ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT AZOJETE December 2022. Vol. 18(4):669-682 Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria. Print ISSN: 1596-2644, Electronic ISSN: 2545-5818 www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 669 ORIGINAL RESEARCH ARTICLE OPTIMAL LOAD SCHEDULING OF POWER PLANTS IN A GRID CONSIDERING THE PLANT’S CAPACITY AND LINE LOSSES P. I. Obi*, O. Oputa and E. A. Amako Department of Electrical & Electronic Engineering, Michael Okpara University of Agriculture Umudike, Abia State, Nigeria. *Corresponding author’s email address: patndyobi@gmail.com 1.0 Introduction Short-term gas – hydro – thermal power plants coordination consists of determining the optimal usage of available gas, hydro and thermal resources during a scheduling period of time ranging from one (1) day to one (1) week (Wood et al., 2013; Hossain and Shiblee, 2017). This is to determine optimally, which of the generating units should run at any point in time as well as the power generated by the gas, hydro and thermal plants so that the total cost is minimized. Minimizing the total cost in this optimization problem is subject of many control and operational constraints and can be obtained using different optimization methods that include Lagrangian relaxation and Benders decomposition-based methods, Mixed-integer and Dynamic programming among others (Farhat and El-Hawary, 2009; Kovalev et al., 2011). These are however conventional optimization methods which use gradients for the search of optimum values. Evolutionary optimization methods have become an alternative to conventional optimization techniques for solving real world problems having non-convexity, ARTICLE INFORMATION ABSTRACT In a grid or part of a grid operated by a single service provider with various types of power plants, the total load demand on the grid can be serviced by the plants’ outputs in such a way that a minimum cost of energy is spent to generate this total demanded load thereby maximizing profit for the service providers without placing exorbitant tariff on customers. This paper considered how this can be archived in the Niger- Delta region of Nigeria (as a case study). We assumed that the power generated by the plants in the region is also consumed by the region. Using the Lagrnge multiplier optimization method in our analysis and for a total power demand/generation of 2170 MW, a total of 752,435 × 104cal/hr of fuel will be burnt. But with the system of running cost minimization developed in this paper, only about 575,521 × 104cal/hr of fuel was burnt in generating that same quantity of power demanded resulting in a net savings of 176,914 × 104 cal/hr. This is equivalent to 2.057 × 103 kWh; with energy sold at ₦48/kWh in Nigeria, a total of ninety-eight thousand, seven hundred and thirty-six Naira (₦98,736.00) only will be saved each hour resulting in a net savings of eight hundred and sixty-four million, nine hundred and twenty-seven thousand, three hundred and sixty naira (₦864,927,360.00) only per annum. This is, therefore, recommended for implementation in running Nigeria’s power system. © 2022 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved. Submitted 6 April, 2022 Revised 13 June, 2022 Accepted 20 June, 2022 Keywords: Optimization Energy Cost Savings and Profitability Grid Plant’s Capacity http://www.azojete.com.ng/ mailto:patndyobi@gmail.com mailto:patndyobi@gmail.com Arid Zone Journal of Engineering, Technology and Environment, December, 2022; Vol. 18(4):669-682 ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 670 non-differentiability and discontinuity. In this way, evolutionary methods have been successfully applied to power system problems as well. Though, the evolutionary methods have been suitable for power system problems, the premature convergence and stagnation they exhibit poise a problem when using them directly (Attaviriyanupap, et al., 2002). A hybrid algorithm combines the conventional and evolutionary optimization techniques for solving power system problems; it can be used for solving the problems having non-convexity and non-smoothness in the function space. This hybrid optimization technique solves problems with multi objectives (Jafari et al., 2019; Lei et al., 2020; Panda et al., 2020). A cost-based approach has also been used to determine the optimal operational strategy that yields a minimum operating cost. The optimal operational strategy is achieved through the estimation of the hourly generated power, the amount of thermal power recovered from the one type of power plant in the grid e.g fuel cell power plants (FCPP) to satisfy the thermal load, the amount of power trade with the local grid, and the amount of by-products (hydrogen) that can be generated from the plant type which is useful to the