supplied to Tianjin. For Tianjin, 5.0 GW of power-supply capacity will be added to Tianjin and the share of imported electricity will be reached more than 30%. For Hebei, it protects the whole external power-supply security and bears approximately 70% of electricity shortage for meeting summer peak in Beijing. 839

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(a)

(b) Joint cumulative distribution function

Joint cumulative distribution function

0.9 0.8 1 0.7

0.8

0.6

0.6

0.5

0.4

0.4

0.2

0.3

0 7 2

Tianjin

-3

-4

0

-2

6

4

2

0.8 1

0.7

0.8 0.6 0.6 0.5 0.4 0.4 0.2 0.3 0

0.2

0.2

7 7

2

0.1

Hebei

Beijing

-3

2 -8

-3

12 0.1

Tianjin

(c)

Joint cumulative distribution function

0.8 1

0.7

0.8

0.6

0.6

0.5

0.4 0.4 0.2 0.3 0 0.2

7 2 -3

Hebei

-8

-4

-2

0

2

4

6 0.1

Beijing

Fig. 3. Joint cumulative distribution functions for annual electricity consumption growth rates of Beijing, Tianjin and Hebei using Gumbel-Hougaard Copula.

3.2. Data collection and scenario design

Table 1 Scenario design. Abbreviation

The planning horizon is five years. The relevant cost parameters were collected from Beijing Statistical Yearbook, Tianjin Statistical Yearbook and Hebei Economic Yearbook; technical data and other parameters were obtained from government official reports, 13th Fiveyear-plan on “Jing-Jin-Ji region” development and literature survey; fuzzy member functions, marginal and joint probability distribution functions (PDF) were based on the experts through survey questionnaires, expert consultations or derived from published papers [54–56]. For example, the historical data of electricity consumptions were used for determining the marginal and joint cumulative distribution functions (CDF) and could be obtained from Beijing Statistical Yearbook, Tianjin Statistical Yearbook and Hebei Economic Yearbook, respectively. Kolmogorov–Smirnov tests were used for evaluating the generated probability distributions if they were best fitted; Pearson’s linear correlation tests were used for confirming the random variables if they were mutually correlated [33]. The Nash–Sutcliffe efficiencies (NSE) for joint cumulative probabilities obtained through using the Gumbel-Hougaard Copula were 0.84, 0.79 and 0.81, respectively. Results indicated good agreements between observed and modeled probabilities, especially for probabilities exceeding 0.5 [57–59]. The Gumbel-Hougaard Copula was suitable for representing the joint distribution of electricity demands among Beijing, Tianjin and Hebei province. The joint CDF for annual electricity consumption growth rates of Jing-Jin-Ji region were respectively obtained through using GumbelHougaard Copula (Fig. 3). In this study, a number of scenarios (i.e. H1-H9; B1-B9; T1-T9) with different joint and individual constraint-violation levels (p,p1 ,p2 ,p3 ) were analyzed. The amounts of continuous variables and binary variables for the study system were 52,650 and 32,400, respectively. The optimal solutions corresponding to various groups of individual constraint-violation levels and credibility levels could be obtained through

Joint and individual constraint-violation levels

(p,p1 ,p2 ,p3 ) Known individual constraint-violations levels of Beijing and Tianjin, unknown Hebei H1 (0.100, 0.020, 0.020, 0.098) H2 (0.100, 0.020, 0.050, 0.089) H3 (0.100, 0.020, 0.070, 0.076) H4 (0.100, 0.050, 0.020, 0.089) H5 (0.100, 0.050, 0.050, 0.080) H6 (0.100, 0.050, 0.070, 0.064) H7 (0.100, 0.070, 0.020, 0.065) H8 (0.100, 0.070, 0.050, 0.064) H9 (0.100, 0.070, 0.070, 0.039) Known individual constraint-violations levels of Tianjin and Hebei, unknown Beijing B1 (0.100, 0.098, 0.020, 0.020) B2 (0.100, 0.089, 0.020, 0.050) B3 (0.100, 0.075, 0.020, 0.070) B4 (0.100, 0.089, 0.050, 0.020) B5 (0.100, 0.081, 0.050, 0.050) B6 (0.100, 0.066, 0.050, 0.070) B7 (0.100, 0.075, 0.070, 0.020) B8 (0.100, 0.066, 0.070, 0.050) B9 (0.100, 0.050, 0.070, 0.070) Known individual constraint-violations levels of Beijing and Hebei, unknown Tianjin T1 (0.100, 0.020, 0.098, 0.020) T2 (0.100, 0.020, 0.089, 0.050) T3 (0.100, 0.020, 0.075, 0.070) T4 (0.100, 0.050, 0.089, 0.020) T5 (0.100, 0.050, 0.079, 0.050) T6 (0.100, 0.050, 0.064.0.070 T7 (0.100, 0.070, 0.075, 0.020) T8 (0.100, 0.070, 0.064, 0.050) T9 (0.100, 0.070, 0.041, 0.070)

