近年來，由於尖峰用電的需求不斷攀升，以及環保意識抬頭，再生能源
的比例逐漸提高，使得電力網路如何能夠更有智慧. 更有效率的運轉. 調度
之需求已迫在眉睫。自從1990 年代起，相量量測單元(phasor measurement
unit，PMU) 的問世，其高速的資料取樣速率以及透過GPS 同步，使得即
時監控電力網路成為可能。這些相量量測單元帶來的大數據啟發了許多廣
域量測. 保護與控制(wide area monitoring protection and control，WAMPAC)
等題目的探討; 而其中，電力網路的可觀性(observability) 更是實現智慧電
網的重要基石。本論文將探討如何以最符合經濟效益的可觀性配置，並
且同時滿足通訊上. 拓樸上等等的限制條件，如在電力領域當中被廣泛討
論的最佳化相量量測單元佈建問題(optimal PMU placement，OPP)，此一
NPcomplete
的組合最佳化問題。
在本論文中，將探討過去研究最佳化相量量測單元佈建問題的三種主流
演算法:(1) 以圖形理論文基礎的演算法，(2) 人工智慧演算法以及(3) 數學
規劃法。這三種演算法分別各有其優缺點與適合求解的最佳化相量量測單
元佈建問題模型，本論文除了介紹這三種演算法以外，更將結合圖形理論
文基礎的演算法與人工智慧演算法的優點，提出一混合型的二階段演算法，
利用圖形法求得的近似解作為人工智慧演算法的初始解，可大幅節省計算
時間與疊代次數；此外，本論文提出一理論分析以佐證此演算法確實可以
有效得到最佳解。在模型的開發上，過去絕大多數的模型中忽略了無限制
可觀性傳遞的問題，在這些可觀性相依的特性之下，無限制的可觀性傳遞，
將使得系統高度暴露於不可觀的風險之下; 因此，本論文提出了一個全新的
模型，於此模型當中，一樣是求解最符合經濟效益的可觀性配置問題，但同時加入了有限的可觀性傳遞的條件，實驗結果顯示可以有效的提升系統
的可觀性。在分散式能源是未來智慧電網的趨勢的潮流下，我們也探討如
何以樹分解(tree decomposition) 電力網路，並且應用動態規劃演算法於此樹
狀結構的樹分解來求解最佳化智慧電網可觀性配置問題。

With increasing global concerns regarding energy management, the concept of
the smart grid has become a particularly important interdisciplinary research topic.
In order to continually monitor a power utility system and efficiently observe all
of the states of electric nodes and branches on a smart grid, placing phasor measurement
units (PMUs) at selected nodes on the grid can monitor the operation
conditions of the entire power grid. The problem of monitoring a power grid with
minimum installation costs of PMUs can be transformed into the optimal PMU
placement (OPP) problem where the system is monitored under power observation
rules. The combinatorial optimization problem has been shown to be NPcomplete.
In this work, three algorithmic approaches: graphtheoretic
approaches,
metaheuristic
algorithms and mathematical programming (MP) are proposed and
discussed.
A new hybrid twophase
algorithm combining both merits of graphtheoretic
approaches and metaheuristic
algorithms is first proposed for the classical OPP
problem, in which the objective is to simultaneously minimize the number of PMUs
and ensure the complete observability of the whole power grid. The numerical
studies on various IEEE power test systems demonstrate the superior performance
of the proposed algorithm in regard to computational time and solution quality.
When the power grid encounters unexpected contingencies, conventional OPP formulations
with unlimited observability propagations may increase the risk of losing
the observability. In order to overcome this difficulty, a new observabilityenhanced
OPP formulation with considering bounded observability propagation constraints is proposed in by introducing the concept of observability propagation
depth (OPD). The OPD can characterize the depth of observability propagations applied
to observe a bus from its nearest PMU measurements. To further explore the
robustness of measurement systems under severe contingencies, a bilevel optimization
framework with a linearized probabilistic observability model is proposed. In
order to accommodate the distributed control architecture for the Distributed energy
resource (DER), another twophase
solution algorithm is also proposed by
exploring sparsity of largescale
power grids and observability rules derived from
power engineering practices. In the first phase, a tree decomposition techniques is
proposed to decompose the original graph topology of the power grid into a treelike
structure and a dynamic programming (DP) approach is applied to the tree in a
bottomup
manner in the second phase. One advantage of the proposed algorithm
is that it seems to be more flexible since some buses can be recommended to install
PMUs from topology characteristics of the power grid. These proposed algorithms
have shown their effectiveness from both the theoretical and practical perspectives.