society (El-Sharkh et al., 2006). As such, the cost-based optimization is archived by minimizing the objective function (OF) as given in Equation (1) 𝑂𝐹 = 𝑀𝑖𝑛 (∑ 𝐶𝑜𝑠𝑡𝑖 − ∑ 𝐼𝑛𝑐𝑜𝑚𝑒𝑖𝑖𝑖 ) (1) Where ∑ 𝐶𝑜𝑠𝑡𝑖𝑖 is the sum all the cost i, of production (in ₦ or any other currency), ∑ 𝑖𝑛𝑐𝑜𝑚𝑒𝑖𝑖 sum of all i source of income (in ₦ or any other currency). However, in grids (like Nigeria’s) where none of the plant’s by-product can be a source of income, such method cannot be adopted. Power systems operate as a grid (Obi et al., 2017) with several power generating plants that include hydropower plants, steam and gas turbine power plants (among others). Suppose the generation and transmission stages of such grid is ran by a single service provider, then the service provider can maximize profit by reducing the cost of production which include generating power and getting the power generated to load centers to the barest minimum (Obi and Offor, 2012). Reduction of production cost can come from the following (Avalos, 2008): 1. Reduction of salaries of technical and non-technical staffs. 2. Reduction of losses along the power lines (before the generated power gets to the consumers so that there will be more power at the load centers). 3. Economic combination of various power plants on the grid to feed the total grid demand (based on their respective running cost). In this research, the item 3 option given above shall be investigated and we shall assume that the power plants feed only loads in the Nigeria regions under study which are the south – south and south – east regions of Nigeria. Also, we shall assume that the total load demand is less than the total power the various plant combinations can generate. At the moment, there are nine (9) functional power plants in the region and their functional generating capacities are given in Table 1. file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:patndyobi@gmail.com Obi et al: Optimal Load Scheduling of Power Plants in a Grid Considering the Plant’s Capacity and Line Losses. AZOJETE, 18(4):669-682. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 671 Table 1: List of power plants in the region of Nigeria under analysis Power station Location Type Capacity Status Aba Power Station Aba Combined cycle gas turbine (more of steam) 340 MW Operational (260MW) Afam I – V Power Station Afam Combined cycle gas turbine (more of steam) 726 MW Partially Operational (313MW) Afam VI Power Station Afam Combined cycle gas turbine (more of gas) 624 MW Operational (600MW) Alaoji Power Station(NIPP) Aba Combined cycle gas turbine (more of steam) 1074 MW Partially operational (250MW) Okpai Power Station Okpai Combined cycle gas turbine (more of gas) 480 MW Operational (470MW) Omoku Power Station Omoku Combined cycle gas turbine (more of steam) 450 MW Operational (340MW) Sapele Power Station Sapele Gas-fired steam turbine and Simple cycle gas turbine (more of steam) 1020 MW Partially Operational (335 MW) Sapele Power Station(NIPP) Sapele Combined cycle gas turbine (more of gas) 450 MW Operational (420MW) Delta - Ughelli Power Station Ughelli Combined cycle gas turbine (more of gas) 900 MW Partially Operational (360 MW) Source: (Nigerian Agip Oil Company, 2003; Nigerian Agip Oil Company and Rivers state Government, 2003; Shell Petroleum Development Company, 2003; Geometric Power Limited, 2005; Shell Petroleum Development Company, 2005; Federal Government of Nigeria National Integrated Power Project, 2012; Transnational Corporation of Nigeria Plc, 2012; Niger Delta Power Holding Company and Federal Government of Nigeria, 2013; Eurafric Power Limited, 2014; Oputa, 2015). They are all combined circle power plants generating power from gas and steam respectively. The heat from the gas turbine section is used to heat up water to steam and eventually used to drive another turbine. The more of steam combined circle plant generate more power with the steam driven alternator than the gas. The combined operation of the power plants in Table 1 was carried out without considering the various plant’s limit or capacity (Oputa, 2015; Oputa et al., 2019). The authors optimized the power plant’s operations by using only the respective plant’s incremental fuel cost without considering their limit and power line losses. 