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A fuzzy-grey multicriteria decision making model for district heating system
Accepted Manuscript Research Paper A fuzzy-grey multicriteria decision making model for district heating system Haichao Wang, Lin Duanmu, Risto Lahdelma, Xiangli Li PII: DOI: Reference:
S1359-4311(17)32705-9 http://dx.doi.org/10.1016/j.applthermaleng.2017.08.048 ATE 10930
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
21 April 2017 25 July 2017 8 August 2017
Please cite this article as: H. Wang, L. Duanmu, R. Lahdelma, X. Li, A fuzzy-grey multicriteria decision making model for district heating system, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/ j.applthermaleng.2017.08.048
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A fuzzy-grey multicriteria decision making model for district heating system Haichao Wang1,2, Lin Duanmu1, Risto Lahdelma2, Xiangli Li1,* 1
Institute of Building Environment and Facility Engineering, School of Civil Engineering, Dalian University of Technology, Dalian
116024, China 2
Department of Energy Technology, Aalto University School of Engineering, P.O. BOX 14100, FI-00076, Aalto, Finland
Abstract: District heating (DH) is playing an indispensable role in the energy supply all over the world. The high share of DH based on combined heat and power (CHP) indicates the energy efficiency of the local heating systems. In the future, the optimal planning and design of a DH system should consider not only the techno-economic feasibility but also the capability to improve energy efficiency and environment protection. This means that single objective optimization model for the planning of DH system is limited in this regard. Therefore, a multicriteria decision making (MCDM) model for decision support on the planning and designing of DH system is developed in this paper. This is a typical problem with uncertainty and imprecision both in the criteria measurements and the weights. In view of this, we adopted the fuzzy set theory and grey relational analysis to develop a fuzzy grey multicriteria decision making (FG-MCDM) model for DH systems. Sensitivity analysis was also conducted to study the influence of weight vectors on the evaluation results. The model can take into account energy, economy and environment concerns synthetically and thus facilitates more judicious decision making on DH systems. Keywords: District Heating; Combined Heat and Power; Multicriteria Decision Making; Weight. Nomenclature Abbreviations AHP
Analytic Hierarchy Process
CHP
Combined Heat and Power
CI
Consistency Index
CJM
Complementary Judgment Matrix
CR
Consistency Ratio
DH
District Heating
DM
Decision Maker
FG-MCDM
Fuzzy-Grey Multicriteria Decision Making
GCC
Grey Correlation Coefficient
*Corresponding Author: Tel: +86 13940860527; Email address: [email protected].
1
GCD
Grey Correlation Degree
GRA
Grey Relational Analysis
IEA
International Energy Agency
IR
Inconsistency Ratio
LP
Linear Programming
LSQ
Least Square
MAUT
Multi-Attribute Utility Theory
MCDM
Multicriteria Decision Making
MOO
Multicriteria Objective Optimization
MSD
Mean Spatial Distribution
RI
Random Index
TFN
Triangular Fuzzy Number
Symbols A
fuzzy subset
Ai
alternative i
A
complementary judgment matrix
aij
element of a complementary judgment matrix
aij
element of an AHP judgment matrix
Ci
criterion i
e
inconsistency error of complementary judgment matrix
G
criterion performance matrix
R
normalized criterion performance matrix
r
overall GCD
sk
criterion score for alternative k
w
weight
ŵ
weight vector of (n-1) dimension
Z0
reference alternative with possible optimal criteria
Zw
reference alternative with possible worst criteria
Zi
criterion performance for alternative i
α(Z0j, Zij)
GCC between a criteria performance sequence and the best sequence
β(Zwj, Zij) ( Z 0 , Zi )
GCC between a criteria performance sequence and the worst sequence GCD between a criteria performance sequence and the best sequence
( Z w , Zi )
GCD between a criteria performance sequence and the worst sequence
β
basic heat load ratio 2
1. Introduction International energy agency (IEA) reported that heating and cooling account for 46% of the total global energy use in 2012 [1], and district heating (DH) is becoming increasingly important for providing better comfortable conditions and services in buildings. More energy is to be consumed for heating, for example over 24% of the total building energy demand is consumed for residential heating in China in 2013 [2]; the situation is similar in Europe, e.g. in Finland the share of space heating in relation to the total end use of energy was 21% in 2005 and it gradually increased to 25% in 2012 [3, 4], even though the specific energy consumption of DH is gradually reduced according to Helsinki Energy company. Nevertheless, under the circumstance of ongoing worldwide economy recession, decision makers (DMs), policy makers, stakeholders and end users have to face the problem of how to make the energy supply more cost-efficient, not only by human behaviours but also by system optimization. Meanwhile, environmental concerns, and the concept of ecosustainability are widespread in the DH sector [5]. For these reasons, DH should be more energy, economic and environmental efficient in order to promote sustainable technologies and judicious decision making under different situations. However, different heating have their own characteristics in relation to a host of criteria. Therefore, multicriteria decision making (MCDM) [6] methods should be used in the decision support for DH system. MCDM is a general term for methods that provides a systematic quantitative approach to support decision making in problems involving multiple criteria and alternatives [7]. The aim is to help the DM make more consistent decisions by taking important objective and subjective factors into account. The implementation of MCDM in DH can be helpful in the following aspects: 1) planning or retrofitting of DH system; 2) evaluation and selection of heat production technologies & DH scenarios and 3) optimization of the design, operation & regulation of DH system. There are many MCDM methods for example summarized by Wang et al. [8,9] and Anastaselos et al. [10], Pohekar and Ramachandran [11]. Some of the methods are suitable in energy planning, for example, outranking models ELECTRE [12] and PROMETHEE [13] have been applied in the evaluation and planning of DH systems in North America. In addition, other methods have also been widely used in the heating sector, such as multiple objective optimization (MOO) [14], linear programming (LP) [15], analytic hierarchy process (AHP) [16, 17], multi-attribute utility theory (MAUT) [18] and so on. It can be found that MCDM methods are becoming more extensively used and the abovementioned models work well for specific problems, but not all of them are suitable for decision support of DH systems due to the coexistence of the following issues: 1) the selected evaluation criteria cannot reflect the key points of the problem; 2) misuse of data, e.g. qualitative indicators are used even when quantitative values can be originally obtained; 3) few of the above studies deal with uncertainties of weighting. In fact, the MCDM of DH system is a typical problem with incomplete information, because some criteria are ordinal and cannot be measured precisely, e.g. the regulation convenience of the DH system is usually measured by a qualitative indicator. Some other criteria cannot be determined accurately or even missing, for example, net heating cost, reliability of DH system and impact of 3
pollutant emissions should be obtained through corresponding sub-models [19], but not directly acquired from the DH system. In some extreme cases, the parameters for solving these sub-models are missing, e.g. the outdoor temperature and wind speed for calculating the heat load and the air dispersion can be inconsecutive or even missing, which will bring certain extent of greyness (or incompleteness) to the problem. Therefore, this is a problem characterized by ‘fuzziness’ and ‘greyness’ that should be carefully addressed. Fuzzy mathematics [20] and grey system theory [21] provide powerful tools to deal with the two imprecise properties. For this reason, we integrate the fuzzy set theory and grey relational analysis (GRA) together and develop a MCDM model for decision support of DH system, named fuzzy-grey multicriteria decision making (FG-MCDM) model. The FG-MCDM model configuration is detailed in section 2. Section 3 shows an application of the model in planning of a combined district heating system. We also present the sensitivity analysis to show how the weight can affect the MCDM conclusions and provide a further discussion on the comparison with the general multi-level AHP method. Finally we draw the overall conclusions in section 4.
2. Methods 2.1 Fuzzy set and triangular fuzzy number (TFN) The multicriteria decision making in DH sector is affected by the uncertainty and imprecision information, which is suitable to be addressed by the fuzzy set theory introduced by Zadeh [22]. Specifically, a fuzzy subset A is defined by a membership function fA(x), which indicates the degree of x in A. The degree to which an element belongs to a set is defined by the value between 0 and 1. If x fully belongs to A, fA(x) = 1, and fully not, fA(x) = 0; the higher fA(x) is, the greater is the degree of membership for x in A. There are many kinds of membership functions with different shapes, in this paper we introduce the triangular fuzzy number (TFN) M [8] using a piecewise linear membership function defined in (1),
xm 1 (x l) lxm (m l ) f A ( x) , (r x) m x r ( r m) otherwise 0
Fig. 1. The membership function of triangular fuzzy number M [8].
4
(1)
where m [l, r], l and r are the upper and lower bounds of the triangular fuzzy number, and then TFN M can be expressed by (l, m, r). (r-l) implies the degree of fuzziness of a TFN, shown in Fig. 1. If r=l, then the TFN M degenerates to a real number m and the degree of fuzziness is thus zero. With TFN, we can describe the uncertainties in the MCDM as intervals instead of real numbers, which overcome the shortcomings of using deterministic values to represent uncertain criteria. Besides, basic manipulation laws should be mentioned to facilitate the use of TFN in the MCDM of DH system. Let M1 = (l1, m1, r1) and M2 = (l2, m2, r2) are two TFNs, then the triangular fuzzy number operations can be expressed as [8], (1) TFN addition: M1 M2 = (l1+ l2, m1+ m2, r1+ r2); (2) TFN multiplication: M1 M2 ≈ (l1l2, m1m2, r1r2); (3) TFN division:
M 1 l1 m1 r1 , , ; M 2 r2 m2 l2
(4) TFN multiplies real number: λ R, λM1 = (λl1, λm1, λr1). (5) Ranking of TFN: R(M)=(l+4m+r)/6 [23], M1>M2 R(M1)>R(M2), M1=M2 R(M1)=R(M2).
2.2 Grey relational analysis (GRA) In 1980s, grey system theory was developed by Professor Julong Deng and has been widely used in many fields afterwards [21]. It is a suitable method to unascertained problems with ‘few data’ or ‘poor information’. Grey relational analysis (GRA) is a system analysis method for problems having multiple influencing factors. It can indicate the relevance of the system structure quantitatively and thus reveal the relevance degree of each alternative to the ideal scheme. The grey correlation coefficient (GCC) is used to measure the relevance degree of each alternative to the ideal scheme, the larger GCC is, the closer one alternative is to the ideal scheme. We need to calculate the distance between a pair of sequences or called schemes, in order to compare and evaluate their performances. Normalization is required to facilitate this comparison process. Let Δmax(x) and Δmin(x) be the maximum and minimum absolute difference between two schemes or two sequences of criteria performances xi and xj for alternative i and j, then,
0 min ( x) ij ( x) max ( x) ,
(2)
Namely,
0
( x) min ( x) ij 1, max ( x) max ( x)
(3)
where Δij(x) is the distance between two schemes on criteria x. It can be concluded from (3) that the larger Δij(x)/Δmax(x) is, the weaker similarity of xi and xj is observed, and vice versa. Then, we can reverse Δij(x)/Δmax(x) and normalize it to the interval [0, 1] by, min ( x) / max ( x) , (4) ij ( x) / max ( x) But Δmin(x) can be zero at times, therefore (4) is then defined as, 5
min ( x) / max ( x) ij ( x) / max ( x)
ij ( x) ,
(5)
where ρ [0, 1], and (5) can take form,
ij ( x)
min ( x) max ( x) , ij ( x) max ( x)
(6)
where ξij(x) is the GCC of alternatives i and j on criterion x. According to (6), ρ is able to control the impact of Δmax(x) on ξij(x) and it is called the coefficient of distinction which is usually set to be 0.5.