[1] A. Aazami, ”Domination in Graphs with Bounded Propagation: Algorithms, Formulation,
and Hardness Results”, J. Comb. Opt., vol. 19, no. 4, pp. 429456,
2010.
[2] A. Aazami and M. D. Stilp, ”Approximation Algorithms and Hardness for Domination
with Propagation”, SIAM J. Discrete Math., 23(3), 1382–1399, 2009.
[3] T. A. Alexopoulos, N. M. Manousakis, and G. N. Korres, ”Fault location observability
using phasor measurements units via semidefinite programming”, IEEE Access, vol. 4,
pp. 5187–5195, 2016.
[4] F. Aminifar, M. FotuhiFiruzabad,
A. Safdarian, A. Davoudi and M. Shahidehpour, ”Synchrophasor
measurement technology in power systems: panorama and stateoftheart”,
IEEE Access, vol. 2, pp. 16071628,
2015.
[5] F. Aminifar, M. FotuhiFiruzabad,
A. Safdarian, and M. Shahidehpour, ”Observability of
hybrid AC/DC power systems with variablecost
PMUs,” IEEE Trans. Power Del., vol.
29, no. 1, pp. 345352,
2014.
[6] F. Aminifar, M. FotuhiFiruzabad,
M. Shahidehpour and A. Khodaei, ”Probabilistic multistage
PMU placement in electric power systems”, IEEE Trans. Power Del., vol. 26, no. 2,
pp. 841–849, 2011.
[7] F. Aminifar, M. FotuhiFiruzabad,
M. Shahidehpour, A. Khodaei, ”Observability enhancement
by optimal PMU placement considering random power system outages”, Energy
Syst., vol. 2, 2011.
[8] F. Aminifar, A. Khodaei, M. FotuhiFiruzabad,
M. Shahidepour, ”ContingencyConstrained
PMU Placement in Power Networks”, IEEE Trans. on Power Systems,
vol. 25, no. 1, 516–523, 2010.
[9] F. Aminifar, C. Lucas, A. Khodaei and M. FotuhiFiruzabad,
”Optimal Placement of Phasor
Measurement Units Using Immunity Genetic Algorithm”, IEEE Trans. on Power Delivery,
24(3), 1014–1020, 2009.
[10] G. Andersson, P. Donalek, R. Farmer, N. Hatziargyriou, I. Kamwa, P. Kundur, N. Martins,
J. Paserba, P. Pourbeik, J. SanchezGasca,
R. Schulz, A. Stankovic, C. Taylor and
V. Vittal, ”Causes of the 2003 Major Grid Blackouts in North America and Europe, and
Recommended Means to Improve System Dynamic Performance”, IEEE Trans. on Power
Systems, vol. 20, no. 4, pp. 1922–1928, Nov. 2005.
[11] B. Appasani, and D. K. Mohanta, ”CoOptimal
Placement of PMUs and Their Communication
Infrastructure for Minimization of Propagation Delay in the WAMS”, IEEE Trans.
Ind. Inform., vol. 14, no.5 pp. 2120–2132, 2018.
[12] S. Azizi, A. S. Dobakhshari, S. A. Sarmadi, and A. M. Ranjbar, ”Optimal PMU Placement
by an Equivalent Linear Formulation for Exhaustive Search”, IEEE Trans. Smart Grid, vol.
3, no. 1, pp. 174–182, Mar. 2012.
[13] T. L. Baldwin, L. Mili, M. B. Boisen Jr. and R. Adapa, ”Power system observability with
minimal phasor measurement placement”, IEEE Trans. on Power System, 8(2), 707–715,
1993.
[14] B. Basturk and D. Karaboga, ”An artificial bee colony (abc) algorithm for numeric function
optimization”, Proc. IEEE Swarm Intelligence Symp. Indianapolis Indiana, 2006.
[15] E. E. Bernabeu, J. S. Thorp and V. A. Centeno, ”Methodology for a security/dependability
adaptive protection scheme based on data mining”, IEEE Trans. Power Del., vol. 27, no.