2. Materials and Methods In this paper, we developed mathematical models showing the relationship between the power generated by the various power plants in the grid and the cost of fuel burnt to generate such power. We shall then use programs in MATLAB to solve the models developed as they are complex models that will be difficult to solve manually. The method of equal incremental fuel cost for all power plants in the region under analysis was employed. Hence, we assumed that all 9 plants run with equal incremental fuel cost. We however did not consider the effect of the by-product of each plant to the environment. http://www.azojete.com.ng/ mailto:patndyobi@gmail.com http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/Aba,_Abia http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam&action=edit&redlink=1 http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam&action=edit&redlink=1 http://en.wikipedia.org/wiki/Combined_cycle_gas_turbine http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/National_Integrated_Power_Project http://en.wikipedia.org/wiki/Abia_state http://en.wikipedia.org/wiki/Combined_cycle_gas_turbine http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Okpai&action=edit&redlink=1 http://en.wikipedia.org/wiki/Combined_cycle_gas_turbine http://en.wikipedia.org/w/index.php?title=Omoku_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Omoku_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/Omoku http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/Sapele http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/National_Integrated_Power_Project http://en.wikipedia.org/wiki/Sapele http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine http://en.wikipedia.org/w/index.php?title=Delta_-_Ughelli_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Delta_-_Ughelli_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/Ughelli http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine Arid Zone Journal of Engineering, Technology and Environment, December, 2022; Vol. 18(4):669-682 ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 672 If a quantity of fuel burns for one hour liberating an energy 𝑓𝑖 (in calorie) to generate a particular power output, 𝑃𝐺𝑖 (in MW) in that one hour, the relationship between the quantity of fuel burnt and power generated is given as (Gupta, 2009; Bommirani and Thenmalar, 2013). 𝑓𝑖 = 𝑎𝑖 + 𝑏𝑖𝑃𝐺𝑖 + 𝑐𝑖𝑃𝐺𝑖 2 (2) 𝑎𝑖 (𝐶𝑎𝑙), 𝑏𝑖 (𝐶𝑎𝑙/𝑀𝑤), 𝑎𝑛𝑑 𝑐𝑖 (𝐶𝑎𝑙/𝑀𝑊2) are constants of the ith power plant. For n plants in the system (nine (9) in this case), the total fuel burnt by all the (9) plants is 𝐹𝑇 = ∑ (9 𝑖=1 𝑎𝑖 + 𝑏𝑖𝑃𝐺𝑖 + 𝑐𝑖𝑃𝐺𝑖 2 ) (3) Reducing the cost of generating the total power generated by all the plants can be archived by minimizing Equation (3); this equation is thus the operating function (OF). The minimization of the OF is subject to some factors (constraints). These include: 1. Real power balance: Real power in supply, 𝑃𝐺𝑖 must be equal to real power in demand, 𝑃𝐷 (plus real power losses, 𝑃𝐿). − ∑ 𝑃𝐺𝑖 + 𝑃𝐷 + ∑ 𝑃𝐿 = 09 𝑖=1 (4) 2. Spare capacity constraint: Some load predictions at load centers are inaccurate; there are also sudden changes in load demand as well as inadvertent losses of schedule generation in the Nigeria power system. The spare capacity constraint can be used to account for these and more. This ensures that the total generation available at any time should be in excess of total anticipated load demand and total system loss by an amount not less than a specific minimum, called the spare capacity, 𝑃𝑆𝑃 (Bogdan et al., 2007; Jinchao et al., 2012). ∑ 𝑃𝐺𝑖 9 𝑖=1 = ∑ 𝑃𝐿 + 𝑃𝑆𝑃 + 𝑃𝐷 (5) PL is calculated in its simplest quadratic for as 𝑃𝐿 = ∑ ∑ 𝑃𝐺𝑖𝐵𝑖𝑗𝑃𝐺𝑗 9 𝑗=1 9 𝑖=1 (6) Where, 𝐵𝑖𝑗 are the loss coefficients connecting the ith and jth buses of the transmission network. It can also be calculated by a more general formula containing a linear and constant term as shown in Equation (7), this is known as the Kron’s loss formula (Saadat, 1999). 𝑃𝐿 = ∑ ∑ 𝑃𝐺𝑖𝐵𝑖𝑗𝑃𝐺𝑗 + ∑ 𝐵0𝑖𝑃𝐺𝑖 + 𝐵00 𝑛 𝑖=1 9 𝑗=1 9 𝑖=1 (7) Generation Capacity Constraint: in carrying out the optimization of the OF, the generating capacity of each plant must not be exceeded. Hence, 𝑃𝐺𝑖 𝑛𝑖𝑚 ≤ 𝑃𝐺𝑖 ≤ 𝑃𝐺𝑖 𝑚𝑎𝑥 (8) 3. Grid Capacity Constraint: For secure operation, the actual transmission capacity must be restricted by its upper limit as 𝑆𝑖𝑖(𝑃𝐺𝑖) ≤ 𝑆𝑙𝑖 𝑚𝑎𝑥 . 𝑖 = 1, 𝑏2, … , 𝑛𝑙 (9) Where nl is the number of transmission lines; Sli is the electric power flow of the ith transmission line which is influenced by the PGi; and 𝑆𝑙𝑖 𝑚𝑎𝑥 is the upper limit of the ith transmission line (Gupta, 2009). file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:patndyobi@gmail.com Obi et al: Optimal Load Scheduling of Power Plants in a Grid Considering the Plant’s Capacity and Line Losses. AZOJETE, 18(4):669-682. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 673 4. Minimum Emission of Pollutant from fossil-fuel Et Constraint: The minimization of the OF should also ensure that the environment is not over polluted with bye-products of the respective plant’s operations. Hence, the OF must be minimized in such a way that Et is minimum (Obi et al., 2017). 