Multi-region dynamic economic dispatch of solar–wind–hydro–thermal power system incorporating pumped hydro energy storage

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Multi-region dynamic economic dispatch of solar–wind–hydro–thermal power system incorporating pumped hydro energy storage✩ M. Basu Department of Power Engineering, Jadavpur University, Kolkata 700098, India

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Keywords: Cascaded reservoirs Wind power uncertainty Solar power uncertainty Pumped hydro energy storage Ramp-rate limits Proscribed workable area

ABSTRACT This paper suggests chaotic fast convergence evolutionary programming (CFCEP) for solving multi-region dynamic economic dispatch (MRDED) problem, multi-region economic dispatch (MRED) problem and economic dispatch (ED) problem. MRDED problem is based on multi-reservoir cascaded hydro plant with time delay, thermal plants with nonsmooth fuel cost function, wind and solar power units with uncertainty and pumped hydro energy storage. MRED problem deals with tie line constraints, transmission losses, valve point effect and proscribed workable area of thermal generators. ED problem deals with valve point effect, proscribed workable area and ramp rate limits of thermal generators. In the recommended technique, chaotic sequences have been pertained for acquiring the dynamic scaling factor setting in fast convergence evolutionary programming (FCEP). The efficacy of the proposed technique has been verified on convoluted three-area system for MRDED problem, four-area system for MRED problem and 140-unit Korean system for ED problem. Test results acquired from the suggested CFCEP technique have been fit to that acquired from FCEP, differential evolution (DE) and particle swarm optimization (PSO). It has been observed from the comparison that the recommended CFCEP technique has the capability to bestow with better-quality solution.

1. Introduction Economic dispatch (ED) seeks out the generation of all dedicated generators most cost-effectively at the same time fulfilling a variety of physical and operational constraints in a single area system. Generally, generators are segregated into a number of generation areas interconnected by tie-lines. Multi-region economic dispatch (MRED) is an escalation of single region economic dispatch. MRED seeks out the generation level and interchange power between areas to minimize cost in all regions at the same time fulfilling power balance constraints, generating limits constraints and tie-line capacity constraints. A variety of techniques (Shoults et al., 1980; Romano et al., 1981; Helmick and Shoults, 1985; Wang and Shahidehpour, 1992; Streiffert, 1995; Yalcinoz and Short, 1998; Jayabarathi et al., 2000; Chen and Chen, 2001; Manoharan et al., 2009; Wang and Singh, 2009; Sharma et al., 2011; Somasundaram and Jothi Swaroopan, 2011; Ghasemi et al., 2016) have been discussed for solving MRED problem. Multi-region dynamic economic dispatch (MRDED) is an escalation of multi-region economic dispatch problem. MRDED seeks out the generation level of dedicated generators and interchange power between areas with the forecasted load demands over a certain period of time so as to minimize cost in all areas at the same time fulfilling all operational constraints. Multi-region dynamic economic dispatch (MADED) with

renewable energy resources has been discussed in Soroudi and Rabiee (2013). The hydrothermal scheduling problem has been explored for a number of decades. Most of the techniques that have been utilized for solving the hydrothermal coordination problem generate a number of simplifying suppositions to aid the optimization problem well broughtup. Various classical techniques, Nilsson and Sjelvgren (1996), Ferrero et al. (1998), Yang and Chen (1989), Wood and Wollenberg (1996), Xia et al. (1988), Braennlund et al. (1986), Pereira and Pinto (1982), Yan et al. (1993), Al-Agtash and Renjeng (1998) and Ruzic and Rajakovic (1998) have been effectively utilized for solving this problem. With the emergence of evolutionary computation techniques, attention has been gradually shifted to significance of such technologybased approaches to knob the difficulty engrossed in actual world problems. Stochastic search algorithms such as simulated annealing technique (Wong and Wong, 1994), evolutionary programming technique (Yang et al., 1996), (Sinha et al., 2003), genetic algorithm (Orero and Irving, 1998; Gil and Bustos, 2003), differential evolution (Lakshminarasimman and Subramanian, 2006), particle swarm optimization (Hota et al., 2009), clonal selection algorithm (Swain et al., 2011), teaching learning based optimization (Roy, 2013) etc. have been pertained for optimal hydrothermal scheduling problem and eluded the

✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.engappai.2019.09.001. E-mail address: [email protected].

https://doi.org/10.1016/j.engappai.2019.09.001 Received 25 October 2018; Received in revised form 17 June 2019; Accepted 2 September 2019 Available online xxxx 0952-1976/© 2019 Elsevier Ltd. All rights reserved.

M. Basu

Nomenclature ( ) 𝑓𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗 𝑡 𝑊𝑖𝑗,𝑎𝑣 max Pmin 𝑤𝑖𝑗 , P𝑤𝑖𝑗

P𝑤𝑟𝑖𝑗 N𝑤𝑖 K𝑤𝑖𝑗 KP𝑤𝑖𝑗 K𝑟𝑤𝑖𝑗 𝑘𝑠 𝑐 𝑣𝑖𝑛 𝑣𝑟

( ) 𝑓𝑃 𝑉 𝑖𝑗 P𝑡𝑃 𝑉 𝑖𝑗 P𝑡𝑃 𝑉 𝑖𝑗 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 P𝑃 𝑉 𝑟𝑖𝑗 𝐺 𝐺𝑠𝑡𝑑 𝑅𝑐 K𝑠𝑖𝑗 KP𝑠𝑖𝑗 K𝑟𝑠𝑖𝑗 N𝑃 𝑉 𝑖 P𝑡𝑔ℎ𝑖 P𝑡𝑝ℎ𝑖 Pmin , Pmax 𝑔ℎ𝑖 𝑔ℎ𝑖 Pmin , Pmax 𝑝ℎ𝑖 𝑝ℎ𝑖 ( ) 𝑄𝑡𝑔ℎ𝑖 P𝑡𝑔ℎ𝑖 ( ) 𝑄𝑡𝑝ℎ𝑖 P𝑡𝑝ℎ𝑖 𝑡 𝑉𝑟𝑒𝑠,𝑖

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

min , 𝑉 max 𝑉𝑟𝑒𝑠,𝑖 𝑟𝑒𝑠,𝑖

Cost function of 𝑗th wind generator in area 𝑖 at time 𝑡 Scheduled power output of 𝑗th wind generator in area 𝑖 at time 𝑡 Available wind power of 𝑗th wind power generator in area 𝑖 at time 𝑡 Lower and upper generation limits for 𝑗th wind power generator in area 𝑖 Rated wind power of 𝑗th wind generator in area 𝑖 Number of committed wind generators in area 𝑖 Direct cost coefficient of 𝑗th wind power generator in area 𝑖 Penalty cost coefficient of 𝑗th wind power generator in area 𝑖 Reserve cost coefficient of the 𝑗th wind power generator in area 𝑖 Shape factor at a given location Scale factor at a given location Cut in wind speed Rated wind speed

𝑠𝑡𝑎𝑟𝑡 , 𝑉 𝑒𝑛𝑑 𝑉𝑟𝑒𝑠,𝑖 𝑟𝑒𝑠,𝑖

( ) 𝑓𝑠𝑖𝑗 P𝑡𝑠𝑖𝑗

Minimum and maximum upper reservoir storage limits of pumped storage plant in area 𝑖 Specified starting and final stored water volumes in upper reservoir of pumped storage plant in area 𝑖

Cost function of 𝑗th thermal generator in area 𝑖 at time 𝑡 P𝑡𝑠𝑖𝑗 Power output of 𝑗th thermal generator in area 𝑖 at time 𝑡 max Lower and upper generation limits for 𝑗th Pmin , P 𝑠𝑖𝑗 𝑠𝑖𝑗 thermal generator in area 𝑖 N𝑠𝑖 Number of committed thermal generators in area 𝑖 𝑎𝑠𝑖𝑗 , 𝑏𝑠𝑖𝑗 , 𝑐𝑠𝑖𝑗 , 𝑑𝑠𝑖𝑗 , 𝑒𝑠𝑖𝑗 Cost coefficients of 𝑗th thermal generator in area 𝑖 𝑈 𝑅𝑖𝑗 , 𝐷𝑅𝑖𝑗 :, Ramp-up rate limit and ramp-down rate limit of 𝑗th thermal generator in area 𝑖 𝐶1𝑖𝑗 , 𝐶2𝑖𝑗 , 𝐶3𝑖𝑗 , 𝐶4𝑖𝑗 , 𝐶5𝑖𝑗 , 𝐶6𝑖𝑗 Power generation coefficients of 𝑗th hydro unit in area 𝑖 I𝑡ℎ𝑖𝑗 Inflow rate of 𝑗th reservoir in area 𝑖 at time 𝑡 𝑄𝑡ℎ𝑖𝑗 Water discharge rate of 𝑗th reservoir in area 𝑖 at time 𝑡 max , 𝑄 Minimum and maximum water discharge 𝑄min ℎ𝑖𝑗 ℎ𝑖𝑗 rate of 𝑗th reservoir in area 𝑖 𝑅𝑢𝑖𝑗 Number of upstream units directly above 𝑗th hydro plant in area 𝑖 𝑡 𝑆ℎ𝑖𝑗 Spillage of 𝑗th reservoir in area 𝑖 at time 𝑡 𝜏𝑖𝑙𝑗 Water transport delay from reservoir 𝑙 to 𝑗 in area 𝑖 𝑡 𝑉ℎ𝑖𝑗 Storage volume of 𝑗th reservoir in area 𝑖 at time 𝑡 min , 𝑉 max Minimum and maximum storage volume of 𝑉ℎ𝑖𝑗 ℎ𝑖𝑗 𝑗th reservoir in area 𝑖 0 𝑉ℎ𝑖𝑗 Initial storage volume of 𝑗th reservoir in area 𝑖 T 𝑉ℎ𝑖𝑗 Final storage volume of 𝑗th reservoir in area 𝑖 P𝑡ℎ𝑖𝑗 Output power of 𝑗th hydro unit in area 𝑖 at time 𝑡 Pmin , Pmax Lower and upper generation limits for 𝑗th ℎ𝑖𝑗 ℎ𝑖𝑗 hydro unit in area 𝑖 Nℎ𝑖 ∶ Number of committed hydro generating units in area 𝑖 M Number of areas P𝑡𝐷𝑖 Power demand of area 𝑖 at time 𝑡 P𝑡𝐿𝑖 Transmission line loss in area 𝑖 at time 𝑡 T𝑡𝑖𝑙 Tie line real power transfer from area 𝑖 to area 𝑙 at time 𝑡 B𝑖𝑗 Transmission loss coefficient 𝑡, T Time index and scheduling period T𝑔𝑒𝑛 Set that contains all time intervals where pumped storage plant operated in generation mode T𝑝𝑢𝑚𝑝 Set that contains all time intervals where pumped storage plant operated in pumping mode

Cost function of 𝑗th solar PV plant in area 𝑖 at time 𝑡 Scheduled power output of 𝑗th solar PV plant in area 𝑖 at time 𝑡 Power output from 𝑗th solar plant in area 𝑖 at time 𝑡 Rated power output of 𝑗th solar PV plant in area 𝑖 Solar irradiation forecast Solar irradiation in the standard environment A certain irradiation point. Direct cost coefficient for 𝑗th solar PV plant in area 𝑖 Penalty cost coefficient of 𝑗th solar PV plant in area 𝑖 Reserve cost coefficient of the 𝑗th solar PV plant in area 𝑖 Number of committed solar PV plants in area 𝑖 Power generation of pumped storage plant in area 𝑖 at time 𝑡 Pumping power of pumped storage plant at time 𝑡 Minimum and maximum power generation limits of pumped storage plant in area 𝑖 Minimum and maximum pumping power limits of pumped storage plant in area 𝑖 Discharge rate of pumped storage plant in area 𝑖 at time 𝑡 Pumping rate of pumped storage plant in area 𝑖 at time 𝑡 Water volume in upper reservoir of pumped storage plant in area 𝑖 at time 𝑡

limitations. Hydrothermal scheduling integrating wind power has been discussed in Dubey et al. (2016) and hydrothermal scheduling integrating solar power has been discussed in Singh Patwal et al. (2018).