2.3 The fuzzy-grey multicriteria decision making (FG-MCDM) model 2.3.1 Criteria aggregation In a complicated MCDM problem with many influencing factors, it is better to establish a hierarchical structure criteria aggregation system. The hierarchy consists of several different criteria levels: the first level should be the objective level and the rest of the levels should show the criteria meanings from general to specific, and then followed by the scheme level. In our study, economy, technology, environment and energy have been chosen as first-level criteria, while each of them can be divided into corresponding second-level criteria with different properties, shown in Fig. 2. Multicriteria decision making for DH system
Objective
First level criteria
Second level criteria
Economy
Technology
Environment
Energy
Net heating cost
Reliability
Regulation convenience
NOx
SO2
PM10
CO2
Energy efficiency
C1
C2
C3
C4
C5
C6
C7
C8
Scheme level β=0.50
β=0.55
β=0.60
β=0.65
β=0.70
β=0.75
β=0.80
β=0.85
β=0.90
β=0.95
Fig. 2. The criteria aggregation hierarchy of multicriteria decision making for DH systems.
β=1.00
Combined district heating system
The current hierarchical structure is practical for the decision support of DH systems. Nevertheless, the criteria can be expanded or changed for some other heating technologies. The normalization of judgment matrices is conducted after obtaining criteria measurements.
2.3.2 Judgment matrix normalization If a problem consists of m alternatives with respect to n criteria, the judgment matrix can take form,
6
C1 G Gij
mn
C2 ...
G11 G12 G 21 G22 Am Gm1 Gm 2
A1 A2
Cn
... G1n ... G2 n , ... Gmn
(7)
where A1, A2, …, Am are alternatives; C1, C2, …, Cn are criteria and Gij is the performance of alternative Ai on criterion Cj. Furthermore, if Cj is a cardinal criterion, then Gij can be listed into the judgment matrix directly, otherwise we need to use TFNs, e.g. shown in Fig. 3 and Table 1 to describe the ordinal measurements. It is possible to use other fuzziness degrees for different problems. f A(x) 11
0
2
3
4
5
6
7
8
9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
10
11
0.9 1.0 x
Fig. 3. The membership functions of TFNs denoting 11 different ordinal levels. Table 1. TFNs for 11 different ordinal levels Ordinal levels
1
2
3
4
5
6
TFNs
(0.0, 0.0, 0.1)
(0.0, 0.1, 0.2)
(0.1, 0.2, 0.3)
(0.2, 0.3, 0.4)
(0.3, 0.4, 0.5)
(0.4, 0.5, 0.6)
Ordinal levels
7
8
9
10
11
TFNs
(0.5, 0.6, 0.7)
(0.6, 0.7, 0.8)
(0.7, 0.8, 0.9)
(0.8, 0.9, 1.0)
(0.9, 1.0, 1.0)
Before normalizing the judgment matrix, note that any real number can be interpreted as a TFN having a zero fuzziness degree. Therefore, it is convenient to denote the judgment matrix as an overall TFN matrix. On this basis, the normalization process can be uniformly carried out as follows. If Cj is a positive (benefit) criterion, then,
Rij
( gijl g lj ), ( gijm g mj ), ( gijr g rj ) gijl g lj gijm g mj gijr g rj , , 1 ,(8) r r m m l l G j G j ( g lj g lj ), ( g mj g mj ), ( g rj g rj ) g j g j g j g j g j g j Gij G j
If Cj is a negative (cost) criterion, then,
( g lj gijl ), ( g mj gijm ), ( g rj gijr ) g lj gijl g mj gijm g rj gijr Rij , m , l 1 ,(9) r r m l l l m m r r G j G j ( g j g j ), ( g j g j ), ( g j g j ) g j g j g j g j g j g j G j Gij
G j max Gij i 1, 2,..., m g lj , g mj , g rj G j min Gij i 1, 2,..., m g lj , g mj , g rj
7
,
(10)
where, Rij is the normalized performance of alternative Ai on criterion Cj; ( gijl , gijm , gijr ) is the TFN for Gij. It can be concluded that the Rij is also a TFN, in which the r value is less or equal to 1.
2.3.3 Weight determination At each node of the criteria hierarchy (see Fig. 2), the DM performs pairwise comparisons between each pair of criteria or sub-criteria to express their relative importance. At the bottom level, the DM is asked to compare the relative performance of each pair of alternatives with respect to each (sub-) criterion. The DM expresses his/her preferences by choosing among different verbal preference statements corresponding to complementary comparison values (aij, aji) presented in Table 2. Intermediate comparison values may be used if the DM’s preference falls between the listed statements. The comparison values of CJM have a very natural interpretation. Considering only two criteria at a time, the DM can interpret the comparison values as tradeoff weights that he/she assigns to the criteria. The DM could express these weights either as a normalized complementary pair (e.g. 0.8, 0.2) or as two positive numbers (e.g. 4, 1) that are normalized to satisfy the complementarity condition. Similarly, when comparing two alternatives with respect to a criterion, the DM is in effect distributing partial value between the two alternatives. Table 2. Verbal preference statements and corresponding comparison values in CJM (aij,aij). Verbal statement aij aji i is equally important/good as j 0.5 0.5 i is a little more important/better than j 0.6 0.4 i is moderately more important/better than j 0.7 0.3 i is much more important/better than j 0.8 0.2 i is extremely more important/better than j 0.9 0.1
At each node of the hierarchy, the comparison values are organized into a complementary judgment matrix, 0.5 a12 a 0.5 A 21 an1 an 2
a1n a2 n , 0.5
(11)
where the diagonal elements are 0.5 to denote that each criterion is of equal importance with itself. The comparison values are related to criteria weights (trade-off ratios). If the weights of ith and jth criterion are wi and wj respectively, then, aij a ji wi w j , i,j {1,…,n}.
Substituting aji =1 - aij into (3) and solving aij gives, wi aij , i,j {1,…,n}. wi w j
(12)
(13)
The comparison matrix A contains more information than necessary to determine the weights uniquely. The redundant information may serve detecting inaccuracies and possible errors in the preference statements. If the judgment matrix is fully consistent, we can find weights that satisfy 8
each equation (3) as equality. The matrix is consistent if (aij/aji)(ajk/akj) = (aik/aki) for each i, j, k. In practice, the matrix may contain some level of inconsistency [24]. This means that we can solve the weights from the equations (3) only in the least squares (LSQ) sense. Different techniques to solve the weighs have been presented in literature. Here we present a technique that is a little simpler but more efficient than past methods. To make the errors associated to more important weights smaller; we scale both sides of (3) by (wi+wj) to obtain the system, (aij 1)wi aij w j 0 , i,j {1,…,n}.
(14)
In addition, to get a unique solution, we consider the normalization condition of the weights wj = 1. To solve the system we solve (arbitrarily) the last weight from the normalization equation, n 1
wn 1 w j ,
(15)
j 1
and substitute it into (4). This yields a linear equation system with (n-1) variables and n2 equations. The system can be easily reduced. Firstly, the n equations corresponding to i = j hold trivially and they can therefore be omitted. Secondly, due to symmetry, the error in the equation for aij is the complement to that for aji. Therefore it is necessary to consider only either the upper or lower triangle of aij equations. The resulting linear equation system with (n-1) variables and (n2-n)/2 constraints is of form, Hŵ = b, (16) where ŵ = [w1, …, wn-1]. When n = 2, there is only a single equation and a consistent solution (w1=b1/H1) trivially found. When n 3, the system is over-determined and the LSQ solution is, ŵ = (HTH)-1HTb. After solving ŵ, the last weight wn is solved from (5).
(17)
Observe that this method of solving the weights does not require a complete set of comparisons. A sufficient requirement is that the graph formed by pairwise comparisons between entities is connected. At minimum, (n-1) comparisons may be sufficient, in which case there is no redundant information and the comparisons are consistent. This gives great flexibility for the DM in particular in large problems, where comparing every pair of entities would be too laborious. After the weights have been computed at each node of the criteria hierarchy, a score sk is computed for each alternative k. At the lowest level criteria nodes t, the criterion score for each alternative equals its weight. At the higher level nodes, the score for each alternative is computed as a weighted average of the scores at the lower level. Writing the node identifier as superscript for scores and weights, we have, wkt when t at lowest level t (18) sk wtj skj when t at higher level , jS (t ) where S(t) refers to the set of sub-nodes of node t in the hierarchy. The overall score for each alternative is the score computed at the top node. The redundant information provided by pairwise comparisons serves two purposes in the CJM method. Firstly, the weights solved from the over-determined system Eqs. (15)-(16) can provide 9
more accurate preference information compared to the case where only a minimal number of comparisons are made. Secondly, large inconsistency may indicate that the DM has expressed his/her preferences incorrectly. The DM may be encouraged to revise his/her comparisons if too large inconsistency is detected. Xu suggests that the AHP inconsistency ratio (IR) is computed also in the CJM method to detect excess inconsistency [25]. Before this method can be applied, it is necessary to transform the CJM into the corresponding reciprocal matrix [ãij] of AHP. The elements of the reciprocal matrix are ãij = aij/aji = aij/(1-aij). Then a consistency index CI = (max-n)/(n-1) is computed, where max is the principal eigenvalue and n is the dimension of the matrix. Finally, IR = CI/RI, where the random index RI is the average consistency index of a large number of random reciprocal matrices. If IR exceeds 10%, the DM is urged to revise his/her comparisons. We suggest here a different technique for the CJM method. We simply compute the inconsistency errors in Eq. (13) based on expressed aij and weights from the LSQ solution, wi eij aij , i,j {1,…,n}. (19) wi w j If the absolute value |eij| is too large for any of the comparisons, the DM is asked to reconsider his/her comparisons. Observe that eij = -eji. The DM may state how accurately he/she should be able to state the comparisons. A reasonable threshold is ±0.1, corresponding to one step uncertainty on the verbal scale (Table 2). We believe that such a threshold may be easier for the DMs to understand than the IR which is based on more complex computations with many criteria. Also, the inconsistency errors can be used to pinpoint the comparisons that are most likely incorrect.
