Generating or converting electricity is quite difficult and requires a lot of time and resources. Thus, losing much of it is worrisome. For efficient performance of transmission lines, transmission losses should be highly minimized, this minimization of losses entails taking into account the equipment or material responsible for the losses, which is accounted for Ohms loss, whereas the environmental factors (air density, disruptive voltage) leading to loss of power in the transmission line is accounted by Corona loss. Power losses whether reactive or active could be curtailed at the design and testing stages [1] [2] . However, Ohms loss could be attributed to increased transmission current, the resistivity of the conducting materials, and the length of transmission lines [3] . Also decreasing the cross-section area of the conducting materials could also boost power losses in the transmission line [3] [4] . Further losses are known to be induced by the appreciable disparity between the phase and disruptive voltages, transmission frequency, and diameter (or radius) of the conducting materials [4] [5] . Conversely, decreasing air density increases losses due to environmental or weather conditions [7] [8] .

Thus, transmission losses could be categorized into material losses; transmission equipment (capacitors and inductors in the transmission circuits), inherent losses (due to the nature of the current; thermal, electrical, and electromagnetic effects/atmospheric disasters [9] [10] [11] . These outrageous factors have to do with the inherent nature as part of the power is converted into heat and lost to the surroundings, and due to forceful earthling of the power by thunderstorms through the airs around the transmission lines as they lose their dielectric properties [7] [12] . However, the installation of thunder arresters could be a remedy or preventive measure [13] [14] . The use of aluminum (animal grades) is prone to power losses relative to copper which has better electrical and mechanical properties than aluminum. Equipment losses could be mitigated by minimizing reactive power in the transmission equipment and suppressing the magnitude of the eddy current developed in the inductors by designing the core of the inductor as laminates parallel to the magnetic field [15] [16] [17] .

Notably, Non-gradient techniques of optimization—artificial intelligence methods. Patil et al. [18] presented Particle Swarm Optimization (PSO) and Salp Swarm Algorithm (SSA) techniques have been employed in finding optimum losses based on the active and reactive power losses; Monshizadeh et al. [19] recommended PSO as best algorithm for minimizing power loss in transmission lines; Baharozu et al. [20] improved PSO by choosing the load bus voltages, generator active and reactive powers, line flow capacities as penalty functions in the objective function. PSO and IPSO-based optimal power flow solutions are compared with each other on IEEE 48, 118 and 300 bus systems [21] . It was found that the IPSO-based solution has fewer power losses than the PSO-based solution; Nadeem et al. [22] employed Compensators and Controllers in Flexible AC Transmission lines to minimize power losses. Naima et al. [23] and Kumar [24] combined genetic algorithm technology, and the particle swarm optimization in the western Algerian power system to reduce power losses in transmission lines; Kuru et al. [25] considered that Chaotic Artificial Bee Colony (CABC) Algorithm is more effective in minimizing active power losses than Artificial Bee Colony (ABC) Algorithm. CABC minimized up to 84,891 MW while ABC 86,203 MW; Nusair and Alomoush [26] and Jagun et al. [27] explored the Teaching Learning Based Optimization Algorithm (TLBO) technique for solving optimal reactive power dispatch and active power losses. TLBO has the efficiency to reduce the active power losses reasonably without violating any constraints, it also had excellent convergence compared to PSO-CM techniques; Badar et al. [28] employed the PSO algorithm with dynamic weights to minimize active power losses. The proposed technique showed better convergence and was better at reducing power losses compared to the existing techniques. The technique reduced losses from 11.61 MW to 8.7036 MW, which was a 25% reduction in total active power losses. Nevertheless, the optimization techniques based on artificial intelligence and deep neural network are widely employed in the minimization of power losses in the transmission lines, but are faced with problems of identifying the global minimum/maximum amidst minima/maxima.

