Anti-islanding selection for grid-connected solar photovoltaic system applications: A MCDM based distance approach

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ScienceDirect Solar Energy 110 (2014) 519–532 www.elsevier.com/locate/solener

Anti-islanding selection for grid-connected solar photovoltaic system applications: A MCDM based distance approach Asim Datta a,⇑, Debasree Saha b, Amitava Ray c, Priyanath Das b a

Department of Electrical & Electronics Engg., National Institute of Technology Meghalaya, Shillong 793003, Meghalaya, India b Department of Electrical Engg., National Institute of Technology, Agartala 799055, India c Department of Mechanical Engg., National Institute of Technology Silchar, Assam 788010, India Received 2 May 2014; received in revised form 3 September 2014; accepted 28 September 2014

Communicated by: Associate Editor Takhir M. Razykov

Abstract The main objectives of this paper are to identify the major challenges of islanding-detection and suggest an appropriate islandingdetection method (IDM) for grid-connected solar photovoltaic system (GCSPVS) application using multi-criteria decision-making (MCDM) analysis. Related articles appearing in the field of islanding-detections for GCSPVS as well as other types of distributed generators are studied and analyzed so that the following questions can be answered: Which factors dominate the selection of IDM? Which evaluating factors are the crucial ones? Which evaluating criteria are paid more attention to for a particular application? In this research, the preferences of the criteria with their correlations have been evaluated by the analytic network process (ANP). And, these criteria preferences are made involved in the decision matrix of the technique for order preference by similarity to ideal solution (TOPSIS) which is a distance based optimization technique. Unlike hierarchy based approach, the proposed approach takes into account the interdependence relationships among all the constraint of IDM selection and the sensitivity analysis of the model indicates the robustness of the selection. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Analytic network process; TOPSIS; Islanding-detection method; Grid-connected solar photovoltaic application

1. Introduction A new paradigm of electrical network is with the significant penetration of distributed generations (DGs). A numerous problems are to be tackled when DG units are connected to the electrical networks. Islandingdetection is one of the primary concerns in the interconnected DGs. Islanding is defined as the continued operation of a DG unit or a group of DG units energizing a portion of network at the time of loss of utility grid. The islanding situation jeopardizes public security or endangers ⇑ Corresponding author.

E-mail address: [email protected] (A. Datta). http://dx.doi.org/10.1016/j.solener.2014.09.042 0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.

maintenance workers. Several researches on unintentional islanding-detection have been accomplished to ensure the system operation as per the requirements of the different standards. A grid-connected solar photovoltaic system (GCSPVS) basically consists of a photovoltaic generator (set of arrays) and a power conversion stage (inverter) (Yu et al., 2010). The basic of a GCSPVS is depicted in Fig. 1. A DC smoothing capacitor optionally with the components for a boost or buck–boost type DC/DC converter is used as DC interface between the SPV array and DC/AC converter. The DC/AC converter stage involves switching devices (e.g., IGBTs, MOSFETs) in a push–pull or bridge-type fashion which converts the DC power into

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Fig. 1. Basic block diagram of a GCSPVS.

the AC power (Ahmad et al., 2013). Typically, the AC interface unit consists of a low-pass filter, a line-frequency transformer or inductor and protective devices. The main control and monitoring algorithm with the proper gate driving logics of semiconductor switches is incorporated in the control unit (Trujillo et al., 2010). As islanding impacts on operation and power quality of the system, personnel and equipment safety, etc., GCSPVSs should be equipped to anticipate it (Llaria et al., 2010; Ye et al., 2004). Recently, several control and monitoring schemes for DGs have been devised to effectively detect the islanding condition and quickly disconnect DG from the network (Mahat et al., 2008; Teoh and Tan, 2011; Yu et al., 2010). Standards-making bodies and solar inverter manufacturers are used to give a lot of efforts in deciding an appropriate anti-islanding scheme for a GCSPVS application. It is impeded due to many reasons. Several technologically established islandingdetection methods (IDMs) are available posing both merits and demerits. Also, a number of criteria as well as their preference order by the decision-maker influence an effective IDM selection. Therefore, there should be an appropriate method to analyze different types of IDMs for evaluating suitability in present GCSPVSs and forecasting for future applications. A suitable MCDM analysis should be applied in this situation for considering all constraints directly and indirectly influencing the anti-islanding selection (Datta et al., 2011; Germano and Roulet, 2006). MCDM is applied in decision-making problems deriving a way to come in a transparent process when faced with numerous and conflicting evaluations (Luna-Rubio et al., 2012). Different MCDM methods such as preference ranking organization method for enrichments evaluations (PROMETHEE) (Alsayed et al., 2014; Chatzimouratidis and Pilavachi, 2012), evacuation management decision support system (EMDSS) (Lin et al., 2010), elimination and choice expressing reality (ELECTRE) (Beccali et al., 1998; Chatzimouratidis and Pilavachi, 2012), technique for order preference by similarity to ideal solution (TOPSIS) (Lai et al., 1994), analytic hierarchy process (AHP) (Germano and Roulet, 2006; Saaty, 2006), analytic network process (ANP) (Saaty, 1996), and fuzzy sets (Yu and Dexter, 2010), among others, are already reported in different decision-making problems. This paper has analyzed and evaluated the operation of the various widely used IDMs, namely, rate of change of

output frequency (RCF), phase-jump detection (PJD), harmonic detection (HD), impedance measurement (IM), slipmode frequency shift (SMS), and Sandia frequency shift (SFS). Comprehensive analysis of the main functional characteristics of IDMs like suitability for inverter based DGs (SIDG), non-detection zone (NDZ), implementation cost (IC), suitability for multiple DG units system (SMDGS), operating time (OT), degradation of power quality (DPQ) and reliability (R), yields evaluation of IDM for an application. It is desirable to have an appropriate method for evaluating the anti-islanding methods in order to get the proper selection for an application. AHP is a hierarchy method transforming qualitative to quantitative analysis of criteria in order to generate a set of priorities for alternatives (Saaty, 1996). The analytic network process (ANP) is a generalization of the AHP for considering mutually influential factors in the hierarchy (Luna-Rubio et al., 2012). TOPSIS is a potential optimization method to identify solution from a finite set of choices. Alone TOPSIS does not take care of interdependence relationships among the elements. Nevertheless, the interdependencies may occur among the elements in the same cluster or in the different clusters in a complex decision-making model. Therefore, in this research, an ANP integrated TOPSIS technique for finding out the best anti-islanding technique GCSPVS application has been devised out. The following section discuses the literature survey on anti-islanding detections. Section 3 explains on the performance analysis of different IDMs. Section 4 highlights the ANP and TOPSIS methodology. The proposed methodology has been discussed in Section 5. Sections 6 and 7 present validation of the proposed model and sensitivity analysis, respectively. Finally, discussion and conclusion are included in Section 8. 2. Review of IDMs Over the years, various techniques have been developed to detect islanding in different types of DG system. Islanding-detection methods, which have been proposed, are generally classified into following four categories: passive methods, active methods, hybrid methods and communication-based methods. The passive detection methods monitor quantities such as voltage, phase angle, frequency, and harmonic distortion, at the output of the DG to judge whether there is an islanding operation. A GCSPVS equipped with an overvoltage relay (OVR), an undervoltage relay (UVR), an overfrequency relay (OFR), and an underfrequency relay (UFR) has the preliminary islanding-detection capability (Ahmad et al., 2013; Chowdhury et al., 2009; Hung et al., 2003). However, if output and load consumption of a DG unit approach a balance, the changes in voltage and frequency of the system are not enough for enabling islanding-detection by these relays. PJD method searches the rapid changes of the phase difference between the