1, pp. 104111,
2012.
[16] J. Bertsch, C. Carnal, D. Karlsson, J. Cdaniel and K. Vu, ”Widearea
protection and power
system utilization”, Proc. IEEE, 93(5), pp. 997–1003, 2005.
[17] D. J. Brueni and L. S. Heath, ”The PMU placement problem”, SIAM J. Discrete Math.,
19(3), 744–761, 2005.
[18] S. Chakrabarti and E. Kyriakides, ”Optimal placement of phasor measurement units for
power system observability”, IEEE Trans. Power Syst., 23(3), 1433–1440, 2008.
[19] S. Chakrabarti, E. Kyriakides and D. G. Eliades, ”Placement of Synchronized Measurements
for Power System Observability”, IEEE Trans. on Power Delivery, 24(1), 12–19,
2009.
[20] M. L. Chan, W. H. Crouch, Jr., ”An integrated load management, distribution automation
and distribution SCADA system for old dominion electric cooperative”, IEEE Trans. on
Power Delivery, Vol. 5, No. 1, pp. 384–390, 1990.
[21] J. Chen and A. Abur, ”Placement of PMUs to enable bad data detection in state estimation,”
IEEE Trans. Power Syst., vol. 21, no. 4, pp. 1608–1615, 2006.
[22] I. Contreras, E. Fernndez and A. Marn, ”The tree of hubs location problem”, European
Journal of Operational Research, 202(2), 390–400, 2010.
[23] B. Daehyun, G. Michailidis and M. Devetsikiotis, ”Towards improved scalability in smart
grid modeling: Simplifying generator dynamics analysis via spectral graph sparsification”,
Proc. 2011 IEEE First International Workshop on Smart Grid Modeling and Simulation
(SGMS), 2011.
[24] J. E. Dagle, ”North american synchrophasor initiative”, Proc. of the 41th Hawaii Int. Conf.
on System Sciences, 2008.
[25] I. Dobson, ”Voltages across an area of a network”, IEEE Trans. on Power Systems, vol.
27, no. 2, pp. 9931002,
2012.
[26] M. Donnelly, M. Ingram and J. R. Carroll, ”Eastern Interconnection Phasor Project”, Proc.
of the 39th Hawaii Int. Conf. on System Sciences, 336–342, 2006.
[27] D. Dua, S. Dambhare, R. K. Gajbhiye and S. A. Soman, ”Optimal multistage scheduling
of PMU placement: An ILP approach”, IEEE Trans. Power Delivery, 23(4), 1812–1820,
2008.
[28] R. Emami and A. Abur, ”Robust Measurement Design by Placing Synchronized Phasor
Measurements on Network Branches”, IEEE Trans. Power Syst., vol. 25, no. 1, pp. 38–43,
2010.
[29] M. Esmaili, K. Gharani and H. A. Shayanfar, ”Redundant observability PMU placement in
the presence of flow measurements considering contingencies,” IEEE Trans. Power Syst.,
vol. 28, no. 4, pp. 3765–3773, 2013.
[30] European Commission. ”European technology platform SmartGrids: vision and strategy
for Europe’s Electricity Networks of the Future”, available: www.smartgrids.eu
[31] GAMS (General Algebraic Modeling System) Software Package. [Online]. Available:
http://www.gams.com
[32] A. Ghasemkhani, H. Monsef, A. RahimiKian
and A. AnvariMoghaddam,
”Optimal Design
of a Wide Area Measurement System for Improvement of Power Network Monitoring
Using a Dynamic Multiobjective Shortest Path Algorithm”, IEEE Syst. J., vol. 11, no. 4,
pp. 23032314,
Dec. 2017.
[33] K. H. Gjermundr, D. E. Bakken, C. H. Hauser and A. Bose, ”GridStat: A flexible QoSmanaged
data dissemination framework for the power grid”, IEEE Trans. Power Del.,
24(1), 136–143, 2009.
[34] M. Göl and A. Abur, ”A hybrid state estimator for systems with limited number of PMUs”
IEEE trans. Power Syst., vol. 30, no. 3, pp. 1511–1517, 2015.
[35] F. R. Gomez, A. D. Rajapakse, U. D. Annakkage and I. T. Fernando, ”Support vector
machinebased
algorithm for postfault
transient stability status prediction using synchronized
measurements,” IEEE Trans. Power Syst., vol. 26, no. 3, pp. 1474–1483, 2011.