The quick increase of electric power demand, day by day depletion of fossil fuel and the global warming caused by fossil fuel fired electric power plants have shoved energy based research in the direction of 183

M. Basu

T𝑐ℎ𝑎𝑛𝑔𝑒_𝑜𝑣𝑒𝑟

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

This paper suggests chaotic fast convergence evolutionary programming (CFCEP) for solving multi-region dynamic economic dispatch (MRDED) problem comprising solar–wind–hydro–thermal power system incorporating pumped hydro energy storage, multi-region economic dispatch (MRED) problem and economic dispatch (ED) problem. Each area of MRDED problem comprises multi-reservoir cascaded hydro plant with time delay, thermal plants with nonsmooth fuel cost functions, wind power generating units with wind power uncertainty, solar PV plant with solar power uncertainty and pumped hydro energy storage. Three-area test system for MRDED, four-area test system for MRED and 140-unit Korean system for ED is exploited here. Simulation outcomes have been matched up to those acquired by fast convergence evolutionary programming (FCEP), differential evolution (DE) and particle swarm optimization (PSO). It has been observed from the comparison that the developed CFCEP gives better solution.

Set that contains all time intervals where pumped storage plant operated in idle mode i.e. in between generating mode and pumping mode

utilization of green energy across the globe. Due to rising concern on climate change and clean energy, solar and wind power are gaining acceptance for meeting energy demand at low cost without any harmful emissions. The incorporation of climate-driven electric power sources i.e. solar and wind power sources has upshot in larger uncertainties. Solar irradiation and wind velocity are uncertain and their availability is immaterial to the load variation. The variability and intermittency of these resources produce significant challenges to be trounced in the generation scheduling problem. This blinking nature may have harmful effect on the entire grid. This can be trounced by integrating pumped hydro energy storage which alleviates fluctuations in generation and supply. The pumped-storage-hydraulic (PSH) unit is acquiring the enormous concentration throughout the globe (Perez-Diaz and Jim, 2016) mostly because of its energy storage feature. The major role of pumpedstorage hydraulic (PSH) units (Fadil and Urazel, 2013) in electric power systems is to hoard low-cost surplus electric energy that is obtainable during off-peak load levels as hydraulic potential energy which is done by pumping water from the lower reservoir of the unit into its upper reservoir. The stored hydraulic potential energy is then used to generate electric energy during peak load levels. The PSH unit is usually worked over daily or weekly periods. Operation of a PSH unit over a period can decrease the total fuel cost in a power system. Lagrangian multiplier and gradient search techniques (Wood and Wollenberg, 1984) is used to find the optimum hydrothermal generation scheduling with pumped-storage-hydraulic unit under practical constraints. Khandualo et al. (2013) have discussed evolutionary programming technique for solving the generation/pumping scheduling problem of hydrothermal system with pumped storage plants. Ma et al. (2015) have discussed the pumped hydro storage system for solar energy infiltration and mainly for small autonomous systems in remote areas. Evolutionary algorithms are populace-based self-adaptive parallel searching techniques. Evolutionary programming (EP) is one of the most reliable evolutionary algorithms based on the human inbred chromosome operation. It creates the global or close to the global optima of an optimization problem by generating many populaces over a number of iterations. EP has three phases which are initialization, creation of off-spring by mutation and competition and selection. In fast convergence evolutionary programming (FCEP) (Basu, 2017), Gaussian and Cauchy mutations to create offspring and one-to-one challenge are initiated in evolutionary programming (EP) to augment the convergence speed and quality of solution. However, FCEP has some disadvantage. The triumphant application of FCEP mostly depends on its scaling factor which is constant all through the whole search procedure. So, it is tricky to decide appropriate value of scaling factor in FCEP without the fine-tuning development. In recent years, a number of triumphant applications of evolutionary algorithms jointed by chaotic sequences in optimization have been described in Caponetto and Fortuna (2003). It is observed that chaotic sequences utilized in evolutionary algorithms are capable for raising the exploitation ability of the evolutionary algorithm in the searching space and boost the convergence property due to the ergodicity, stochastic chattels and irregularity of the chaotic technique. In turn to shun the disadvantage of FCEP, chaotic fast convergence evolutionary programming (CFCEP) rooted in Tent equation for acquiring dynamic scaling factor control method is recommended.

2. Problem formulation 2.1. Multi-region dynamic economic dispatch The objective of multi-region dynamic economic dispatch (MRDED) problem of solar–wind–hydro–thermal power system incorporating pumped hydro energy storage is devised to minimize the total cost of supplying loads to all areas while fulfilling both equality and inequality constraints. 2.1.1. Objective function With the insignificant marginal cost of hydroelectric plant, operational cost of a solar–wind–hydro–thermal system with pumped hydro energy storage essentially reduces to that of the fuel cost for thermal plants along with the cost of wind power generating units and solar PV plants. The total cost can be stated as 𝐹𝐶 =

[ M {N T 𝑠𝑖 ∑ ∑ ∑ 𝑡=1

𝑖=1

N𝑤𝑖 𝑃𝑉 𝑖 ( ) ∑ ( ) N∑ ( ) 𝑓𝑠𝑖𝑗 P𝑡𝑠𝑖𝑗 + 𝑓𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗 + 𝑓𝑃 𝑉 𝑖𝑗 P𝑡𝑃 𝑉 𝑖𝑗

𝑗=1

𝑗=1

}]

𝑗=1

(1) The fuel cost function of 𝑗th committed thermal generator in area 𝑖 at 𝑡th time taking into consideration the valve-point effect (Walters and Sheble, 1993), is stated as ( )2 { ( )}| ( ) | + ||𝑑𝑠𝑖𝑗 × sin 𝑒𝑠𝑖𝑗 × Pmin − P𝑡𝑠𝑖𝑗 || 𝑓𝑠𝑖𝑗 P𝑡𝑠𝑖𝑗 = 𝑎𝑠𝑖𝑗 + 𝑏𝑠𝑖𝑗 P𝑡𝑠𝑖𝑗 + 𝑐𝑠𝑖𝑗 P𝑡𝑠𝑖𝑗 𝑠𝑖𝑗 | |

(2) The cost of wind power comprises three terms, a direct cost, an under estimation penalty cost (𝐶𝑝𝑖𝑗 ) for not using all the available wind power and a reserve cost (𝐶𝑟𝑖𝑗 ) due to over estimation of wind power when available wind power is less than the scheduled wind power. So the wind power cost of 𝑗th wind power generating unit in area 𝑖 at 𝑡th time can be calculated as (Hetzer et al., 2008): ( ) {( ) ( ) 𝑡 𝑓𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗 = K𝑤𝑖𝑗 × P𝑡𝑤𝑖𝑗 + 𝐶𝑝𝑤𝑖𝑗 𝑊𝑖𝑗,𝑎𝑣 − P𝑡𝑤𝑖𝑗 ( )} 𝑡 +𝐶𝑟𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗 − 𝑊𝑖𝑗,𝑎𝑣 (3) ( ) 𝑡 𝐶𝑝𝑤𝑖𝑗 𝑊𝑖𝑗,𝑎𝑣 − P𝑡𝑤𝑖𝑗 = KP𝑤𝑖𝑗 × ( ) 𝑡 𝐶𝑟𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗 − 𝑊𝑖𝑗,𝑎𝑣 = K𝑟𝑤𝑖𝑗 ×

P𝑤𝑟𝑘

∫P𝑡

𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗

∫0

( ) 𝑤 − P𝑡𝑤𝑖𝑗 𝑓𝑤 (𝑤) 𝑑𝑤 (

) P𝑡𝑤𝑖𝑗 − 𝑤 𝑓𝑤 (𝑤) 𝑑𝑤

( ) ) 𝑘 −1 ⎧ ⎡( ⎫ ⎡ ⎤ 𝑠 𝑘𝑠 ⎤ ⎪ ⎢ 1 + ℎ𝑤 ⎪ 1 + Pℎ𝑤 𝑣𝑖𝑛 ⎥ 𝑣 𝑡 ⎢ ⎥ 𝑖𝑛 P𝑤𝑖𝑗 𝑤𝑟𝑖𝑗 𝑘𝑠 ℎ𝑣𝑖𝑛 ⎢ ⎪ ⎢ ⎪ ⎥ ⎥ 𝑓𝑤 (𝑤) = × exp ⎨− ⎬ ⎥ ⎢ ⎥ P𝑤𝑟 𝑐 ⎢ 𝑐 𝑐 ⎪ ⎢ ⎪ ⎢ ⎥ ⎥ ⎪ ⎣ ⎣ ⎦ ⎦⎪ ⎩ ⎭ 184

(4)

(5)

(6)

M. Basu

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

Solar power model The power output (Liang and Liao, 2007) from PV cell is expressed by

The wind power characterization is done by utilizing Weibul pdf, 𝑣 𝑓𝑤 (𝑤). Here ℎ = 𝑣 𝑟 − 1. Detail description can be found in Hetzer 𝑖𝑛 et al. (2008). The cost of solar power (Liang and Liao, 2007) comprises three terms, a direct cost, an under estimation penalty cost (𝐶𝑝𝑠𝑖𝑗 ) for not using all the available solar power and a reserve cost (𝐶𝑟𝑠𝑖𝑗 ) due to over estimation of solar power when available solar power is less than the scheduled solar power. The solar power characterization is done by utilizing lognormal pdf (Tian-Pau, 2010). So the solar power cost of 𝑗th solar power plant in area 𝑖 at 𝑡th time can be calculated as ( ) {( ) 𝑓𝑃 𝑉 𝑖𝑗 P𝑡𝑃 𝑉 𝑖𝑗 = K𝑠𝑖𝑗 × P𝑡𝑃 𝑉 𝑖𝑗 ( ) ( )} +𝐶𝑝𝑠𝑖𝑗 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 − P𝑡𝑃 𝑉 𝑖𝑗 + 𝐶𝑟𝑠𝑖𝑗 P𝑡𝑃 𝑉 𝑖𝑗 − P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗

(

) 𝐺2 , for 0 < 𝐺 < 𝑅𝑐 𝐺𝑠𝑡𝑑 𝑅𝑐 ( ) 𝐺 = P𝑃 𝑉 𝑟𝑖𝑗 , for 𝐺 > 𝑅𝑐 𝐺𝑠𝑡𝑑

P𝑡𝑃 𝑉 𝑖𝑗 = P𝑃 𝑉 𝑟𝑖𝑗 P𝑡𝑃 𝑉 𝑖𝑗

(15)

Total transmission loss P𝑡𝐿𝑖 can be computed by using B-coefficient stated as P𝑡𝐿𝑖 =

N𝑡 N𝑡 ∑ ∑

P𝑡𝑖𝑘 B𝑖𝑗 P𝑡𝑖𝑗 +

𝑘=1 𝑗=1

N𝑡 ∑

B0𝑖 P𝑡𝑖𝑗 + B00

(16)

𝑗=1

Here, total number of plants N𝑡𝑖 = N𝑠𝑖 + Nℎ𝑖 + N𝑤𝑖 + N𝑃 𝑉 𝑖 and P𝑡𝑖𝑗 is the respective thermal, hydro, wind and solar power generation in area 𝑖 at time 𝑡.