However, the gradient method of minimizing power losses in the transmission line has the capacity to identify the global minimum/maximum. Pertinently, Bamigbola et al. [29] examined a flow model (mathematical Equation) to predict current and voltage parameters in the transmission lines, Revolutionized and simulated transmission model to obtain valid transmission losses; Sharma et al. [31] , and Bisiriyu et al. [32] observe that Corona loss can be reduced by phase change in double circuit lines; Kuchanskyy and Zaitsev [33] [34] achieved loss minimization in transmission line through optimum length and radius of the transmission cable; Ali and Siddique [35] examined high Corona loss during wet or moisture in the atmosphere and low Corona loss in the dry atmosphere, degradation of the conducting material also causes Corona loss; Omeje [36] and Yaabari et al. [37] examined minimization of Corona loss at a very low transmission current and large spacing between bundled conductors; Llamo-Laborí and Santos-Fuentefria [38] observed that increase in spacing for the conductor bundles minimizes and balance Corona loss; Tonmitr et al. [6] conducted simulative research on minimizing the power losses using various conductor parameters, including conductor radius and system voltage; Ogar et al. [39] investigated distance between generating station, frequency and load center contributes to Corona loss based on Peek’s formula and Peterson; Yahaya et al. [4] examined the relationship between power losses from Corona and Corona characteristics using Matlab programs. Radio interference, audible noise, gaseous effluents, and shock potential are some of the impacts of Corona electrical surface gradients, and the performance of the conductor is influenced by conductor voltage, diameter and shape, dust, water drops, surface abnormalities like scratches; Al_issa et al. [40] used different approximation methods to examine active power loses through voltage and weather conditions; Al-Hamouz [41] and Hünnekens et al. [42] Compared Ohms and Corona loss under different loading conditions and found out that the percentage in moderate and light loadings could rise to 100% under deep Corona.

Notably, the application of the gradient method of optimization using the Corona and Ohms transmission loss model is not feasible as optimum variables of the objective function cannot be derived nor the optimum values possible. Expectantly, the gradient method of optimization guarantees global minimum/ maximum. However, this could not be achieved using the traditional Corona and Ohms transmission loss model. Therefore, there is a silent need to develop Corona and Ohms models, which could be suitable for the gradient method of optimization in order to obtain the optimum objective function and optimum variables with the corresponding minimization values. Thus, this paper is concerned with the modification of the traditional power losses objective function by introducing a third dimensionally consistent term, a product of Corona and Ohms losses, and to validate the optimality of the unmodified and modified models on different power transmission lines.

This section presents the materials and methods used in this study.

The tables and figures resulting from the study are presented in this section. The explanation of the tables and figures from the results is follows.

A modified model for power loss minimization in the transmission lines of Bujagali-Kawanda and Masaka West-Mbarara North in Uganda has been developed. The optimum variables and values of transmission current, phase voltage and air density were determined using the gradient optimization technique that is the first derivative of the modified model and is feasible with the modified model. An optimality test was also carried out using Hessian matrix both on the unmodified and modified models. The optimal results for both modified and unmodified established that the modified model has a high potential to minimize power losses in the transmission lines. Consequently, this paper recommends that a modified model should be implemented in the transmission line in the design and operation of transmission lines, due to its ability to minimize power losses in the transmission lines. The results of this study are not limited peculiar to Uganda Electricity Transmission Lines, they could be implemented by other transmission lines outside Uganda.

The authors wish to acknowledge Uganda Electricity Transmission Lines for availing input data used in the analysis of the local transmission lines.

No conflict of interest of any kind.

Living, O., Nnamchi, S.N., Ukagwu, K.J., Abdulkarim, A. and Jagun, Z.O. (2023) Minimization of Electric Power Losses on 132 kV and 220 kV Uganda Electricity Transmission Lines. Energy and Power Engineering, 15, 127-150. https://doi.org/10.4236/epe.2023.152006

Optimum variables

The optimum variables are obtained by differentiating Equation (4) with respect to these variables, equating the differential to zero and making the optimum variables the subject of the formula in Equations (10)-(12). Thus, differentiating Equation (4) with respect to (wrt) I gives optimum current, I * Equation (10)

∂ p ˜ ∂ I = 2 I ρ A − 0.0123 I ρ ( V − V 0 ) 2 ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ l ;

I * = 0.00615 r d d γ l ρ A r ( f + 25 ) ( V − V 0 ) r d d γ l ρ (10)

Differentiating Equation (4) wrt transmission phase voltage gives optimum phase transmission voltage, V * (V) in Equation (11)

∂ p ˜ ∂ V = 121 ( V − V 0 ) ( f + 25 ) r d 25000 γ l − 0.0123 I 2 ρ ( V − V 0 ) ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ l ;

V * = 1 r d A ( f + 25 ) ( 8.264 × 10 − 9 ( 1.21 × 10 8 A r d f V             + 4.84 × 10 8 A r d V + 3.025 × 10 9 r d A V             − 3.0746 × 10 8 A r d f γ l ρ I 2 + 25 A r d γ l ρ I 2 ) ) (11)

Differentiating Equation (4) wrt air density factor gives optimum air density factor, γ * (kg/m³) in Equation (12)

∂ p ˜ ∂ γ = − 121 ( V − V 0 ) 2 ( f + 25 ) r d 50000 l γ 2 + 0.00615 I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ 2 l ;