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Table 2 Selected values of joint cumulative distribution and marginal cumulative distributions as well as corresponding values of random variables under scenarios of H1-H9. Scenarios

Hx ,y,z (x ,y,z )

Fx (x )

Gy (y )

Hx ,y (x ,y )

Wz (z )

x (%)

y (%)

z (%)

H1 H2 H3 H4 H5 H6 H7 H8 H9

0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900

0.980 0.980 0.980 0.950 0.950 0.950 0.930 0.930 0.930

0.980 0.950 0.930 0.980 0.950 0.930 0.980 0.950 0.930

0.973 0.947 0.928 0.947 0.932 0.917 0.918 0.917 0.905

0.902 0.911 0.924 0.911 0.921 0.936 0.935 0.936 0.961

6.602 6.602 6.602 5.287 5.287 5.287 4.744 4.744 4.744

10.016 8.022 7.197 10.016 8.022 7.197 10.016 8.022 7.197

7.804 8.107 8.626 8.107 8.489 9.188 9.112 9.188 10.623

taking different (p,p1 ,p2 ,p3 ) and λ levels. The detailed descriptions for these scenarios were illustrated in Table 1, where p denoted the joint constraint-violation level; p1, p2 and p3 denoted the individual constraint-violation levels corresponding to the annual electricity growth rates of Beijing, Tianjin and Hebei, respectively. Tables 2–4 depict the selected values of joint cumulative distribution and marginal cumulative distributions as well as corresponding values of random variables through using Gumbel-Hougaard Copula under different scenarios. Moreover, five credibility levels (i.e. λ = 0.6, 07, 0.8, 0.9 and 1) of fuzzy constraints were considered in association with each of the representative scenario. The fitting simulated random values would be used as the inputs of CSFP to achieve the desired REPS planning results.

current domestic electricity production continues far less than the increasing electricity demand. Importing electricity from adjacent power grids (i.e. Shaanxi, Shandong and Inner Mongolia) or from the region’s domestic electricity production (i.e. Hebei) become indispensable to offset the enormous power shortfalls. Fig. 5 depicts the imported electricity under different scenarios as well as various credibility levels. Summarily, for the each individual EPS, more of imported electricity would be needed to satisfy the electricity demand of Beijing, followed by the Hebei and Tianjin. This is because, after the upgrading and reconstruction of coal-fired boilers, all the coal-fired power plants in Beijing have been transformed to gas-fired power plants. Electricity of outsourcing took the primary role in bearing the tremendous electricity shortage owing to the limited installed gas-fired power capacities and available renewable energy resources. Since Tianjin is a self-supporting based city, the imported electricity in Tianjin would be lower than that in Heber. Besides, different scenarios of individual constraint-violation levels would correspond to different electricity demands, thus leading to changed power imports. In addition, the imported electricity in each individual EPS would be reduced along with the increased credibility level in response to the constrictive pollutant-emission requirement. Fig. 6 presents emissions of multiple pollutants (i.e. SO2, NOx and PM10) under different credibility levels. Summarily, the coal-fired power in Hebei accounted for the dominated role of total pollutant emissions, followed by the gas-fired power in Tianjin and gas-fired power in Beijing, as well as coal-fired power in Tianjin, respectively. This is because lots of installed coal-fired power capacities had been constructed and used for supplying electricity to satisfy the electricity demand of Hebei; although Tianjin had numbers of installed coal-fired power capacities, gas-fired power took the primary role of pollutant emissions owing to the lower pollutant discharge coefficient, constrictive environmental requirement and increasing power import; after the completely reform of coal-fired boilers, gas-fired power became the primary pollutant emitter in Beijing. In addition, for each planning year, the pollutant emissions would be reduced with time in response to the implement of air pollution controls, reform of coal-burning boils, stimulation of renewable energy resources, as well as improvement of energy transmission capacities and efficiencies in the REPS. Moreover, since the credibility levels were employed to the constraints of pollutant-emission requirement, a higher credibility level would correspond