[36] B. Gou, ”Generalized integer linear programming formulation for optimal PMU placement,”
IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1099–1104, 2008.
[37] B. Gou, ”Optimal placement of PMUs by integer linear programming,” IEEE Trans. Power
Syst., vol. 23, no. 3, pp. 1525–1526, 2008.
[38] B. Gou and A. Abur, ”A direct numerical method for observability analysis”, IEEE Trans.
Power Syst., vol. 15, no. 2, pp. 625630,
2000.
[39] J. Guo, R. Niedermeier, and D. Raible, “Improved Algorithms and Complexity Results
for Power Domination in Graphs,＂Algorithmica, vol. 52, no. 2, pp. 177–202, 2008.
[40] X. C. Guo, C. S. Liao, and C. C. Chu, ”Optimal PMU placements under propagation depth
constraints by mixed integer linear programming”, Proc. 7th IEEE Int. Conf. Smart Grid
Commun. (SmartGridComm), 2016.
[41] X. C. Guo, C. S. Liao, and C. C. Chu, ”Enhanced Optimal PMU Placements Limited
Observability Propagations”, IEEE Access, vol. 8, pp.22515–22524, 2020.
[42] I. Gurobi Optimization, 2012[Online]. Gurobi Optimizer reference manual. Avaialable:
http://www.gurobi.com.
[43] M. Hajian, A. M. Ranjbar, T. Amraee and B. Mozafari, ”Optimal placement of PMUs to
maintain network observability using a modified BPSO algorithm”, Int. J. Elec. Power
Energy Syst. vol. 33, no. 1, pp. 28–34, 2011.
[44] T. W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, and M.A. Henning, ”Domination in
graphs applied to electric power networks”, SIAM J. Discrete Math., vol. 15, no. 4, pp.
519–529, 2002.
[45] L. Huang, Y. Sun, J. Xu, W. Gao, J. Zhang and Z. Wu, ”Optimal PMU placement considering
controlled islanding of power system”, IEEE Trans. Power Syst., vol. 29, no. 2, pp.
742–755, Mar. 2014.
[46] I. Kamwa, R. Grondin and Y. Hebert, ”Widearea
measurement based stabilizing control
of large power systemsa
decentralized/hierarchical approach,” IEEE Trans. Power Syst.,
vol. 16, no. 1, pp. 136–153, 2001.
[47] P. Kansal and A. Bose, ”Bandwidth and latency requirements for smart transmission grid
applications”, IEEE Trans. on Smart Grid, vol. 3, no. 3, pp. 13441352,
2012.
[48] D. Karaboga and B. Akay, ”A Comparative Study of Artificial Bee Colony Algorithm”,
Applied Mathematics and Computation, 108–132, 2009.
[49] R. Kateb, P. Akaber, M. Tushar, A. ALBarakati,
M. Debbabi, and C. Assi, ”Enhancing
WAMS communication network against delay attacks”, IEEE Trans. Smart Grid, to be
published.
[50] R. Kavasseri and S. K. Srinivasan, ”Joint placement of phasor and power flow measurements
for observability of power systems”, IEEE Trans. on Power Systems, vol. 26, no. 4,
pp. 1929–
1936, 2011.
[51] V. Kekatos, G. B. Giannakis and B. Wollenberg, ”Optimal Placement of Phasor Measurement
Units via Convex Relaxation”, IEEE Trans. Power Syst., vol. 27, no. 3, pp. 1521–
1530, 2012.
[52] K. G. Khajeh, E. Bashar, A. M. Rad, and G. B. Gharehpetian ”Integrated model considering
effects of zero injection buses and conventional measurements on optimal PMU
placement”, IEEE Trans. Smart Grid, vol, 8, no. 2, pp. 1006–1013, 2017.
[53] V. Khiabani, O. P. Yadav and R. Kavesseri, ”Reliabilitybased
placement of phasor measurement
units in power systems”, J. Risk Reliab., vol. 226, no. 1, pp.109–117, 2012.
[54] J. Kneis, D. Molle, S. Richter and P. Rossmanith, ”Parameterized power domination complexity”,
Inform. Proc. Letters 98(4), 145–149, 2006.