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inverter output voltage and current in order to detect islanding condition (Kunte and Gao, 2008). Under grid failure, for the same rate of load change, the rate of change of output power (RCOP) of DG becomes much greater than before (Llaria et al., 2010). The rate of change of frequency (RCF) relay monitors the frequency of PCC voltage and trips if the rate of change of frequency becomes higher than the threshold value for longer than the preset time-delay. For a small power mismatch between the DG output and local load power, the rate of change of frequency over power (RCFOP) is more compared to the rate of change of frequency over time (Mahat et al., 2008). Monitoring the source impedance time to time is a technique for islanding-detection (Chowdhury et al., 2009). In three phase systems checking voltage imbalance is an option for the islanding-detection as there is a high possibility of disturbing of load balance under the occurrence of islanding-condition. For inverter based DG, the total harmonic distortion (THD) at the DG output voltage becomes higher at the time of islanding-condition; therefore, monitoring the change in THD at the DG output voltage islanding condition can be detected (Jang and Kim, 2004). Recently, several new passive methods, using intelligent techniques for detecting power islands, have been proposed. Wavelet-transform based techniques attempt to simultaneously comprehending time and frequency information of the measured signals, such as voltages, and currents. Faqhruldin et al. (2014) proposed a universal islandingdetection method for both inverter and synchronous-based DG extracting a group of features from measured data and using random forest (RF) classification technique. Chen and Li (2014) presented a rapid islanding-detection strategy for DGs introducing a reactive power reference to maintain the consistency of the frequency variation trends caused due to both active and reactive power mismatches. Active methods intentionally introduce disturbances at the output of the DG unit in order to determine whether they affect voltage, frequency or impedance parameters to confirm islanding condition (Rani et al., 2013; Kunte and Gao, 2008; Trujillo et al., 2010). They can detect islanding condition even at the perfect balance of DG output and load power and have a low NDZ (Mahat et al., 2008). However, active methods introduce perturbation and degrade the power quality in the system. Active IDMs based on perturbation in phase angle or frequency may disturb the synchronization and power sharing with the grid. For the synchronous generator based DG, islanding can be detected by motoring the change in voltage and reactive power at PCC with the variation in the field excitation (Laghari et al., 2013; Llaria et al., 2010). A high frequency signal is injected at the DG unit terminal through a voltage-divider from time to time and the signal becomes more significant under the occurrence of islanding (Tsang and Chan, 2014). In multiple DG units system, interference between inverters can occur if multiple inverters inject the high-frequency signal. In master–slave configuration, master inverter is responsible of injecting a high-frequency

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voltage signal in the system and the rest of (slave) inverters use the signal to detect islanding (Reigosa et al., 2014). The slip-mode frequency shift (SMS) method uses positive feedback by making the current–voltage phase angle of the inverter as a function of the PCC voltage frequency (Hung et al., 2003). In active frequency drift (AFD) method, the waveform of the injected current from the inverter into the PCC is made slightly distorted with respect to the utility voltage so that the frequency of utility voltage drifts up or down augmenting natural frequency under islanding condition (Kunte and Gao, 2008; Lopes and Zhang, 2007). Sandia frequency shift (SFS) method utilizes positive feedback to the frequency of voltage at PCC and acts to reinforce the frequency deviation under islanding condition (Bower and Ropp, 2002; Vahedi and Karrari, 2013). The hybrid methods are a combination of the active and passive methods. These involve two stages of detecting procedures while the active technique is applied only if islanding is suspected by the passive technique. Examples of hybrid methods are voltage unbalance and frequency set point (Menon and Nehrir, 2007), rate of voltage change and real power shift (Mahat et al., 2009), etc. Remote methods rely on external communication link between the DGs and the utility side. They monitor the state of circuit breakers and switches and have very low NDZ. These methods are more expensive than other methods due to the need of carrier signal based communication infrastructure (Mahat et al., 2008; Xu et al., 2007). IDMs are not a completely established technology and some challenges are still exist (Hung et al., 2003). 3. Performance analysis of IDMs Passive methods are simple, low cost and easily can be applied; however, they are not reliable and having large NDZ. Active methods are greatly emphasized in most of the research and development works for islanding protection. Many active methods lose effectiveness if there are a number of DG units feeding the same island (Ye et al., 2004). In this research, some widely used IDMs have been taken into consideration: 3.1. Rate of change of output frequency (RCF) The RCF can be given by (Chowdhury et al., 2009; Laghari et al., 2013): df DP f ð1Þ ¼ dt 2HG where DP is the power deviation at DG output, H is the moment of inertia of DG unit, and G is the nominal generation of the DG unit. RCF relay monitors the voltage frequency and operates if the measured rate of change of frequency higher than a cut-off value for longer than a preset delay. The relay setting is chosen such that it trips only for fluctuations under

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the grid failure, but not for those fluctuations governed by the utility time constants. The method is highly reliable if there is a large mismatch in DG output and load powers. But it cannot consistently discriminate frequency changes due to grid failure or by other turbulences. 3.2. Phase-jump detection (PJD) In this method, the phase of inverter current is instantaneously synchronized at zero-crossing with phase of PCC voltage utilizing a phase-locked-loop (PLL) (Bower and Ropp, 2002; Teoh and Tan, 2011). At the incident of islanding, PCC voltage no longer remains constant and a phase-error is occurred which is the indication for islanding-detection. Under islanding condition, PJD method solely depends on the power factor at the DG output. Thus, NDZ are dependent on the nature of local load (Kunte and Gao, 2008). 3.3. Harmonic detection (HD) The change of total harmonic distortion (THD) of the DG output voltage is continuously monitored as the change in THD gives a picture to confirm islanding condition (Bower and Ropp, 2002; Jang and Kim, 2004; Kunte and Gao, 2008). Under islanding condition, the harmonic currents produced by the inverter flow into the load and these harmonic currents interacting with the larger load impedance produce larger harmonics. Sensing this voltage harmonics or the change in the level of voltage harmonics islanding can be detected (Jang and Kim, 2004). This technique is highly suitable for the inverter based DG like GSCPVS. 3.4. Impedance measurement (IM) A perturbation in the DG output current (DI) is continuously imposed in order to detect islanding condition (Bower and Ropp, 2002; Mahat et al., 2008). This results variations in voltage (DV) and power (DP) which depend on utility resistance (R) as: rffiffiffi DP R DV ¼ ð2Þ 2 P When grid is disconnected, the change in DV/DI is detectable to prevent islanding. The method is very effective in single DG system. In fact, it can detect islanding in the perfect balance of DG output and load power condition. In case of multiple DGs system, there is a necessity for synchronization in the perturbation, otherwise inaccurate impedance estimation may occur for leading misoperation (Kunte and Gao, 2008). 3.5. Slip-mode frequency shift (SMS) Usually, solar inverters operate nearby at unity power factor, so that the phase angle between the inverter output