3.3. Result analysis For the REPS of Jing-Jin-Ji, fossil energies occupied more than 80% of the total domestic electricity production while renewable energies accounted for minor share. As shown in Fig. 4, under λ = 0.6, renewable energies of the entire region occupied [13.8, 14.2]% of the total electricity generation; the percent of electricity generated by renewable energies would be [11.1, 12.3]% for Beijing, [4.1, 4.4]% for Tianjin and [16.5, 17.1]% for Hebei, respectively. Although the region is dominated by fossil fuels over the planning horizon, the regional renewable energies would still have slight increment based on series of renewable energy development plans. For instance, over the planning horizon, the share of renewable energies would increase [13.3, 13.5]% for Beijing, [2.6, 2.7]% for Tianjin, [12.0, 12.2]% for Hebei and [10.0, 10.1]% for Jing-Jin-Ji, respectively. Besides, since the fuzzy-credibility level was employed to the constraint of pollutant-emission requirement, different credibility levels would lead to different pollutant-emission requirements and then result in different electricity-supply patterns. For the Jing-Jin-Ji region, the share of fossil-based electricity would reduce with a ratio of 1.4% (λ = 0.6 to 1). Moreover, with the continuing cleaner production of fossil power and increasing investment in upgrading electricity distribution and transmission infrastructures, the region’s future electricity-supply pattern would gradually tend to transit from fossil energies to renewable energies. Although Jing-Jin-Ji region has its effectiveness in supplying large amounts of electricity through the self-supporting generation, the

Table 3 Selected values of joint cumulative distribution and marginal cumulative distributions as well as corresponding values of random variables under scenarios of B1-B9. Scenarios

Hx ,y,z (x ,y,z )

Gy (y )

Wz (z )

Hy,z (y,z )

Fx (x )

x (%)

y (%)

z (%)

B1 B2 B3 B4 B5 B6 B7 B8 B9

0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900

0.980 0.980 0.980 0.950 0.950 0.950 0.930 0.930 0.930

0.980 0.950 0.930 0.980 0.950 0.930 0.980 0.950 0.930

0.974 0.949 0.929 0.949 0.936 0.921 0.929 0.921 0.911

0.903 0.912 0.925 0.912 0.920 0.934 0.925 0.934 0.951

4.166 4.340 4.629 4.340 4.506 4.832 4.629 4.832 5.303

10.016 10.016 10.016 8.022 8.022 8.022 7.197 7.197 7.197

12.378 9.914 8.895 12.378 9.914 8.895 12.378 9.914 8.895

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Table 4 Selected values of joint cumulative distribution and marginal cumulative distributions as well as corresponding values of random variables under scenarios of T1-T9. Scenarios

Hx ,y,z (x ,y,z )

Fx (x )

Wz (z )

Hx ,z (x ,z )

Gy (y )

x (%)

y (%)

z (%)

T1 T2 T3 T4 T5 T6 T7 T8 T9

0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900 0.900

0.980 0.980 0.980 0.950 0.950 0.950 0.930 0.930 0.930

0.980 0.950 0.930 0.980 0.950 0.930 0.980 0.950 0.930

0.973 0.948 0.928 0.948 0.933 0.918 0.928 0.918 0.906

0.903 0.911 0.925 0.911 0.921 0.937 0.925 0.937 0.959

6.602 6.602 6.602 5.287 5.287 5.287 4.744 4.744 4.744

6.320 6.581 7.014 6.581 6.882 7.442 7.014 7.442 8.476

12.378 9.914 8.895 12.378 9.914 8.895 12.378 9.914 8.895

increased credibility levels corresponding to the raised environmental mitigation cost.