[55] A. Konak, ”Network design problem with relays: a genetic algorithm with a pathbased
crossover and a set covering formulation”, European Journal of Operational Research,
218(3), 829–837, 2012.
[56] M. Korkali and A. Abur, ”Placement of PMUs with channel limits”, Proc. IEEE Power
Eng. Soc. General Meeting, 2009.
[57] M. Korkali and A. Abur, ”Impact of network sparsity on strategic placement of phasor
measurement units with fixed channel capacity,” Proc. IEEE Int. Symp. on Circuits and
Syst., pp. 34453448,
2010.
[58] G. N. Korres, N. M. Manousakis, T. C. Xygkis, and J. Löfberg, ”Optimal phasor measurement
unit placement for numerical observability in the presence of conventional measurements
using semidefinite programming,” IET Gener. Transm. Distrib., vol. 9, no. 15, pp.
2427–2436, 2015.
[59] N. C. Koutsoukis, N. M. Manousakis, P. S. Georgilakis, G. N. Korres, ”Numerical observability
method for optimal phasor measurement units placement using recursive Tabu
search method”, IET Generat. Transmiss. Distrib., vol. 7, no. 4, pp. 347356,
2013.
[60] G.R. Krumpholz, K.A. Clements, and P.W. Davis, ”Power System Observability: A Practical
Algorithm Using Network Topology”, IEEE Trans. on Power App. and Syst., vol.
PAS99,
pp. 15341542,
1980.
[61] W. T. Li, C. K. Wen, J. C. Chen, K. K. Wong, J. H. Teng, and C. Yuen, ”Location identification
of power line outages using PMU measurements with bad data”, IEEE Trans. on
Power Systems, vol. 31, no. 5, pp. 3624 3635,
2015.
[62] C. S. Liao, ”Power domination with bounded time constraints” Journal of Combinatorial
Optimization, vol. 31, pp. 725–742,
2016.
[63] C. S. Liao, T. J. Hsieh, X. C. Guo, J. H. Liu, and C. C. Chu, ”Hybrid search for the optimal
PMU placement problem on a power grid”, Eur. J. Oper. Res., vol. 243, no. 3, pp.985–994,
2015.
[64] C. Lu, Z. Wang, M. Ma, R. Shen and Y. Yu, ”An Optimal PMU Placement With Reliable
Zero Injection Observation”, IEEE Access, vol. 6, pp. 54417–54426, 2018.
[65] T. K. Maji and P. Acharjee, ”A prioritybased
multistage PMU installation approach for
direct observability of all network buses”, IEEE Systems Journal, vol. pp, no. 99, pp. 1–9,
2018.
[66] N. M. Manousakis and G. N. Korres, ”A weighted least squares algorithm for optimal
PMU placement”, IEEE Trans. Power Syst., vol. 28, no. 3, pp. 34993500,
2013.
[67] N. M. Manousakis, G. N. Korres and P. S. Georgilakis, ”Taxonomy of PMU placement
methodologies”, IEEE Trans. Power Syst., vol. 27, no. 2, pp. 10701077,
2012.
[68] R. Marques , J. Rocha , S. Rafael and J. F. Martins ”Design and implementation of a
reconfigurable remote laboratory, using oscilloscope/PLC network for WWW access”,
IEEE Trans. Ind. Electron., vol. 55, no. 6, pp.2425–2432, 2008.
[69] F. J. Martin, F. GarciaLagos,
G. Joya and F. Sandoval, ”Genetic algorithms for optimal
placement of phasor measurement units in electric networks”, IEEE Electronics Letters,
39(19), 1403–1405, 2003.
[70] S. M. Mazhari, H. Monsef, H. Lesani and A. Fereidunian, ”A MultiObjective
PMU
Placement Method Considering Measurement Redundancy and Observability Value Under
Contingencies”, IEEE Trans. Power Syst., vol. 28, no. 3, pp. 2136–2146, 2013.
[71] B. Milosevic and M. Begovic, ”Nondominated sorting genetic algorithm for optimal phasor
measurement placement”, IEEE Trans. on Power Systems, 18(1), 69–75, 2003.
[72] M. NazariHeris
and B. MohammadiIvatloo,
”Application of heuristic algorithms to optimal
PMU placement in electric power systems: An updated review”, Renewable and
Sust. Eng. Reviews, vol. 50, pp. 214–228, 2015.