current and PCC voltage is zero or close to zero (Kunte and Gao, 2008). SMS method applies positive feedback to the phase of the voltage and controls the phase angle as a function of PCC voltage frequency. Generally, a SMS curve is designed in such a way that its slope is greater than that of the phase of the load in the unstable region. SMS is effective in terms of NDZ for islanding-detection compared to other active techniques. However, as it based on phase shift perturbation can lead for power fluctuation in the system. 3.6. Sandia frequency shift (SFS) Sandia frequency shift (SFS) uses a positive feedback of the frequency of the PCC voltage (Bower and Ropp, 2002; Vahedi and Karrari, 2013). Under grid-connected condition, the method detects and tries to amplify small changes in frequency, but the presence of the grid prevents it. But, during the islanded condition, the change in frequency approaches the threshold of the over-frequency relay (OFR) to proceed for tripping (Kunte and Gao, 2008). Passive methods are simpler and cause no degradation to power quality. However, passive methods have large NDZ and the difficulties in threshold-setting (Teoh and Tan, 2011). When the output and load consumption of a DG system approach a balance, the variations in voltage and frequency become very are small even in the islanding condition. Active methods can detect islanding even under such equilibrium state, which are generally not possible by the passive methods. In active method, the response time tends to be prolonged as external turbulence is being imposed (Trujillo et al., 2010). The IC is the main prominent factor in determining reasonable method for islanding-detection. IC for different IDMs in 10 kW rated inverter is approximated in Table 1. Many anti-islanding schemes require additional control circuit to impart the adequate turbulences which increase the complexity for implementation. Additional circuits may cause unpredicted effects to the electric power quality, such as the deterioration of the power quality, and system instability. Multiple DGs units connected to a PCC is an obstacle in determining the threshold value of operating quantities and thus reduces the sensitivity of islandingdetection. Many things are to be considered to select an appropriate anti-islanding scheme. Following criteria,

Table 1 Approximate added cost for implementing of anti-islanding methods in a 10 kW grid-connected inverter. Sl. No.

IDMs

Cost (US$)

1 2 3 4 5 6

RCF PJD HD IM SFS SMS

10 20 15 80 70 70

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directly and indirectly, influence an effective IDM selection for a DG application. 3.7. Suitability for inverter based DG (SIDG) GCSPVS is an inverter based DG system. Therefore, some anti-islanding methods suitable for rotating machine based DG, such as wind generation, and micro-hydro generation, but they are not equally suitable for the GCSPVS (Mahat et al., 2008). Islanding-detection based on change in terminal voltage or reactive power with the variation of excitation of the generator is not suitable for the GCSPVS (Laghari et al., 2013; Yu et al., 2010). Frequency shift based methods (SMS, SFS, etc.) are mostly suitable to inverter-based DG units (Llaria et al., 2010; Reigosa et al., 2014; Tsang and Chan, 2014).

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the same characteristics are installed in the same PCC, active signals of islanding detection can be synchronized there; it is preferable to make the automatic voltage regulator (AVR) voltage setting variation at the same frequency and in the same phase. But, DG units with different characteristics, the AVR setting should be made independently of each other to cope with the frequency variations. Active IDMs may fail to detect islanding in multiple DG unit islands if all the units do not have identical active antiislanding tactics (Lopes and Zhang, 2007). In the active interharmonic injection system, it is necessary to have the different order of the injected harmonics for different DG units. Moreover, it is necessary to use a band-pass filter in order that the injected harmonic of a DG unit does not affect another (Tsukamoto et al., 2001). 3.11. Operating time (OT)

3.8. Non-detection zone (NDZ) NDZ is the range in terms of power mismatch or load parameter in which an islanding-detection scheme fails to detect the islanding condition (Rani et al., 2013; Ye et al., 2004). Passive methods present large NDZ. However, the NDZ can be reduced if the sensitivity of relay is increased, but a misoperation is likely to occur as a result. Active methods have lower NDZ compared to passive methods and communication based methods very low NDZ (no NDZ in theory). Frequency shift active methods (e.g., SMS, SFS) also have considerable NDZ for high Q-factor loads (large capacitance and small inductance) (Ahmad et al., 2013; Vahedi and Karrari, 2013). 3.9. Implementation cost (IC)

Islanding is harmful to the utility personnel, system components and utility loads Therefore, operating time of anti-islanding devices is a critical factor and should not be prolonged (Ahmad et al., 2013). The operating time of passive methods depend on time-setting of the protective relays. Active method requires time to give an external disturbance and to detect change in quantity due to the external disturbance. Thus, the response time is more for active methods compared to passive methods. Compared to any other methods, remote methods based on carrier signal are very fast. 3.12. Degradation of power quality (DPQ)

The IC is one of the important factors of IDM selection in order to achieve nominal compromising between the cost and system quality (Teoh and Tan, 2011). As voltage and frequency protective relays of the solar inverter are also considered for the function of passive islanding-detection, added cost for islanding-detection by them is less. The need of additional components for some active methods leads to high IC and system complexity. Remote islanding methods require scheme for carrier signal which incurs to high implementation complexity and costs (Mahat et al., 2008).

With power generation requirements, DGs should meet the power quality requirements also. Power quality problems include frequency deviation, voltage fluctuation, harmonic distortions, and EMI (Rani et al., 2013; Reigosa et al., 2014). When the DG systems are operated in parallel with utility grid, especially with active islanding detection, the power quality problems can be unpredictable and significant (Tsukamoto et al., 2001). Like other power converters, a solar inverter also produces some current harmonics in its AC output current. And, as per the international standards, a grid-connected inverter should not generate more than 5% THD of its full rated current.

