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With the increasingly largescale interconnection of power system, the object of this paper was to analyze the fragility of the fault of IEEE14 nodes based on the mixed entropy measure.The mixed entropy approach was adopted to quantify the fragile links in the system, which is made up by the flow entropy and the risk entropy. The simulation experiments involved in the study were implemented in the MATPOWER toolbox of the MATLAB platform. The results obtained in this study include that the flow entropy is the key factor on unbalanced distribution of power grid, furthermore the safety of the whole power grid can be achieved a quantitative assessment by the risk entropy. The simulation experiments of IEEE14 node involved in the study were implemented in the MATPOWER toolbox of the MATLAB platform. The results were presented in the form of data and histograms. The impacts of the obtained results are that the transfer entropy is modified by distribution entropy of power flow. the findings of this study may do good to the power network vulnerability analysis with large nodes.
mixed entropy, chain failures, vulnerability, reliability analysis
Power systems are the largescale interconnected systems consisting of subsystems with unknown parameters. Chained failures may cause large scale blackout and lead to serious consequences. Moreover, it is rather difficult to search the modes of chained failures and analyze the consequences. In order to deal with chained failures of power grid and reasonable and effective evaluation on power system reliability, many researchers pay much more attention on reliability analysis on power system or power network (Thomasian and Blaum, 2006; Creen et al., 2003; Christopher et al., 2014; Iacoboaiea et al., 2016; Blažej and Juraj, 2014 Carvalho et al., 2018). Furthermore, securityconstrained power flow optima and redistribution of power flow plays an important role in the propagation of chain failures (Kazemdehdashti et al., 2018; Wang et al., 2018; Fang et al., 2017; Barocio et al., 2017).
There are a large number of studies on analysis on solving securityconstrained optimal power flow (SCOPF) with the help of Monte Carlo simulation (Monticelli et al., 1987; Stott et al., 1987; Wood et al., 2014; Momoh, 2009; Zhu, 2009). While the major shortcoming of randomgradientbased methods is that the power flow quantitative evaluation and the reliability analysis cannot be reached on small sample. Moreover, the list popular evolutionary algorithm methods (e.g. genetic algorithms, evolution strategies, differential evolution, artificial immunological systems, etc.) is not a global optimization method. The system stability and vulnerability analysis on power grid is influenced by the initial iteration value and the randomgradient direction (Shahidehpour et al., 2002; Capitanescu, 2011; Capitanescu and Wehenkel, 2012). Conversely, the mixed entropy method will be introduced to cope with the power network vulnerability analysis with large nodes (Phan and Kalagnanam, 2012; MaranoMarcolini et al., 2012; Wang et al., 2013).
The basic idea for using mixed entropy method in network vulnerability analysis is the nonlinear combination of power flow entropy and risk entropy (Ardakani and Bouffard, 2013; Platbrood et al., 2014; Wang et al., 2018). On one hand, network risk entropy plays an important role in assessment on system symmetry and topological structure of the whole network. On the other hand, the power flow entropy is the combination of power flow transfer factor and power flow distribution factor. The former is connected with the branch outage, while the latter is connected with the chain failures. The difference between network risk entropy and power flow entropy is shown in Table 1. Finally, the proposed method of mixed entropy measure of IEEE14 node involved in the study were implemented in the MATPOWER toolbox of the MATLAB platform. The results were presented in the form of data and histograms.
Table 1. Comparison between different entropy
Type of entropy 
Power flow transfer factor 
Power flow distribution factor 
Risk entropy 
Emphasis point 
Potential fault with branch outage 
Outage resistance 
Uncertain of system outage 
Advantage 
Transfer connected with power flow 
Chain failure connected with branch 
Reliability analysis connected with unbalance grid 
Disadvantage 
Reliability analysis is ignored 
Network unbalance is ignored 
Power flow transfer is ignored 
The reminder of this paper is organized as follows. Section 2 presents the methodology introduction for the mixed entropy measure. Section 3 describes the power flow fluctuation of load side and generation side. Simulation and analysis are studied in Section 4. The conclusions are drawn in Section 5.
There are three subsections are made up in this Section. In the first one, the basic entropy theory is described. In the second subsection, the basic concept of IEEE 14 node framework is introduced. Finally, in the third subsection, load fluctuations under three different conditions are discussed.