2.3.4 FG-MCDM model (1) Determine the criteria performance sequences for all alternatives and the reference sequences The reference sequences are the artificial alternatives’ performances consisting of the best or worst criteria measurements among all alternatives in terms of each criterion and thus can be referred to as the best or worst reference sequence, which stand for the possible optimal alternative or the worst alternative in theory. They are denoted by Z0 and Zw,
Z0 {Z0 j j 1, 2,..., n} ,
(20)
Z w {Z wj j 1, 2,..., n} ,
(21)
Other sequences are the criteria performances for each alternative,
Zi {Zij i 1, 2,..., m ; j 1, 2,..., n} ,
(22)
(2) Calculate the grey correlation coefficient (GCC) matrix The GCC between a criteria performance sequence and the best sequence is,
( Z 0 j , Zij )
min i min j Z 0 j Zij max i max j Z 0 j Zij Z 0 j Zij max i max j Z 0 j Zij
The GCC between a criteria performance and the worst sequence is,
10
,
(23)
( Z wj , Zij )
min i min j Z wj Zij max i max j Z wj Zij Z wj Zij max i max j Z wj Zij
,
(24)
where ρ is the coefficient of distinction, ρ = 0.5. (3) Calculate the grey correlation degree (GCD) considering the weight vectors The GCD between a criteria performance sequence and the best sequence is, n
( Z 0 , Zi ) ( Z 0 j , Zij )w j ,
(25)
j 1
The GCD between a criteria performance sequence and the worst sequence is, n
( Z w , Zi ) ( Z wj , Zij ) w j ,
(26)
j 1
where wj is the weight of criterion j. Therefore, the overall GCD of Zi and Z0 is defined as, 1 . r ( Z 0 , Z w , Zi ) 1 [ ( Z w , Zi ) / ( Z 0 , Zi )]2
(27)
It can be seen from Eq. (27) that the larger r(Z0, Zw, Zi) (overall GCD) is, the higher rank that alternative i receives. Therefore, we can use the overall GCD to rank the alternatives as it can indicate the distances from one alternative to the theoretical best and worst schemes comprehensively.
3. Results and discussion 3.1 The example combined district heating system A combined district heating system is a kind of widely used DH system that has two or more heat sources, including base and peak heat sources in the same network. For a combined district heating system, an important parameter named base heat load ratio (β) [26], which indicates the ratio of base load and the design heat load of the whole system should be highlighted. In this study, we demonstrate the FG-MCDM model for decision support by assessing combined district heating alternatives having different base heat load ratios in a DH system of a city in China. The base heat plants are CHPs and peak heat sources are gas-fired boilers. Some relevant parameters of this combined DH system are shown in Table 3 and more detailed data can be found in Wang et al. [5, 26]. There are eleven combined district heating alternatives with β ranging from 0.5 to 1 with an interval of 0.05, to be evaluated. For example, the alternative β = 0.5 means that the base load ratio of CHP for this combined district heating alternative is 0.5; β = 1 means that the base load plant provides all needed heat and there is no peak shaving gas-fired boilers. In fact, these eleven combined district heating alternatives shown in Table 4 are the schemes at the bottom of the criteria aggregation system in Fig. 2.
11
Table 3. Design parameters of the combined district heating system Item
Value
Unit
Heat load
616
MW
Heating area
8.6
million m2
Heating substations
50
Specific fractional resistance of main pipelines
30–70
Pa/m
Local resistance rate
30
%
Design supply and return water temperature
130/80
°C
Design outdoor temperature
－26
°C
Design indoor temperature
18
°C
Heating period
181
d
Table 4. Criteria measurements at different base heat load ratios First level
Economy
Technology
Environment
Energy
criteria
Second
C1
C2
C3
C4
C5
C6
C7
C8
▼
▲
▲
▼
▼
▼
▼
▼
quant.
quant.
qual.
quant.
quant.
quant.
quant.
quant.
ensuring
Regulation
Cmsd-NOx
Cmsd-SO2
Cmsd-PM10
CO2
Equivalent
coefficient
convenience
(μg/m3)
(μg/m3)
(μg/m3)
emission
electricity
(Mt)
(108 kWh)
level criteria
Reliability Net heating cost
1
(108 Yuan)
(%)
β=0.50
2.923
61.11
(0.0, 0.0, 0.1)
0.3601
0.1283
0.0371
2.025
25.215
β=0.55
2.657
55.00
(0.0, 0.1, 0.2)
0.4255
0.1553
0.0436
2.035
25.570
β=0.60
2.439
48.89
(0.1, 0.2, 0.3)
0.4801
0.1795
0.0495
2.044
25.888
β=0.65
2.269
42.78
(0.2, 0.3, 0.4)
0.5334
0.2008
0.0543
2.052
26.169
β=0.70
2.157
36.67
(0.3, 0.4, 0.5)
0.5349
0.2027
0.0543
2.059
26.413
β=0.75
2.085
30.56
(0.4, 0.5, 0.6)
0.5583
0.2137
0.0566
2.066
26.620
β=0.80
2.086
24.44
(0.5, 0.6, 0.7)
0.5760
0.2221
0.0583
2.071
26.792
β=0.85
2.115
18.33
(0.6, 0.7, 0.8)
0.5890
0.2282
0.0595
2.074
26.927
β=0.90
2.195
12.22
(0.7, 0.8, 0.9)
0.6008
0.2340
0.0607
2.077
27.026
β=0.95
2.333
6.11
(0.8, 0.9, 1.0)
0.5986
0.2377
0.0611
2.079
27.091
β=1.00
2.507
0
(0.9, 1.0, 1.0)
0.5986
0.2394
0.0601
2.080
27.122
1
Chinese currency RMB, 1US dollar ≈ 6.8 RMB.
In Table 4, there are seven quantitative criteria and one qualitative criterion described by TFNs. In order to determine the judgment matrix, sub-models should be employed to obtain measurements for quantitative criteria, for example, techno-economic appraisal model to minimize the net heating 12
cost can be found in [26]; reliability ensuring coefficient that indicates the system reliability under the worst possible accident situation is introduced in [27]; environmental related issues are discussed in [5] by modelling the MSD (Mean Spatial Distribution) of each pollutant; the concept of equivalent electricity is based on calculating the total energy consumption (in electricity equivalent) considering the qualities of different energy forms. Consequently, the normalized criteria judgment matrix used in the FG-MCDM model can be calculated according to (8)-(10).
3.2 Weighting analysis The complementary judgment matrices for the example combined district heating system were obtained by questionnaire survey. The respondents include scholars at universities, the operation and management staff of the heating system, and even heat users. Although it is quite difficult to make all respondents understand the objective and procedure of giving a complementary judgment matrix, twelve valid feedbacks have still been collected. Subsequently, these judgment matrices were used to compute the weight vectors using the CJM method. The weight vectors of first and second-level criteria with respect to the objective are shown in Table 5. Table 5. Average weights of the first and second-level criteria (%) C1
C2
C3
C4
C5
C6
C7
C8
Criteria Economy
Technology
Environment
Energy
First level criteria
42.55
11.43
18.18
27.84
Second level criteria
42.55
8.00
3.43
3.90
5.87
5.87
2.54
27.84
Table 5 shows that the net heating cost (C1) is the dominant factor with a weight of 42.55% followed by the energy efficiency criterion (C8), which has a weight percentage of 27.84%. The remaining criteria have weight percentages lower than 1/8. Criteria weights provide the interface with which DMs’ preference can be reflected in the model, but introduce the subjectivity to the model at the same time. This problem can be handled with the sensitivity analyses in section 3.3.
3.3 Results and sensitivity analysis It is widely acknowledged that the weight information can have great impact to the multicriteria evaluation results. Therefore, we propose a sensitivity analysis on the weight vectors, and check the influences of different weight vectors on the evaluation results. In the sensitivity analysis, we used five extreme weight combinations in addition to the initial weight vector. They are: 1) even weight distribution; 2) economy only; 3) technology only; 4) environment only; and 5) energy only. These weight combinations are shown in Table 6. The second-level criteria weights for all weight combination scenarios can be calculated based on Table 5 and shown in Fig. 4.
13
Table 6. Weight combination scenarios for sensitive analysis. First-level criteria Economy
Technology
Environment
Energy
Initial weight vector
42.55%
11.43%
18.18%
27.84%
Even weight
25%
25%
25%
25%
Economy only
100%
0
0
0
Technology only
0
100%
0
0
Environment only
0
0
100%
0
Energy only
0
0
0
100%
50 45 40 35 30 25 20 15 10 5 0
% 42.55%
27.84%
8.00%
C1
C2
3.43%
3.90%
5.87%
C3
C4
C5
5.87%
Weight percentage
Weight percentage
Weight combinations
2.54%
C6
C7
C8
50 45 40 35 30 25 20 15 10 5 0
%
25.00%
25.00% 17.50% 7.50%
C1
C2
C3
(a) Initial weight vector
C3
C4
C5
3.50%
C7
C8
C6
C7
60 50 40 30.00%
30 20 10 0
C8
C1
C2
C3
C4
C5
C6
C7
C8
Second-level criteria
(c) Economy only
(d) Technology only
%
%
32.30%
Weight percentage
Weight percentage
C6
70.00%
Second-level criteria
50 45 40 35 30 25 20 15 10 5 0
C5
%
70
Weight percentage
Weight percentage
80 100.00%
C2
8.08%
(b) Even weight
%
C1
C4
8.08%
Second-level criteria
Second-level criteria
100 90 80 70 60 50 40 30 20 10 0
5.36%
32.30%
21.43% 13.97%
C1
C2
C3
C4
C5
C6
C7
C8
100 90 80 70 60 50 40 30 20 10 0
100.00%
C1
Second-level criteria
C2
C3
C4
C5
C6
C7
C8
Second-level criteria
(e) Environment only
(f) Energy only
Fig. 4. Weight combination scenarios used in the sensitivity analysis.
Figure 5(a) shows the FG-MCDM results corresponding to the initial weight vector. It was found that GCDs of different combined district heating alternatives to the best and worst reference 14
sequences have the same variation pattern before β=0.75; but they are developing towards different directions afterwards, i.e. getting away from the best sequence and approaching the worst sequence, leading to very low overall GCDs. Then we found that the overall GCD firstly increase slightly
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Grey correlation degree (GCD)
Grey correlation degree (GCD)
along with β and reach the maximum at β=0.75, after which it decreases dramatically. Namely the combined district heating alternative with β=0.75 is the most preferred scheme with initial weight vector.