3.10. Suitability for multiple DG units system (SMDGS)

3.13. Reliability (R)

In a multiple DG units system, the primary drawback is large NDZ. The choice of IDM in a multi-DG system mainly depends on type of generator and generation capacity ratio (Ahmad et al., 2013; Laghari et al., 2013). There can be a diversity of combinations of multiple DG units and, according, it is necessary for evaluating the detecting sensitivity of IDMs (Choudhry and Khan, 2010). If two DG units with different capacities are connected to same point, the larger capacity unit dominates the behavior of the islanded operation. If multiple DG units with almost

Islanding-detection scheme should not lead for false tripping; there should not be any fault in the component itself. Islanding-detection scheme should differentiate between overloaded condition and islanding condition. Mal-operation can occur if there is a sudden load change. Therefore, the settings of anti-islanding devices must be carefully selected to avoid mal-operation during these conditions. It is very difficult to select trip threshold in the passive methods. The sensitivity of the relay should not be set too high, in consideration of the stability of the relay, and

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thus, it is convenient to use an active detection mode together with a passive detection mode to enhance the reliability (Kamel et al., 2010; Yu et al., 2010). 4. Overview on ANP and TOPSIS ANP is a decision support tool developed by Saaty in which intangible factors are qualified based on subjective criteria to provide a numeric scale for prioritizing decision alternatives by considering the dependences between the elements of the hierarchy (Aragone´s-Beltra´n et al., 2014). An ANP model consists of an objective, criteria, subcriteria, alternatives treating all as elements and interdependencies between the mentioned decision components. An overview of ANP process is depicted Fig. 2. The proposed network model for ANP implementation is shown in Fig. 3, and which includes the follows:  The overall goal of the problem is to select the best antiislanding method.  Factors or criteria for the decision, i.e. SIDG, NDZ, IC, SMDGS, OT, DPQ and R.  The decision alternatives, i.e. RCF, PJD, HD, IM, SMS and SFS. The TOPSIS is a distance based optimization technique incorporating two reference points: positive ideal solution and negative ideal solution. Its basic principle is that the chosen alternative should be the closest to the positive ideal solution and the farthest to the negative ideal solution (Lai et al., 1994). The method does not consider the relative importance of the distances from the ideal solution points. Basic steps of TOPSIS method is depicted in Fig. 4. 5. Research design In this paper, authors apply an ANP integrated TOPSIS technique to select the best anti-islanding method for

GCSPVS applications by taking multiple uncertainties into account. The methodology of the ANP integrated TOPSIS technique being expressed in a series of steps: Step 1. Construct a network model. The problem is decomposed into a rational system like a network where criteria, sub-criteria and alternatives are considered as elements connected as per their interdependencies. Depending on homogeneity, set of elements constitute clusters and clusters are considered as nodes of the network. Step 2. Construct a decision matrix. Cluster-wise comparison matrix is constructed with pair-wise comparison on the elements reflecting the decision maker’s judgment of the relative importance. Each element of the matrix is based on Saaty’s nine-point scale (shown in Table 2). The pair-wise comparison is made in such a way that the criterion in row i (i = 1, 2, 3. . .n) is ranked relative to each of the criteria represented by the n columns. Develop a normalized matrix by dividing each number in a column of the pair-wise comparison matrix by its column sum. The normalized value nij is calculated as: .qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xm nij ¼ xij x2 ; ðj ¼ 1; . . . ; m; i ¼ 1; . . . nÞ ð3Þ j¼1 ij Average each row of the normalized matrix. These row averages form the priority vector (PV) of cluster with respect to a particular criterion/alternative. To determine whether or not a level of consistency is reasonable, a quantifiable for the comparison matrix is to be developed. If w is the column vector of the relative weight wi (i = 1, 2, 3 . . . n), comparison matrix A is consistent if Aw = nw. Compute the consistency ratio as CR = CI/RI, where CI = consistency index of comparison matrix = (nmax  n) /(n  1) and RI = random inconsistency = 1.987(n  2)/n. If the consistency ratio is <10%, the level of inconsistency

Fig. 2. Description of the ANP process.

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Fig. 3. Proposed ANP model for anti-islanding method selection.

Fig. 4. Overview of TOPSIS.

is acceptable. If else, the inconsistency A is high, and the elements of the decision matrix must be revised. Step 3. Develop supermatrix. Supermatrix is a partitioned matrix, where each matrix segment represents a relationship between two nodes (clusters) in the network. Local PVs of all clusters obtained from Step. 2 are entered in the appropriate columns of supermatrix. An element in the matrix represents the influence priority of a criterion/alternative positioned on the left of the

matrix with respect to a particular control criterion positioned at the top of the column. Generally, each column of this supermatrix is not normalized or column sum is not equal to one, which is known as ‘unweighted’. In an ANP network, supermatrix is represented as:

Table 2 The nine-point scale of pair-wise comparison. Intensity of relative importance

Definition

1 3 5 7 9 2, 4, 6, 8

Equally important Moderately preferred Strongly preferred Very strongly preferred Extremely preferred Intermediate judgment between two adjacent judgments

ð4Þ

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Aþ ¼ fðmax vij jj 2 J Þ; ðmin vij jj 2 J 0 Þ;

where 2

W xy

y1 x1

W 6 y1 6W 6 x2 ¼6 6 6 4 W y1 xnx

W

y2 x1



W y2 x2 ..

.. . W y2 xnx

.



W

yny x1

þ þ ¼ fvþ 1 ; v2 ; . . . ; vm g

yn 7 W x2y 7 7 7 .. 7 . 7 5

W

i ¼ 1; . . . ; ng

i

i

3

A ¼ fðmin vij jj 2 J Þ; ðmax vij jj 2 J 0 Þ; i

ð5Þ

i ¼ 1; . . . ; ng

i

  ¼ fv 1 ; v2 ; . . . ; vm g

ð7Þ 0

yny xnx

Ci is the ith cluster, exy is the element y of cluster x, nx is the number of elements of cluster x, and yk is the kth element of cluster y (where k = 1, . . ., ny). Step 4. Develop the weighted supermatrix. Supermatrix is then should be transformed to the weighted supermatrix to make it stochastic, i.e., each column of the matrix sums to unity. This is developed by dividing each element in a column (of the supermatrix formed in Step 3) by its column sum. Thus, sum of the elements of each column of the weighted supermatrix is one and the matrix becomes column stochastic. This feature allows to converge the weighted supermatrix to a limit supermatrix. Step 5. Calculate the limit supermatrix. Limit supermatrix is computed by raising weighted supermatrix (Wwt) to powers to give the long-term relative influences of the elements on each other as follows (Horenbeek and Pintelon, 2014): 2kþ1 W Limit ¼ limW xwt or W wt

ð6Þ

x!0

where J is a set of the benefit attributes and J is a set of the cost attributes. v+ and v are the set of the positive and negative ideal solutions, respectively. Step 8. Calculate the separation measures, using the mdimensional Euclidean distance. The separation of each alternative from the positive ideal solution is given as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xm 2 i ¼ 1; . . . ; n ð8Þ Sþ ðvij  vþ i Þ ; i ¼ j¼1 Similarly, separation from the negative ideal solution is given as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Xm  2 ; i ¼ 1; . . . ; n ð9Þ S ¼ ðv  v Þ ij i i i¼1 Step 9. Calculate the relative closeness to the ideal solution. The relative closeness of the alternative Ai with respect to A+ is defined as: C i ¼

ðS þ j

C i ¼ 1

S i ; 0 < Cþ i < 1; þ S j Þ if Ai ¼ Aþ



where k is an arbitrarily large number. To achieve a convergence on the importance weights with an iterative process by raising to the power of (2k + 1), the weighted supermatrix is transformed to limit supermatrix. In the iterative process, the solution is found when all the elements of (Wwt)2k+1 are equal to corresponding all the elements of (Wwt)2k with a certain precision. Step 6. Separate (alternatives  criteria) submatrix. From the limit supermatrix (formed in Step 5), (alternatives  criteria) submatrix is separated and normalized. Step 7. Find the positive and negative ideal solutions. The positive ideal solution (A+) and the negative ideal solution (A) corresponding to all criteria are found as:

¼A ;

and

i ¼ 1; . . . ; n:

C i ¼ 0

if Ai

hence C i 2 ½0; 1

ð10Þ

Step 10. Rank the preference order. The set of alternatives can now be preference ranked according to the descending order of C*i . 6. Validation of the model Having identified the criteria and the alternatives, they are placed into an ANP network model (Step 1) as shown in Fig. 3, which is then used to construct pair-wise comparison matrix (PCM) for each cluster based on relative assigning value (on nine-point scale as shown in Table 2). Normalized PCM is computed for each cluster, and CR computed according to each PCM with the calculated

Table 3 PCM with respect to goal: criteria  criteria. Initial

SIDG NDZ IC SMDGS OT DPQ R

Normalized

SIDG

NDZ

IC

SMDGS

OT

DPQ

R

SIDG

NDZ

IC

SMDGS

OT

DPQ

R

1 3 2 7 5 1 1/2

1/3 1 1/2 4 3 1/3 1/4

1/2 2 1 5 3 1/2 1/3

1/7 1/4 1/5 1 1/3 1/6 1/8

1/5 1/3 1/3 3 1 1/3 1/4

1 3 2 6 3 1 1/2

2 4 3 8 4 2 1

0.0513 0.1538 0.1026 0.3590 0.2564 0.0513 0.0256

0.0354 0.1062 0.0531 0.4248 0.3186 0.0354 0.0265

0.0405 0.1622 0.0811 0.4054 0.2432 0.0405 0.0270

0.0644 0.1127 0.0902 0.4509 0.1503 0.0751 0.0564

0.0367 0.0612 0.0612 0.5505 0.1835 0.0612 0.0459

0.0606 0.1818 0.1212 0.3636 0.1818 0.0606 0.0303

0.0833 0.1667 0.1250 0.3333 0.1667 0.0833 0.0417

PV

CR

0.0532 0.1349 0.0906 0.4125 0.2144 0.0582 0.0362

0.0294

A. Datta et al. / Solar Energy 110 (2014) 519–532

527

Table 4 PCM with respect to individual criterion: alternative  alternative. Initial RCF

Normalized PJD

Criterion I: suitability for inverter RCF 1 2 PJD 1/2 1 HD 1/3 1/2 IM 4 5 SFS 3 4 SMS 2 3

HD

IM

SFS

based DG (SIDG) 3 1/4 1/3 2 1/5 1/4 1 1/6 1/4 6 1 2 4 1/2 1 3 1/3 1/2

PV

CR

SMS

RCF

PJD

HD

IM

SFS

SMS

1/2 1/4 1/3 3 2 1

0.0923 0.0462 0.0308 0.3692 0.2769 0.1846

0.1290 0.0645 0.0323 0.3226 0.2581 0.1935

0.1579 0.1053 0.0526 0.3158 0.2105 0.1579

0.1020 0.0816 0.0680 0.4082 0.2041 0.1361

0.0769 0.0577 0.0577 0.4615 0.2308 0.1154

0.0706 0.0353 0.0471 0.4235 0.2824 0.1412

0.1048 0.0651 0.0481 0.3835 0.2438 0.1548

0.0175

Criterion II: non-detectable zone (NDZ) RCF 1 2 1/3 PJD 1/2 1 1/3 HD 3 3 1 IM 1/3 1/2 1/4 SFS 1/3 1/3 1/5 SMS 1/4 1/4 1/6

3 2 4 1 1/3 1/4

3 3 5 3 1 1/2

4 4 6 4 2 1

0.1846 0.0923 0.5538 0.0615 0.0615 0.0462

0.2824 0.1412 0.4235 0.0706 0.0471 0.0353

0.1460 0.1460 0.4380 0.1095 0.0876 0.0730

0.2835 0.1890 0.3780 0.0945 0.0315 0.0236

0.1935 0.1935 0.3226 0.1935 0.0645 0.0323

0.1905 0.1905 0.2857 0.1905 0.0952 0.0476

0.2134 0.1587 0.4003 0.1200 0.0646 0.0430

0.0455

Criterion III: implementation cost RCF 1 1/2 PJD 2 1 HD 1/3 1/3 IM 4 4 SFS 3 2 SMS 4 3

1/4 1/4 1/6 1 1/4 1/3

1/3 1/2 1/3 4 1 2

1/4 1/3 1/4 3 1/2 1

0.0698 0.1395 0.0233 0.2791 0.2093 0.2791

0.0462 0.0923 0.0308 0.3692 0.1846 0.2769

0.1500 0.1500 0.0500 0.3000 0.1500 0.2000

0.1111 0.1111 0.0741 0.4444 0.1111 0.1481

0.0408 0.0612 0.0408 0.4898 0.1224 0.2449

0.0469 0.0625 0.0469 0.5625 0.0938 0.1875

0.0775 0.1028 0.0443 0.4075 0.1452 0.2228

0.0502

6 4 3 6 2 1

0.4082 0.2041 0.1361 0.1020 0.0816 0.0680

0.2927 0.1463 0.0488 0.4390 0.0366 0.0366

0.2338 0.2338 0.0779 0.3896 0.0390 0.0260

0.6723 0.0560 0.0336 0.1681 0.0420 0.0280

0.3030 0.2424 0.1212 0.2424 0.0606 0.0303

0.2727 0.1818 0.1364 0.2727 0.0909 0.0455

0.3638 0.1774 0.0923 0.2690 0.0585 0.0391

0.0788

Criterion IV: suitability RCF 1 PJD 1/2 HD 1/3 IM 1/4 SFS 1/5 SMS 1/6

(IC) 3 3 1 6 3 4

for multi-DG systems (SMDGS) 2 3 4 5 1 3 1/3 4 1/3 1 1/5 2 3 5 1 4 1/4 1/2 1/4 1 1/4 1/3 1/6 1/2

Criterion V: operating time (OT) RCF 1 1/2 PJD 2 1 HD 1/2 1/3 IM 3 3 SFS 4 4 SMS 3 2

1/3 1/3 1/4 1 3 1/2

1/4 1/4 1/6 1/3 1 1/3

1/3 1/2 1/3 2 3 1

0.0741 0.1481 0.0370 0.2222 0.2963 0.2222

0.0462 0.0923 0.0308 0.2769 0.3692 0.1846

0.1053 0.1579 0.0526 0.2105 0.3158 0.1579

0.0615 0.0615 0.0462 0.1846 0.5538 0.0923

0.1071 0.1071 0.0714 0.1429 0.4286 0.1429

0.0465 0.0698 0.0465 0.2791 0.4186 0.1395

0.0734 0.1061 0.0474 0.2194 0.3971 0.1566

0.0340

Criterion VI: degrading power quality (DPQ) RCF 1 3 2 1/4 PJD 1/3 1 1/2 1/5 HD 1/2 2 1 1/4 IM 4 5 4 1 SFS 6 8 7 3 SMS 3 4 3 1/3