2.1. Basic theory of entropy on power flow
The definition of entropy is:
$H=\sum\limits_{i=1}^{N}{I{}_{i}\ln }I{}_{i}$ (1)
2.1.1. Power flow entropy Q_{E}
Q_{E} is made up by power flow transfer factor Q_{Ti}and power flow distribution factor Q_{Di. }Q_{E}=Q_{Ti}Q_{D}_{i}
$\Delta {{P}_{ji}}={{P}_{ji}}{{P}_{j0}},j\ne i$ (2)
where ∆P_{ji} is transfer number of branch j to branch i, δ_{ji} is transfer impact rate, E_{T} is the power flow entropy factor.
$\Delta {{P}_{ia}}={{P}_{ia}}{{P}_{i0}}$ (3)
$\Delta {{P}_{a}}=\sum\limits_{i=1}^{N}{({{P}_{ia}}{{P}_{i0}})}$ (4)
${{\delta }_{ia}}=\frac{\Delta {{P}_{ia}}}{\Delta {{P}_{a}}}$ (5)
${{E}_{Dia}}={{\delta }_{ia}}\ln {{\delta }_{ia}}$ (6)
${{E}_{Di}}={{E}_{DiaG}}{{E}_{DiaL}}={{\delta }_{iaL}}\ln {{\delta }_{iaL}}{{\delta }_{iaG}}\ln {{\delta }_{iaG}}$ (7)
${{Q}_{Di}}=\frac{1}{2}\frac{1}{{{N}_{G}}{{N}_{L}}}\sum\limits_{{{a}_{G}}\in G}{\sum\limits_{{{a}_{L}}\in L}{\left\{ {{E}_{Di}}+\max {{E}_{Di}} \right\}}}$ (8)
Where ∆P_{ia} is the increment of power flow, ∆P_{a} is the sum of the increment of power flow for node a, δ_{ia} is the distribution impact rate from node a to branch I, E_{Dia} is the power flow entropy between node a and brunch I, E_{Di} is the sum distribution impact rate of brunch I.
2.1.2. Power flow risk entropy H_{R}
The risk entropy H_{R} and V_{R} between branch j and branch i is defined as:
${{V}_{R}}=\frac{{{H}_{Rj}}{{H}_{\min }}}{{{H}_{\max }}{{H}_{\min }}}$ (9)
$\Delta {{P}_{ij}}=\left {{P}_{ij}}{{P}_{i0}} \right$ (10)
$\Delta {{P}_{j}}=\sum\limits_{i=1}^{L}{\left {{P}_{ij}}{{P}_{i0}} \right}$ (11)
${{\eta }_{ij}}=\frac{\Delta {{P}_{ij}}}{\Delta {{P}_{j}}}$ (12)
${{H}_{Rj}}=\sum\limits_{i=1}^{L}{{{\eta }_{ij}}\ln }{{\eta }_{ij}}$ (13)
where ∆P_{ij}is the real power variation between node j and brunch I, ∆P_{j} is the total real power variation of node j, h_{ij} is the relative change rate between node j and node i, H_{R} is the risk entropy of node i, and V_{R} is the index for reliability analysis.
2.2. power flow calculation of IEEE 14
As is shown in Figure 1, the whole frame of IEEE14 system is made up by 5 generator bus (1,2,3,6,8) and 11 load nodes (2,3,4,5,6,9,10,11,12,13,14), and the active power and real power parameters are shown in table 2 and table 3.
Figure 1. The whole frame of IEEE14 system
Table 2. Power flow calculation parameters of IEEE 14
Node Number 
P(MW) 
Q(MVar) 
S(MVA) 
Node 2 
21.70 
12.70 
25.14 
Node 3 
94.20 
19.00 
96.10 
Node 4 
47.80 
3.90 
47.96 
Node 5 
7.60 
1.60 
7.77 
Node 6 
11.20 
7.50 
13.48 
Node 9 
29.50 
16.60 
33.85 
Node 10 
9.00 
5.80 
10.71 
Node11 
3.50 
1.80 
3.94 
Node12 
6.10 
1.60 
6.31 
Node13 
13.50 
5.80 
14.69 
Node14 
14.90 
5.00 
15.72 
Table 3. Upper limit active power value and real power value of generators
Generator number 
P_{max} 
P_{min} 
Q_{max} 
Q_{min} 
Generator 1 
332.4 
0 
10 
0 
Generator 2 
140 
0 
50 
40 
Generator 3 
100 
0 
40 
0 
Generator 6 
100 
0 
24 
6 
Generator 8 
100 
0 
24 
6 
2.3. load fluctuation under three different conditions
In order to simplify the calculation process results, there are only three load fluctuation conditions considered in this paper. And the generators reactive power constraints are shown in Table 4. And the load fluctuations under different conditions are shown from Table 5 to Table 7. Condition A: the reactive power is increased and the real power is constant. Condition B: the real power is increased and the reactive power is constant. Condition C: the reactive power and the real power are increased in the same proportion.