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β
Combined district heating systems with different β
(b) Even weight
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Grey correlation degree (GCD)
Grey correlation degree (GCD)
(a) Initial weight vector 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β
Combined district heating systems with different β
(d) Technology only Grey correlation degree (GCD)
Grey correlation degree (GCD)
(c) Economy only 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β
Combined district heating systems with different β
(e) Environment only GCD to the best reference sequence ( Z 0 , Zi )
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
(f) Energy only GCD to the worst reference sequence (Z w , Zi )
Overall GCD r ( Z 0 , Z w , Z i )
Fig. 5. Sensitivity analysis on weights of the combined heating system’s FG-MCDM results.
It can be concluded from Fig. 5 that, the weight vectors have great influences on the multicriteria evaluation results. We found that unless the economy criterion is extremely favored, the combined district heating alternative with β=1.00 is the worst one. Even though the economy are the only criterion, β=1.00 is not the best alternative, instead β=0.75 is the optimal choice as shown in Fig. 15
5(c). Although peak-shaving heat sources usually have higher operating cost, they are necessary in the DH system. Furthermore, the combined district heating alternatives with proper base heat load ratio β can be better from the economic viewpoint compared to the traditonal DH system without peak-shaving heat sources. For the weight combination scenarios (b), (d), (e) and (f), the overall GCDs are deacreasing with the increasing value of β. This implies that the combined district heating alternatives have better performances in terms of technology, environment and energy criteria, provided gas-fired boilers are used as the peak heat sources. It can also be found that the variation of GCDs to the best and worst references are not opposite all the time, on the contrary
Overall GCD
they have the same developing trend sometimes, e.g. before β=0.75 in Fig. 5(a) and (b). Therefore, it is not reasonable to consider either only the GCDs to the best or to the worst references, instead we should consider the overall GCD to evaluate the alternatives. This is because we need to find the optimum that is most far from the worst reference and most close to the best reference at the same time. Initial weight Technology only
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.50
0.55
0.60
0.65
Even weight Environment only
0.70
0.75
0.80
Economy only Energy only
0.85
0.90
0.95
1.00
Combined district heating systems with different β Fig. 6. Overall GCDs for combined DH scenarios with all weight combination scenarios.
Figure 6 shows the overall GCDs for all weight combination scenarios. It is interesting that the combined district heating alternative with β=0.60 is the most insensitive alternative whose overall GCD changes very little even with different extreme weight vectors. In addition, its overall GCD varies around a high value of 0.6, which means that the combined district heating alternative with β=0.60 is a quite good compromise alternative if there is no weight information at all or DMs’ preferences differs too much, since it is quite insensitive to the change of weight vectors.
3.4 Discussion In order to evaluate the applicability of the propsoed FG-MCDM model, we comopare it with the general multi-level AHP method. Specifically, the first-level criteria and environmental secondlevel criteria were judged with 1~9 scale, and the jugdment matrices used in the comparision are shown in Table 7.
16
Table 7. AHP Judgment matrix for the first-level criteria. Criteria
Economy
Technology
Environment
Energy
Economy
1
7
3
5
Technology
1/7
1
1/5
1/3
Environment
1/3
5
1
3
Energy
1/5
3
1/3
1
Table 8. AHP Judgment matrix for the environmental second-level criteria. Criteria
NOx
SO2
PM10
CO2
NOx
1
1/3
1/2
3
SO2
3
1
1/3
5
PM10
5
3
1
5
CO2
1/3
1/5
1/5
1
The CI (consistency index) for these two matrices are 0.039 and 0.066, and their CR (consistency ratio) values are 0.04 and 0.07, which are smaller than 0.1. Therefore, these two judgment matrices comply with the consistency check. In addition, the technology first-level criterion only has two second-level criteria, so that the judgment matrix is naturally fully consistent, and the importance of reliability compared to regulation convenience is deemed to be 3. The eigenvalue method was used to determine the first-level weight vector based on these judgment matrices, and the result is
W 0.564, 0.055, 0.263, 0.118
T
(28)
Similarly, the weight vector of the environmental second-level criteria is T 0.123, 0.275, 0.540, 0.062 . Therefore, the weight vector based on the AHP method can be C1 C2 calculated and shown in Fig. 7.
60
C3
C4
C5
C6
C7
C8
%
60
56.38%
Weight percentage
50
50
CJM
42.55%
AHP
40
40
30
27.84%
30
20
20 14.23% 11.78% 8.00%
10
4.13%
0
C1
C2
7.23% 5.87% 3.43% 3.23% 3.90% 1.38%
C3
C4
C5
10
5.87% 1.64% 2.54%
C6
C7
Second-level criteria
Fig. 7. Comparision of the weight vectors obtained by CJM and AHP.
17
0
C8
Figure 7 compares the AHP weight vector with the initial weight vector obtained from the proposed CJM method. We find that the weight vectors from both methods are similar, but with some small differences in some criteria. Specifically, AHP gives higher weights to the net heating cost (C1, 56.38%), local pollutants like PM10 (C6, 14.23%) and SO2 (C5, 7.23%). But the CJM method apparently pays more attention to energy efficiency (C8, 27.84%) and reliatility (C2, 8.00%) compared to AHP. Figure 8 shows the comparision of the evaluation results based on AHP and the FG-MCDM model. Note that the AHP overall evaluation results are similar to the grey correlation degree (GCD) to the best reference, because they all prefer the alternatives with β =0.50-0.55 most, and also alternatives with β =0.70-0.75. But our proposed FG-MCDM model can also give a measurement considering the GCD to the worst reference. The motivation is that even if an alternative has a very high overall evaluation, it can be one of the worst cases, especially when the weight information varies too much from the initial determined weight vector. 0.9
GCD to the best reference Overall GCD
GCD to the worst reference AHP results
0.8
0.15
0.7 0.6
0.10
0.5 0.4 0.3
0.05
0.2 0.1 0.0
AHP overall evaluation results
Grey correlation degree (GCD)
1.0
0.00 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β Fig. 8. Comparison of the evaluation results from AHP and FG-MCDM model.
In the case study, our proposed model can give more reallistic results because the combined district heating should not have such a low base load ratio (0.50-0.55) in reality. For example, in Finland, the based load ratio for the CHP based district heating system is usually around 0.7. Furthermore, the lower the base load ratio is, the higher the operating cost will be, this also means that the economy criterion will be much worse than envisaged in the general MCDM models. This is also why we propose to use GCDs to the best and worst references and give an overall GCD based on them. For these reasons, FG-MCDM model is suitable for general problems that other MCDM methods can be applied to, and could have better applicability due to the consideration of the GCD to the worst reference. Through the comparision study, we found that the calcualtion demands for the propsoed FG-MCDM model is apparently more than the typical multi-level AHP method, but it can be easily solved on a computer with Intel core i5 CPU, 2GB RAM, and Windows 7 operating system. 18
4. Conclusion In this paper, we developed a fuzzy grey multicriteria decision making (FG-MCDM) model for the district heating (DH) system. In the model, we integrated the fuzzy set theory and grey relational analysis (GRA) method because the MCDM of DH system is a typical problem with uncertain or imprecise information. Therefore, we used triangular fuzzy numbers (TFNs) with some fuzzy degrees to describe criteria uncertainties and performed sensitivity analysis of extreme weight combinations to examine how the weight vectors can affect the multicriteria evaluation results. Then the FG-MCDM model was demonstrated in planning a combined district heating system in a city of China. In the case study, economy, technology, environment and energy criteria are considered. It is concluded that the developed FG-MCDM model works well for planning the district heating systems. In addition, it is not reasonable to consider either only the grey correlation degrees (GCDs) to the best or to the worst references; instead we should consider the overall GCD to evaluate the alternatives. This is because we need to find the optimal alternative which is most far from the worst reference and most close to the best reference at the same time. Further, according to the sensitivity analysis of weight vectors on overall GCDs of different combined district heating alternatives, the optimal alternative under initial weight vector is the combined district heating alternative with base heat load ratio equal 0.75, but the combined district heating alternative with base heat load ratio of 0.60 is a quite good compromise alternative if there is no weight information or DMs’ preferences differs too much. Because it is the most insensitive alternative who has a relatively high overall GCD that changes very little even with different extreme weight vectors.
Acknowledgments This work was supported by the China national key research and development program – ChinaFinland intergovernmental cooperation in science and technology innovation (Funding NO. 2016YFE0114500) and Academy of Finland funding (grant number NO. 299186). We would also like to thank the ‘Xinghai’ talent project of Dalian University of Technology.
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[6] Carlsson C., Tackling an MCDM-problem with the help of some results from fuzzy set theory. Eur. J. Oper. Res. 1982;10(3):270-81. [7] Clemen R.T., Making hard decisions – an introduction to decision analysis, 2nd edition, CA: Duxbury Press at Wadsworth Publishing Company, Belmont; 1996. [8] Wang J.J., Jing Y.Y., Zhang C.F., Weighting methodologies in multi-criteria evaluations of combined heat and power systems. Int. J. Energ. Res. 2009;33(12):1023-39. [9] Wang J.J., Jing Y.Y., Zhang C.F., Zhao J.H., Review on multicriteria decision analysis aid in sustainable energy decision-making. Renew. Sust. Energ. Rev. 2009;13(9):2263-78. [10] Anastaselos, D., Theodoridou, I., Papadopoulos, A.M., Hegger, M. Integrated evaluation of radiative heating systems for residential buildings. Energy, 2011 36(7): 4207–4215. [11] Pohekar S.D., Ramachandran M., Application of multi-criteria decision making to sustainable energy planning – A review. Renew. Sust. Energ. Rev. 2004;8(4):365-81. [12] Mróz T.M., Planning of community heating systems modernization and development. Appl. Therm. Eng. 2008;28(14-15):1844-52. [13] Ghafghazi S., Sowlati T., Sokhansanj S., Melin S., A multicriteria approach to evaluate district heating system options. Appl. Energ. 2010;87(4):1134-40. [14] Molyneaux A, Leyland G, Favrat D., Environomic multi-objective optimisation of a district heating network considering centralized and decentralized heat pumps. Energy 2010(35):7518. [15] Rong A.Y., Lahdelma R., Efficient algorithms for combined heat and power production planning under the deregulated electricity market. Eur. J. Oper. Res. 2007;176(2):1219-45. [16] Pilavachi P.A, Roumpeas C.P, Minett S, Afgan N.H., Multi-criteria evaluation for CHP system options. Energ. Convers. Manage. 2006;47(20):3519-29. [17] Xu G., Yang Y.P., Lu S.Y., Li L., Song X.N., Comprehensive evaluation of coal-fired power plants based on grey relational analysis and analytic hierarchy process. Energ. Policy 2011;39(5):2343-51. [18] Keeny R.L., Raiffa H., Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: Wiley; 1976. [19] Wang, H.C. Development and application of a multicriteria decision support framework for planning or retrofitting district heating systems. Aalto University publication series Doctoral dissertations, 2013. [20] Li H.J., Fundamentals of fuzzy mathematics and practical algorithms. Beijing: Science Press; 2005. [21] Deng J.L., Grey control system. Beijing: Science Press; 1993. [22] Zadeh L.A., Fuzzy set. Inform. Control 1965, 8(3):338-353. [23] Chou C.C. Canonical Representation of Multiplication Operation on Triangular Fuzzy Numbers. Computers and Mathematics with Applications, 2003, 45, 1601-1610. [24] Brunelli M. On the conjoint estimation of inconsistency and intransitivity of pairwise comparisons. Operations Research Letters, 2016, 44, 672-675.