1/6 1/8 1/7 1/3 1 1/4

1/3 1/4 1/3 3 4 1

0.0674 0.0225 0.0337 0.2697 0.4045 0.2022

0.1304 0.0435 0.0870 0.2174 0.3478 0.1739

0.1143 0.0286 0.0571 0.2286 0.4000 0.1714

0.0497 0.0397 0.0497 0.1987 0.5960 0.0662

0.0826 0.0619 0.0708 0.1652 0.4956 0.1239

0.0374 0.0280 0.0374 0.3364 0.4486 0.1121

0.0803 0.0374 0.0559 0.2360 0.4488 0.1416

0.0429

Criterion VII: reliability RCF 1 PJD 1/3 HD 1/3 IM 1/8 SFS 1/7 SMS 1/9

7 3 5 1/2 1 1/3

9 5 9 2 3 1

0.4888 0.1629 0.1629 0.0611 0.0698 0.0543

0.3854 0.1285 0.3854 0.0321 0.0428 0.0257

0.6236 0.0693 0.2079 0.0346 0.0416 0.0231

0.3721 0.1860 0.2791 0.0465 0.0930 0.0233

0.4158 0.1782 0.2970 0.0297 0.0594 0.0198

0.3103 0.1724 0.3103 0.0690 0.1034 0.0345

0.4327 0.1496 0.2738 0.0455 0.0684 0.0301

0.0367

(R) 3 1 3 1/4 1/3 1/5

2 3 1 4 6 3

3 1/3 1 1/6 1/5 1/9

8 4 6 1 2 1/2

value of CI and RI (Step 2). Table 3 represents the PCM and the relative PV for seven criteria based on assumed end-user’s preference for the goal. Table 4 presents the PCM and the relative PV for six alternatives on each criterion. Table 5 presents PCM and relative PV of criteria considering their interdependency relationships among themselves. For each PCM, CR is <0.1 or 10%. The

supermatrix formation in this anti-islanding selection problem with three levels (i.e., goal (G), criteria (C), and alternatives (A)) model is being represented as:

ð11Þ

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A. Datta et al. / Solar Energy 110 (2014) 519–532

Table 5 Interdependency relation based PCM: critera  criteria. Initial NDZ

Normalized SMDGS

OT

DPQ

CR

R

NDZ

IC

SMDGS

OT

DPQ

R

Criterion I: suitability for inverter based DG (SIDG) NDZ 1 3 2 6 4 IC 1/3 1 1/3 3 2 SMDGS 1/2 3 1 4 3 OT 1/6 1/3 1/4 1 1/2 DPQ 1/4 1/2 1/3 2 1 R 1/3 2 1/2 3 2

3 1/2 2 1/3 1/2 1

0.3871 0.1290 0.1935 0.0645 0.0968 0.1290

0.3051 0.1017 0.3051 0.0339 0.0508 0.2034

0.4528 0.0755 0.2264 0.0566 0.0755 0.1132

0.3158 0.1579 0.2105 0.0526 0.1053 0.1579

0.3200 0.1600 0.2400 0.0400 0.0800 0.1600

0.4091 0.0682 0.2727 0.0455 0.0682 0.1364

0.3650 0.1154 0.2414 0.0489 0.0794 0.1500

0.0218

Criterion II: non-detectable zone (NDZ) SIDG 1 1/2 1/5 IC 2 1 1/4 SMDGS 5 4 1 OT 3 3 1/3 DPQ 1 1/2 1/5 R 1/2 1/2 1/5

1/3 1/3 3 1 1/3 1/4

1 2 5 3 1 1/2

2 2 5 4 2 1

0.0800 0.1600 0.4000 0.2400 0.0800 0.0400

0.0526 0.1053 0.4211 0.3158 0.0526 0.0526

0.0916 0.1145 0.4580 0.1527 0.0916 0.0916

0.0635 0.0635 0.5714 0.1905 0.0635 0.0476

0.0800 0.1600 0.4000 0.2400 0.0800 0.0400

0.1250 0.1250 0.3125 0.2500 0.1250 0.0625

0.0821 0.1214 0.4272 0.2315 0.0821 0.0557

0.0280

Criterion III: implementation cost (IC) SIDG 1 1/2 1/4 NDZ 2 1 1/4 SMDGS 4 4 1 OT 3 3 1/3 DPQ 1/2 1/5 1/3 R 1/2 1/3 1/5

1/3 1/3 3 1 1 1/4

1 2 5 3 2 1/2 1/2

2 3 5 41

0.0536 0.1071 0.4286 0.3214 0.0536 0.0357

0.1119 0.1119 0.4478 0.1493 0.0896 0.0896

0.0635 0.0635 0.5714 0.1905 0.0635 0.0476

0.0800 0.1600 0.4000 0.2400 0.0800 0.0400

0.1176 0.1765 0.2941 0.2353 0.1176 0.0588

0.0856 0.1322 0.4150 0.2329 0.0819 0.0525

0.0309

1

0.0870 0.1739 0.3478 0.2609 0.0870 0.0435

2 3 4 4 2 1

0.1053 0.2105 0.2105 0.3158 0.1053 0.0526

0.0857 0.1714 0.0857 0.5143 0.0857 0.0571

0.0690 0.2759 0.1379 0.4138 0.0690 0.0345

0.1290 0.1290 0.1290 0.3871 0.1290 0.0968

0.1053 0.2105 0.2105 0.3158 0.1053 0.0526

0.1250 0.1875 0.2500 0.2500 0.1250 0.0625

0.1032 0.1975 0.1706 0.3661 0.1032 0.0594

0.0306

Criterion IV: suitability SIDG 1 NDZ 2 IC 2 OT 3 DPQ 1 R 1/2

IC

PV

for multi-DG systems (SMDGS) 1/2 1/2 1/3 1 1 2 1/3 2 1/2 1 1/3 2 3 3 1 3 1/2 1/2 1/3 1 1/3 1/4 1/4 1/2

Criterion V: operation time (OT) SIDG 1 1/3 1/2 NDZ 3 1 2 IC 2 1/2 1 SMDGS 5 4 4 DPQ 2 1/2 1/2 R 1/2 1/3 1/3