γ * = 0.000759 r d A ( V 2 f − 2 V f V 0 + f V 0 2 + 25 V 2 − 50 V V 0 + 25 V 0 2 ) I 2 ρ l (12)

H = [ ∂ 2 ( p ˜ ) ∂ 2 I 2 ∂ 2 ( p ˜ ) ∂ I ∂ V ∂ 2 ( p ˜ ) ∂ I ∂ γ ∂ 2 ( p ˜ ) ∂ V ∂ I ∂ 2 ( p ˜ ) ∂ 2 V 2 ∂ 2 ( p ˜ ) ∂ V ∂ γ ∂ 2 ( p ˜ ) ∂ γ   ∂ I ∂ 2 ( p ˜ ) ∂ γ   ∂ V ∂ 2 ( p ˜ ) ∂ 2 γ 2 ] (13)

Derivatives of the Hessian matrix

The diagonal elements of the Hessian matrix in Equation (13) is given in Equations (14)-(16). The second derivative of power losses, p wrt I is given in Equation (14)

∂ 2 p ˜ ∂ I 2 = 2 ρ A + 0.0123 I 2 ρ 2 ( V − V 0 ) 4 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l ) 3 / 2 A 2 γ 2 l 2       − 0.0123 ρ ( V − V 0 ) 2 ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ l (14)

The second derivative of power losses, p wrt V is given in Equation (15)

∂ 2 p ˜ ∂ V 2 = 121 ( f + 25 ) r d 25000 γ l + 0.0123 I 4 ρ 2 ( V − V 0 ) 2 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l ) 1.5 A 2 γ 2 l 2       − 0.0123 I 2 ρ ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ l (15)

The second derivative of power losses, p wrt γ is given in Equation (16)

∂ 2 p ˜ ∂ γ 2 = 121 ( V − V 0 ) 2 ( f + 25 ) r d 25000 γ 3 l + 0.00307 I 4 ρ 2 ( V − V 0 ) 4 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l ) 3 / 2 A 2 γ 4 l 2       − 0.0123 I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ 3 l (16)

Upper diagonal elements of Hessian matrix

Upper diagonal elements are derived in Equations (17)-(19). The second derivative of power losses wrt I and V is given in Equation (17)

∂ 2 p ˜ ∂ I ∂ V = 0.0123 I 3 ρ 2 ( V − V 0 ) 3 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ ) 1.5 A 2 γ 2 l 2       − 0.0246 I ρ ( V − V 0 ) ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ A γ l (17)

The second derivative of power losses wrt I and γ is given in Equation (18)

∂ 2 p ˜ ∂ I ∂ γ = − 0.00615 I 3 ρ 2 ( V − V 0 ) 4 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ ) 1.5 A 2 γ 3 l 2       + 0.0123 ρ ( V − V 0 ) 2 ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ 2 l (18)

The second derivative of power losses w.r.t V and γ is given in Equation (19)

∂ 2 p ˜ ∂ V ∂ γ = − 121 ( V − V 0 ) r d 25000 l γ 2       − 0.00615 I 4 ρ 2 ( V − V 0 ) 3 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ ) 1.5 A 2 γ 3 l 2       + 0.0123 I 2 ρ ( V − V 0 ) ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ 2 l (19)

Lower diagonal elements of Hessian matrix

The lower diagonal elements of the Hessian matrix are obtained as the reciprocal of Equations (20)-(22).

The second derivative of power losses wrt V and I is given in Equation (20)

∂ 2 p ˜ ∂ V ∂ I = 0.0123 I 3 ρ 2 ( V − V 0 ) 3 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ ) 1.5 A 2 γ 2 l 2       − 0.0246 I ρ ( V − V 0 ) ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ A γ l (20)

The second derivative of power losses wrt γ and I is given in Equation (21)

∂ 2 p ˜ ∂ γ ∂ I = − 0.00615 I 3 ρ 2 ( V − V 0 ) 4 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ ) 1.5 A 2 γ 3 l 2       + 0.0123 I ρ ( V − V 0 ) 2 ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ 2 l (21)

The second derivative of power losses wrt γ and V is given in Equation (22)

∂ 2 p ˜ ∂ γ ∂ V = − 121 ( V − V 0 ) r d 25000 l γ 2 − 0.00615 I 4 ρ 2 ( V − V 0 ) 3 ( f + 25 ) 2 ( r d ) ( I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A l γ ) 1.5 A 2 γ 3 l 2       + 0.0123 I 2 ρ ( V − V 0 ) ( f + 25 ) r d I 2 ρ ( V − V 0 ) 2 ( f + 25 ) r d A γ l A γ 2 l (22)