(7) ( ) ( ) 𝑡 𝑡 𝑡 𝑡 𝐶𝑝𝑠𝑖𝑗 P𝑃 𝑉 𝑎𝑣,𝑖𝑗 − P𝑃 𝑉 𝑖𝑗 = K𝑝𝑠𝑖𝑗 ∗ 𝑓𝑠 P𝑃 𝑉 𝑎𝑣,𝑖𝑗 > P𝑃 𝑉 𝑖𝑗 [ ( ) ] ∗ 𝐸 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 > P𝑡𝑃 𝑉 𝑖𝑗 − P𝑡𝑃 𝑉 𝑖𝑗 (8) ( ) 𝑓𝑠 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 > P𝑡𝑃 𝑉 𝑖𝑗 is the probability of solar power surplus than the ( ) scheduled power and 𝐸 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 > P𝑡𝑃 𝑉 𝑖𝑗 is the expectation of solar

(ii) Pumped-storage constraints Here, pure pumped-storage-hydraulic (PSH) unit is used which relies entirely on water that has been pumped to an upper reservoir from a lower reservoir. When the PSH unit operates in the generating mode and decides to change its situation to pumping mode or vice-versa, the unit should be off for an hour due to the physical limitation of the PSH unit and this is known as changeover time. ( ) (𝑡+1) 𝑡 = 𝑉𝑟𝑒𝑠,𝑖 + 𝑄𝑡𝑝ℎ𝑖 P𝑡𝑝ℎ𝑖 , 𝑖 ∈ M and 𝑡 ∈ T𝑝𝑢𝑚𝑝 𝑉𝑟𝑒𝑠,𝑖 (17)

power than the scheduled power. ( ) ( ) 𝐶𝑟𝑠𝑖𝑗 P𝑡𝑃 𝑉 𝑖𝑗 − P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 = K𝑟𝑠𝑖𝑗 ∗ 𝑓𝑠 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 < P𝑡𝑃 𝑉 𝑖𝑗 [ ( )] ∗ P𝑡𝑃 𝑉 𝑖𝑗 − 𝐸 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 < P𝑡𝑃 𝑉 𝑖𝑗 (9) ( ) 𝑓𝑠 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 < P𝑡𝑃 𝑉 𝑖𝑗 is the probability of solar power shortage than the ( ) scheduled power and 𝐸 P𝑡𝑃 𝑉 𝑎𝑣,𝑖𝑗 < P𝑡𝑃 𝑉 𝑖𝑗 is the expectation of solar power below the scheduled power. 2.1.2. Constraints

( ) (𝑡+1) 𝑡 = 𝑉𝑟𝑒𝑠,𝑖 − 𝑄𝑡𝑔ℎ𝑖 P𝑡𝑔ℎ𝑖 , 𝑖 ∈ M and 𝑡 ∈ T𝑔𝑒𝑛 𝑉𝑟𝑒𝑠,𝑖

(18)

(𝑡+1) 𝑡 𝑉𝑟𝑒𝑠,𝑖 = 𝑉𝑟𝑒𝑠,𝑖 , 𝑖 ∈ M and 𝑡 ∈ T𝑐ℎ𝑎𝑛𝑔𝑒_𝑜𝑣𝑒𝑟

(19)

𝑡 max Pmin 𝑔ℎ𝑖 ≤ P𝑔ℎ𝑖 ≤ P𝑔ℎ𝑖 𝑖 ∈ M and 𝑡 ∈ T𝑔𝑒𝑛

(20)

𝑡 max Pmin 𝑝ℎ𝑖 ≤ P𝑝ℎ𝑖 ≤ P𝑝ℎ𝑖 𝑖 ∈ M and 𝑡 ∈ T𝑝𝑢𝑚𝑝

(21)

min 𝑡 max 𝑉𝑟𝑒𝑠,𝑖 ≤ 𝑉𝑟𝑒𝑠,𝑖 ≤ 𝑉𝑟𝑒𝑠,𝑖 , 𝑖 ∈ M, 𝑡 ∈ T

(22)

(i) Power balance constraints: N𝑠𝑖 ∑

P𝑡𝑠𝑖𝑗

𝑗=1

+

+

∑

N𝑤𝑖 ∑

P𝑡𝑤𝑖𝑗

+

𝑗=1

T𝑡𝑖𝑙 , 𝑖

Nℎ𝑖 ∑

N𝑃 𝑉 𝑖

P𝑡ℎ𝑖𝑗

+

𝑗=1

∑

P𝑡𝑃 𝑉 𝑖𝑗

+ P𝑡𝑔ℎ𝑖

=

P𝑡𝐷𝑖

+ P𝑡𝐿𝑖

𝑗=1

∈ M and 𝑡 ∈ T𝑔𝑒𝑛

(10)

Since the initial and final water volume of the upper reservoir of the PSH unit are taken as the same in this problem, the total net water amount utilized by the PSH unit must be equal to zero.

𝑙,𝑙≠𝑖 N𝑠𝑖

∑

N𝑤𝑖

P𝑡𝑠𝑖𝑗 +

𝑗=1

+

∑

∑

Nℎ𝑖

P𝑡𝑤𝑖𝑗 +

𝑗=1

∑

N𝑃 𝑉 𝑖

P𝑡ℎ𝑖𝑗 +

𝑗=1

∑

P𝑡𝑃 𝑉 𝑖𝑗 = P𝑡𝐷𝑖 + P𝑡𝑝ℎ𝑖 + P𝑡𝐿𝑖

𝑗=1

T𝑡𝑖𝑙 , 𝑖 ∈ M and 𝑡 ∈ T𝑝𝑢𝑚𝑝

0 T 𝑠𝑡𝑎𝑟𝑡 𝑒𝑛𝑑 𝑉𝑟𝑒𝑠,𝑖 = 𝑉𝑟𝑒𝑠,𝑖 = 𝑉𝑟𝑒𝑠,𝑖 = 𝑉𝑟𝑒𝑠,𝑖

(11)

𝑙,𝑙≠𝑖 N𝑠𝑖 ∑

P𝑡𝑠𝑖𝑗 +

𝑗=1

+

∑

N𝑤𝑖 ∑

P𝑡𝑤𝑖𝑗 +

𝑗=1

Nℎ𝑖 ∑ 𝑗=1

(iii) Generation limits:

N𝑃 𝑉 𝑖

P𝑡ℎ𝑖𝑗 +

∑

P𝑡𝑃 𝑉 𝑖𝑗 = P𝑡𝐷𝑖 + P𝑡𝐿𝑖

𝑗=1

T𝑡𝑖𝑙 , 𝑖 ∈ M and 𝑡 ∈ T𝑐ℎ𝑎𝑛𝑔𝑒_𝑜𝑣𝑒𝑟

(12)

𝑙,𝑙≠𝑖

where T𝑡𝑖𝑙 is the tie line real power transfer from area 𝑖 to area 𝑙. T𝑡𝑖𝑙 is positive when power flows from area 𝑖 to area 𝑙 and T𝑡𝑖𝑙 is negative when power flows from area 𝑙 to area 𝑖. The hydroelectric generation is a function of water discharge rate and reservoir water head, which in turn is a function of storage. ( )2 ( )2 𝑡 𝑡 𝑡 P𝑡ℎ𝑖𝑗 = 𝐶1𝑖𝑗 𝑉ℎ𝑖𝑗 + 𝐶2𝑖𝑗 𝑄𝑡ℎ𝑖𝑗 + 𝐶3𝑖𝑗 𝑉ℎ𝑖𝑗 𝑄𝑡ℎ𝑖𝑗 + 𝐶4𝑖𝑗 𝑉ℎ𝑖𝑗 + 𝐶5𝑖𝑗 𝑄𝑡ℎ𝑖𝑗 + 𝐶6𝑖𝑗 , 𝑗 ∈ Nℎ𝑖 𝑖 ∈ M, 𝑡 ∈ T

P𝑡𝑤𝑖𝑗 P𝑡𝑤𝑖𝑗

for 𝑣𝑤𝑡 < 𝑣𝑖𝑛 and 𝑣𝑤𝑡 > 𝑣𝑜𝑢𝑡 ) 𝑣𝑤𝑡 − 𝑣𝑖𝑛 , for 𝑣𝑖 ≤ 𝑣𝑤𝑡 ≤ 𝑣𝑟 = P𝑤𝑟𝑖𝑗 𝑣𝑟 − 𝑣𝑖𝑛 = P𝑤𝑟𝑖𝑗 , for 𝑣𝑟 ≤ 𝑣𝑤𝑡 ≤ 𝑣𝑜𝑢𝑡

𝑡 max Pmin ℎ𝑖𝑗 ≤ Pℎ𝑖𝑗 ≤ Pℎ𝑖𝑗

𝑗 ∈ Nℎ𝑖 , 𝑖 ∈ M, 𝑡 ∈ T

(24)

𝑡 max Pmin 𝑠𝑖𝑗 ≤ P𝑠𝑖𝑗 ≤ P𝑠𝑖𝑗

𝑗 ∈ N𝑠𝑖 , 𝑖 ∈ M, 𝑡 ∈ T

(25)

𝑗 ∈ N𝑤𝑖 , 𝑖 ∈ M, 𝑡 ∈ T

(26)

𝑡 max Pmin 𝑤𝑖𝑗 ≤ P𝑤𝑖𝑗 ≤ P𝑤𝑖𝑗

(iv) Thermal generator ramp rate limits constraints The ramp rate limits of each thermal generator should be within its ramp-up rate limit 𝑈 𝑅𝑖𝑗 , and ramp-down rate limit 𝐷𝑅𝑖𝑗 , so that P𝑡𝑠𝑖𝑗 − P(𝑡−1) 𝑠𝑖𝑗 ≤ 𝑈 𝑅𝑖𝑗 ,

𝑗 ∈ N𝑠𝑖 𝑖 ∈ M, 𝑡 ∈ T

(27)

𝑡 P(𝑡−1) 𝑠𝑖𝑗 − P𝑠𝑖𝑗 ≤ 𝐷𝑅𝑖𝑗 ,

𝑗 ∈ N𝑠𝑖 𝑖 ∈ M, 𝑡 ∈ T

(28)

(13)

Wind power model The power output (Hetzer et al., 2008) of 𝑘th wind power generating unit at time 𝑡 for a given wind speed is expressed as P𝑡𝑤𝑖𝑗 = 0,

(23)

(v) Hydraulic network constraints The hydraulic outfitted constraints comprise the water balance equations for each hydro unit in addition to the bounds on reservoir storage and release targets. These bounds are decided by the physical reservoir and plant limitations as well as the multipurpose necessity of the hydro system. These constraints comprise:

(

(14)

185

M. Basu

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

(iii) Power generation capacity constraints The power generation capacity of each generator must not exceed max its minimum limit Pmin 𝑖𝑗 and maximum limit P𝑖𝑗 , so that

(a) Physical limitations on reservoir storage volumes and discharge rates, min 𝑡 max 𝑉ℎ𝑖𝑗 ≤ 𝑉ℎ𝑖𝑗 ≤ 𝑉ℎ𝑖𝑗

𝑗 ∈ Nℎ , 𝑖 ∈ M, 𝑡 ∈ T

(29)

max Pmin 𝑖𝑗 ≤ P𝑖𝑗 ≤ P𝑖𝑗

𝑄min ℎ𝑖𝑗

≤

𝑄𝑡ℎ𝑖𝑗

≤

𝑄max ℎ𝑖𝑗

𝑗 ∈ Nℎ , 𝑖 ∈ M, 𝑡 ∈ T

𝑅

𝑢𝑗 ∑

( ( ) ( )) 𝑡−𝜏 𝑡−𝜏 𝑄ℎ𝑖𝑙 𝑖𝑗𝑙 + 𝑆ℎ𝑖𝑙 𝑖𝑗𝑙 ,

𝑙=1

𝑙 Pmin 𝑖𝑗 ≤ P𝑖𝑗 ≤ P𝑖𝑗,1

(31)

𝑗 ∈ Nℎ , 𝑖 ∈ M, 𝑡 ∈ T

P𝑢𝑖𝑗,𝑛 ≤ P𝑖𝑗 ≤ Pmax 𝑖𝑗 𝑖𝑗

where 𝑚 represents the number of proscribed workable area. P𝑢𝑖𝑗,𝑚−1 is the maximum limit of (𝑚 − 1)th proscribed workable area of 𝑗 the generator in area 𝑖. P𝑙𝑖𝑗,𝑚 is the minimum limit of 𝑚th proscribed workable area of 𝑗 the generator in area 𝑖. Total number of proscribed workable areas of 𝑗 the generator in area 𝑖 is 𝑛𝑖𝑗 .

(32)

Tmax 𝑖𝑙

where is the power flow limit from area 𝑖 to area 𝑙 and the power flow limit from area 𝑙 to area 𝑖.