1/5 1/4 1/4 1 1/5 1/6

1/2 2 2 5 1 1/2

2 3 3 6 2 1

0.0741 0.2222 0.1481 0.3704 0.1481 0.0370

0.0500 0.1500 0.0750 0.6000 0.0750 0.0500

0.0600 0.2400 0.1200 0.4800 0.0600 0.0400

0.0968 0.1210 0.1210 0.4839 0.0968 0.0806

0.0455 0.1818 0.1818 0.4545 0.0909 0.0455

0.1176 0.1765 0.1765 0.3529 0.1176 0.0588

0.0740 0.1819 0.1371 0.4570 0.0981 0.0520

0.0305

Criterion VI: degrading SIDG 1 NDZ 3 IC 2 SMDGS 5 OT 4 R 1/2

power quality (DPQ) 1/3 1/2 1/5 1 2 1/4 1/2 1 1/5 4 5 1 3 4 1/3 1/4 1/2 1/6

1/4 1/3 1/4 3 1 1/3

2 4 2 6 3 1

0.0645 0.1935 0.1290 0.3226 0.2581 0.0323

0.0367 0.1101 0.0550 0.4404 0.3303 0.0275

0.0385 0.1538 0.0769 0.3846 0.3077 0.0385

0.0930 0.1163 0.0930 0.4651 0.1550 0.0775

0.0484 0.0645 0.0484 0.5806 0.1935 0.0645

0.1111 0.2222 0.1111 0.3333 0.1667 0.0556

0.0654 0.1434 0.0856 0.4211 0.2352 0.0493

0.0515

Criterion VII: reliability SIDG 1 NDZ 3 IC 2 SMDGS 6 OT 4 DPQ 1/2

(R) 1/3 1 1/2 4 3 1/3

1/4 1/3 1/3 3 1 1/4

2 3 2 5 4 1

0.0606 0.1818 0.1212 0.3636 0.2424 0.0303

0.0364 0.1091 0.0545 0.4364 0.3273 0.0364

0.0417 0.1667 0.0833 0.4167 0.2500 0.0417

0.0775 0.1163 0.0930 0.4651 0.1550 0.0930

0.0484 0.0645 0.0645 0.5806 0.1935 0.0484

0.1176 0.1765 0.1176 0.2941 0.2353 0.0588

0.0637 0.1358 0.0890 0.4261 0.2339 0.0514

0.0413

1/2 2 1 5 3 1/2

1/6 1/4 1/5 1 1/3 1/5

Where W21 is a vector that represents the impact of goal on the criteria and formed with the PV calculated in Table 3. W22 is the sub-matrix that represents the dependency of criteria among themselves and formed with the PV calculated in Table 5. W23 is the sub-matrix that represents performance of alternatives on each criterion and values are directly assigned in the supermatrix. W32 is the

sub-matrix that represents the impact of the criteria on each of the alternatives and formed with PV calculated in Table 4. I is the identity matrix of alternatives. The complete unweighted supermatrix is being shown in Table 6 (Step 3). For convergence to be occurred, the unweighted supermatrix is transformed in weighted supermatrix making normalized with column stochastic as shown in Table 7

A. Datta et al. / Solar Energy 110 (2014) 519–532

529

Table 6 Unweighted supermatrix of anti-islanding method selection.

Goal SIDG NDZ IC SMDGS OT DPQ R RCF PJD HD IM SFS SMS

Goal

SIDG

NDZ

IC

SMDGS

OT

DPQ

R

RCF

PJD

HD

IM

SFS

SMS

0 0.0532 0.1349 0.0906 0.4125 0.2144 0.0582 0.0362 0 0 0 0 0 0

0 0 0.3650 0.1154 0.2414 0.0489 0.0794 0.1500 0.1048 0.0651 0.0481 0.3835 0.2438 0.1548

0 0.0821 0 0.1214 0.4272 0.2315 0.0821 0.0557 0.2134 0.1587 0.4003 0.1200 0.0646 0.0430

0 0.0856 0.1322 0 0.4150 0.2329 0.0819 0.0525 0.0775 0.1028 0.0443 0.4075 0.1452 0.2228

0 0.1032 0.1975 0.1706 0 0.3661 0.1032 0.0594 0.3638 0.1774 0.0923 0.2690 0.0585 0.0391

0 0.0740 0.1819 0.1371 0.4570 0 0.0981 0.0520 0.0734 0.1061 0.0474 0.2194 0.3971 0.1566

0 0.0654 0.1434 0.0856 0.4211 0.2352 0 0.0493 0.0803 0.0374 0.0559 0.2360 0.4488 0.1416

0 0.0637 0.1358 0.0890 0.4261 0.2339 0.0514 0 0.4327 0.1496 0.2738 0.0455 0.0684 0.0301

0 0.2000 0.1000 0.2000 0.1000 0.2000 0.1500 0.0500 1.0000 0 0 0 0 0

0 0.2000 0.1000 0.1500 0.1000 0.1500 0.2500 0.0500 0 1.0000 0 0 0 0

0 0.2000 0.1000 0.2000 0.1500 0.1500 0.1500 0.0500 0 0 1.0000 0 0 0

0 0.1000 0.2000 0.1000 0.1000 0.1500 0.1000 0.2500 0 0 0 1.0000 0 0

0 0.1500 0.2000 0.1500 0.2000 0.1000 0.0500 0.1500 0 0 0 0 1.0000 0

0 0.1000 0.250 0.0500 0.2500 0.1000 0.0500 0.2000 0 0 0 0 0 1.0000

Table 7 Weighted supermatrix of anti-islanding method selection.

Goal SIDG NDZ IC SMDGS OT DPQ R RCF PJD HD IM SFS SMS

Goal

SIDG

NDZ

IC

SMDGS

OT

DPQ

R

RCF

PJD

HD

IM

SFS

SMS

0 0.0532 0.1349 0.0906 0.4125 0.2144 0.0582 0.0362 0 0 0 0 0 0

0 0 0.1825 0.0577 0.1207 0.0244 0.0397 0.0750 0.0524 0.0325 0.0240 0.1917 0.1219 0.0774

0 0.0410 0 0.0607 0.2136 0.1157 0.0410 0.0278 0.1067 0.0793 0.2001 0.0600 0.0323 0.0215

0 0.0428 0.0661 0 0.2075 0.1164 0.0409 0.0262 0.0387 0.0514 0.0221 0.2037 0.0726 0.1114

0 0.0516 0.0987 0.0853 0 0.1830 0.0516 0.0297 0.1819 0.0887 0.0461 0.1345 0.0292 0.0195

0 0.0370 0.0909 0.0685 0.2285 0 0.0490 0.0260 0.0367 0.0530 0.0237 0.1097 0.1985 0.0783

0 0.0327 0.0717 0.0428 0.2105 0.1176 0 0.0246 0.0401 0.0187 0.0279 0.1180 0.2244 0.0708

0 0.0319 0.0679 0.0445 0.2130 0.1170 0.0257 0 0.2163 0.0748 0.1369 0.0227 0.0342 0.0150

0 0.1000 0.0500 0.1000 0.0500 0.1000 0.0750 0.0250 0.5000 0 0 0 0 0

0 0.1000 0.0500 0.0750 0.0500 0.0750 0.1250 0.0250 0 0.5000 0 0 0 0

0 0.1000 0.0500 0.1000 0.0750 0.0750 0.0750 0.0250 0 0 0.5000 0 0 0

0 0.0500 0.1000 0.0500 0.0500 0.0750 0.0500 0.1250 0 0 0 0.5000 0 0

0 0.0750 0.1000 0.0750 0.1000 0.0500 0.0250 0.0750 0 0 0 0 0.5000 0

0 0.0500 0.1250 0.0250 0.1250 0.0500 0.0250 0.1000 0 0 0 0 0 0.5000

Table 8 Limit supermatrix of anti-islanding method selection.