Table 4. Reactive power constraints of generators

generator1 
generator2 
generator3 
generator6 
generator8 
NODE 2 
11.68* 
47.55 
25.04 
12.72 
17.61 
NODE3 
5.57 
26.44 
68.48* 
12.61 
17.40 
NODE4 
2.85* 
49.60 
33.71 
18.61 
21.26 
NODE5 
14.46* 
43.88 
25.59 
13.45 
17.86 
NODE6 
15.16* 
40.18 
24.32 
16.00 
17.57 
NODE9 
11.74* 
36.09 
23.63 
18.32 
20.22 
NODE10 
15.22* 
40.99 
24.51 
14.41 
18.07 
NODE11 
16.8* 
42.52 
24.85 
13.75 
17.72 
NODE12 
15.78* 
41.73 
24.68 
15.71 
17.68 
NODE13 
14.78* 
39.54 
24.19 
17.88 
17.73 
NODE14 
14.32* 
39.55 
24.30 
17.52 
18.72 
NODE WITH RECATIVE POWER 
53.72* 
33.09 
75.69 
49.15 
27.30 
Table 5. Load fluctuations under condition A
NODE NUMUBER 
RESULTS 
generator 
load fluctuations factor 
NODE2 
bus2 P: 2.169630e+01 Q: 1.270000e+01 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE3 
bus3 P: 9.420303e+01 Q: 19 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE4 
bus4 P: 4.780117e+01 Q: 3.900000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE5 
bus5 P: 7.603479e+00 Q: 600000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE6 
bus6 P: 1.120091e+01 Q: 7.500000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE9 
bus9 P: 2.950021e+01 Q: 1.660000e+01 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE10 
bus10 P: 9.003560e+00 Q: 5.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE11 
bus11 P: 33504797e+00 Q: 1.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE12 
bus12 P:6.103778e+00 Q: 600000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE13 
bus13 P: 1.349652e+01 Q: 5.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE14 
bus14 P: 1.1490364e+01 Q: 5+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE NUMUBER 
RESULTS 
generator 
load fluctuations factor 
NODE2 
bus2 P: 2.17000e+01 Q: 270000e+01 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE3 
bus3 P: 9.420000e+01 Q: 19 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE4 
bus4 P: 4.780000e+01 Q: 3.900000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE5 
bus5 P: 7.600000e+00 Q: 1.600000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE6 
bus6 P: 1.120000e+01 Q: 7.500000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE9 
bus9 P: 2.950000e+01 Q:1.660000e+01 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE10 
bus10 P: 9 Q: 5.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE11 
bus11 P: 3.500000e+00 Q: 1.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE12 
bus12 P: 6.100000e+00 Q: 1.600000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE13 
bus13 P: 1.350000e+01 Q: 5.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE14 
bus14 P: 1.490000e+01 Q: 5 The generator of number 1 has violated Q constraints 
1 
(0~1] 
Table 7. Load fluctuations under condition C
NODE NUMUBER 
RESULTS 
generator 
load fluctuations factor 
NODE2 
bus2 P: 2.170000e+01 Q: 1.270000e+01 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE3 
bus3 P: 9.420000e+01 Q: 19 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE4 
bus4 P: 4.780000e+01 Q: 3.900000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE5 
bus5 P: 7.600000e+01 Q: 1.600000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE6 
bus6 P: 1.120000e+01 Q: 7.500000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE9 
bus9 P: 2.950000e+01 Q: 1.660000e+01 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE10 
bus10 P: 9 Q: 5.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE11 
bus11 P: 3.500000e+00 Q: 1.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE12 
bus12 P: 6.100000e+01 Q: 1.600000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE13 
bus13 P: 1.350000e+01 Q: 5.800000e+00 The generator of number 1 has violated Q constraints 
1 
(0~1] 
NODE14 
bus14 P: 1.490000e+01 Q: 5 The generator of number 1 has violated Q constraints 
1 
(0~1] 
Compared table 3 with table 4 and table 5, we can draw a conclusion that the maximum load fluctuations factor is 1 under three different conditions. Otherwise, reactive power constraints of generators will be happened.
In order to verify the efficiency of mixed entropy measure on IEEE14 network, the different set of experiments are carried out in the MATPOWER toolbox of the MATLAB platform. Moreover the flow charts of power flow transfer entropy and power flow distribution entropy are shown in Figure 2 and Figure 4. Moreover, the experiment results of two entropy methods under MATPOWER toolbox are shown in Figure 3 and Figure 5.