to a tightened environment requirement, thus leading to a lower pollutant emission; while a lower credibility level would correspond to a relaxed environment requirement, thus resulting in a higher pollutant emission. Uncertainties related to different scenarios would result in different electricity-supply structures and then generate various system costs. Fig. 7 shows system costs under different scenarios as well as different credibility levels. For example, at the same credibility level (e.g., λ = 0.6), the system cost would be changed from $ 4.64 × 1012 to $ 4.89 × 1012 in scenarios of H1-H9; while in scenarios of B1-B9, the system cost would be varied from $ 4.68 × 1012 to $ 5.21 × 1012; compared to scenarios of H-1H9 and B1-B9, the system cost would be changed from $ 4.69 × 1012 to $ 5.16 × 1012 in scenarios of T1-T9. Summarily, the system cost would be varied under different scenarios associated with different individual constraint-violation levels, and the minimum system cost would be $ 4.64 × 1012 under the scenario H2 while the maximum system would be $ 5.21 × 1012 under B1. This is because the region’s lowest power import would occur in scenario H2 (i.e. 1.53 × 106 GW h), while the price of imported electricity was lower than that in domestic electricity production corresponding to a lower electricity shortage; contrarily, the highest power import of the region would be 1.78 × 106 GW h under scenario B1, which led to the maximum system cost. Besides, different credibility levels would result in varied system costs, and the system cost would be ascended with the (a) Beijing

1.3%

1.4%

4. Discussion The study problem could be formulated as stochastic fuzzy-credibility programming (SFP) if the “copula” technique was not considered. In SFP, each individual constraint-violation level ( p1, p2 and p3 ) was equal to joint probability level ( p ). When λ = 1 and p = .05, the system cost obtained from SFP would be $ 5.06 × 1012; in comparison, the system cost obtained from CSFP would range from $ 4.77 × 1012 to $ 5.40 × 1012. Obviously, the system cost from SFP would be in the range of value from CSFP. It is mainly because the objective of SFP was to minimize the system cost, without the consideration of interactions among different electricity demands of each individual EPS. Besides, SFP couldnot analyze the individual and interactive effects on the system cost, leading to the loss of uncertain information and the reduction of solution robustness. In CSFP, random variables might have any same or different forms of probability distributions and they might be correlated with each other, and the interactive relationships of all the groups of individual constraint-violation levels could be obtained based on the copula analysis.Additionally, three factors ( p1, p2 and p3 ) at three levels (low, medium and high) were considered to visualize influence of the factors on the response and to compare the relative

(b) Tianjin

2.0%

2.0%

(c) Hebei

0.8%

0.1%

0.8%

0.1%

1.9% 2.2%

0.9%

0.6% 0.4% 0.4% 0.4% 0.4% 0.3% 0.8% 0.7%

0.9%

0.0%

1.6%

10.8%

2.3% 2.3%

0.8%

0.1%

0.9%

0.1%

2.3%

2.6%

0.5% 0.7% 8.6%

0.4% 0.8%

0.6% 0.3% 0.7%

8.5%

0.9%

0.6% 0.5% 0.4% 0.4% 0.5% 0.3% 0.9% 0.8%

2.7%

0.0%

12.0%

0.0%

12.0%

2.7%

2.3%

2.5%

2.8%

Hydro

Wind

Photovoltaic

3.6%

4.4%

1.0%

2.5%

3.5%

4.3%

1.7%

3.0%

0.5%

2.4%

2.4%

1.5%

10.9%

0.0%

2.6%

1.9%

3.3%

4.0%

2.3%

2.6%

3.2%

3.9%

1.4% 2.1%

(d) Total

Biomass

Waste

Pumped-storage

Fig. 4. Electricity supply patterns by renewable energies.

842

0.5%

0.6% 0.7% 0.5% 9.4% 9.4%

0.9%

0.7% 0.4% 0.8%

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(a)

(c)

(b)

Imported electricity (106 GWh) Imported electricity (106 GWh)

0 1.2

0.8

0.4

0 1.2 0.8

λ = 0.8

0.4

0 1.2

0.8

λ = 0.9

Imported electricity (106 GWh)

0.4

λ = 0.7

0.4

0 1.2 0.8

λ=1

Imported electricity (106 GWh)

0.8

λ = 0.6

Imported electricity (106 GWh)

1.2

0.4

0 H1

H2

H3

H4

H5

H6

H7

H8

H9

B1

B2

B3

B4

Beijing

B5

B6

Tianjin

B7

B8

B9

T1

T2

T3

T4

T5

T6

T7

T8

T9

Hebei

Fig. 5. Imported electricity under different scenarios.