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[25] Xu, Z S. Two approaches to improving the consistency of complementary judgment matrix. Applied Mathematics - a Journal of Chinese Universities, 2002, Ser. B 17(2), 227-235.. [26] Wang H.C., Jiao W.L., Lahdelma R., Zou P.H., Techno-economic analysis of a coal-fired CHP based combined district heating system with gas-fired boilers for peak load compensation. Energ. Policy 2011;39(12):7950-62. [27] Wang H.C., Zou P.H., Jiao W.L., Reliability assessment of combined district heating system using quota heating coefficient. Adv. Mat. Res. 2011;243-249:5017-20.
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Highlights A Fuzzy-Grey multicriteria decision making model was developed for DH systems. Uncertainty and imprecision in criteria measurements and weights can be handled. The model can facilitates more judicious decision making on DH systems. The model works well for DH systems and can be extended to other energy systems.
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S1359-4311(17)32705-9 http://dx.doi.org/10.1016/j.applthermaleng.2017.08.048 ATE 10930
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
21 April 2017 25 July 2017 8 August 2017
Please cite this article as: H. Wang, L. Duanmu, R. Lahdelma, X. Li, A fuzzy-grey multicriteria decision making model for district heating system, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/ j.applthermaleng.2017.08.048
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A fuzzy-grey multicriteria decision making model for district heating system Haichao Wang1,2, Lin Duanmu1, Risto Lahdelma2, Xiangli Li1,* 1
Institute of Building Environment and Facility Engineering, School of Civil Engineering, Dalian University of Technology, Dalian
116024, China 2
Department of Energy Technology, Aalto University School of Engineering, P.O. BOX 14100, FI-00076, Aalto, Finland
Abstract: District heating (DH) is playing an indispensable role in the energy supply all over the world. The high share of DH based on combined heat and power (CHP) indicates the energy efficiency of the local heating systems. In the future, the optimal planning and design of a DH system should consider not only the techno-economic feasibility but also the capability to improve energy efficiency and environment protection. This means that single objective optimization model for the planning of DH system is limited in this regard. Therefore, a multicriteria decision making (MCDM) model for decision support on the planning and designing of DH system is developed in this paper. This is a typical problem with uncertainty and imprecision both in the criteria measurements and the weights. In view of this, we adopted the fuzzy set theory and grey relational analysis to develop a fuzzy grey multicriteria decision making (FG-MCDM) model for DH systems. Sensitivity analysis was also conducted to study the influence of weight vectors on the evaluation results. The model can take into account energy, economy and environment concerns synthetically and thus facilitates more judicious decision making on DH systems. Keywords: District Heating; Combined Heat and Power; Multicriteria Decision Making; Weight. Nomenclature Abbreviations AHP
Analytic Hierarchy Process
CHP
Combined Heat and Power
CI
Consistency Index
CJM
Complementary Judgment Matrix
CR
Consistency Ratio
DH
District Heating
DM
Decision Maker
FG-MCDM
Fuzzy-Grey Multicriteria Decision Making
GCC
Grey Correlation Coefficient
*Corresponding Author: Tel: +86 13940860527; Email address: [email protected].
1
GCD
Grey Correlation Degree
GRA
Grey Relational Analysis
IEA
International Energy Agency
IR
Inconsistency Ratio
LP
Linear Programming
LSQ
Least Square
MAUT
Multi-Attribute Utility Theory
MCDM
Multicriteria Decision Making
MOO
Multicriteria Objective Optimization
MSD
Mean Spatial Distribution
RI
Random Index
TFN
Triangular Fuzzy Number
Symbols A
fuzzy subset
Ai
alternative i
A
complementary judgment matrix
aij
element of a complementary judgment matrix
aij
element of an AHP judgment matrix
Ci
criterion i
e
inconsistency error of complementary judgment matrix
G
criterion performance matrix
R
normalized criterion performance matrix
r
overall GCD
sk
criterion score for alternative k
w
weight
ŵ
weight vector of (n-1) dimension
Z0
reference alternative with possible optimal criteria
Zw
reference alternative with possible worst criteria
Zi
criterion performance for alternative i
α(Z0j, Zij)
GCC between a criteria performance sequence and the best sequence
β(Zwj, Zij) ( Z 0 , Zi )
GCC between a criteria performance sequence and the worst sequence GCD between a criteria performance sequence and the best sequence
( Z w , Zi )
GCD between a criteria performance sequence and the worst sequence
β
basic heat load ratio 2
1. Introduction International energy agency (IEA) reported that heating and cooling account for 46% of the total global energy use in 2012 [1], and district heating (DH) is becoming increasingly important for providing better comfortable conditions and services in buildings. More energy is to be consumed for heating, for example over 24% of the total building energy demand is consumed for residential heating in China in 2013 [2]; the situation is similar in Europe, e.g. in Finland the share of space heating in relation to the total end use of energy was 21% in 2005 and it gradually increased to 25% in 2012 [3, 4], even though the specific energy consumption of DH is gradually reduced according to Helsinki Energy company. Nevertheless, under the circumstance of ongoing worldwide economy recession, decision makers (DMs), policy makers, stakeholders and end users have to face the problem of how to make the energy supply more cost-efficient, not only by human behaviours but also by system optimization. Meanwhile, environmental concerns, and the concept of ecosustainability are widespread in the DH sector [5]. For these reasons, DH should be more energy, economic and environmental efficient in order to promote sustainable technologies and judicious decision making under different situations. However, different heating have their own characteristics in relation to a host of criteria. Therefore, multicriteria decision making (MCDM) [6] methods should be used in the decision support for DH system. MCDM is a general term for methods that provides a systematic quantitative approach to support decision making in problems involving multiple criteria and alternatives [7]. The aim is to help the DM make more consistent decisions by taking important objective and subjective factors into account. The implementation of MCDM in DH can be helpful in the following aspects: 1) planning or retrofitting of DH system; 2) evaluation and selection of heat production technologies & DH scenarios and 3) optimization of the design, operation & regulation of DH system. There are many MCDM methods for example summarized by Wang et al. [8,9] and Anastaselos et al. [10], Pohekar and Ramachandran [11]. Some of the methods are suitable in energy planning, for example, outranking models ELECTRE [12] and PROMETHEE [13] have been applied in the evaluation and planning of DH systems in North America. In addition, other methods have also been widely used in the heating sector, such as multiple objective optimization (MOO) [14], linear programming (LP) [15], analytic hierarchy process (AHP) [16, 17], multi-attribute utility theory (MAUT) [18] and so on. It can be found that MCDM methods are becoming more extensively used and the abovementioned models work well for specific problems, but not all of them are suitable for decision support of DH systems due to the coexistence of the following issues: 1) the selected evaluation criteria cannot reflect the key points of the problem; 2) misuse of data, e.g. qualitative indicators are used even when quantitative values can be originally obtained; 3) few of the above studies deal with uncertainties of weighting. In fact, the MCDM of DH system is a typical problem with incomplete information, because some criteria are ordinal and cannot be measured precisely, e.g. the regulation convenience of the DH system is usually measured by a qualitative indicator. Some other criteria cannot be determined accurately or even missing, for example, net heating cost, reliability of DH system and impact of 3
pollutant emissions should be obtained through corresponding sub-models [19], but not directly acquired from the DH system. In some extreme cases, the parameters for solving these sub-models are missing, e.g. the outdoor temperature and wind speed for calculating the heat load and the air dispersion can be inconsecutive or even missing, which will bring certain extent of greyness (or incompleteness) to the problem. Therefore, this is a problem characterized by ‘fuzziness’ and ‘greyness’ that should be carefully addressed. Fuzzy mathematics [20] and grey system theory [21] provide powerful tools to deal with the two imprecise properties. For this reason, we integrate the fuzzy set theory and grey relational analysis (GRA) together and develop a MCDM model for decision support of DH system, named fuzzy-grey multicriteria decision making (FG-MCDM) model. The FG-MCDM model configuration is detailed in section 2. Section 3 shows an application of the model in planning of a combined district heating system. We also present the sensitivity analysis to show how the weight can affect the MCDM conclusions and provide a further discussion on the comparison with the general multi-level AHP method. Finally we draw the overall conclusions in section 4.
2. Methods 2.1 Fuzzy set and triangular fuzzy number (TFN) The multicriteria decision making in DH sector is affected by the uncertainty and imprecision information, which is suitable to be addressed by the fuzzy set theory introduced by Zadeh [22]. Specifically, a fuzzy subset A is defined by a membership function fA(x), which indicates the degree of x in A. The degree to which an element belongs to a set is defined by the value between 0 and 1. If x fully belongs to A, fA(x) = 1, and fully not, fA(x) = 0; the higher fA(x) is, the greater is the degree of membership for x in A. There are many kinds of membership functions with different shapes, in this paper we introduce the triangular fuzzy number (TFN) M [8] using a piecewise linear membership function defined in (1),
xm 1 (x l) lxm (m l ) f A ( x) , (r x) m x r ( r m) otherwise 0
Fig. 1. The membership function of triangular fuzzy number M [8].