−Tmax 𝑖𝑙

(38)

P𝑢𝑖𝑗,𝑟−1 ≤ P𝑖𝑗 ≤ P𝑙𝑖𝑗,𝑟 ; 𝑟 = 2, 3, … , 𝑛𝑖𝑗

(vi) Tie line capacity constraints The tie line power transfer T𝑡𝑖𝑙 at time 𝑡 from area 𝑖 to area 𝑙 should not exceed the tie line transfer capacity for security consideration. −Tmax ≤ T𝑡𝑖𝑙 ≤ Tmax 𝑖𝑙 𝑖𝑙

(37)

(iv) Proscribed workable area The feasible operating regions of the 𝑗 the generator in area 𝑖 can be stated as follows:

(b) The continuity equation for the hydro reservoir network (𝑡+1) 𝑡 𝑡 𝑉ℎ𝑖𝑗 = 𝑉ℎ𝑖𝑗 + I𝑡ℎ𝑖𝑗 − 𝑄𝑡ℎ𝑖𝑗 − 𝑆ℎ𝑖𝑗 +

𝑖 ∈ M and 𝑗 ∈ N𝑠𝑖

(30)

is

2.2. Multi-region economic dispatch

2.3. Economic dispatch

The goal of multi-region economic dispatch (MRED) is to minimize the cost of all areas by loading of all on line generators in such a manner that the power balance constraints, generation limits constraints and tie-line capacity constraints are fulfilled.

Economic dispatch (ED) minimizes the cost of an electric power generating station simultaneously rewarding various constraints. The ED problem digs up valve-point effect, proscribed workable area and ramp rate borders together with the load demand, transmission loss and working capability limits.

2.2.1. Objective function The total fuel cost 𝐹𝑡 of all thermal generators of all areas taking into consideration valve point effect can be stated as follows: 𝐹𝑡 =

N𝑠𝑖 M ∑ ∑

2.3.1. Objective function The ED problem can be avowed as:

M ∑ 𝑠𝑖 [ ( ) ∑ 𝑎𝑠𝑖𝑗 + 𝑏𝑠𝑖𝑗 P𝑖𝑗 + 𝑐𝑠𝑖𝑗 P2𝑖𝑗 𝐹𝑖𝑗 P𝑖𝑗 = N

Min

𝑖=1

𝑖=1 𝑗=1

𝑖=1 𝑗=1

N𝑠 ∑

{ ( )}|] | | + ||𝑑𝑖𝑗 × sin 𝑒𝑖𝑗 × Pmin − P (33) 𝑠𝑖𝑗 | 𝑠𝑖𝑗 | | ( ) where 𝐹𝑖𝑗 P𝑖𝑗 is the cost function of 𝑗th generator in area 𝑖, P𝑖𝑗 is the real power output of 𝑗th generator in area 𝑖.

N

𝑠 ( ) ∑ { ( )}| | 𝐹𝑖 P𝑖 = 𝑎𝑖 + 𝑏𝑖 P𝑖 + 𝑐𝑖 P2𝑖 + |𝑑𝑖 × sin 𝑒𝑖 × Pmin − P𝑖 | 𝑖 | |

(39)

𝑖=1

where 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖 and 𝑒𝑖 are the fuel cost coefficients of the 𝑖th thermal generator, P𝑖 is the power output of the 𝑖th generator and N𝑠 is the total number of committed thermal generators. 2.3.2. Constraints (i) Power balance constraint:

2.2.2. Constraints

N𝑠 ∑

(i) Power balance constraint: In area 𝑖, the total power generation of all generators must be equal the area power demand, transmission loss with the consideration of imported and exported power and can be stated as follows: N𝑠𝑖 ∑

P𝑖𝑗 = P𝐷𝑖 + P𝐿𝑖 +

∑

The transmission loss P𝐿 can be avowed as P𝐿 =

T𝑖𝑘

𝑖∈M

(34)

N𝑠𝑖 M ∑ ∑

P𝑖𝑗 B𝑖𝑙𝑗 P𝑠𝑖𝑙 +

𝑙=1 𝑗=1

N𝑠𝑖 ∑

B0𝑖𝑗 P𝑖𝑗 + B00𝑖

N𝑠 ∑

B0𝑖 P𝑖 + B00

(41)

𝑖=1

(ii) Generating capability border constraints The power output of the thermal generator must be within its lower and upper borders such that

(35)

𝑗=1

where P𝐷𝑖 is the real power demand of area 𝑖, T𝑖𝑘 is the tie line real power transfer from area 𝑖 to area 𝑘. T𝑖𝑘 is positive when power flows from area 𝑖 to area 𝑘 and T𝑖𝑘 is negative when power flows from area 𝑘 to area 𝑖.

Pmin ≤ P𝑖 ≤ Pmax 𝑖 𝑖 , 𝑖 ∈ N𝑠

(42) Pmax 𝑠𝑖

Pmin 𝑠𝑖

where is the lower border and is the upper border of the 𝑖th thermal generator. (iii) Proscribed workable area The workable area of a thermal generator with proscribed workable area can be avowed as:

(ii) Tie line capacity constraints Tie line power transfer T𝑖𝑘 from area 𝑖 to area 𝑘 must be within the tie line power transfer capacity limits. max −Tmax 𝑖𝑘 ≤ T𝑖𝑘 ≤ T𝑖𝑘

P𝑙 B𝑙𝑖 P𝑖 +

where P𝐷 is load demand and B𝑙𝑖 , B0𝑖 and B00 are the B-coefficients or loss coefficients.

The transmission loss P𝐿𝑖 of area 𝑖 can be stated as follows: P𝐿𝑖 =

N𝑠 N𝑠 ∑ ∑ 𝑙=1 𝑖=1

𝑘,𝑘≠𝑖

𝑗=1

(40)

P𝑖 − P 𝐷 − P 𝐿 = 0

𝑖=1

Pmin ≤ P𝑖 ≤ P𝑙𝑖,1 𝑖

(36)

P𝑢𝑖,𝑟−1 ≤ P𝑖 ≤ P𝑙𝑖,𝑟 ,

where Tmax is the power flow limit from area 𝑖 to area 𝑘 and -Tmax is 𝑖𝑘 𝑖𝑘 the power flow limit from area 𝑘 to area 𝑖.

P𝑢𝑖,𝑟 ≤ P𝑖 ≤ Pmax 𝑖 𝑖

186

𝑟 = 2, 3, … , 𝑛𝐢 𝑖 ∈ N𝑠

(43)

M. Basu

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

where 𝑟 signifies the number of proscribed workable area of the 𝑖 the thermal generator. P𝑢𝑖,𝑟−1 is the upper border of (𝑟 − 1)th proscribed workable area and P𝑙𝑖,𝑟 is the lower border of 𝑟th proscribed workable area. Total number of proscribed workable area of the 𝑖th thermal generator is 𝑛𝑖 .

3.3. Selection ( ) Perform assortment for each parent X𝑖 by comparing (its objective ) ∕

function value with that of the corresponding offspring X𝑖 which has the with lesser objective function value among the two offsprings generated by Gaussian mutation and Cauchy mutation. The populace that has lesser objective function value between parent and offspring will stay alive for the next iteration.

(iv) Ramp rate boundary constraints The power P𝑖 , produced by the 𝑖th thermal generator in definite period should neither go beyond that of the preceding period P𝑖0 by above a definite quantity 𝑈 𝑅𝑖 , the up-ramp border nor below that of the preceding period by above a definite quantity 𝐷𝑅𝑖 , the down-ramp border of the generator. These create the subsequent constraints. As generation boosts P𝑖 − P𝑖0 ≤ 𝑈 𝑅𝑖

{ X𝑖 =

( ) ( ) ∕ ∕ X𝑖 , if 𝑓 X𝑖 ≤ 𝑓 X𝑖 X𝑖 , otherwise

, 𝑖 ∈ NP

(51)

(44)

The process is repetitive until the maximum number of iterations or no improvement is seen in the best individual after many iterations.

(45)

4. Chaotic fast convergence evolutionary programming (CFCEP)

As generation reduces P𝑖0 − P𝑖 ≤ 𝐷𝑅𝑖 and

Calculate the objective function value 𝑓𝑖 of each populace and maximum objective function value 𝑓max .

Chaos theory is extremely useful in numerous engineering applications. Chaos is the crucial character of a nonlinear system. It has a number of properties such as randomicity, ergodicity and regulation, etc. With the deep influence to the development of science, the chaos has been pioneered into the evolutionary computation to build new intelligence algorithms which conquer prematurity during search procedure and boost the convergence property. The concert of FCEP depends on the scaling factor which is constant throughout the process. Therefore, it is tricky to decide appropriate value of scaling factor without the fine-tuning development which would generally need high computational time, due to the multiple runs for scaling factor optimization. Also, it often suffers from premature convergence to local optima. So, a dynamic scaling factor control method rooted in chaotic sequences to adjust the scaling factor value setting of FCEP adaptively throughout the search process is proffered in this work. The use of chaotic sequences for adjusting the scaling factor value setting of FCEP adaptively has two merits: firstly, as there is no requirement of trial and fine-tuning procedure, so no additional computational time is needed which perk up optimization effectiveness. Lastly, as the scaling factor value setting is adjusted adaptively, the variety of the populace can be perked up which boosts convergence property. The dynamic scaling factor control method is rooted in the Tent equation (Caponetto and Fortuna, 2003), which is a well-known chaos system. The Tent equation can be stated as: { 2𝑐𝑥𝑔 0 < 𝑐𝑥𝑔 < 0.5 𝑐𝑥𝑔+1 = (52) 2 (1 − 𝑐𝑥𝑔 ) 0.5 < 𝑐𝑥𝑔 < 1

3.2. Mutation

𝑔 = 1, 2, … ., 𝑔max

( ) ( max ) max Pmin 𝑖 , P𝑖0 − 𝐷𝑅𝑖 ≤ P𝑖 ≤ min P𝑖 , P𝑖0 + 𝑈 𝑅𝑖

(46)

3. Overview of fast convergence evolutionary programming In fast convergence evolutionary programming (FCEP), Gaussian and Cauchy mutations to generate offspring (Basu, 2017) and one-toone contest have been initiated in evolutionary programming (EP) to enhance the convergence speed and quality of solution. This one-toone contest produces faster convergence speed as an offspring competes one-to-one with that of the matching parent. The phases are initialization, mutation and selection. These can be affirmed as: 3.1. Initialization The initial populace (X0𝑖 ) of the control variable is selected at random avowed as: ( ) max 𝑥0𝑖,𝑗 ∼ 𝑈 𝑥min , 𝑗 ∈ 𝑛, 𝑖 ∈ NP (47) 𝑗 , 𝑥𝑗 where 𝑛 is the number of decision variables in an individual, NP is the populace size; 𝑥0𝑖,𝑗 signifies the initial 𝑗th variable of the 𝑖th populace; 𝑥min and 𝑥max are the lower and upper leaps of the 𝑗th 𝑗 ) (𝑗 , 𝑥max indicates a uniform random variable decision variable; 𝑈 𝑥min 𝑗 𝑗 [ ] max min ranging over 𝑥𝑗 , 𝑥𝑗 .

where 𝑔 symbolizes the iteration number, 𝑔max denotes the maximum iteration, 𝑐𝑥𝑔 specifies the chaotic variable at 𝑔th iteration and its value is between [0, 1]. It is seen from equation (38 that during the iterative process, the value of 𝑐𝑥𝑔 will be dealt out (between [0, ) 1], under the condition that the initial 𝑐𝑥0 ∈ (0, 1), 𝑐𝑥0 ∉ 41 , 21 , 23 , 43 This paper recommends Tent equation based dynamic scaling factor control method to perk up the computational effectiveness of FCEP. The value setting of scaling factor 𝛽 of Eq. (50) in CFCEP technique is attuned dynamically by Eq. (53) throughout the process. ) ( 1 1 2 3 𝛽 0 ∈ (0, 1) , 𝛽 0 ∉ , , , 4 2 3 4

Two offsprings are generated from each parent one by utilizing Gaussian mutation and other by utilizing Cauchy mutation. Each se( ) ( ) ∕ lected parent X𝑖 is mutated to generate offspring X𝑖 by utilizing Gaussian mutation and Cauchy mutation avowed as: ′

𝑥𝑖,𝑗 = 𝑥𝑖,𝑗 + 𝜎𝑗 × 𝑁 (0, 1) , 𝑗 ∈ 𝑛, 𝑖 ∈ NP ′

𝑥𝑖,𝑗 = 𝑥𝑖,𝑗 + 𝜎𝑗 × 𝐶 (0, 1) , 𝑗 ∈ 𝑛, 𝑖 ∈ NP ( ) 𝑓 − 𝑥min ,𝑗 ∈ 𝑛 𝜎𝑗 = 𝛽 × 𝑖 × 𝑥max 𝑗 𝑗 𝑓max

(48) (49) (50)

where 𝑁 (0, 1) symbolizes a Gaussian random variable with mean 0 and standard deviation 1; 𝐶 (0, 1) symbolizes to a Cauchy random variable with scaled parameter 𝑡 = 1 centered at zero; 𝑓max is the maximum objective function value of the last iteration and 𝛽 is the scaling factor. Calculate the objective function value of each offspring. Objective function values of two offsprings are compared and offspring with lesser objective function value selected for assortment process.