costs between prediction and actual value under λ = 0.6. High R2 values suggested that the regression curves could be well fitted based on the obtained optimal solutions. Such regression functions would be considered as the inverse of the membership functions of the selected scenarios and the objective-function value. Under this case, the solution for each scenario under any credibility level can be obtained directly through the membership functions instead of solving the CSFP-REPS model again. It would significantly enhance the computation efficiency for the optimization model and thus extend the model’s applicability to various REPS planning problems. Summarily, based on the synergistic effect of Jing-Jin-Ji region, approximately [4.8, 5.2]% of electricity-shortage would be offset in terms of region’s domestic power supply (i.e. Zhangjiakou and Weixian), and [3.2, 3.3]% of system cost would be reduced simultaneously. Besides, based on the main effect analysis, the individual and interactive effects on the system cost would be completely reflected, and the most significant parameter to system cost would be ascribed to the EPS of Hebei owing to its abundant electricity demand. Therefore, multiple joint planning strategies of Jing-Jin-Ji region and capital

magnitude of the effects. Table 5 illustrates the contributions of individual and interactive parameters to system cost. It indicated that p3 was the most significant parameter to system cost, with a largest contribution of 94.8%; while the other parameters accounted for the minor contribution. As shown in Fig. 8, the plot also shows that the response changes remarkably depending on the levels of factor p3 with the steepest line. It indicated that p3 had the largest main effect on system cost; while p1 and p2 had smaller contributions to the system cost because their lines were almost horizontal in the main effects plot.According to Table 5 and Fig. 8, the most pronounced factor on system cost would be ascribed to p3 (the electricity-demand violation level of Hebei) owing to its abundant electricity demand. Results indicate that the system cost of REPS would go ascend with the reduction of joint cumulative constraint-violation level of Beijing and Tianjin (i.e. C (1−p1 ,1−p2 ) ) while go against with the increment of p3 . Referring back to Table 1 and Fig. 7, a function between individual constraintviolation levels ( p1, p2 and p3) of each random value and system cost was generated through statistical regression methods based on the optimal solutions. For instance, Fig. 9 depicts the comparative system

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(a) Lower bound

(b) Upper bound

λ = 0.6

1.6% 0.4% 0.3% 0.1% 9.3%

456.2

67.9%

3.1%

67.2%

17.0% 0.0% 0.2% 1.7% 0.4% 0.3% 8.9%

SO2

16.7 NOx

0.0%

SO2

0.1%

0.4% 8.8% 0.1%

433.9

0.0%

SO2

NOx

527.4

18.0

12.3

0.0%

PM10

SO2

NOx

PM10

0.2%

0.2%

1.7% 0.5%

0.4% 8.6% 0.1% 0.2%

8.5% 406.6

2.3%

66.4%

19.0% SO2

0.4% 0.2% 0.2%

NOx

497.1

2.1% 19.8%

14.9

10.2

0.0%

16.9

11.5

0.0%

PM10

0.2% Beijing Gas-fired Tianjin Waste

PM10

2.5%

18.8%

15.9

1.8% 0.4%

66.9%

NOx

0.2% 66.9%

2.7%

17.8%

18.8

12.8

PM10

0.2% 1.6% 0.5%

10.9

λ=1

2.8% 18.0%

0.2% 67.6%

552.8

0.2%

0.2%

11.4

λ = 0.8

1.6% 0.5% 0.4% 0.1% 9.1%

SO2

NOx

PM10

0.2% Beijing Biomass Hebei Coal-fired

Beijing Waste Hebei Gas-fired

Tianjin Coal-fired Hebei Biomass

Tianjin Gas-fired Hebei Waste

Tianjin Biomass

Fig. 6. Air-pollutant emissions.

REPS of Jing-Jin-Ji region. Several findings can be disclosed as follows: (a) the region’s future electricity-supply pattern would gradually tend to transit from fossil energies to renewable energies and the share of renewable energies would increase approximately 10% over the planning horizon; (b) importing electricity from adjacent power grids would occupy primary role in bearing tremendous electricity shortage owing to the limited installed gas-fired power capacities and available renewable energy resources; (c) reform of coal-burning boils, stimulation of renewable energy resources, constriction of pollutant-emission requirement, as well as improvement of energy transmission capacities and efficiencies in REPS would have significant effects on reducing pollutant emissions. Based on the synergistic effect scheme, [4.8, 5.2]% of electricity shortage would be offset, and [3.2, 3.3]% of system cost would be reduced. Moreover, based on the main effect analysis and regression analysis, multiple joint planning strategies would be achieved in terms of CSFP-REPS model in a cost- and computation-effective way. However, there are several assumptions for formulating the CSFPREPS model, which may lead to some limitations for planning REPS. Firstly, Eqs. (3b) and equation (6b) are generally nonlinear, and the set of feasible constraints is convex only for some particular distributions and certain levels of pi,t such as the case when ai,k,t are deterministic and bi,t are random (for all pi,t values). Secondly, capacity expansion of each power-conversion facility in the planning horizon is limited to the condition of finance investment and facility service life, the binary