𝑀𝑖𝑛𝐸𝑡 = ∑ 𝐸𝑖(𝑃𝐺𝑖)𝑁𝑐 𝑖=1 (10) or 𝑀𝑖𝑛𝐸𝑡 = ∑ (𝛼𝑖𝑃𝐺𝑖 2 + 𝛽𝑖𝑃𝐺𝑖 + 𝛾𝑖 + 𝜉𝑖𝑒𝑥𝑝(𝜏𝑖𝑃𝐺𝑖))𝑁𝑐 𝑖=1 (11) where αi, βi, γi, ζi, and 𝜏I, are coefficients of the ith generator’s emission characteristics, 𝑁𝑐 is the number of power plant in the grid. Our task is therefore to minimize Equation (3) subject to Equations (4) to (10). However, we shall not consider the effect of emission of pollutant from fossil-fuel and the grid capacity constant. Hence, we shall minimize the OF as shown in Equation (3) subject to constrains as shown in Equations (5), (6) and (8) alone. Hence, we are to minimize the total cost fuel burnt 𝐹𝑇 = ∑ (9 𝑖=1 𝑎𝑖 + 𝑏𝑖𝑃𝐺𝑖 + 𝑐𝑖𝑃𝐺𝑖 2 ) subject to (i) − ∑ 𝑃𝐺𝑖 + 𝑃𝐷 + ∑ 𝑃𝐿 = 09 𝑖=1 (ii) ∑ 𝑃𝐺𝑖 9 𝑖=1 = ∑ 𝑃𝐿 + 𝑃𝑆𝑃 + ∑ 𝑃𝐷𝑖 9 𝑖=1 (iii) 𝑃𝐺𝑖 𝑛𝑖𝑚 ≤ 𝑃𝐺𝑖 ≤ 𝑃𝐺𝑖 𝑚𝑎𝑥 again, we can also neglect the spare capacity in the constrain in (i) above. By combining the OF and constrains functions and applying Lagrange multiplier (ℒ) (Saadat, 1999). ℒ = 𝐹𝑇 + 𝜆 ((𝑃𝐷 + ∑ 𝑃𝐿 − ∑ 𝑃𝐺𝑖 9 𝑖=1 ) + ∑ µ𝑖(max)(𝑃𝐺𝑖 − 𝑃𝐺𝑖(𝑚𝑎𝑥))9 𝑖=1 ) + ∑ µ𝑖(min)(𝑃𝐺𝑖 −9 𝑖=1 𝑃𝐺𝑖(𝑚𝑖𝑛)) (12) where µ𝑖(max) = 0 when 𝑃𝐺𝑖 < 𝑃𝐺𝑖(𝑚𝑎𝑥) and µ𝑖(min) = 0 when 𝑃𝐺𝑖 > 𝑃𝐺𝑖(𝑚𝑖𝑛). Thus, if the constraint is not violated, its associate µ variable is zero and the corresponding term in Equation (12) does not exist. Equation (12) is minimum at 𝜕ℒ 𝜕𝑃𝑖 = 0 (13) 𝜕ℒ 𝜕𝜆 = 0 (14) 𝜕ℒ 𝜕µ𝑖(𝑚𝑎𝑥) = 𝑃𝐺𝑖 − 𝑃𝐺𝑖(𝑚𝑎𝑥) = 0 (15) 𝜕ℒ 𝜕µ𝑖(𝑚𝑖𝑛) = 𝑃𝐺𝑖 − 𝑃𝐺𝑖(𝑚𝑖𝑛) = 0 (16) The solution of the Equation (12) is therefore given as 𝑑𝑓𝑖 𝑑𝑃𝐺𝑖 + 𝜆 𝜕𝑃𝐿 𝜕𝑃𝐺𝑖 = 𝜆 (17) From Equation (7), 𝜕𝑃𝐿 𝜕𝑃𝐺𝑖 = 2 ∑ 𝐵𝑖𝑗𝑃𝐺𝑗 + 𝐵0𝑖 9 𝑗=1 (18) http://www.azojete.com.ng/ mailto:patndyobi@gmail.com Arid Zone Journal of Engineering, Technology and Environment, December, 2022; Vol. 18(4):669-682 ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 674 Putting (15) and (2) into (14), (14) can be rewrite as 𝑏𝑖𝑃𝐺𝑖 + 2𝑐𝑖𝑃𝐺𝑖 + 2 ∑ 𝐵𝑖𝑗𝑃𝐺𝑗 + 𝐵0𝑖 9 𝑗=1 = 𝜆 (19) Or ( 𝑐𝑖 𝜆 + 𝐵𝑖𝑖) 𝑃𝐺𝑖 + ∑ 𝐵𝑖𝑗𝑃𝐺𝑗 9 𝑗=1 𝑗≠𝑖 = 1 2 (1 − 𝐵0𝑖 − 𝑏𝑖 𝜆 ) (20) The optimal dispatch for an estimated 𝜆1 by solving Equation (20) using Iterative process by gradient method. 𝑃𝐺𝑖 and its value at the kth iteration is given as (Saadat, 1999) 𝑃𝐺𝑖 𝑘 = 𝜆𝑘(1−𝐵0𝑖)−𝑏𝑖−2𝜆𝑘 ∑ 𝐵𝑖𝑗𝑗≠𝑖 𝑃𝐺𝑗 𝑘 2(𝑐𝑖+𝜆𝑘𝐵𝑖𝑖) (21) Total power generated must be equal to the total power demanded and total power lost. ∑ 𝑃𝑖 = 𝑃𝐷 + 𝑃𝐿 9 𝑖=1 (22) Implementing Equation (21) in (22), 𝑓𝜆𝑘 = 𝑃𝐷 + 𝑃𝐿 𝑘 (23) Where 𝑓𝜆𝑘 = ∑ 𝑃𝐺𝑖 𝑘 = 𝜆𝑘(1−𝐵0𝑖)−𝑏𝑖−2𝜆𝑘 ∑ 𝐵𝑖𝑗𝑗≠𝑖 𝑃𝐺𝑗 𝑘 2(𝑐𝑖+𝜆𝑘𝐵𝑖𝑖) 9 𝑖=1 (24) Using Taylor series on the RHS of (21) 𝑓𝜆𝑘 + ( 𝑑𝑓𝜆 𝑑𝜆 ) 𝑘 ∆𝜆𝑘 = 𝑃𝐷 + 𝑃𝐿 𝑘 (25) ∆𝜆𝑘 = ∆𝑃𝑘 ( 𝑑𝑓𝜆 𝑑𝜆 ) 𝑘 = ∆𝑃𝑘 ∑( 𝜕𝑃𝐺𝑖 𝜕𝜆 ) 𝑘 (26) where ∑ ( 𝜕𝑃𝐺𝑖 𝜕𝜆 ) 𝑘 9 𝑖=1 = ∑ 𝑐𝑖(1−𝐵0𝑖)+𝐵𝑖𝑖𝑏𝑖−2𝜆𝑘 ∑ 𝐵𝑖𝑗𝑗≠𝑖 𝑃𝐺𝑗 𝑘 2(𝑐𝑖+𝜆𝑘𝐵𝑖𝑖) 2 9 𝑖=1 (27) Note that for any iteration, ∆𝑃𝑘 = 𝑃𝐷 + 𝑃𝐿 𝑘 − ∑ 𝑃𝐺𝑖 𝑘9 𝑖=1 (28) The iterative process is continued until ∆𝑃𝑘 is less than a specific accuracy limit. Using the simplest/approximate loss formula 𝑃𝐿 = ∑ 𝐵𝑖𝑖𝑃𝐺𝑖 29 𝑖=1 (29) Using Equation (29), the common incremental fuel cost of all power plants 𝜆 as 𝐵𝑖𝑗 = 𝐵00 = 0. Then Equations (21) and (27) respectively becomes 𝑃𝐺𝑖 𝑘 = 𝜆𝑘−𝑏𝑖 2(𝑐𝑖+𝜆𝑘𝐵𝑖𝑖) (30) ∑ ( 𝜕𝑃𝐺𝑖 𝜕𝜆 ) 𝑘 9 𝑖=1 = ∑ 𝑐𝑖+𝐵𝑖𝑖𝑏𝑖 2(𝑐𝑖+𝜆𝑘𝐵𝑖𝑖) 2 9 𝑖=1 (31) Equations (28) to (31) were used to analyze the system. 3. Results and Discussion Data collected from respective power plants’ data books show the energy or fuel burnt in generating a corresponding amount of power (Oputa, 2015) and they are given in Table 2. file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:patndyobi@gmail.com Obi et al: Optimal Load Scheduling of Power Plants in a Grid Considering the Plant’s Capacity and Line Losses. AZOJETE, 18(4):669-682. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 675 Table 2: Fuel burnt/Power generation Characteristics of each plant Power Station Energy (Fuel) Burnt/Hour [(Cal) × 104/ℎ𝑟] Plant Output Power (Mw) Power Station Energy (Fuel) Burnt/Hour [(Cal) × 104/ ℎ𝑟] Plant Output Power (Mw) Aba Power Station 2.3 0 Delta – 5.1 0 9,200 80 Ughelli Power 102,000 300 11,430 90 Station 105,385 305 13,900 100 115,845 320 16,600 110 134,385 345 Afam I – V Power Station 3.1 0 Sapele Power 3.2 0 22,500 150 (Station NIPP) 147,200 400 28,560 170 150,865 405 35,345 190 158,330 415 42,840 210 161,360 419 Afam VI Power Station 2.8 0 Sapele Power 4.0 0 254,000 500 Station 8,725 80 285,145 530 13,505 100 295,925 540 15,140 106 329,460 570 19,330 120 Alaoji Power Station (NIPP) 4.2 0 Omoku 2.1 0 36,725 180 Power 8,885 80 40,857 190 Station 14,810 105 45,205 200 18,200 117 47,462 205 21,580 128 Okpai Power Station 3.3 0 163,200 400 175,548 415 195,350 438 219,950 465 Source: (Nigerian Agip Oil Company, 2003; Nigerian Agip Oil Company and Rivers state Government, 2003; Shell Petroleum Development Company, 2003; Geometric Power Limited, 2005; Shell Petroleum Development Company, 2005; Federal Government of Nigeria National Integrated Power Project, 2012; Transnational Corporation of Nigeria Plc, 2012; Niger Delta Power Holding Company and Federal Government of Nigeria, 2013; Eurafric Power Limited, 2014). The quantity of energy (or quantity of fuel burnt) by each plant when generating