{ 𝛽 𝑔+1 =

2𝛽 𝑔 2 (1 − 𝛽 𝑔 )

0 < 𝛽 𝑔 < 0.5 0.5 < 𝛽 𝑔 < 1

(53)

Fig. 1 depicts the flowchart of chaotic fast convergence evolutionary programming. 187

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Engineering Applications of Artificial Intelligence 86 (2019) 182–196

Step (2) Compute dynamic scaling factor (𝛽) using Tent equation. Step (3) Perform mutation for each parent vector as described in Section 3.2. The mutated vector should satisfy the constraints given by Eqs. (10)–(12) and (17)–(32). Step (4) Perform selection for each parent vector, by comparing its total cost with that of the corresponding offspring which has the with lesser objective function value among the two offsprings generated by Gaussian mutation and Cauchy mutation. The populace that has lesser objective function value between parent and offspring will stay alive for the next iteration. Step (5) Stop if the maximum number of iterations is reached otherwise go to Step 2. 6. Simulation results The recommended chaotic fast convergence evolutionary programming (CFCEP) has been pertained for solving MRDED, MRED and ED. Simulation results have been utilized to match up the efficacy of the recommended CFCEP with that of fast convergence evolutionary programming (FCEP) technique (Basu, 2017) differential evolution (DE) and particle swarm optimization (PSO). The developed CFCEP, FCEP, DE and PSO are utilized by using MATLAB 7.0 on Intel (R), Core (TM) i7-4790 CPU 3.66 GHz and 16 GB RAM, 64-bit operating system. 6.1. Case study for MRDED problem A three-area test system is considered here. Each area comprises a multi-chain cascade of four reservoir hydro plants, one equivalent wind power generating unit, one equivalent solar PV plant, ten thermal power plants with nonsmooth fuel cost function and one pumped hydro energy storage unit. The entire scheduling period is 1 day and divided into 24 intervals. The detailed parameters of hydro power plant are taken from Lakshminarasimman and Subramanian (2006). The detailed parameters of thermal generators are listed in Table A.1. Load demands are listed in Table A.2. The direct cost coefficient for the wind power generating unit is chosen as K𝑤 = 1. The reserve cost and penalty cost for the wind power generator are taken as 2 and 1 respectively. The rating of the wind power generator is P𝑤𝑟 = 175 MW. The cut in, cut out and rated wind speeds are 𝑣𝑖𝑛 = 5 m/s, 𝑣𝑜 = 25 m/s and 𝑣𝑟 = 15 m/s respectively The rating of solar PV generator is P𝑃 𝑉 𝑟 = 175 MW. The direct cost coefficient (K𝑠 ) for the solar PV generator is taken 3.5. The reserve cost and penalty cost for the solar PV generator are taken as 2 and 1 respectively. The solar radiation in the standard environment (𝐺𝑠𝑡𝑑 ) and a certain radiation point (𝑅𝑐 ) are taken as 1000 W/m2 and 150 W/m2 . The upper and lower forecast limits of solar irradiation and the wind velocity are reported in Figs. A.1 and A.2 respectively. The power flow limit from area 1 to area 2 or from area 2 to area 1, from area 1 to area 3 or from area 3 to area 1, from area 2 to area 3 or from area 3 to area 2 is 200 MW. The pumped storage hydro plant has the following characteristics: Generating mode 𝑄𝑔ℎ𝑡 is positive when generating, P𝑔ℎ𝑡 is positive and 0 ≤ P𝑔ℎ𝑡 ≤ ( ) 100 MW, 𝑄𝑔ℎ𝑡 P𝑔ℎ𝑡 = 70 + 2P𝑔ℎ𝑡 acre-ft/hr Pumping mode 𝑄𝑝ℎ𝑡 is negative when pumping, P𝑝ℎ𝑡 is negative and −100 MW < ( ) P𝑝ℎ𝑡 ≤ 0 MW, 𝑄𝑝ℎ𝑡 P𝑝ℎ𝑡 = −200 acre-ft/h with P𝑝ℎ𝑡 = −100 MW Operating limitations: The pumped hydro plant will be permitted to work only at −100 MW when pumping. The reservoir starts at 3000 acre-ft and must be at 3000 acre-ft at the end of the 24 h. The water inflow rate is neglected and the spillage is not considered. The problem is solved by utilizing developed CFCEP, FCEP, DE and PSO. Here, parameter is chosen as NP = 100 for both CFCEP and FCEP. Scaling factor (𝛽) is 1 for FCEP. In case of PSO the parameters are taken as NP = 100, 𝑤max = 0.25, 𝑤min = 0.05, 𝑐1 = 0.5 and 𝑐2 = 0.5. In case

Fig. 1. Flowchart of chaotic fast convergence evolutionary programming.

5. Implementation of CFCEP algorithm for MRDED problem In this section, an algorithm based on CFCEP for solving MRDED problem is described below. Step (1) Let [{( ) ( ) ( ) 𝑝𝑘 = P111 , … , P11N , … , P1𝑖1 , … , P1𝑖𝑗 , ..P1𝑖N , … , P1M1 , … , P1MN , 𝑖 𝑖 M } {( ) ( 1 ) ( ) T12 , … , T11M , T123 , … , T12M , … , T1(M−1)M , ...., P𝑡11 , … , P𝑡1N , … , 1 ( ) ( ) ( ) P𝑡𝑖1 , … , P𝑡𝑖𝑗 , … , P𝑡𝑖N , … , P𝑡M1 , … , P𝑡MN , T𝑡12 , … , T𝑡1M , 𝑖 M } {( ) ( 𝑡 ) T23 , … , T𝑡2M , … , T𝑡(M−1)M , … , PT11 , … , PT1N , … , 1 ( ) ( ) ( ) PT𝑖1 , … , PT𝑖𝑗 , … , PT𝑖N , … , PTM1 , … , PTMN , TT12 , … , TT1M , 𝑖 M ] }]∕ [ ( T ) and P𝑡𝑖𝑗 = P𝑡𝑠𝑖𝑗 , P𝑡ℎ𝑖𝑗 , P𝑡𝑤𝑖𝑗 , P𝑡𝑃 𝑉 𝑖𝑗 , P𝑡𝑔ℎ𝑖 T23 , … , TT2M , … , TT(M−1)M 𝑝𝑘 be the k th parent vector of a population to be evolved and 𝑘 = 1, 2, … , NP . The elements of 𝑝𝑛 are real power outputs of the committed generators of all areas and tie line real power flows. The elements of 𝑝𝑘 should satisfy the constraints given by Eqs. (10)–(12) and (17)–(32). Total cost of each parent vector 𝑝𝑘 is calculated. 188

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Engineering Applications of Artificial Intelligence 86 (2019) 182–196

Fig. 2. Hydro plant discharges (× 104 m3 ) of three area system acquired from FCEP.

Fig. 3. Hydro reservoir storage volumes (× 104 m3 ) of three area system acquired from CFCEP.

189

M. Basu

Table 1 Hydro–wind–solar–thermal generation (MW) of area 1 for multi-region dynamic economic dispatch acquired from CFCEP.

190

Pℎ1

Pℎ2

Pℎ3

Pℎ4

P𝑤

P𝑃 𝑉

P𝑔ℎ

P𝑠1

P𝑠2

P𝑠3

P𝑠4

P𝑠5

P𝑠6

P𝑠7

P𝑠8

P𝑠9

P𝑠10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

60.2044 58.7571 83.4267 75.4086 66.3176 100.4152 73.3932 96.7588 54.4416 87.5973 71.2409 62.2948 96.5233 72.1360 68.5159 66.3749 90.7196 80.4923 79.6767 61.2892 63.6449 93.3430 67.2994 69.0322

51.9379 82.2884 48.5004 51.3228 84.0007 70.5455 48.7674 69.5699 68.8833 53.1999 52.0161 70.3945 62.1969 58.0788 56.9350 75.3350 53.6423 60.2030 43.3934 40.1653 44.6168 49.3695 49.6770 43.1287

59.0048 54.6938 12.0682 52.3723 49.0789 53.4598 54.7313 55.3248 54.4425 38.4810 36.6233 58.4286 59.1175 58.3498 54.8611 46.5323 60.7246 62.2038 18.0122 38.7076 20.6364 29.6416 56.6483 38.1100

198.7294 147.5719 165.4866 201.1968 203.7217 138.1519 199.3498 195.5886 127.8626 121.1581 176.9932 229.7531 202.0247 179.8979 187.3391 245.1966 190.5130 242.2661 189.7149 205.0865 266.6392 258.6084 267.3222 233.3705

86.0259 160.9692 175.0000 133.7662 175.0000 175.0000 175.0000 175.0000 64.7131 33.1080 46.9247 0 112.5075 55.1940 38.6445 175.0000 9.1080 65.6036 16.9838 20.3724 65.7692 103.5763 57.9581 42.9843

0 0 0 0 2.6517 44.1323 74.6892 103.2115 113.3116 126.8874 159.9325 155.1647 117.2901 109.1089 87.5498 38.1558 37.3232 28.6607 0.0464 0 0 0 0 0

−100.0000 −100.0000 −100.0000 −100.0000 −100.0000 −100.0000 −100.0000 0 76.0797 93.4066 95.2457 69.6306 69.4918 74.0286 36.0556 63.3310 65.3893 48.0912 24.2288 0 −100.0000 −100.0000 −100.0000 −100.0000

83.3102 75.1095 45.2488 60.0269 86.3616 110.5930 111.3867 82.9981 56.8395 66.3987 69.5681 92.3130 70.0009 91.7335 101.9915 101.0641 78.6951 69.8820 85.1678 87.0622 89.1668 77.4856 44.4073 61.3134

43.7087 53.2253 49.8748 77.6942 88.9184 55.6270 61.3283 69.6182 42.7241 73.7335 96.4739 78.9988 95.7597 97.4497 108.4965 83.8434 93.1406 94.8934 61.8180 82.1874 112.4498 94.3898 66.3290 86.5180

80.8245 96.0702 96.5796 66.3255 83.4129 86.3164 111.5881 99.4003 76.3786 76.9896 102.9246 81.2993 109.7160 105.6416 105.6446 102.1775 84.7380 90.0920 102.5538 76.3674 90.4975 97.9842 60.8979 72.1662

107.3900 141.3502 189.3115 162.1212 167.5139 161.2397 110.6585 114.3707 172.4607 187.7781 166.3954 181.7408 161.3562 183.2716 124.9506 158.5272 166.4977 158.4988 175.0269 187.9116 176.5638 119.2385 94.4308 151.5572

71.5346 86.7825 94.0719 76.7836 90.9000 64.2900 64.3805 53.7534 58.3538 59.0704 54.4293 50.4811 47.6128 58.2920 78.8159 56.9815 78.1077 72.5931 70.1137 49.1068 73.5906 66.3230 57.9084 54.8554

125.5159 99.5725 75.4054 97.0276 91.6367 69.8592 90.2043 98.8922 94.6951 128.9054 98.4606 130.7902 94.4076 70.8975 90.4881 90.9109 121.6040 99.3758 107.8571 88.2233 126.7975 137.9388 129.8745 135.0347

238.1593 230.3242 180.0673 188.7279 205.7476 182.3017 251.7295 255.4518 298.6316 243.1377 231.7075 286.0838 280.6690 287.6661 291.7482 231.4768 216.3611 199.0457 259.2774 288.0718 287.1927 228.5192 299.8312 273.2049

256.7122 185.8091 233.5274 177.6185 176.6260 241.6769 223.6967 268.7381 216.8885 236.3403 231.8307 219.2056 212.3006 210.9613 165.6637 213.8192 230.3668 280.2210 272.2285 277.2921 295.9423 245.0772 174.4022 172.1895

146.9637 218.1509 139.1280 138.0223 211.0033 209.5648 207.9867 237.1465 296.1763 279.5139 245.2098 241.9125 284.2342 209.4153 204.2248 205.2286 173.7860 135.4004 187.9708 157.3875 203.1115 183.7902 217.1753 203.8481

278.9070 252.6840 291.8421 283.7870 212.0715 161.2161 192.9409 203.7674 139.0616 166.1684 186.4559 155.2473 131.4350 194.3055 166.4514 200.2924 240.5238 298.4241 280.5546 246.5782 211.2191 233.0946 221.5998 238.8089

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

Hour

M. Basu

Table 2 Hydro–wind–solar–thermal generation (MW) of area 2 for multi-region dynamic economic dispatch acquired from CFCEP.