investment can be obtained in terms of CSFP-REPS model in a cost- and computation-effective way. 5. Conclusions In this study, a copula-based stochastic fuzzy-credibility programming (CSFP) method has been advanced for the joint planning of regional-scale electric power systems (REPS). CSFP cannot only convert the nonlinear model into an “equivalent” deterministic one using copula-based stochastic programming (CSP) approach, but also tackle multiple uncertainties through integrating fuzzy-credibility constrained programming (FCP) and interval-parameter programming (IPP). It can also reflect uncertain interactions among multiple random variables in REPS. Compared to the conventional stochastic programming approaches, CSFP can effectively reflect uncertain interactions among random variables on system cost even when the random variables follow different probability distributions and have previously unknown correlations. It can also more effectively analyze the individual and interactive effects on system cost, avoiding loss of uncertain information and reduction of solution robustness. In detail, in CSFP, the random variables may have any same or different forms of probability distributions and they may be correlated with each other, and the interactive relationships of all the groups of individual constraint-violation levels can be obtained based on the copulas. The developed CSFP-REPS model has been applied to planning the

844

Applied Energy 212 (2018) 834–849 λ = 0.6

λ = 0.7

λ = 0.8

λ = 0.9

1

1

1

1

12

9 4.89 4.67

2

9 4.93 4.71

4.68 3 84.77

4.72 6

4.71

7 4.82

4

4.65

4.76 6

5

12

System costs ($ 10 )

4.80

7 5.13 4.69 4.82

4.84

7 5.17 4.73 4.86 6

12

System costs ($ 10 )

4.69 7 5.13 6

4.88

4.83

5.13 4.80

4

4.87

4.74 4.84

3 8

4.71

7 5.17

4.84

6

5

3

4.75

4.73

4.93 7 5.26 4.83 6

8

5.17 4

4.92

4.79

4.80

4.89 4.77

7 5.21

5

4.87

4.94 3

4.99 7 5.32 4.88

5

6

4.89

6

5.21 4

7 5.26 4.82 4.94 6

5

Lower bound

4.85

4.94

5.34 4

5

5

5.35

9

2

4.97

4.84

3 8

5.00

1 5.30

9

2

2

5.06

8

5.28 4

4.95

5.39

1

5.25

4

5

9

4.88 3

5

9

4.86 4.84

4.90

1 2 5.01

4.81 8

5.23 4

4.90

6

2

4.87 3

6

5.34

1 5.20

7 4.97

4 5

9

4.83 3

5

9

2

2

4.96

7 5.21 4.77

8 4.91

1

8

5.19 4

4.79

4.85 6

5.29 4.76

3

3

4.80

4.91

5

9

1 5.16

4.70 4.80

2

4.79

5

7

4.83

4.78 4.82

1

4.72

5.15 4

4

4.80 4.74

2

4.89 3 5.27 4

8

2 5.02 4.90

4.99 75.32

4.87 6

4.99

3 5.32 4

Scenarios of T1-T9

8

4.75

4.86

4.91

1 9

7

4

6

5.25

3 8

4.75

5.03 4.80

4.77 3 8 4.86

5

9

2

4.87

6

8 4.81

1 9 5.08 4.86

2

Scenarios of B1-B9

5.21 4.68

8

4.97 4.75

1

1 9

4.69

9

2 4.73

4.72 3

4.67

74.78

9

4.68

4.64 8 4.73

2

λ=1

Scenarios of H1-H9

System costs ($ 10 )

L. Yu et al.

5

Upper bound

Fig. 7. System costs under different scenarios.

substation can be considered in the future study [60,61]. Besides, several potential limitations and further improvements should be addressed in future study: (a) in CSFP, marginal and joint probability distributions of random variables need to be determined firstly, which may lead to increased complexities, improved data requirement and reduced computation efficiency; (b) in CSFP, when many random variables in the individual chance constraints exist, a large number of possibilistic combinations will have to be examined. Therefore, improvements would be desirable in further investigations to mitigate these limitations.