4
(1)
where m [l, r], l and r are the upper and lower bounds of the triangular fuzzy number, and then TFN M can be expressed by (l, m, r). (r-l) implies the degree of fuzziness of a TFN, shown in Fig. 1. If r=l, then the TFN M degenerates to a real number m and the degree of fuzziness is thus zero. With TFN, we can describe the uncertainties in the MCDM as intervals instead of real numbers, which overcome the shortcomings of using deterministic values to represent uncertain criteria. Besides, basic manipulation laws should be mentioned to facilitate the use of TFN in the MCDM of DH system. Let M1 = (l1, m1, r1) and M2 = (l2, m2, r2) are two TFNs, then the triangular fuzzy number operations can be expressed as [8], (1) TFN addition: M1 M2 = (l1+ l2, m1+ m2, r1+ r2); (2) TFN multiplication: M1 M2 ≈ (l1l2, m1m2, r1r2); (3) TFN division:
M 1 l1 m1 r1 , , ; M 2 r2 m2 l2
(4) TFN multiplies real number: λ R, λM1 = (λl1, λm1, λr1). (5) Ranking of TFN: R(M)=(l+4m+r)/6 [23], M1>M2 R(M1)>R(M2), M1=M2 R(M1)=R(M2).
2.2 Grey relational analysis (GRA) In 1980s, grey system theory was developed by Professor Julong Deng and has been widely used in many fields afterwards [21]. It is a suitable method to unascertained problems with ‘few data’ or ‘poor information’. Grey relational analysis (GRA) is a system analysis method for problems having multiple influencing factors. It can indicate the relevance of the system structure quantitatively and thus reveal the relevance degree of each alternative to the ideal scheme. The grey correlation coefficient (GCC) is used to measure the relevance degree of each alternative to the ideal scheme, the larger GCC is, the closer one alternative is to the ideal scheme. We need to calculate the distance between a pair of sequences or called schemes, in order to compare and evaluate their performances. Normalization is required to facilitate this comparison process. Let Δmax(x) and Δmin(x) be the maximum and minimum absolute difference between two schemes or two sequences of criteria performances xi and xj for alternative i and j, then,
0 min ( x) ij ( x) max ( x) ,
(2)
Namely,
0
( x) min ( x) ij 1, max ( x) max ( x)
(3)
where Δij(x) is the distance between two schemes on criteria x. It can be concluded from (3) that the larger Δij(x)/Δmax(x) is, the weaker similarity of xi and xj is observed, and vice versa. Then, we can reverse Δij(x)/Δmax(x) and normalize it to the interval [0, 1] by, min ( x) / max ( x) , (4) ij ( x) / max ( x) But Δmin(x) can be zero at times, therefore (4) is then defined as, 5
min ( x) / max ( x) ij ( x) / max ( x)
ij ( x) ,
(5)
where ρ [0, 1], and (5) can take form,
ij ( x)
min ( x) max ( x) , ij ( x) max ( x)
(6)
where ξij(x) is the GCC of alternatives i and j on criterion x. According to (6), ρ is able to control the impact of Δmax(x) on ξij(x) and it is called the coefficient of distinction which is usually set to be 0.5.
2.3 The fuzzy-grey multicriteria decision making (FG-MCDM) model 2.3.1 Criteria aggregation In a complicated MCDM problem with many influencing factors, it is better to establish a hierarchical structure criteria aggregation system. The hierarchy consists of several different criteria levels: the first level should be the objective level and the rest of the levels should show the criteria meanings from general to specific, and then followed by the scheme level. In our study, economy, technology, environment and energy have been chosen as first-level criteria, while each of them can be divided into corresponding second-level criteria with different properties, shown in Fig. 2. Multicriteria decision making for DH system
Objective
First level criteria
Second level criteria
Economy
Technology
Environment
Energy
Net heating cost
Reliability
Regulation convenience
NOx
SO2
PM10
CO2
Energy efficiency
C1
C2
C3
C4
C5
C6
C7
C8
Scheme level β=0.50
β=0.55
β=0.60
β=0.65
β=0.70
β=0.75
β=0.80
β=0.85
β=0.90
β=0.95
Fig. 2. The criteria aggregation hierarchy of multicriteria decision making for DH systems.
β=1.00
Combined district heating system
The current hierarchical structure is practical for the decision support of DH systems. Nevertheless, the criteria can be expanded or changed for some other heating technologies. The normalization of judgment matrices is conducted after obtaining criteria measurements.
2.3.2 Judgment matrix normalization If a problem consists of m alternatives with respect to n criteria, the judgment matrix can take form,
6
C1 G Gij
mn
C2 ...
G11 G12 G 21 G22 Am Gm1 Gm 2
A1 A2
Cn
... G1n ... G2 n , ... Gmn
(7)
where A1, A2, …, Am are alternatives; C1, C2, …, Cn are criteria and Gij is the performance of alternative Ai on criterion Cj. Furthermore, if Cj is a cardinal criterion, then Gij can be listed into the judgment matrix directly, otherwise we need to use TFNs, e.g. shown in Fig. 3 and Table 1 to describe the ordinal measurements. It is possible to use other fuzziness degrees for different problems. f A(x) 11
0
2
3
4
5
6
7
8
9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
10
11
0.9 1.0 x
Fig. 3. The membership functions of TFNs denoting 11 different ordinal levels. Table 1. TFNs for 11 different ordinal levels Ordinal levels
1
2
3
4
5
6
TFNs
(0.0, 0.0, 0.1)
(0.0, 0.1, 0.2)
(0.1, 0.2, 0.3)
(0.2, 0.3, 0.4)
(0.3, 0.4, 0.5)
(0.4, 0.5, 0.6)
Ordinal levels
7
8
9
10
11
TFNs
(0.5, 0.6, 0.7)
(0.6, 0.7, 0.8)
(0.7, 0.8, 0.9)
(0.8, 0.9, 1.0)
(0.9, 1.0, 1.0)
Before normalizing the judgment matrix, note that any real number can be interpreted as a TFN having a zero fuzziness degree. Therefore, it is convenient to denote the judgment matrix as an overall TFN matrix. On this basis, the normalization process can be uniformly carried out as follows. If Cj is a positive (benefit) criterion, then,
Rij
( gijl g lj ), ( gijm g mj ), ( gijr g rj ) gijl g lj gijm g mj gijr g rj , , 1 ,(8) r r m m l l G j G j ( g lj g lj ), ( g mj g mj ), ( g rj g rj ) g j g j g j g j g j g j Gij G j
If Cj is a negative (cost) criterion, then,
( g lj gijl ), ( g mj gijm ), ( g rj gijr ) g lj gijl g mj gijm g rj gijr Rij , m , l 1 ,(9) r r m l l l m m r r G j G j ( g j g j ), ( g j g j ), ( g j g j ) g j g j g j g j g j g j G j Gij
G j max Gij i 1, 2,..., m g lj , g mj , g rj G j min Gij i 1, 2,..., m g lj , g mj , g rj
7
,
(10)
where, Rij is the normalized performance of alternative Ai on criterion Cj; ( gijl , gijm , gijr ) is the TFN for Gij. It can be concluded that the Rij is also a TFN, in which the r value is less or equal to 1.
2.3.3 Weight determination At each node of the criteria hierarchy (see Fig. 2), the DM performs pairwise comparisons between each pair of criteria or sub-criteria to express their relative importance. At the bottom level, the DM is asked to compare the relative performance of each pair of alternatives with respect to each (sub-) criterion. The DM expresses his/her preferences by choosing among different verbal preference statements corresponding to complementary comparison values (aij, aji) presented in Table 2. Intermediate comparison values may be used if the DM’s preference falls between the listed statements. The comparison values of CJM have a very natural interpretation. Considering only two criteria at a time, the DM can interpret the comparison values as tradeoff weights that he/she assigns to the criteria. The DM could express these weights either as a normalized complementary pair (e.g. 0.8, 0.2) or as two positive numbers (e.g. 4, 1) that are normalized to satisfy the complementarity condition. Similarly, when comparing two alternatives with respect to a criterion, the DM is in effect distributing partial value between the two alternatives. Table 2. Verbal preference statements and corresponding comparison values in CJM (aij,aij). Verbal statement aij aji i is equally important/good as j 0.5 0.5 i is a little more important/better than j 0.6 0.4 i is moderately more important/better than j 0.7 0.3 i is much more important/better than j 0.8 0.2 i is extremely more important/better than j 0.9 0.1
At each node of the hierarchy, the comparison values are organized into a complementary judgment matrix, 0.5 a12 a 0.5 A 21 an1 an 2
a1n a2 n , 0.5
(11)
where the diagonal elements are 0.5 to denote that each criterion is of equal importance with itself. The comparison values are related to criteria weights (trade-off ratios). If the weights of ith and jth criterion are wi and wj respectively, then, aij a ji wi w j , i,j {1,…,n}.
Substituting aji =1 - aij into (3) and solving aij gives, wi aij , i,j {1,…,n}. wi w j
(12)
(13)
The comparison matrix A contains more information than necessary to determine the weights uniquely. The redundant information may serve detecting inaccuracies and possible errors in the preference statements. If the judgment matrix is fully consistent, we can find weights that satisfy 8
each equation (3) as equality. The matrix is consistent if (aij/aji)(ajk/akj) = (aik/aki) for each i, j, k. In practice, the matrix may contain some level of inconsistency [24]. This means that we can solve the weights from the equations (3) only in the least squares (LSQ) sense. Different techniques to solve the weighs have been presented in literature. Here we present a technique that is a little simpler but more efficient than past methods. To make the errors associated to more important weights smaller; we scale both sides of (3) by (wi+wj) to obtain the system, (aij 1)wi aij w j 0 , i,j {1,…,n}.
(14)
In addition, to get a unique solution, we consider the normalization condition of the weights wj = 1. To solve the system we solve (arbitrarily) the last weight from the normalization equation, n 1
wn 1 w j ,
(15)
j 1
and substitute it into (4). This yields a linear equation system with (n-1) variables and n2 equations. The system can be easily reduced. Firstly, the n equations corresponding to i = j hold trivially and they can therefore be omitted. Secondly, due to symmetry, the error in the equation for aij is the complement to that for aji. Therefore it is necessary to consider only either the upper or lower triangle of aij equations. The resulting linear equation system with (n-1) variables and (n2-n)/2 constraints is of form, Hŵ = b, (16) where ŵ = [w1, …, wn-1]. When n = 2, there is only a single equation and a consistent solution (w1=b1/H1) trivially found. When n 3, the system is over-determined and the LSQ solution is, ŵ = (HTH)-1HTb. After solving ŵ, the last weight wn is solved from (5).