191

Pℎ1

Pℎ2

Pℎ3

Pℎ4

P𝑤

P𝑃 𝑉

P𝑔ℎ

P𝑠1

P𝑠2

P𝑠3

P𝑠4

P𝑠5

P𝑠6

P𝑠7

P𝑠8

P𝑠9

P𝑠10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

95.5531 68.1685 59.6132 95.8168 54.7173 79.5160 71.0603 59.7279 86.5249 63.8006 56.8660 68.7197 66.7643 67.8078 54.4037 63.1110 105.4402 64.1757 102.6035 99.3875 99.5504 65.6048 58.4279 74.2150

49.5073 63.1263 51.2782 80.9621 74.6011 58.0716 55.0947 57.6469 57.1488 73.7064 64.1110 70.3723 54.3511 73.3072 53.6651 54.5801 85.1979 54.9245 50.0178 68.1935 61.3201 51.1879 51.3931 64.5740

20.1001 54.4697 54.7693 55.7074 46.1936 55.7985 57.5798 55.0121 58.2573 16.2547 57.9338 22.7013 56.6963 23.6220 12.8945 17.1265 53.5344 2.5175 13.7956 52.4112 54.6828 3.4430 53.9163 55.4722

132.6283 185.1484 138.3565 236.9171 198.0628 216.0524 107.8195 131.0829 198.9576 174.2319 203.8939 221.5624 146.6013 222.0404 219.1360 160.3422 241.8134 194.2010 238.0108 209.2615 240.9596 163.6788 281.0744 270.1677

82.6269 173.9379 175.0000 119.1934 175.0000 175.0000 175.0000 171.4966 140.0301 47.9821 53.6319 0 16.6903 96.9442 63.4497 175.0000 15.3005 25.5335 160.3152 47.8526 24.0400 94.2003 100.7672 0

0 0 0 0 5.3859 43.1541 75.8501 95.0092 119.1643 125.4099 167.0797 153.8492 132.6474 106.1960 88.6069 33.8150 38.2036 22.8224 3.8898 0 0 0 0 0

−100.0000 −100.0000 −100.0000 −100.0000 −100.0000 −100.0000 −100.0000 0 48.2982 85.4697 75.4897 12.7266 69.4322 86.6227 47.6409 88.4064 23.8728 90.2831 86.9811 0 −100.0000 −100.0000 −100.0000 −100.0000

348.3961 308.7321 310.7898 255.1305 351.3812 318.4956 252.9097 224.4675 315.5768 246.8117 174.4014 183.5225 116.8590 171.1836 191.0525 126.6947 97.8677 137.5464 233.3495 184.0506 109.1967 183.4912 95.6879 118.3512

123.3815 94.0955 157.9966 94.2490 190.1550 142.0344 106.3773 190.0036 103.0710 195.9357 258.4945 349.3696 255.0875 177.3923 157.4863 101.9376 151.5345 148.5396 225.6649 250.6754 327.1323 323.5576 232.5955 133.4729

226.9411 198.2785 246.4192 282.1747 294.7334 200.2268 142.5727 221.6896 303.6840 273.4299 314.5157 390.6337 350.9019 428.2163 469.7685 389.9865 327.0175 362.4556 370.1626 460.6496 417.5972 331.4480 283.7694 274.0573

235.1534 162.5139 143.8435 171.6522 131.2601 169.9329 187.9893 176.7823 236.8047 319.5702 310.9615 339.6089 281.9351 244.9265 227.3223 229.8581 310.3132 230.1370 214.3273 284.5982 317.6606 379.0979 307.5412 336.3541

161.0839 187.7515 143.1099 137.9140 154.1262 128.9544 198.3223 188.3971 172.6106 233.0730 219.9545 225.7745 325.6162 337.8514 384.8247 364.8118 430.9860 493.5389 470.9330 431.8899 378.8757 342.6042 244.9581 278.9177

184.3881 136.2417 209.2236 294.3208 328.5782 388.2939 339.6815 266.9400 331.0739 398.0696 378.4350 373.1049 397.4258 302.7375 295.0467 218.0378 146.0967 210.1152 300.7100 235.4467 153.6597 220.3892 213.4491 129.3384

317.3093 296.0942 338.2316 360.5808 389.3510 365.8338 458.7870 418.1496 410.8999 408.5676 493.2233 397.1473 415.9077 464.4846 374.9480 456.4863 439.3825 367.7234 388.7566 473.7364 396.7765 470.4212 485.8466 394.8465

220.7780 274.4394 245.1361 244.2115 252.3734 252.9274 334.7212 290.0101 376.5966 431.6302 380.1398 457.8013 486.0000 409.2327 320.1576 224.1420 279.3985 304.7781 355.7015 387.2630 332.0702 349.2765 416.5172 371.0468

345.5437 305.8902 345.4624 256.7991 307.9103 400.5033 373.8862 307.6230 355.8510 362.2502 379.7431 462.7372 385.3097 287.0755 360.1513 294.2903 242.7814 321.2145 384.3957 305.4494 288.1689 251.4305 307.2521 248.6094

524.1233 541.8344 476.9107 494.4799 428.0944 425.0100 452.8073 433.6231 410.0316 383.2921 359.4623 430.5097 509.6076 487.1917 435.4507 373.6287 397.3178 333.0246 399.0225 381.5535 414.7432 385.7708 397.5485 487.8336

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

Hour

M. Basu

Table 3 Hydro–wind–solar–thermal generation (MW) of area 3 for multi-region dynamic economic dispatch acquired from CFCEP.

192

Pℎ1

Pℎ2

Pℎ3

Pℎ4

P𝑤

P𝑃 𝑉

P𝑔ℎ

P𝑠1

P𝑠2

P𝑠3

P𝑠4

P𝑠5

P𝑠6

P𝑠7

P𝑠8

P𝑠9

P𝑠10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

90.7922 52.8342 89.9731 61.9008 94.7925 53.1617 84.2469 56.8325 62.3887 80.7418 55.9065 97.9173 80.5967 68.9859 59.0129 65.0788 99.1796 55.4416 96.6298 68.7265 92.9470 60.2595 98.2624 74.0330

49.1945 72.6543 63.4401 82.2577 78.2020 66.6088 50.9621 53.0837 63.5370 47.6188 60.4749 44.9718 55.8910 51.4958 68.0112 63.9581 46.7103 76.3747 43.4118 48.5136 47.6622 53.4261 66.1804 54.2629

25.1145 17.1734 48.0089 50.8449 49.6445 43.9805 46.6491 55.7586 27.1699 33.6658 55.0179 56.1199 38.1897 13.6384 14.9536 55.2631 10.6911 53.9903 53.4908 48.9849 56.6905 59.0869 37.1536 58.3491

231.6524 192.7284 228.2001 115.7765 143.2765 178.5539 166.7009 112.5116 158.6977 194.7913 216.4785 213.5928 229.9822 238.5177 239.7679 234.4248 255.2594 244.6299 239.3991 256.5813 257.7100 223.4388 192.9050 243.6043

148.4045 172.6103 174.7939 133.2645 175.0000 175.0000 171.3207 129.1024 91.1629 27.7823 28.4592 0 70.9958 107.9975 61.9494 175.0000 20.3312 124.1808 141.5794 26.7368 33.5616 87.7939 37.5254 25.9695

0 0 0 0 0.0822 43.3373 74.3936 107.9095 116.7664 132.6690 161.5347 156.7327 122.3652 103.9313 94.1319 37.2821 36.2544 20.2875 5.6202 0 0 0 0 0

−100.0000 −100.0000 −100.0000 −100.0000 −100.0000 −100.0000 −100.0000 0 59.0758 53.1715 65.6198 61.9228 49.3601 88.5933 70.3100 36.5307 59.2645 86.8995 84.6965 0 −100.0000 −100.0000 −100.0000 −100.0000

269.0238 271.7480 276.4017 312.1228 288.5158 257.5475 285.4127 277.2985 331.9174 367.3191 409.7957 490.0463 412.3488 329.3355 348.5900 435.2994 358.7192 375.5866 306.0284 389.7846 376.2244 276.6716 290.7442 347.7616

302.3128 297.7895 301.1287 397.8705 304.6109 361.2909 420.0372 463.9730 519.3009 526.9816 523.6244 549.5093 477.9719 384.8749 362.7896 346.9163 324.0050 394.1095 432.2815 449.8164 383.1609 288.1018 314.9902 358.6874

259.5523 316.7621 353.8671 351.9683 292.9237 366.7774 447.3242 429.1137 451.8383 547.7084 547.0760 511.7890 427.9453 388.1618 441.3721 448.0959 358.8923 358.3456 447.2946 481.3428 437.0525 356.5667 446.7887 400.8274

283.8908 348.5348 368.1303 365.5452 367.9695 390.0924 470.9801 534.5780 532.2277 549.0425 498.8388 535.1430 438.2385 349.4226 377.4602 310.4630 273.8598 364.6363 443.1143 524.1476 541.8836 486.6528 489.5127 458.2058

255.3489 316.4001 348.3526 331.6963 321.5005 324.0465 406.3622 327.6451 331.6276 418.5180 504.2703 537.0476 456.2706 381.6838 306.4479 367.9687 451.0793 496.4224 404.4403 421.4204 464.1354 435.1881 353.3461 263.4647

15.5975 43.5445 21.8209 33.7406 62.3747 93.7391 67.9885 34.4349 66.1077 33.3758 36.2609 67.3216 23.6206 26.6505 13.5729 35.0413 56.8192 72.4451 44.6086 69.3954 86.6301 101.0664 63.8493 49.7640

28.8328 10.3504 50.9591 47.4886 34.8307 47.5733 78.9064 71.3245 39.9638 33.0898 44.6056 71.9597 93.8816 72.6684 98.2671 100.5168 110.3416 92.4979 78.2651 97.4428 78.6392 36.9188 11.2683 47.3056

24.5437 21.4312 25.8805 56.2440 37.4075 74.1596 104.1923 126.0832 116.3792 100.4402 81.1569 69.1461 31.0793 24.2411 71.0009 27.9334 74.1761 31.5092 14.4925 43.9009 81.8977 83.7865 58.7711 19.0562

89.2245 66.4223 48.9948 48.9963 63.6071 91.3020 95.7339 73.3110 66.8694 94.3714 94.7536 75.3502 56.7680 63.7854 54.7008 83.9908 54.4650 74.3028 96.1790 75.6771 69.0452 70.0685 76.7063 70.7136

291.8725 274.4754 279.8291 265.0124 338.1655 388.1743 481.9788 467.8282 446.4245 500.2527 526.1067 481.8401 423.3074 327.7759 406.2198 343.9848 433.9127 480.8125 455.9462 394.1208 464.3478 546.1918 482.0605 483.4162

Engineering Applications of Artificial Intelligence 86 (2019) 182–196

Hour

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Engineering Applications of Artificial Intelligence 86 (2019) 182–196 Table 4 Comparison of performance.

of DE, the population size (NP ), scaling factor (𝐹 ) and crossover constant (𝐶𝑅 ) have been selected 100, 0.75 and 1 respectively. Maximum iteration number has been selected as 300 for all these techniques. Hydro–wind-solar–thermal-pumped storage generations for three areas matching to the best cost acquired from recommended CFCEP, are summarized in Tables 1, 2, and 3 respectively. The best, average and worst cost and average CPU time among 100 runs of solutions acquired from recommended CFCEP, FCEP, DE and PSO are summarized in Table 4. The optimal hourly discharges of four hydro plants for three areas matching to the best cost acquired from recommended CFCEP are portrayed in Fig. 2(a), (b) and (c) respectively. Fig. 3(a), (b) and (c) respectively portray the reservoir storage volumes of four hydro plants for three areas matching to the best cost acquired from recommended CFCEP. The pumped storage hydro (PSH) plant discharges for three areas matching to the best cost acquired from recommended CFCEP are depicted in Fig. 4. Tie-line power flows of three area system matching to the best cost acquired from recommended CFCEP are portrayed in Fig. 5. Fig. 6 depicts cost convergence characteristics acquired from CFCEP, FCEP, DE and PSO. Due to space limitation, hydro–wind–solar–thermal generations, hourly discharges of four hydro plants and the reservoir storage volumes of four hydro plants acquired from FCEP, DE and PSO are not given here.