Table 5 Contributions of individual and interactive parameters to system cost. Parameter

Effect on system cost

p1 p2 p3 p1 ∗p2 p1 ∗p3 p2 ∗p3 p1 ∗p2 ∗p3

1.000% 3.894% 94.807% 0.060% 0.060% 0.060% 0.119%

Acknowledgements

p1

This research was supported by Beijing Natural Science Foundation of China (L160011), State Grid Science & Technology Project (5217L017000N) and the State Scholarship Fund of China (201606730023). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.

p2

Fig. 8. The main effect of each parameter on system cost.

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System costs ($ 10 )

variables are used for indicating whether a particular facility development or expansion option needs to be undertaken. Thirdly, in this study, we assumed that the current capacity of transmission lines could be acceptable for the REPS over the planning horizon, such that the expansion cost for electric power transmissions are neglected. The investment in the transmission line and the cost for the transformer

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Fig. 9. Comparison of prediction and actual values (λ = 0.6).

0.1 (a) Upper - R2 = 0.994

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Actual value

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Predicttion

Appendix A. Solution method of CSFP According to an interactive algorithm [62], a two-step method was generated to solve the CSFP method. The CSFP method can be converted into two deterministic submodels, which correspond to the lower and upper bounds of the objective-function value. E − as the lower bound of the objective-function can be first generated in order to minimize the system cost. Submodel (1) I

Min E − =

k1

T

∑∑∑

I

di−,k,t x i−,k,t +

i=1 k=1 t=1

K

T

∑ ∑ ∑

di−,k,t x i+,k,t

(A.1a)

i = 1 k = k1+ 1 t = 1

Subject to: k1

K

|ai±,k,t |+ Sign (ai±,k,t ) x i−,k,t +

∑ k=1

∑

( p )−,i = 1,2,…,k1 ,

|ai±,k,t |− Sign (ai±,k,t ) x i+,k,t ⩽ bi,ti,t

∀ t (A.1b)

k = k1+ 1

C (1−p1,t ,1−p2,t ,…,1−pi,t ) = 1−pt , ∀ i,t k1

(A.1c)

K

|ai±,k,t |+ Sign (ai±,k,t ) x i−,k,t +

∑ k=1

∑

|ai±,k,t |− Sign (ai±,k,t ) x i+,k,t ⩽ ei,t + (1−2λi+,t )(ei,t − ei,t ),i = k1 + 1,k1 + 2,…,I ,∀ t

k = k1+ 1

(A.1d)

x i−,k,t

⩾ 0,k = 1,2,…,k1 , ∀ i,t

(A.1e)

x i+,k,t

⩾ 0,k = k1 + 1,k1 + 2,…,K , ∀ i,t

(A.1f)

Based on the solution of submodel (1), the second submodel corresponding to E + can be achieved. Submodel (2) I

Min E + =

k1

T

∑∑∑

I

di+,k,t x i+,k,t +

i=1 k=1 t=1

K

T

∑ ∑ ∑

di+,k,t x i−,k,t (A.2a)

i = 1 k = k1+ 1 t = 1

Subject to: k1

∑ k=1

K

|ai±,k,t |− Sign (ai±,k,t ) x i+,k,t +

∑

( p )+,i = 1,2,…,k1 ,

|ai±,k,t |+ Sign (ai±,k,t ) x i−,k,t ⩽ bi,ti,t

k = k1+ 1

C (1−p1,t ,1−p2,t ,…,1−pi,t ) = 1−pt , ∀ i,t

∀ t (A.2b) (A.2c)

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∑

K

|ai±,k,t |− Sign (ai±,k,t ) x i+,k,t +

k=1

x i+,k,t

⩾

x i−,k,t opt ,k

∑

|ai±,k,t |+ Sign (ai±,k,t ) x i−,k,t ⩽ ei,t + (1−2λi−,t )(ei,t − ei,t ),i = k1 + 1,k1 + 2,…,I ,∀ t

k = k1+ 1

(A.2d)

= 1,2,…,k1 , ∀ i,t

(A.2e)

x i+,k,t opt ⩾ x i−,k,t ⩾ 0,k = k1 + 1,k1 + 2,…,K , ∀ i,t

(A.2f)

Through solving the two submodels, the final solutions for the CSFP model (i.e. credibility level can be obtained.