(17)
Observe that this method of solving the weights does not require a complete set of comparisons. A sufficient requirement is that the graph formed by pairwise comparisons between entities is connected. At minimum, (n-1) comparisons may be sufficient, in which case there is no redundant information and the comparisons are consistent. This gives great flexibility for the DM in particular in large problems, where comparing every pair of entities would be too laborious. After the weights have been computed at each node of the criteria hierarchy, a score sk is computed for each alternative k. At the lowest level criteria nodes t, the criterion score for each alternative equals its weight. At the higher level nodes, the score for each alternative is computed as a weighted average of the scores at the lower level. Writing the node identifier as superscript for scores and weights, we have, wkt when t at lowest level t (18) sk wtj skj when t at higher level , jS (t ) where S(t) refers to the set of sub-nodes of node t in the hierarchy. The overall score for each alternative is the score computed at the top node. The redundant information provided by pairwise comparisons serves two purposes in the CJM method. Firstly, the weights solved from the over-determined system Eqs. (15)-(16) can provide 9
more accurate preference information compared to the case where only a minimal number of comparisons are made. Secondly, large inconsistency may indicate that the DM has expressed his/her preferences incorrectly. The DM may be encouraged to revise his/her comparisons if too large inconsistency is detected. Xu suggests that the AHP inconsistency ratio (IR) is computed also in the CJM method to detect excess inconsistency [25]. Before this method can be applied, it is necessary to transform the CJM into the corresponding reciprocal matrix [ãij] of AHP. The elements of the reciprocal matrix are ãij = aij/aji = aij/(1-aij). Then a consistency index CI = (max-n)/(n-1) is computed, where max is the principal eigenvalue and n is the dimension of the matrix. Finally, IR = CI/RI, where the random index RI is the average consistency index of a large number of random reciprocal matrices. If IR exceeds 10%, the DM is urged to revise his/her comparisons. We suggest here a different technique for the CJM method. We simply compute the inconsistency errors in Eq. (13) based on expressed aij and weights from the LSQ solution, wi eij aij , i,j {1,…,n}. (19) wi w j If the absolute value |eij| is too large for any of the comparisons, the DM is asked to reconsider his/her comparisons. Observe that eij = -eji. The DM may state how accurately he/she should be able to state the comparisons. A reasonable threshold is ±0.1, corresponding to one step uncertainty on the verbal scale (Table 2). We believe that such a threshold may be easier for the DMs to understand than the IR which is based on more complex computations with many criteria. Also, the inconsistency errors can be used to pinpoint the comparisons that are most likely incorrect.
2.3.4 FG-MCDM model (1) Determine the criteria performance sequences for all alternatives and the reference sequences The reference sequences are the artificial alternatives’ performances consisting of the best or worst criteria measurements among all alternatives in terms of each criterion and thus can be referred to as the best or worst reference sequence, which stand for the possible optimal alternative or the worst alternative in theory. They are denoted by Z0 and Zw,
Z0 {Z0 j j 1, 2,..., n} ,
(20)
Z w {Z wj j 1, 2,..., n} ,
(21)
Other sequences are the criteria performances for each alternative,
Zi {Zij i 1, 2,..., m ; j 1, 2,..., n} ,
(22)
(2) Calculate the grey correlation coefficient (GCC) matrix The GCC between a criteria performance sequence and the best sequence is,
( Z 0 j , Zij )
min i min j Z 0 j Zij max i max j Z 0 j Zij Z 0 j Zij max i max j Z 0 j Zij
The GCC between a criteria performance and the worst sequence is,
10
,
(23)
( Z wj , Zij )
min i min j Z wj Zij max i max j Z wj Zij Z wj Zij max i max j Z wj Zij
,
(24)
where ρ is the coefficient of distinction, ρ = 0.5. (3) Calculate the grey correlation degree (GCD) considering the weight vectors The GCD between a criteria performance sequence and the best sequence is, n
( Z 0 , Zi ) ( Z 0 j , Zij )w j ,
(25)
j 1
The GCD between a criteria performance sequence and the worst sequence is, n
( Z w , Zi ) ( Z wj , Zij ) w j ,
(26)
j 1
where wj is the weight of criterion j. Therefore, the overall GCD of Zi and Z0 is defined as, 1 . r ( Z 0 , Z w , Zi ) 1 [ ( Z w , Zi ) / ( Z 0 , Zi )]2
(27)
It can be seen from Eq. (27) that the larger r(Z0, Zw, Zi) (overall GCD) is, the higher rank that alternative i receives. Therefore, we can use the overall GCD to rank the alternatives as it can indicate the distances from one alternative to the theoretical best and worst schemes comprehensively.
3. Results and discussion 3.1 The example combined district heating system A combined district heating system is a kind of widely used DH system that has two or more heat sources, including base and peak heat sources in the same network. For a combined district heating system, an important parameter named base heat load ratio (β) [26], which indicates the ratio of base load and the design heat load of the whole system should be highlighted. In this study, we demonstrate the FG-MCDM model for decision support by assessing combined district heating alternatives having different base heat load ratios in a DH system of a city in China. The base heat plants are CHPs and peak heat sources are gas-fired boilers. Some relevant parameters of this combined DH system are shown in Table 3 and more detailed data can be found in Wang et al. [5, 26]. There are eleven combined district heating alternatives with β ranging from 0.5 to 1 with an interval of 0.05, to be evaluated. For example, the alternative β = 0.5 means that the base load ratio of CHP for this combined district heating alternative is 0.5; β = 1 means that the base load plant provides all needed heat and there is no peak shaving gas-fired boilers. In fact, these eleven combined district heating alternatives shown in Table 4 are the schemes at the bottom of the criteria aggregation system in Fig. 2.
11
Table 3. Design parameters of the combined district heating system Item
Value
Unit
Heat load
616
MW
Heating area
8.6
million m2
Heating substations
50
Specific fractional resistance of main pipelines
30–70
Pa/m
Local resistance rate
30
%
Design supply and return water temperature
130/80
°C
Design outdoor temperature
－26
°C
Design indoor temperature
18
°C
Heating period
181
d
Table 4. Criteria measurements at different base heat load ratios First level
Economy
Technology
Environment
Energy
criteria
Second
C1
C2
C3
C4
C5
C6
C7
C8
▼
▲
▲
▼
▼
▼
▼
▼
quant.
quant.
qual.
quant.
quant.
quant.
quant.
quant.
ensuring
Regulation
Cmsd-NOx
Cmsd-SO2
Cmsd-PM10
CO2
Equivalent
coefficient
convenience
(μg/m3)
(μg/m3)
(μg/m3)
emission
electricity
(Mt)
(108 kWh)
level criteria
Reliability Net heating cost
1
(108 Yuan)
(%)
β=0.50
2.923
61.11
(0.0, 0.0, 0.1)
0.3601
0.1283
0.0371
2.025
25.215
β=0.55
2.657
55.00
(0.0, 0.1, 0.2)
0.4255
0.1553
0.0436
2.035
25.570
β=0.60
2.439
48.89
(0.1, 0.2, 0.3)
0.4801
0.1795
0.0495
2.044
25.888
β=0.65
2.269
42.78
(0.2, 0.3, 0.4)
0.5334
0.2008
0.0543
2.052
26.169
β=0.70
2.157
36.67
(0.3, 0.4, 0.5)
0.5349
0.2027
0.0543
2.059
26.413
β=0.75
2.085
30.56
(0.4, 0.5, 0.6)
0.5583
0.2137
0.0566
2.066
26.620
β=0.80
2.086
24.44
(0.5, 0.6, 0.7)
0.5760
0.2221
0.0583
2.071
26.792
β=0.85
2.115
18.33
(0.6, 0.7, 0.8)
0.5890
0.2282
0.0595
2.074
26.927
β=0.90
2.195
12.22
(0.7, 0.8, 0.9)
0.6008
0.2340
0.0607
2.077
27.026
β=0.95
2.333
6.11
(0.8, 0.9, 1.0)
0.5986
0.2377
0.0611
2.079
27.091
β=1.00
2.507
0
(0.9, 1.0, 1.0)
0.5986
0.2394
0.0601
2.080
27.122
1
Chinese currency RMB, 1US dollar ≈ 6.8 RMB.
In Table 4, there are seven quantitative criteria and one qualitative criterion described by TFNs. In order to determine the judgment matrix, sub-models should be employed to obtain measurements for quantitative criteria, for example, techno-economic appraisal model to minimize the net heating 12
cost can be found in [26]; reliability ensuring coefficient that indicates the system reliability under the worst possible accident situation is introduced in [27]; environmental related issues are discussed in [5] by modelling the MSD (Mean Spatial Distribution) of each pollutant; the concept of equivalent electricity is based on calculating the total energy consumption (in electricity equivalent) considering the qualities of different energy forms. Consequently, the normalized criteria judgment matrix used in the FG-MCDM model can be calculated according to (8)-(10).
3.2 Weighting analysis The complementary judgment matrices for the example combined district heating system were obtained by questionnaire survey. The respondents include scholars at universities, the operation and management staff of the heating system, and even heat users. Although it is quite difficult to make all respondents understand the objective and procedure of giving a complementary judgment matrix, twelve valid feedbacks have still been collected. Subsequently, these judgment matrices were used to compute the weight vectors using the CJM method. The weight vectors of first and second-level criteria with respect to the objective are shown in Table 5. Table 5. Average weights of the first and second-level criteria (%) C1
C2
C3
C4
C5
C6
C7
C8
Criteria Economy
Technology
Environment
Energy
First level criteria
42.55
11.43
18.18
27.84
Second level criteria
42.55
8.00
3.43
3.90
5.87
5.87
2.54
27.84
Table 5 shows that the net heating cost (C1) is the dominant factor with a weight of 42.55% followed by the energy efficiency criterion (C8), which has a weight percentage of 27.84%. The remaining criteria have weight percentages lower than 1/8. Criteria weights provide the interface with which DMs’ preference can be reflected in the model, but introduce the subjectivity to the model at the same time. This problem can be handled with the sensitivity analyses in section 3.3.
3.3 Results and sensitivity analysis It is widely acknowledged that the weight information can have great impact to the multicriteria evaluation results. Therefore, we propose a sensitivity analysis on the weight vectors, and check the influences of different weight vectors on the evaluation results. In the sensitivity analysis, we used five extreme weight combinations in addition to the initial weight vector. They are: 1) even weight distribution; 2) economy only; 3) technology only; 4) environment only; and 5) energy only. These weight combinations are shown in Table 6. The second-level criteria weights for all weight combination scenarios can be calculated based on Table 5 and shown in Fig. 4.