CFCEP FCEP DE PSO

Best cost ($)

Average cost ($)

Worst cost ($)

CPU time (s)

2 364 048 2 365 215 2 365 622 2 365 846

2 364 059 2 365 229 2 365 638 2 365 866

2 364 072 2 365 244 2 365 659 2 365 887

109.2537 115.4689 118.8123 120.9053

Fig. 4. Pumped storage hydro plant discharges of three area system acquired from CFCEP.

6.2. Case study for MRED problem The system, comprising forty generators with valve point effect, is segregated into four areas. The data has been adopted from Basu (2013). The total load demand is 10 500 MW. Area 1, area 2, area 3 and area 4 comprises first ten generators and 15% of the total load demand, second ten generators and 40% of the total load demand, third ten generators and 30% of the total load demand and last ten generators and 15% of the total load demand respectively. The power flow limit from area 1 to area 2 or from area 2 to area 1 is 200 MW. The power flow limit from area 1 to area 3 or from area 3 to area 1 is 200 MW. The power flow limit from area 2 to area 3 or from area 3 to area 2 is 200 MW. The power flow limit from area 4 to area 1 or from area 1 to area 4 is 100 MW. The power flow limit from area 4 to area 2 or from area 2 to area 4 is 100 MW. The power flow limit from area 4 to area 3 or from area 3 to area 4 is 100 MW. The problem is solved by utilizing developed CFCEP, FCEP and DE and PSO. Here, parameter is chosen as NP = 100 for both CFCEP and FCEP. Scaling factor (𝛽) is 1 for FCEP. In case of PSO the parameters are taken as NP = 100, 𝑤max = 0.25, 𝑤min = 0.05, 𝑐1 = 0.5 and 𝑐2 = 0.5. In case of DE, the population size (NP ), scaling factor (𝐹 ) and crossover constant (𝐶𝑅 ) have been selected 100, 0.75 and 1 respectively. Maximum iteration number has been selected as 300 for all these techniques. Test results acquired from the best fuel cost among 100 runs of solutions by using developed CFCEP, FCEP, DE are summarized in Table 5. The cost convergence characteristic acquired from developed CFCEP, FCEP, DE and PSO has been portrayed in Fig. 7.

Fig. 5. Tie-line power flow of three area system acquired from CFCEP.

6.3. Case study for ED problem This is 140 unit complicated Korean system of twelve generators with valve-point effect and four generators with proscribed workable area. Here ramp rate limits are considered. The load demand is 49 342 MW. The data has been adopted from Park et al. (2010). The problem is solved by utilizing developed CFCEP, FCEP and DE and PSO. Here, parameter is chosen as NP = 200 for both CFCEP and FCEP. Scaling factor (𝛽) is 1 for FCEP. In case of PSO the parameters are taken as NP = 200, 𝑤max = 0.25, 𝑤min = 0.05, 𝑐1 = 0.5 and 𝑐2 = 0.5. In case of DE, the population size (NP ), scaling factor (𝐹 ) and crossover constant (𝐶𝑅 ) have been selected 200, 0.75 and 1

Fig. 6. Cost convergence characteristics of three-area system.

respectively. Maximum iteration number has been selected as 600 for all these techniques. The generation matching to the best cost among 100 runs obtained from developed CFCEP is revealed in Table 6. The best, average and 193

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Engineering Applications of Artificial Intelligence 86 (2019) 182–196 Table 5 Simulation results for four-area test system. Power (MW)

CFCEP

FCEP

DE

Power (MW)

CFCEP

FCEP

DE

P1,1 P1,2 P1,3 P1,4 P1,5 P1,6 P1,7 P1,8 P1,9 P1,10 P2,1 P2,2 P2,3 P2,4 P2,5 P2,6 P2,7 P2,8 P2,9 P2,10 P3,1 P3,2 P3,3

110.9854 111.2548 97.4013 179.7331 92.2583 140.0000 259.6061 284.6101 284.6352 130.0000 94.0001 243.5996 214.7598 394.2794 394.2794 394.2794 489.2798 489.2795 511.2794 511.2794 523.2794 523.2794 523.27934

111.0210 111.2493 97.4051 179.7467 92.2492 139.9723 259.6164 284.6065 284.6489 130.0838 94.00477 243.6037 214.7675 394.3151 394.2897 394.2725 489.2777 489.2944 511.2936 511.3452 523.2950 523.2812 523.2996

111.8871 111.2192 97.4737 179.7728 96.9686 139.9951 259.6767 284.6154 284.8686 130.0067 94.0508 94.0000 304.6228 394.4612 394.2926 394.2769 489.4047 489.4068 511.3798 534.1247 523.2896 523.5742 523.4098

P3,4 P3,5 P3,6 P3,7 P3,8 P3,9 P3,10 P4,1 P4,2 P4,3 P4,4 P4,5 P4,6 P4,7 P4,8 P4,9 P4,10 T12 T31 T32 T41 T42 T43

523.2804 523.2794 523.2846 10.0000 10.0000 10.0000 89.4042 190.0000 190.0000 190.0000 165.1169 165.0944 165.0935 89.1925 109.9999 90.8172 458.7989 181.4288 20.3305 188.3199 45.6139 93.9360 99.5636

523.3661 523.3068 523.3044 10.0000 10.0267 10.0188 88.9325 189.9994 190.0000 189.9957 165.1003 165.0921 165.0804 89.2760 109.9946 90.76587 458.7993 181.4527 20.2524 188.1426 45.6004 93.9398 99.5636

523.3785 523.4110 523.3050 10.0740 10.0000 10.0009 96.5558 189.9945 189.9911 189.9909 165.2028 165.4539 164.8549 92.2140 109.9894 109.9997 458.8060 200.0000 15.0030 199.9912 63.5131 99.9884 97.9956

Fuel cost ($/h)

121 679.44

121 682.59

121 794.23

CPU time (s)

30.67

33.45

35.38

Fig. 7. Cost convergence characteristics of four-area system.

Fig. 8. Cost convergence characteristics of 140-unit system.

worst cost and average CPU time among 100 runs acquired from developed CFCEP, FCEP, DE and PSO are summed up in Table 7. The cost acquired from improved particle swarm optimization (IPSO) (Park et al., 2010) is also revealed in Table 7. The cost convergence characteristic acquired from developed CFCEP, FCEP, DE and PSO has been portrayed in Fig. 8. 7. Conclusion In this paper, chaotic fast convergence evolutionary programming (CFCEP) has been suggested for solving a real world complicated multiregion dynamic economic dispatch problem of solar–wind–hydro– thermal power system problem with pumped hydro energy storage considering solar and wind power uncertainty, multi-region economic dispatch problem and economic dispatch problem. The recommended technique adopts a new dynamic scaling factor control method rooted in Tent chaotic sequences for adjusting the scaling factor value of FCEP adaptively. Three different test systems for three different problems are solved by utilizing the recommended CFCEP, FCEP, DE and PSO. It has been observed that the recommended CFCEP technique carries out better than FCEP, DE and PSO. CFCEP can be further perked up by utilizing Tent chaotic mapping for population generation instead of

Fig. A.1. The upper and lower forecast limits of solar irradiation.

random initialization. The future scope of this research is to consider outage possibility of generating units.

194

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Engineering Applications of Artificial Intelligence 86 (2019) 182–196 Table A.1 Thermal generator characteristics.

Fig. A.2. The upper and lower forecast limits of wind speed.

Table 6 Generation (MW) of 140-unit system acquired from CFCEP. Unit

GEN

Unit

GEN

Unit

GEN

Unit

GEN

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

118.4000 163.9999 190.0000 189.9999 168.5398 189.9999 490.0000 490.0000 496.0000 496.0000 496.0000 496.0000 506.0000 509.0000 506.0000 505.0000 506.0000 506.0000 505.0000 505.0000 505.0000 505.0000 505.0000 505.0000 537.0000 537.0000 549.0000 549.0000 501.0000 490.0002 506.0000 506.0000 506.0000 506.0000 500.0000

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

500.0000 241.0000 241.0000 774.0000 769.0000 3.0000 3.0000 249.9999 249.9999 250.0000 249.9999 249.9999 249.9999 250.0000 249.9999 165.0000 165.0000 165.0000 165.0001 180.0000 180.0000 103.0000 198.0000 312.0000 308.5969 163.5998 95.0000 510.9999 511.0000 490.0000 256.7389 489.9998 490.0000 130.0000 339.4397

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

141.5000 388.3272 195.0000 196.2280 196.2569 257.8500 400.9899 330.0000 531.0000 531.0000 542.0000 56.0000 115.0000 115.0000 116.0000 207.0000 207.0000 175.0000 175.0000 180.4329 175.0001 575.4000 547.5000 836.8000 836.5003 682.0000 720.0000 718.0000 720.0000 964.0000 958.0000 947.9000 934.0000 935.0000 876.5000

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140

889.9000 873.7000 877.4000 871.7000 864.8000 882.0000 94.0000 94.0000 94.0000 244.0000 244.0000 244.0000 95.0000 95.0000 116.0000 175.0000 2.0000 4.0000 15.0000 9.0000 12.0000 10.0000 112.0000 4.0000 5.0000 5.0000 50.0000 5.0000 42.0000 42.0000 41.0000 17.0000 7.0000 7.0000 26.0000

Best cost ($) Average cost ($) Worst cost ($) CPU time (s)

CFCEP FCEP DE PSO IPSO (Park et al., 2010)

1 657 923 1 657 924 1 657 989 1 658 348 1 657 962

1 657 924 1 657 926 1 657 994 1 658 359 1 657 962

1 657 925 1 657 931 1 657 998 1 658 372 1 657 962

𝑏𝑖𝑗 𝑐𝑖𝑗 𝑑𝑖𝑗 𝑒𝑖𝑗 𝑈 𝑅𝑖𝑗 𝐷𝑅𝑖𝑗 $/MWh $/(MW)2 h $/h rad/MW MW/h MW/h

1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 1-10 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10

6.73 6.73 7.07 8.18 5.35 8.05 8.03 6.99 6.60 12.9 12.9 12.8 12.5 8.84 8.84 8.84 7.97 7.95 7.97 7.97 6.63 6.63 6.66 6.66 7.10 7.10 3.33 3.33 5.35 3.33

36 36 60 80 47 68 110 135 135 130 94 94 125 125 125 125 220 220 242 242 254 254 254 254 254 254 10 10 47 10

114 114 120 190 97 140 300 300 300 300 375 375 500 500 500 500 500 500 550 550 550 550 550 550 550 550 150 150 97 150

94.705 94.705 309.540 369.030 148.890 222.330 287.710 391.980 455.760 722.820 635.200 654.690 913.400 1760.40 1760.40 1760.40 647.850 649.690 647.830 647.810 785.960 785.960 794.530 794.530 801.320 801.320 1055.10 1055.10 148.89 1055.10