± Eopt

=

− + [Eopt ,Eopt ]

and

x i±,k,t opt

=

[x i−,k,t opt ,x i+,k,t opt ])

under each

Appendix B. Nomenclatures for variables and parameters Decision variables supply amount of electricity resource under individual EPS i by resource j in period t (TJ) NYLi±,j,t

EGAi±,k,t

electricity generation amount under individual EPS i by power k in period t (GWh)

PEi±,t

imported electricity amount under individual EPS i in period t (GWh)

OEi±,t

exported electricity amount under individual EPS i in period t (GWh)

x i±,k,t

decision variables under individual EPS i by power k in period t (i.e. electricity-generation amount, imported electricity amount, exported electricity amount) 0–1 variables for capacity expansion under individual EPS i by power k in period t

YCi±,k,t Parameters ± ∼ i C Cr j k

AMRi±,k,t ,q

the interval value with lower and upper bounds fuzzy sets individual EPS (where i = 1 for Beijing, i = 2 for Tianjin and i = 3 for Hebei) Gumbel-Hougaard Copula a credibility measure resource type (where j = 1 for coal, j = 2 for crude oil and j = 3 for natural gas) electricity-conversion technology (where k = 1 for coal-fired power, k = 2 for gas-fired power, k = 3 for hydro power, k = 4 for wind power, k = 5 for photovoltaic power, k = 6 for biomass power, k = 7 for waste power and k = 8 for pumped-storage power) credibility levels of pollutant emissions a probability measure pollutant type (where q = 1 for SO2, q = 2 for NOx and q = 3 for PM10) time period (t = 1–5) parameters in the constraints under individual EPS i by power k in period t (i.e. energy consumption rate, electricity-conversion efficiency, pollutant-emission coefficient) pollutant-emission coefficient under individual EPS i by power k in period t of pollutan q (tonne/GWh)

ARi±,j,t

available resource under individual EPS i by resource j in period t (TJ)

p (bi,ti,t )± CapT ±

electricity demands with unknown distribution information under individual EPS i in period t

λ± Pr q t ai±,k,t

CapUk±,t

limitation of total expanded capacity for electricity-conversion technology (GW) limitation of expanded capacity for electricity-conversion technology k in period t (GW)

CEi±,t ,q

pollutant emission cost under individual EPS i in period t of pollutant q ($/GW)

CPi±,t ,q

pollutant control cost under individual EPS i in period t of pollutant q ($/GWh)

CUi±,k,t

electricity transmission cost under individual EPS i by power k in period t ($/GWh)

di±,k,t

Demandi±,t

parameters in the objective function under individual EPS i by power k in period t (i.e. purchasing electricity resource cost, importing electricity cost, service time of electricity-conversion technology, fixed and variable costs for electricity-generation and capacity-expansion) electricity demand under individual EPS i in period t (GWh)

ei,̃ t E± ECi±,k,t

allowed pollutant emissions under individual EPS i in period t expected system cost over the planning horizon ($) expanded capacity for electricity-conversion technology k under individual EPS i in period t (GW)

ESi±,t ,q

allowed pollutant emissions under individual EPS i in period t of pollutant q (tonne)

FEi±,k,t FECi±,k,t FGCi±,k,t OEJi±,t

energy consumption rate under individual EPS i by power k in period t (TJ/GWh)

pi,t

probabilistic violation levels of electricity demands under EPS i in period t

1−pt PECi±,j,t

prescribed joint probability levels of the total electricity demand in REPS in period t purchasing electricity resource cost under individual EPS i by resource j in period t ($/TJ)

PEJi±,t

importing electricity cost under individual EPS i in period t ($/GWh)

RCi±,k,t

residual capacity for electricity-conversion technology k under individual EPS i in period t (GW)

STi±,k,t

service time of electricity-conversion technology k under individual EPS i in period t (h)

fixed cost for electricity expanding capacity under individual EPS i by power k in period t ($/GW) fixed maintenance cost for generating electricity under individual EPS i by power k in period t ($/GW) exporting electricity cost under individual EPS i in period t ($/GWh)

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SUi±,k,t

financial subsidy under individual EPS i by power k in period t ($/GWh)

TEi±,k,t

electricity-conversion efficiency under individual EPS i by power k in period t (%)

VECi±,k,t

variable cost for electricity expanding capacity under individual EPS i by power k in period t ($/GW)

VGCi±,k,t

variable cost for generating electricity under individual EPS i by power k in period t ($/GWh)

ZLi±,k,t

electricity-conversion consumption rate under individual EPS i by power k in period t (%)

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