power is given in Table 2. From the table, the values of the plant constant- ai, bi, ci for each plant can be calculated and their approximate values are given in Table 3. The data in Table 2 were used to obtain the fuel cost – power equation any particular plant. For example, using the figures in the blue highlighted values of Aba power station, 92,000kcal/hr generated 80MWin that 1 hour. Hence, 92000 = 𝑎 + 𝑏(80) + 𝑐(80)2. Also, 139,000 = 𝑎 + 𝑏(100) + 𝑐(100)2 http://www.azojete.com.ng/ mailto:patndyobi@gmail.com http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/National_Integrated_Power_Project http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 Arid Zone Journal of Engineering, Technology and Environment, December, 2022; Vol. 18(4):669-682 ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 676 Having some 3 set of equations for a particular plant, ‘a’, ‘b’ and ‘c’ for that plant were obtained and presented in Table 3. Table 3: Energy equations for various power Plants in the region (a Function of their power Generated) Power station a B c Fuel Eqn Aba Power Station 2.3 19 1.2 2.3 + 19𝑃𝐺 + 1.2𝑃𝐺 2 Afam I – V Power Station 3.1 15 0.9 3.1 + 15𝑃𝐺 + 0.9𝑃𝐺 2 Afam VI Power Station 2.8 8 1 2.8 + 8𝑃𝐺 + 𝑃𝐺 2 Alaoji Power Station(NIPP) 4.2 6 1.1 4.2 + 6𝑃𝐺 + 1.1𝑃𝐺 2 Okpai Power Station 3.3 8 1 3.3 + 8𝑃𝐺 + 𝑃𝐺 2 Omoku Power Station 2.1 15 1.2 2.1 + 15𝑃𝐺 + 1.2𝑃𝐺 2 Sapele Power Station 4 5 1.3 4 + 5𝑃𝐺 + 1.3𝑃𝐺 2 Sapele Power Station(NIPP) 3.2 8 0.9 3.2 + 8𝑃𝐺 + 0.9𝑃𝐺 2 Delta - Ughelli Power Station 5.1 10 1.1 5.1 + 10𝑃𝐺 + 1.1𝑃𝐺 2 The minimum power to be generated by each individual power plant without economically running at a loss is given in Table 4. Table 4: Minimum Power to be generated without economic loss Power station Minimum Power Generation (MW) Plant Operational Capacity (MW) Aba Power Station 20 260 Afam I – V Power Station 38 313 Afam VI Power Station 58 600 Alaoji Power Station(NIPP) 22 250 Okpai Power Station 55 470 Omoku Power Station 45 340 Sapele Power Station 40 335 Sapele Power Station(NIPP) 55 420 Delta-Ughelli Power Station 38 360 Source: (Nigerian Agip Oil Company, 2003; Nigerian Agip Oil Company and Rivers state Government, 2003; Shell Petroleum Development Company, 2003; Geometric Power Limited, 2005; Shell Petroleum Development Company, 2005; Federal Government of Nigeria National Integrated Power Project, 2012; Transnational Corporation of Nigeria Plc, 2012; Niger Delta Power Holding Company and Federal Government of Nigeria, 2013; Eurafric Power Limited, 2014). The minimum power that a particular power plant must be generating to keep the said plant in the system is got from the fact that most plant in the system are combined cycle power plants. Thus, it comprises of more than one unit and some unit’s operation depend on the operations of some other units. From Tables 3 and 4, the respective incremental fuel cost and the plant’s limits are given in Table 5: file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:patndyobi@gmail.com http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Omoku_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Delta_-_Ughelli_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Omoku_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Delta_-_Ughelli_Power_Station&action=edit&redlink=1 Obi et al: Optimal Load Scheduling of Power Plants in a Grid Considering the Plant’s Capacity and Line Losses. AZOJETE, 18(4):669-682. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 677 Table 5: Incremental Fuel Cost model of various power plant in the region S/N Power station Incremental 𝜆 Plant’s Limit 1 Aba Power Station [19 + 2.4𝑃𝐺] 20 ≤ 𝑃𝐺 ≤ 260 2 Afam I – V Power Station [15 + 1.8𝑃𝐺] 38 ≤ 𝑃𝐺 ≤ 313 3 Afam VI Power Station [8 + 2𝑃𝐺] 58 ≤ 𝑃𝐺 ≤ 600 4 Alaoji Power Station(NIPP) [6 + 2.2𝑃𝐺] 22 ≤ 𝑃𝐺 ≤ 250 5 Okpai Power Station [3.3 + 2𝑃𝐺]𝐿5 55 ≤ 𝑃𝐺 ≤ 470 6 Omoku Power Station [15 + 2.4𝑃𝐺] 45 ≤ 𝑃𝐺 ≤ 340 7 Sapele Power Station [5 + 2.6𝑃𝐺] 40 ≤ 𝑃𝐺 ≤ 335 8 Sapele Power Station(NIPP) [8 + 1.8𝑃𝐺]𝐿8 55 ≤ 𝑃𝐺 ≤ 420 9 Delta-Ughelli Power Station [10 + 2.2𝑃𝐺] 38 ≤ 𝑃𝐺 ≤ 360 Taking a case where the 9 power plants generate a total power of 2,170 MW at a particular time from Table 2; we calculated the approximate lost coefficient of the system as given in Table 6 and they are specified in per unit on a 1,000 MWA base. Table 6: Lost coefficient of the system under analysis Line Coefficient Value (× 10−6𝑀𝑊−1 Line Coefficient Value (× 10−6𝑀𝑊−1) 𝐵11 1.46 𝐵22 2.17 𝐵33 1.15 𝐵44 1.73 𝐵55 2.22 𝐵66 2.68 𝐵77 1.99 𝐵88 2.73 𝐵99 1.68 Then the simplest power loss for the system as given in Equation (29); 𝑃𝐿 = 0.0146 ( 𝑃𝐺1 100 ) 2 + 0.0217 ( 𝑃𝐺2 100 ) 2 + 0.0115 ( 𝑃𝐺3 100 ) 2 + 0.0173 ( 𝑃𝐺4 100 ) 2 + 0.0222 ( 𝑃𝐺5 100 ) 2 + 0.0268 ( 𝑃𝐺6 100 ) 2 + 0.0199 ( 𝑃𝐺7 100 ) 2 + 0.0273 ( 𝑃𝐺8 100 ) 2 + 0.0168 ( 𝑃𝐺9 100 ) 