0.00690 0.00690 0.02028 0.00942 0.01140 0.01142 0.00357 0.00492 0.00573 0.00605 0.00515 0.00569 0.00421 0.00752 0.00752 0.00752 0.00313 0.00313 0.00313 0.00313 0.00298 0.00298 0.00284 0.00284 0.00277 0.00277 0.52124 0.52124 0.01140 0.52124

100 100 100 150 120 100 200 200 200 200 200 200 300 300 300 300 300 300 300 300 300 300 300 300 300 300 120 120 120 120

0.084 0.084 0.084 0.063 0.077 0.084 0.042 0.042 0.042 0.042 0.042 0.042 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.077 0.077 0.077 0.077

40 40 40 60 30 50 80 80 80 80 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 50 50 30 50

40 40 40 60 30 50 80 80 80 80 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 50 50 30 50

Table A.2 Hourly load demand (MW) in each area. Hour

P𝐷1

P𝐷2

P𝐷3

Hour

P𝐷1

P𝐷2

P𝐷3

1 2 3 4 5 6 7 8 9 10 11 12

1692.0 1727.6 1772.4 1777.6 1862.6 1927.6 2071.2 2117.6 2221.2 2326.0 2405.0 2492.6

2791.8 2850.5 2924.5 2933.0 3073.3 3180.5 3417.5 3494.0 3665.0 3837.9 3968.3 4112.8

2538.0 2591.4 2658.6 2666.4 2793.9 2891.4 3106.8 3176.4 3331.8 3489.0 3607.5 3738.9

13 14 15 16 17 18 19 20 21 22 23 24

2352.6 2398.8 2123.6 2095.0 2024.4 2133.0 2255.6 2210.8 2148.4 2048.0 1955.8 1872.0

3881.8 3958.0 3503.9 3456.8 3340.3 3519.4 3721.7 3647.8 3544.9 3379.2 3227.1 3088.8

3528.9 3598.2 3185.4 3142.5 3036.6 3199.5 3383.4 3316.2 3222.6 3072.0 2933.7 2808.0

References Al-Agtash, S., Renjeng, S., 1998. Augmented Lagrangian approach to hydro-thermal scheduling. IEEE Trans. Power Syst. 13 (4), 1392–1400. Basu, M., 2013. Artificial bee colony optimization for multi-area economic dispatch. Int. J. Electr. Power Energy Syst. 49, 181–187. Basu, M., 2017. Fast convergence evolutionary programming for economic dispatch problems. IET Gener. Transm. Distrib. 11 (16), 4009–4017. Braennlund, H., Bubenko, J.A., Sjelvgren, D., Andersson, N., 1986. Optimal short term operation planning of a large hydrothermal power system based on a nonlinear network flow concept. IEEE Trans. Power Syst. 1 (4), 75–82. Caponetto, R., Fortuna, L., 2003. Chaotic sequences to improve the performance of evolutionary algorithm algorithms. IEEE Trans. Evol. Comput. 7 (3), 289–304. Chen, C.L., Chen, N., 2001. Direct search method for solving economic dispatch problem considering transmission capacity constraints. IEEE Trans. Power Syst. 16 (4), 764–769. Dubey, H.M., Pandit, M., Panigrahi, B.K., 2016. Ant lion optimization for short-term wind integrated hydrothermal power generation scheduling. Int. J. Electr. Power Energy Syst. 83, 158–174. Fadil, S., Urazel, B., 2013. Solution to security constrained non-convex pumped-storage hydraulic unit scheduling problem by modified sub-gradient algorithm based on feasible values and pseudo water price. Electr. Power Compon. Syst. 41, 111–135.

Table 7 Comparison of performance of 140-unit system. Techniques

Gen 𝑖𝑗 Pmin Pmax 𝑎𝑖𝑗 𝑖𝑗 𝑖𝑗 MW MW $/h

34.5075 36.3742 39.0941 43.6385 –

Appendix See Figs. A.1, A.2 and Tables A.1, A.2. 195

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Engineering Applications of Artificial Intelligence 86 (2019) 182–196 Sharma, Manisha, Pandit, Manjaree, Srivastava, Laxmi, 2011. Reserve constrained multi-area economic dispatch employing differential evolution with time-varying mutation. Int. J. Electr. Power Energy Syst. 33 (3), 753–766. Shoults, R.R., Chang, S.K., Helmick, S., Grady, W.M., 1980. A practical approach to unit commitment, economic dispatch and savings allocation for multiple-area pool operation with import/export constraints. IEEE Trans. Power Appar. Syst. 99 (2), 625–635. Singh Patwal, R., Narang, N., Garg, H., 2018. A novel TVAC-PSO based mutation strategies algorithm for generation scheduling of pumped storage hydrothermal system incorporating solar units. Energy 142, 822–837. Sinha, N., Chakrabarti, R., Chattopadhyay, P.K., 2003. Fast evolutionary programming techniques for short-term hydrothermal scheduling. IEEE Trans. PWRS 18 (1), 214–220. Somasundaram, P., Jothi Swaroopan, N.M., 2011. Fuzzified particle swarm optimization algorithm for multi-area security constrained economic dispatch. Electr. Power Compon. Syst. 39, 979–990. Soroudi, A., Rabiee, A., 2013. Optimal multi-area generation schedule considering renwable resources mix: a real-time approach. IET Gener., Transm. Distrib. 7 (9), 1011–1026. Streiffert, D., 1995. Multi-area economic dispatch with tie line constraints. IEEE Trans. Power Syst. 10 (4), 1946–1951. Swain, R.K., Barisal, A.K., Hota, P.K., Chakrabarti, R., 2011. Short-term hydrothermal scheduling using clonal selection algorithm. Int. J. Electr. Power Energy Syst. 33, 647–656. Tian-Pau, Chang, 2010. Investigation on frequency distribution of global radiation using different probability density functions. Int. J. Appl. Sci. Eng. 8 (2), 99–107. Walters, D.C., Sheble, G.B., 1993. Genetic algorithm solution of economic dispatch with valve point loading. IEEE Trans. Power Syst. 8 (3), 1325–1332. Wang, C., Shahidehpour, S.M., 1992. A decomposition approach to non-linear multi area generation scheduling with tie-line constraints using expert systems. IEEE Trans. Power Syst. 7 (4), 1409–1418. Wang, L., Singh, C., 2009. Reserve-constrained multi-area environmental / economic dispatch based on particle swarm optimization with local search. Eng. Appl. Artif. Intell. 22, 298–307. Wong, K.P., Wong, Y.W., 1994. Short-term hydrothermal scheduling part 1: simulated annealing approach. IEE Proc. Gener., Transm. Distrib. 141 (5), 497–501. Wood, A.J., Wollenberg, B.F., 1984. Power Generation, Operation and Control, second ed. Wiley, New York, pp. 230–239, ch. 7. Wood, A.J., Wollenberg, B.F., 1996. Power Generation, Operation and Control, second ed. Wiely, New York. Xia, Q., Xiang, N., Wang, S., Zhang, B., Huang, M., 1988. Optimal daily scheduling of cascaded plants using a new algorithm of nonlinear minimum cost network flow. IEEE Trans. Power Syst. 3 (3), 929–935. Yalcinoz, T., Short, M.J., 1998. Neural networks approach for solving economic dispatch problem with transmission capacity constraints. IEEE Trans. Power Syst. 13 (2), 307–313. Yan, H., Luh, P.B., Guan, X., Rogan, P.M., 1993. Scheduling of hydrothermal power systems. IEEE Trans. Power Syst. 8 (3), 1358–1365. Yang, J.S., Chen, N.M., 1989. Short term hydrothermal coordination using multi-pass dynamic programming. IEEE Trans. Power Syst. 4 (3), 1050–1056. Yang, P.C., Yang, H.T., Huang, C.L., 1996. Scheduling short-term hydrothermal generation using evolutionary programming techniques. IEE Proc. Gener. Transm. Distrib. 143 (4), 371–376.

Ferrero, R.W., Rivera, J.F., Shahidehpour, S.M., 1998. A dynamic programming twostage algorithm for long-term hydrothermal scheduling of multi-reservoir systems. IEEE Trans. Power Syst. 13 (4), 1534–1540. Ghasemi, M., Aghaei, J., Akbari, E., Ghavidel, S., Li, Li, 2016. A differential evolution particle swarm optimizer for various types of multi-area economic dispatch problems. Energy 107, 182–195. Gil, E., Bustos, J., 2003. Short-term hydrothermal generation scheduling model using a genetic algorithm. IEEE Trans. PWRS 18 (4), 1256–1264. Helmick, S.D., Shoults, R.R., 1985. A practical approach to an interim multi-area economic dispatch using limited computer resources. IEEE Trans. Power Appar. Syst. 104 (6), 1400–1404. Hetzer, J., Yu, D.C., Bhattarai, K., 2008. An economic dispatch model incorporating wind power. IEEE Trans. Energy Convers. 23 (2), 603–611. Hota, P.K., Barisal, A.K., Chakrabarti, R., 2009. An improved PSO technique for short-term optimal hydrothermal scheduling. Electr. Power Syst. Res. 79 (7), 1047–1053. Jayabarathi, T., Sadasivam, G., Ramachandran, V., 2000. Evolutionary programming based multi-area economic dispatch with tie line constraints. Electr. Mach. Power Syst. 28, 1165–1176. Khandualo, S.K., Barisal, A.K., Hota, P.K., 2013. Scheduling of pumped storage hydrothermal system with evolutionary programming. J. Clean Energy Technol. 1 (4), 308–312. Lakshminarasimman, L., Subramanian, S., 2006. Short-term scheduling of hydrothermal power system with cascaded reservoirs by using modified differential evolution. IEE Proc. Gener., Transm. Distrib. 153 (6), 693–700. Liang, Ruey-Hsun, Liao, Jian-Hao, 2007. A fuzzy-optimization approach for generation scheduling with wind and solar energy systems. IEEE Trans. PWRS 22 (4), 1665–1674. Ma, T., Yang, H., Lu, L., Peng, J., 2015. Pumped storage-based standalone photovoltaic power generation system: Modeling and techno-economic optimization. Appl. Energy 137 (1), 649–659. Manoharan, P.S., Kannan, P.S., Baskar, S., Willjuice Iruthayarajan, M., 2009. Evolutionary algorithm solution and KKT based optimality verification to multi-area economic dispatch. Electr. Power Energy Syst. 31 (7–8), 365–373. Nilsson, O., Sjelvgren, D., 1996. Mixed-integer programming applied to short-term planning of a hydro-thermal system. IEEE Trans. Power Syst. 11 (1), 281–286. Orero, S.O., Irving, M.R., 1998. A genetic algorithm modeling framework and solution technique for short term optimal hydrothermal scheduling. IEEE Trans. PWRS 13 (2). Park, J.B., Jeong, Y.W., Shin, J.R., Lee, K.Y., 2010. An improved particle swarm optimization for nonconvex economic dispatch problems. IEEE Trans. Power Syst. 25 (1), 156–166. Pereira, M.V.F., Pinto, L.M.V.G., 1982. A decomposition approach to the economic dispatch of the hydrothermal system. IEEE Trans. Power Appar. Syst. PAS-101 (10), 3851–3860. Perez-Diaz, J., Jim, J., 2016. Contribution of a pumped-storage hydropower plant to reduce the scheduling costs of an isolated power system with high wind power penetration. Energy 109 (1), 92–104. Romano, R., Quintana, V.H., Lopez, R., Valadez, V., 1981. Constrained economic dispatch of multi-area systems using the Dantzig–Wolfe decomposition principle. IEEE Trans. Power Appar. Syst. 100 (4), 2127–2137. Roy, P.K., 2013. Teaching learning based optimization for short-term hydrothermal scheduling problem considering valve point effect and prohibited discharge constraint. Int. J. Electr. Power Energy Syst. 53, 10–19. Ruzic, S., Rajakovic, R., 1998. Optimal distance method for Lagrangian multipliers updating in short-term hydro-thermal coordination. IEEE Trans. Power Syst. 13 (4), 1439–1444.

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