[73] M. NazariHeris
and B. MohammadiIvatloo,
”Optimal placement of phasor measurement
units to attain power system observability utilizing an upgraded binary harmony search
algorithm”, Energy Systems, vol. 6, issue 2, pp. 201–220, 2015.
[74] M. K. Neyestanaki and A. M. Ranjbar, ”An adaptive PMUbased
wide area backup protection
scheme for power transmission lines”, IEEE Trans. Smart Grid, vol. 6, no. 3, pp.
15501559,
May 2015.
[75] R.F. Nuqui and A.G. Phadke, ”Phasor Measurement Unit Placement Techniques for Complete
and Incomplete Observability”, IEEE Trans. Power Del., vol. 20, no. 4, pp. 2381–
2388, 2005.
[76] A. Pal, G. A. SanchezAyala,
V. A. Centeno, and J. S. Thorp, ”A PMU placement scheme
ensuring realtime
monitoring of critical buses of the network”, IEEE Trans. Power Del.,
vol. 29, no. 2, pp. 510517,
2014.
[77] C. Peng, H. Sun and J. Guo, ”Multiobjective
optimal PMU placement using a nondominated
sorting differential evolution algorithm”, Int. J. Elec. Power and Energy Systems,
32(8), 886–892, 2010.
[78] J. Peng, Y. Sun and H. F. Wang, ”Optimal PMU placement for full network observability
using Tabu search algorithm”, Int. J. Elec. Power and Energy Systems, 28, 223–231, 2006.
[79] A. G. Phadke, ”Synchronized phasor measurements in power systems”, IEEE Computer
Applications in Power, vol. 6, no. 2, pp. 10–15, 1993.
[80] A. G. Phadke and J.S. Thorp, ”Synchronized Phasor Measurements and Their Applications”,
Springer, New York, 2008.
[81] Power systems test case archive. [Online]. Available:
http://www.ee.washington.edu/research/pstca/.
[82] D. Raible and H. Fernau, ”An Exact Exponential Time Algorithm for Power Dominating
Set”, Algorithmica, 63, 323–346, 2012.
[83] C. Rakpenthai, S. Premrudeepreechacharn, S. Uatrongjit, and N. R. Watson, ”An optimal
PMU placement method against measurement loss and branch outage,” IEEE Trans. on
Power Del., vol. 22, no. 1, pp. 101–107, Jan. 2007.
[84] Z. H. Rather, Z. Chen, P. Thogerson, P. Lund and B. Kirby, ”Realistic approach for phasor
measurement unit placement: consideration of practical hidden costs”, IEEE Trans. Power
Del., vol. 30, no. 1, pp. 315,
Feb. 2015.
[85] M. Sarailoo and N. Eva Wu, ”CostEffective
Upgrade of PMU Networks for FaultTolerant
Sensing”, IEEE Trans. Power Syst., vol. 33, no. 3, pp. 3052–3063, 2018.
[86] A. Sharma, S. C. Srivastava and S. Chakrabarti, ”An iterative multiarea state estimation
approach using area slack bus adjustment”, IEEE Syst. J., vol. 10, no. 1, pp. 6977,
2014.
[87] K. Sun, S. Likhate, V. Vittal, V. S. Kolluri and S. Mandal, ”An online dynamic security
assessment scheme using phasor measurements and decision trees”, IEEE Trans. Power
Syst., vol. 22, no. 4, pp. 19351943,
2007.
[88] W. Y. Szeto, Y. Wu and S. C. Ho, ”An artificial bee colony algorithm for the capacitated
vehicle routing problem”, European Journal of Operational Research, 215, 126–135, 2011.
[89] V. Terzija, G. Valverde, D. Cai, P. Regulski, V. Madani, J. Fitch, S. Skok, M. M. Begovic,
and A.G. Phadke, ”WideArea
Monitoring, Protection, and Control of Future Electric
Power Networks”, In Proceedings of the IEEE, vol. 99, no. 1, pp. 80–93, 2011.
[90] B. Xu, and A. Abur, ”Observability analysis and measurement placement for systems with
PMUs”, proc. IEEE Power Eng. Soc. Power Systems Conf. Expo., pp. 943946,
2004.
[91] M. Zhao, L. Kang and G. J. Chang, ”Power domination in graphs”, Discrete Math. 306(15),
1812–1816, 2006.