2 (32) Starting with 𝜆1 as 400.0, the values for 𝑃𝐺1 1 , 𝑃𝐺2 1 , … . , 𝑃𝐺9 1 from Equation (30), and 𝑃𝐿 from Equation (32) were computed by a program developed in MATLAB and the results are as presented in Table 7 in MW (where 𝑃𝐺1 1 , 𝑃𝐺2 1 , … . , 𝑃𝐺9 1 are in the order as presented in the Table 5 listing the power plants, i.e 𝑃𝐺1 1 power generated by Aba power station, 𝑃𝐺2 1 is the power generated from Afam I – V Power Station and so on) Table 7: Power plant generating power after 1st iteration Plant generation Value (MW) Plant generation Value (MW) Plant generation Value (MW) 𝑃𝐺1 1 158.6728 𝑃𝐺2 1 213.6828 𝑃𝐺3 1 195.9099 𝑃𝐺4 1 178.9783 𝑃𝐺5 1 195.8261 𝑃𝐺6 1 160.2735 𝑃𝐺7 1 151.8301 𝑃𝐺8 1 217.5139 𝑃𝐺9 1 177.1645 𝑃𝐿 1 0.7215 http://www.azojete.com.ng/ mailto:patndyobi@gmail.com http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Omoku_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Delta_-_Ughelli_Power_Station&action=edit&redlink=1 Arid Zone Journal of Engineering, Technology and Environment, December, 2022; Vol. 18(4):669-682 ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 678 Total power 2,170 MW Real power difference from the 1st iteration is therefore ∆𝑃1 = 2170 − ∑ 𝑃𝐺𝑖 19 𝑖=1 − 𝑃𝐿 1 = 511.6009 From equation (31), ∑ ( 𝜕𝑃𝐺𝑖 𝜕𝜆 ) 1 9 𝑖=1 = 4.1157 ∆𝜆1 = 511.6009 4.1157 = 124.305 𝜆2 = 𝜆1 + 124.305 = 524.305 Repeating the process for the second iteration (using 𝜆2 = 524.305 ) gives the results displayed in Table 8 Table 8: Power plant generating power after 2nd iteration Plant generation Value (MW) Plant generation Value (MW) Plant generation Value (MW) 𝑃𝐺1 2 210.4095 𝑃𝐺2 2 283.5900 𝑃𝐺3 2 257.9969 𝑃𝐺4 2 235.3991 𝑃𝐺5 2 258.8524 𝑃𝐺6 2 211.9622 𝑃𝐺7 2 199.6725 𝑃𝐺8 2 287.3723 𝑃𝐺9 2 234.5880 𝑃𝐿 1 0.8632 Real power difference from the 2nd iteration is therefore ∆𝑃2 = 2170 − ∑ 𝑃𝐺𝑖 29 𝑖=1 − 𝑃𝐿 2 = −4.8632 Again, from Equation (31), ∑ ( 𝜕𝑃𝐺𝑖 𝜕𝜆 ) 2 9 𝑖=1 = 4.0081 ∆𝜆2 = −4.8632 4.0081 = −1.2113 𝜆3 = 𝜆2 − 1.2113 = 523.0937 Repeating the process for the third iteration (using 𝜆3 = 523.0937) gives the results displayed in Table 9 Table 9: Power plant generating power after 3rd iteration Plant generation Value (MW) Plant generation Value (MW) Plant generation Value (MW) 𝑃𝐺1 3 209.9055 𝑃𝐺2 3 282.6187 𝑃𝐺3 3 257.3920 𝑃𝐺4 3 234.8494 𝑃𝐺5 3 258.2481 𝑃𝐺6 3 211.4587 𝑃𝐺7 3 199.1074 𝑃𝐺8 3 286.7098 𝑃𝐺9 3 233.6382 𝑃𝐿 3 0.8461 This iterative process continues until the 6th and 7th iterations. ∆𝑃6 = 2170 − ∑ 𝑃𝐺𝑖 39 𝑖=1 − 𝑃𝐿 6 = −0.4036 Again, from Equation (31), ∑ ( 𝜕𝑃𝐺𝑖 𝜕𝜆 ) 6 9 𝑖=1 = 3.7465 ∆𝜆6 = −0.4036 3.7465 = −0.1077 file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:patndyobi@gmail.com Obi et al: Optimal Load Scheduling of Power Plants in a Grid Considering the Plant’s Capacity and Line Losses. AZOJETE, 18(4):669-682. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 679 𝜆7 = 𝜆6 − 0.1077 = 518.7254 𝑎𝑠 𝜆6 = 518.8331 The 7th iteration gives results presented on Table 10: Table 10: Power plant generating power after 7th iteration Plant generation Value (MW) Plant generation Value (MW) Plant generation Value (MW) 𝑃𝐺1 7 209.1435 𝑃𝐺2 7 282.1315 𝑃𝐺3 7 256.124 𝑃𝐺4 7 234.3539 𝑃𝐺5 7 257.8456 𝑃𝐺6 7 211.2426 𝑃𝐺7 7 198.5624 𝑃𝐺8 7 286.1638 𝑃𝐺9 7 233.5374 𝑃𝐿 7 0.7653 ∆𝑃7 = 2170 − ∑ 𝑃𝐺𝑖 39 𝑖=1 − 𝑃𝐿 7 = −0.3815 We settle for the results on table 9 as our final iterative results since the difference is below 4 kW (and 4 𝑘𝑊 ≪ 1270 𝑀𝑊). The total fuel consumption worth of the total power plants to generate the 2170 MW (including losses) at normal generation and for optimum power generation are given in Equations (33) and (34) respectively. 𝐹𝑇𝑛 = 𝑓1(80) + 𝑓2(150) + 𝑓3(500) + 𝑓4(180) + 𝑓5(400) + 𝑓6(80) + 𝑓7(80) + 𝑓8(400) + 𝑓9(300)(33) 𝐹𝑇𝑒𝑑 = 𝑓1(209.1435) + 𝑓2(282.1315) + 𝑓3(256.624) + 𝑓4(234.3539) + 𝑓5(257.8456) + 𝑓6(211.2426) + 𝑓7(198.5624) + 𝑓8(286.1638) + 𝑓9(233.5374) (34) The power generated by the various plants and fuel/energy burnt per hour when the total generation is shared as highlighted in Table 2 and when shared with the plants running on equal incremental fuel cost is contained in results of Equations (33) and (34) are presented in Table 11. Table 11: Comparing sharing of 2,170 MW by the various plants in grid POWER STATIONS WITH UNEQUAL INCREMENTAL FUEL COST WITH EQUAL INCREMENTAL FUEL COST Fuel Burnt per Hour [× 104Cal/hr ] Plant Output Power [MW] Fuel Burnt per Hour [× 104Cal/hr] Plant Output Power [MW] Aba Power Station 9,200 80 56,413 209.1435 Afam IV-V Power Station 22,500 150 75,299 282.1315 Afam VI Power Station 254,000 500 67,860 256.124 Alaoji Power Station (NIPP) 36,725 180 61,772 234.3539 Okpai Power Station 163,200 400 67,976 257.8456 Omoku Power Station 8,885 80 56,667 211.2426 Sapele Power Station 8,725 80 52,252 198.5624 Sapele Power Station (NIPP) 147,200 400 75,471 286.1638 Delta-Ughelli Power Station 102,000 300 61,811 233.5374 