13
Table 6. Weight combination scenarios for sensitive analysis. First-level criteria Economy
Technology
Environment
Energy
Initial weight vector
42.55%
11.43%
18.18%
27.84%
Even weight
25%
25%
25%
25%
Economy only
100%
0
0
0
Technology only
0
100%
0
0
Environment only
0
0
100%
0
Energy only
0
0
0
100%
50 45 40 35 30 25 20 15 10 5 0
% 42.55%
27.84%
8.00%
C1
C2
3.43%
3.90%
5.87%
C3
C4
C5
5.87%
Weight percentage
Weight percentage
Weight combinations
2.54%
C6
C7
C8
50 45 40 35 30 25 20 15 10 5 0
%
25.00%
25.00% 17.50% 7.50%
C1
C2
C3
(a) Initial weight vector
C3
C4
C5
3.50%
C7
C8
C6
C7
60 50 40 30.00%
30 20 10 0
C8
C1
C2
C3
C4
C5
C6
C7
C8
Second-level criteria
(c) Economy only
(d) Technology only
%
%
32.30%
Weight percentage
Weight percentage
C6
70.00%
Second-level criteria
50 45 40 35 30 25 20 15 10 5 0
C5
%
70
Weight percentage
Weight percentage
80 100.00%
C2
8.08%
(b) Even weight
%
C1
C4
8.08%
Second-level criteria
Second-level criteria
100 90 80 70 60 50 40 30 20 10 0
5.36%
32.30%
21.43% 13.97%
C1
C2
C3
C4
C5
C6
C7
C8
100 90 80 70 60 50 40 30 20 10 0
100.00%
C1
Second-level criteria
C2
C3
C4
C5
C6
C7
C8
Second-level criteria
(e) Environment only
(f) Energy only
Fig. 4. Weight combination scenarios used in the sensitivity analysis.
Figure 5(a) shows the FG-MCDM results corresponding to the initial weight vector. It was found that GCDs of different combined district heating alternatives to the best and worst reference 14
sequences have the same variation pattern before β=0.75; but they are developing towards different directions afterwards, i.e. getting away from the best sequence and approaching the worst sequence, leading to very low overall GCDs. Then we found that the overall GCD firstly increase slightly
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Grey correlation degree (GCD)
Grey correlation degree (GCD)
along with β and reach the maximum at β=0.75, after which it decreases dramatically. Namely the combined district heating alternative with β=0.75 is the most preferred scheme with initial weight vector.
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β
Combined district heating systems with different β
(b) Even weight
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Grey correlation degree (GCD)
Grey correlation degree (GCD)
(a) Initial weight vector 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β
Combined district heating systems with different β
(d) Technology only Grey correlation degree (GCD)
Grey correlation degree (GCD)
(c) Economy only 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β
Combined district heating systems with different β
(e) Environment only GCD to the best reference sequence ( Z 0 , Zi )
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
(f) Energy only GCD to the worst reference sequence (Z w , Zi )
Overall GCD r ( Z 0 , Z w , Z i )
Fig. 5. Sensitivity analysis on weights of the combined heating system’s FG-MCDM results.
It can be concluded from Fig. 5 that, the weight vectors have great influences on the multicriteria evaluation results. We found that unless the economy criterion is extremely favored, the combined district heating alternative with β=1.00 is the worst one. Even though the economy are the only criterion, β=1.00 is not the best alternative, instead β=0.75 is the optimal choice as shown in Fig. 15
5(c). Although peak-shaving heat sources usually have higher operating cost, they are necessary in the DH system. Furthermore, the combined district heating alternatives with proper base heat load ratio β can be better from the economic viewpoint compared to the traditonal DH system without peak-shaving heat sources. For the weight combination scenarios (b), (d), (e) and (f), the overall GCDs are deacreasing with the increasing value of β. This implies that the combined district heating alternatives have better performances in terms of technology, environment and energy criteria, provided gas-fired boilers are used as the peak heat sources. It can also be found that the variation of GCDs to the best and worst references are not opposite all the time, on the contrary
Overall GCD
they have the same developing trend sometimes, e.g. before β=0.75 in Fig. 5(a) and (b). Therefore, it is not reasonable to consider either only the GCDs to the best or to the worst references, instead we should consider the overall GCD to evaluate the alternatives. This is because we need to find the optimum that is most far from the worst reference and most close to the best reference at the same time. Initial weight Technology only
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.50
0.55
0.60
0.65
Even weight Environment only
0.70
0.75
0.80
Economy only Energy only
0.85
0.90
0.95
1.00
Combined district heating systems with different β Fig. 6. Overall GCDs for combined DH scenarios with all weight combination scenarios.
Figure 6 shows the overall GCDs for all weight combination scenarios. It is interesting that the combined district heating alternative with β=0.60 is the most insensitive alternative whose overall GCD changes very little even with different extreme weight vectors. In addition, its overall GCD varies around a high value of 0.6, which means that the combined district heating alternative with β=0.60 is a quite good compromise alternative if there is no weight information at all or DMs’ preferences differs too much, since it is quite insensitive to the change of weight vectors.
3.4 Discussion In order to evaluate the applicability of the propsoed FG-MCDM model, we comopare it with the general multi-level AHP method. Specifically, the first-level criteria and environmental secondlevel criteria were judged with 1~9 scale, and the jugdment matrices used in the comparision are shown in Table 7.
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Table 7. AHP Judgment matrix for the first-level criteria. Criteria
Economy
Technology
Environment
Energy
Economy
1
7
3
5
Technology
1/7
1
1/5
1/3
Environment
1/3
5
1
3
Energy
1/5
3
1/3
1
Table 8. AHP Judgment matrix for the environmental second-level criteria. Criteria
NOx
SO2
PM10
CO2
NOx
1
1/3
1/2
3
SO2
3
1
1/3
5
PM10
5
3
1
5
CO2
1/3
1/5
1/5
1
The CI (consistency index) for these two matrices are 0.039 and 0.066, and their CR (consistency ratio) values are 0.04 and 0.07, which are smaller than 0.1. Therefore, these two judgment matrices comply with the consistency check. In addition, the technology first-level criterion only has two second-level criteria, so that the judgment matrix is naturally fully consistent, and the importance of reliability compared to regulation convenience is deemed to be 3. The eigenvalue method was used to determine the first-level weight vector based on these judgment matrices, and the result is
W 0.564, 0.055, 0.263, 0.118
T
(28)
Similarly, the weight vector of the environmental second-level criteria is T 0.123, 0.275, 0.540, 0.062 . Therefore, the weight vector based on the AHP method can be C1 C2 calculated and shown in Fig. 7.
60
C3
C4
C5
C6
C7
C8
%
60
56.38%
Weight percentage
50
50
CJM
42.55%
AHP
40
40
30
27.84%
30
20
20 14.23% 11.78% 8.00%
10
4.13%
0
C1
C2
7.23% 5.87% 3.43% 3.23% 3.90% 1.38%
C3
C4
C5
10
5.87% 1.64% 2.54%
C6
C7
Second-level criteria
Fig. 7. Comparision of the weight vectors obtained by CJM and AHP.
17
0
C8
Figure 7 compares the AHP weight vector with the initial weight vector obtained from the proposed CJM method. We find that the weight vectors from both methods are similar, but with some small differences in some criteria. Specifically, AHP gives higher weights to the net heating cost (C1, 56.38%), local pollutants like PM10 (C6, 14.23%) and SO2 (C5, 7.23%). But the CJM method apparently pays more attention to energy efficiency (C8, 27.84%) and reliatility (C2, 8.00%) compared to AHP. Figure 8 shows the comparision of the evaluation results based on AHP and the FG-MCDM model. Note that the AHP overall evaluation results are similar to the grey correlation degree (GCD) to the best reference, because they all prefer the alternatives with β =0.50-0.55 most, and also alternatives with β =0.70-0.75. But our proposed FG-MCDM model can also give a measurement considering the GCD to the worst reference. The motivation is that even if an alternative has a very high overall evaluation, it can be one of the worst cases, especially when the weight information varies too much from the initial determined weight vector. 0.9
GCD to the best reference Overall GCD
GCD to the worst reference AHP results
0.8
0.15
0.7 0.6
0.10
0.5 0.4 0.3
0.05
0.2 0.1 0.0
AHP overall evaluation results
Grey correlation degree (GCD)
1.0
0.00 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Combined district heating systems with different β Fig. 8. Comparison of the evaluation results from AHP and FG-MCDM model.
In the case study, our proposed model can give more reallistic results because the combined district heating should not have such a low base load ratio (0.50-0.55) in reality. For example, in Finland, the based load ratio for the CHP based district heating system is usually around 0.7. Furthermore, the lower the base load ratio is, the higher the operating cost will be, this also means that the economy criterion will be much worse than envisaged in the general MCDM models. This is also why we propose to use GCDs to the best and worst references and give an overall GCD based on them. For these reasons, FG-MCDM model is suitable for general problems that other MCDM methods can be applied to, and could have better applicability due to the consideration of the GCD to the worst reference. Through the comparision study, we found that the calcualtion demands for the propsoed FG-MCDM model is apparently more than the typical multi-level AHP method, but it can be easily solved on a computer with Intel core i5 CPU, 2GB RAM, and Windows 7 operating system. 18
4. Conclusion In this paper, we developed a fuzzy grey multicriteria decision making (FG-MCDM) model for the district heating (DH) system. In the model, we integrated the fuzzy set theory and grey relational analysis (GRA) method because the MCDM of DH system is a typical problem with uncertain or imprecise information. Therefore, we used triangular fuzzy numbers (TFNs) with some fuzzy degrees to describe criteria uncertainties and performed sensitivity analysis of extreme weight combinations to examine how the weight vectors can affect the multicriteria evaluation results. Then the FG-MCDM model was demonstrated in planning a combined district heating system in a city of China. In the case study, economy, technology, environment and energy criteria are considered. It is concluded that the developed FG-MCDM model works well for planning the district heating systems. In addition, it is not reasonable to consider either only the grey correlation degrees (GCDs) to the best or to the worst references; instead we should consider the overall GCD to evaluate the alternatives. This is because we need to find the optimal alternative which is most far from the worst reference and most close to the best reference at the same time. Further, according to the sensitivity analysis of weight vectors on overall GCDs of different combined district heating alternatives, the optimal alternative under initial weight vector is the combined district heating alternative with base heat load ratio equal 0.75, but the combined district heating alternative with base heat load ratio of 0.60 is a quite good compromise alternative if there is no weight information or DMs’ preferences differs too much. Because it is the most insensitive alternative who has a relatively high overall GCD that changes very little even with different extreme weight vectors.
Acknowledgments This work was supported by the China national key research and development program – ChinaFinland intergovernmental cooperation in science and technology innovation (Funding NO. 2016YFE0114500) and Academy of Finland funding (grant number NO. 299186). We would also like to thank the ‘Xinghai’ talent project of Dalian University of Technology.
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Highlights A Fuzzy-Grey multicriteria decision making model was developed for DH systems. Uncertainty and imprecision in criteria measurements and weights can be handled. The model can facilitates more judicious decision making on DH systems. The model works well for DH systems and can be extended to other energy systems.
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