Figure 2. Flow chart of power flow transfer entropy
Compared with the Figure 3 and Figure 5, we can draw the conclusion that the permutation of transfer entropy and distribution entropy are shown in table 810. and the sequence of distribution entropy is shown in fig.6.which means that the node 2 is the vulnerabilities in the whole network. The first set is based on IEEE 14 BUS test system. For a comparison between the mixed entropy method and power flow entropy method (or risk entropy method), the different scheduling order are shown in Fig.6 and Fig.7.As can be seen from the Fig.6(mixed entropy method), the node 3,4,5 have the same risk entropy. However, we can only draw the conclusion that the node 2 is the vulnerabilities in the whole network.
Figure 3. Experiment results of transfer entropy under MATPWOER
Figure 4. Flow chart of power flow of distribution entropy
Figure 5. Experiment results of distribution entropy under MATPWOER
Table 8. Transfer entropy
number 
test1 
test2 
test3 
test4 
test5 
average 
node2 
1.298603 
1.298620 
1.298616 
1.298626 
1.298620 
1.30 
node3 
1.929835 
1.929849 
1.929832 
1.929824 
1.929825 
1.93 
node4 
2.112372 
2.112372 
2.112372 
2.112372 
2.112372 
2.11 
node5 
1.828546 
1.828548 
1.828549 
1.828547 
1.828549 
1.83 
node6 
2.633288 
2.633288 
2.633282 
2.633289 
2.633287 
2.63 
node9 
2.739682 
2.739690 
2.739687 
2.739690 
2.739692 
2.74 
node10 
2.683222 
2.683221 
2.683219 
2.683225 
2.683216 
2.68 
node11 
2.532113 
2.532119 
2.532114 
2.532116 
2.532116 
2.53 
node12 
2.742392 
2.742389 
2.742391 
2.742390 
2.742389 
2.74 
node13 
2.716629 
2.716629 
2.716629 
2.716629 
2.716629 
2.72 
node14 
2.698687 
2.698684 
2.698684 
2.698688 
2.698687 
2.70 
Table 9. Distribution entropy
Number 
test1 
test2 
test3 
test4 
test5 
average 
node2 
0.09355856 
0.08894123 
0.07713753 
0.1428928 
0.1347842 
0.11 
node3 
0.6137061 
0.6127401 
0.6151092 
0.6107501 
0.6131079 
0.61 
node4 
0.7826699 
0.7856685 
0.7851088 
0.7717788 
0.7743651 
0.78 
node5 
0.7570722 
0.7396852 
0.7577844 
0.7399870 
0.7579246 
0.75 
node6 
1.335513 
1.339591 
1.340334 
1.331429 
1.333509 
1.34 
node9 
1.432500 
1.433163 
1.430637 
1.427934 
1.431422 
1.43 
node10 
1.365670 
1.350894 
1.350821 
1.362479 
1.354593 
1.36 
node11 
1.285974 
1.295250 
1.287376 
1.291148 
1.291902 
1.29 
node12 
1.444787 
1.446974 
1.448250 
1.449639 
1.451017 
1.45 
node13 
1.426913 
1.423656 
1.421644 
1.422398 
1.426396 
1.42 
node14 
1.384793 
1.392451 
1.390621 
1.388106 
1.388971 
1.39 
Table 10. Measure of distribution entropy
Number 
2 
3 
4 
5 
6 
9 
10 
11 
12 
13 
14 
Measure 
0.14* 
1.18 
1.65 
1.37 
3.52 
3.92 
3.64 
3.26 
3.97 
3.86 
3.75 
Where * is the maximum value
Figure 6. Sequence of distribution entropy
Figure 7. Sequence of risk entropy
In this paper, the mixed entropy measure method is proposed to analyze the fragility of the fault of IEEE14 nodes. Compared with the power flow entropy method or the risk entropy method, the proposed method has the advantage that the sequence of distribution entropy can be quantized with the number of distribution entropy. On the other hand, the proposed method can achieve better solutions for the same computational effort. Further research work includes that a high performance method for the power network vulnerability analysis with large nodes.
This paper is supported by Natural Science Foundation of Fujian Province under grant (grant number 2016H6019, 2016J01267), in part by Scientific and Technological Projects of Fuzhou City (grant number 2016G53), in part by scientific research project of Xiamen City (grant number 3502Z20189033), in part by the Scientific Research Items of MJU [grant number MJY18003]
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