TOTAL 752,435 2170 575,521 2169.6185 http://www.azojete.com.ng/ mailto:patndyobi@gmail.com http://en.wikipedia.org/w/index.php?title=Aba_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_IV-V_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Afam_VI_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Alaoji_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/National_Integrated_Power_Project http://en.wikipedia.org/w/index.php?title=Okpai_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Omoku_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Omoku_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Sapele_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/wiki/National_Integrated_Power_Project http://en.wikipedia.org/w/index.php?title=Delta_-_Ughelli_Power_Station&action=edit&redlink=1 http://en.wikipedia.org/w/index.php?title=Delta_-_Ughelli_Power_Station&action=edit&redlink=1 Arid Zone Journal of Engineering, Technology and Environment, December, 2022; Vol. 18(4):669-682 ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 680 With a total power of 2,170 MW generated by all nine (9) generating units in the region to service the power demand and losses along the power lines, the energy burnt by each plant and the total energy burnt by all the 9 power plants in every hour are presented in Table 10. Using a case where the plant’s individual generation as highlighted on Table 2 equals 2170 MW, their individual fuel burnt are shown in Table 10 and the total fuel burnt by all nine (9) plants in the generation of this power, is 752,435 × 104 𝐶𝑎𝑙/ℎ𝑟. However, when the plants ran with the same incremental fuel cost, the power generated by each plant changed. More generating responsibilities were given to those plants whose generation costs are relatively lower. For example, Afam IV-V and Aba power stations (a more of steam combine circle plants) were given more generating responsibilities as they are relatively cheaper generating with them than most other plants on the grid. Hence, their generations were increased from 150 MW to 282.1315 MW and from 80 MW to 209.1435 MW respectively. However, the same total power generation of approximately 2170 MW was achieved (precisely 2169.6185 MW). The energy used by the individual plant and the total energy used by all nine (9) plants to generate that same total of 2,170 MW power each hour as presented in Table 10 is 575,521 × 104 𝐶𝑎𝑙/ℎ𝑟 . This implies that a total of 176,914 × 104 𝐶𝑎𝑙 (or 7.407 × 109 𝐽𝑜𝑢𝑙𝑒𝑠 ) of energy is saved each hour. This value converted to kWh gives 2.057 × 103 𝑘𝑊ℎ as the energy saved each hour. With energy sold at forty-eight naira only per kilo- watt-hour (₦48/kWh) in Nigeria, a total of ninety-eight thousand, seven hundred and thirty- six naira (₦98,736.00) only will be saved each hour. This means that a total of eight hundred and sixty-four million, nine hundred and twenty-seven thousand, three hundred and sixty naira (₦864,927,360.00) only will be saved per annum. 4. Conclusion This paper showed how the total running cost of different power generating plants in a grid feeding a fixed load can be achieved. This was done by giving more generation task to those power plants whose cost of generating power is relatively low (provided their generation capacity is not exceeded). With this, the same quantity of total power is generated with a relatively lower cost; this can be regarded as comparative cost advantage in power generation cost. As seen from the studies, a total of over eight hundred million (₦800,000,000:00) naira was saved annually when all power plants in the region analyzed were ran with this method (equal incremental fuel cost) over when the load was shared arbitrarily by the plants. This saved money can be used to build more power plants, expand transmission capacities or used for other capital projects in the country. Therefore, optimized combination of power plant running is highly recommended in running the Nigerian power system. References Attaviriyanupap, P., Kita, H., Tanaka, E. and Hasegawa, J. 2002. A hybrid EP and SQP for Dynamic Economic Dispatch with Non-Smooth Fuel Cost Function. IEEE Transaction on Power Systems, 17(2): 411-416. Avalos, MJR. 2008. Analysis and Application of Optimization Techniques to power System Security and Market. PhD Thesis, Department of Electrical and Computer Engineering University of Waterloo, Canada. Bogdan, Z., Cehil, M. and Kopjar, D. 2007. Power System Optimization. Energy, 32: 955-960. Bommirani, B. and Thenmalar, K. 2013. Optimization Techniques for the Economic Dispatch in Power System Operation. International Journal of Information Technology, 2(1): 475-482. file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:patndyobi@gmail.com Obi et al: Optimal Load Scheduling of Power Plants in a Grid Considering the Plant’s Capacity and Line Losses. AZOJETE, 18(4):669-682. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 681 El-Sharkh, MY., Tanrioven, M., Rahman, A. and Alam, MS. 2006. A Study of Cost-Optimized Operation of a Grid-Parallel PEM Fuel Cell Power Plant. IEEE Transactions on Power Systems, 21(3): 1104-1114. Eurafric Power Limited. 2014. 1020MW Gas-fired Steam Turbine and Simple Cycle Gas Turbine Data Book, Sapele, Delta State. Farhat, IA. and El-Hawary, ME. 2009. Optimization Methods Applied for Solving the Short- Term Hydrothermal Coordination Problem. Electric Power Systems Research, 79(9): 1308- 1320. Federal Government of Nigeria National Integrated Power Project (NIPP). 2012. 450MW Combined Cycle Gas Turbine Power Plant Data Book, Sapele, Delta State. Geometric Power Limited. 2005. 340MW Aba Gas Turbine Power Plant Data Book, Aba Abia State. Gupta, JB. 2009. Generation and Economic Considerations, SK Kataria and Sons, Delhi, India. Hossain, SM. and Shiblee, MSAAF. 2017. A Short Review Study on Problems During Hydro- Thermal Optimal Scheduling. International Journal of Advanced Technology and Engineering Exploration, 4(34): 142-147. Jafari, A., Khalili, T., Babaei, E. and Bidram, A. 2019. A Hybrid Optimization Technique Using Exchange Market and Genetic Algorithms. IEEE Access, 8: 2417-2427. Jinchao, L., Jinying, L., Dongxiao, N. and Yunna, W. 2012. A Parallel Adaptive Particle Swarm Optimization Algorithm for Economic/Environmental Power Dispatch. Mathematical Problems in Engineering, 2012: 1-15. Kovalev, GF., Lebedeva, LM. and Krupeniov, DS. 2011. Models and Methods for Estimation and Optimization of Electric Power System Reliability. Reliability: Theory & Applications, 12(2): 92-105. Lei, G., Song, H. and Rodriguez, D. 2020. Power Generation Cost Minimization of the Grid- Connected Hybrid Renewable Energy System through Optimal Sizing Using the Modified Seagull Optimization Technique. Energy Reports, 6: 3365-3376. Niger Delta Power Holding Company (NDPHC) and Federal Government of Nigeria. 2013. National Integrated Power Project (NIPP), 1,072MW Combined Cycle Gas Turbine Power Plant Data Book, Alaoji, Abia State. Nigerian Agip Oil Company (N.A.O.C). 2003. National Integrated Power Project (NIPP), 490MW Okpai Combined Cycle Power Plant Data book, Kwalle, Delta State. Nigerian Agip Oil Company (N.A.O.C) and Rivers state Government. 2003. National Integrated Power Project (NIPP) Omoku 600MW Combined Cycle Power Plant Data book, Omoku, Rivers State. Obi, PI., Iloh, JPI., Ulasi, AJ. and Emeghara, MC. 2017. Decentralized Power Supply Improvement in Africa Economics: A Case Study of Nigeria. Proceedings of the 3rd International Conference on Engineering Adaptation and Policy Reforms (IJEAPR, 2017), Uli, Nigeria, 17 – 18 May 2017, pp. 9-17. Obi, PI. and Offor, KJ. 2012. Power Flow and Contingency Assessment of the Existing 330kV Nigeria Power Grid to Cope with the Proposed Increase in Power Generation 2014. International Journal of Engineering Research and Technology (IJERT), 1(4): 1-9. http://www.azojete.com.ng/ mailto:patndyobi@gmail.com http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine http://en.wikipedia.org/wiki/Simple_cycle_combustion_turbine Arid Zone Journal of Engineering, Technology and Environment, December, 2022; Vol. 18(4):669-682 ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: patndyobi@gmail.com 682 Oputa, O. 2015. Running Cost Minimization of an Electric Power Sector. International Journal of Engineering and Computer Science, 4(7): 13169-13175. Oputa, O., Obi, PI., Okeke, O. and Anyaka, BO. 2019. An Optimized Combination of Power Plants Running in a Grid. Proceedings of the 1st NIEEE Nsukka Chapter Conference on Engineering and Management Technology in the 3rd Industrial Revolution, Nsukka, Nigeria, 28 – 30 November 2019, pp. 26-36. Panda, A., Mishra, U., Ming-Lang, T. and Ali, MH. 2020. Hybrid Power Systems with Emission Minimization: Multi-Objective Optimal Operation. Journal of Cleaner Production, 268: 153-160. Saadat, H. 1999. Power System Analysis. 3rd Edition, McGraw – Hill Company, New York. Shell Petroleum Development Company (S.P.D.C). 2003. Afam I – V Combined Cycle Gas Turbine 750MW Power Plant Data Book, Afam, Rivers State. Shell Petroleum Development Company (S.P.D.C). 2005. Afam VI Combined Cycle Gas Turbine 750MW Power Plant Data Book, Afam, Rivers State. Transnational Corporation of Nigeria Plc, Transcorp’s Power Subsidiary, Transcorp Ughelli Power Limited Ughelli. 2012. General Electric 900MW Gas-Fired Thermal Power Plant Data Book, Ughelli, Delta State. Wood, AJ., Wollenberg, BF. and Sheble, GB. 2013. Power Generation Operation and Control, 3rd Edition, John Wiley & Sons, Inc, New York, USA file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:patndyobi@gmail.com