SIDG NDZ IC SMDGS OT DPQ R RCF PJD HD IM SFS SMS

Goal

SIDG

NDZ

IC

SMDGS

OT

DPQ

R

RCF

PJD

HD

IM

SFS

SMS

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

0.0572 0.0804 0.0641 0.1134 0.0868 0.0497 0.0484 0.1007 0.0615 0.0684 0.1212 0.0951 0.0531

(Step 4). The weighted supermatrix is raised to powers for reaching in stabilization or convergence. The resulting matrix is the limit supermatrix shown in Table 8, which provides the global priority vector (Step 5). From limit supermatrix the (alternatives  criteria) submatrix is separated and normalized as shown in Table 9 (Step 6).

The positive ideal solution (A+) and the negative ideal solution (A) corresponding to all criteria (Step 7) are: Aþ ¼ f0:2424; 0:2424; 0:1062; 0:2424; 0:2424; 0:2424; 0:2424g A ¼ f0:1062; 0:1062; 0:2424; 0:1062; 0:1062; 0:1062; 0:1062g ð12Þ

530

A. Datta et al. / Solar Energy 110 (2014) 519–532

Table 9 Normalized alternatives  criteria matrix.

RCF PJD HD IM SFS SMS

SIDG

NDZ

IC

SMDGS

OT

DPQ

R

0.2014 0.1230 0.1368 0.2424 0.1902 0.1062

0.2014 0.1230 0.1368 0.2424 0.1902 0.1062

0.2014 0.1230 0.1368 0.2424 0.1902 0.1062

0.2014 0.1230 0.1368 0.2424 0.1902 0.1062

0.2014 0.1230 0.1368 0.2424 0.1902 0.1062

0.2014 0.1230 0.1368 0.2424 0.1902 0.1062

0.2014 0.1230 0.1368 0.2424 0.1902 0.1062

Table 10 Separation measures of alternatives. Alternatives

Sþ i

S i

Ci (unnormalized)

Ci (normalized)

Rank

RCF PJD HD IM SFS SMS

0.1384 0.2930 0.2605 0.1362 0.1530 0.3336

0.2368 0.1263 0.1295 0.3336 0.2123 0.1362

0.6311 0.6988 0.6679 0.7100 0.5812 0.2900

0.1763 0.1953 0.1866 0.1984 0.1624 0.0810

4 2 3 1 5 6

Calculated separation measures (Step 8), closeness coefficient (Step 9) are being shown in Table 9. According to the closeness coefficient, ranking order of alternatives is being shown in Table 10 (Step 10). 7. Sensitivity analysis Sensitivity analysis is a realistic way for determining the effects of uncertainties with the variation of the key factors (criteria) of a model and showing the consequential effect on the selection of alternative. This analysis helps to judge the robustness of the results of a model in the presence of uncertainty. As IC is an important factor from the commercial aspects of a solar inverter, sensitivity analysis is performed with respect to cost factor. If the weight assign to the subjective factors is a (0 < a < 1), the objective factors obtain a weight of (1  a). Thus, sensitivity index (SI) for alternative i (Bhattacharya et al., 2005): SI i ¼ kðSFM i Þ þ ð1  kÞðOFM i Þ

calculated using Eq. (14) with the OFC values given in Table 1. A sensitivity plot (shown in Fig. 5) is to analyze the effect of the decision-maker’s preference on the ratio of objective and subjective factors measures, is strongly recommended. Fig. 5 shows a number of break-even points. A break-even point indicates the critical point at which two or more alternatives attain the same priority. Though a number of break-even points in the plot only two significant break-even points, A (a = 0.91) and B (a = 0.99), are clearly identified in Fig. 5, as these two points decides the critical judgement for the highest priority alternatives. At break-even point A, three alternatives RCF, HD and PJD, attain the highest priority and at break-even point B, two alternatives, IM and PJD, attain the highest priority. For, (0 < a < 0.91), RCF gets the highest priority, (0.91 < a < 0.99) PJD gets the highest priority and (0.99 < a < 1) IM gets the highest priority. The choice of a is an important concern and depends on the decision-

ð13Þ

where SFMi is the subjective factor measure for alternative i. OFMi is the objective factor measure for alternative i and defined as: " #1 n X OFM i ¼ OFC i ð1=OFC i Þ ð14Þ i¼1

where, OFCi is the objective factor cost of alternative i. n is the number of the alternatives (n = 6 in the present case). The SFM value, i.e., the global priority of an alternative, is the original measure found using the proposed ANP integrated TOPSIS technique. Thus, the SFM values used in Eq. (13) are the scores of Ci (normalized) found from Table 10. The unit of OFC is in US$, whereas the OFM values are nondimensional quantities. OFM values are

Fig. 5. Sensitivity plot with respect to IC.

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maker’s preference as regards the importance of objective and subjective factor measures. 8. Discussion and conclusion Recently, a large number of anti-islanding methods have been proposed for different types of DGs and selection of a suitable technique for a DG type for an application has become a big challenge. Many issues are involved with the anti-islanding selection such as technological advancement in transmission and distribution system, advancement towards smart grid, and future expandability of both the DG and the utility grid. In addition, energy conversion technique, connection topologies of DG, capacity of the DG, short circuit capacity (SCC) at PCC, and new regulations can have significant influence, directly or indirectly, on anti-islanding selection. All these constraints as well as their interactions are uncertain and cannot be accounted for with deterministic ways. It is difficult in handling by the decision-makers without significant proficiency. Therefore, the anti-islanding selection for GCSPVS is considered as a MCDM problem. In this paper, an ANP integrated TOPSIS technique is applied to select the best IDM for GCSPVS application. In a decision-making problem, the ANP procedure involves multidirectional relationships between goal, criteria, sub-criteria and alternatives. The TOPSIS is based on consideration of displacement from the two reference points: positive and negative ideal solutions. But, TOPSIS does not consider the relative importance of the distances from these points. Therefore, ANP procedure preceded to TOPSIS technique considers all interdependence relationships among criteria and performance of alternatives on each criterion. This gives more appropriate selection in complex multi-criteria decision analysis problems. Applying the proposed MCDM, the prioritization of six IDMs based on subjective factors in a typical IDM selection for GCSPVS application is performed. Also, a sensitivity analysis is performed to determine whether the final solution is robust to change of weights of one or more decision alternatives. The anti-islanding selection is elaborated in this paper exploring only the widely used passive and active techniques. In fact, no single technique is enough to offer an ultimate guarantee. Therefore, hybrid methods, combination of both passive and active techniques, are gaining attention (Mahat et al., 2008; Teoh and Tan, 2011). There is a scope for future research in deciding the best combination of active and passive techniques to instrument effective hybrid detection schemes. Though without inclusion of ANP procedure, it is not possible to account for all the interdependencies in the decision-making problem, care must be taken in the application of the ANP approach due to the some issues (Horenbeek and Pintelon, 2014; Luna-Rubio et al., 2012). Sometimes it is not easy judgement in subjective factors to state preference by the distinct ratio scale. Incorporating

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