Multi‐objective manta ray foraging algorithm for efficient operation of hybrid AC/DC power grids with emission minimisation

Funding information Taif University Researchers Supporting Project, Grant/Award Number: TURSP-2020/86 Abstract The current paper presents a multi-objective manta ray foraging algorithm (MO-MRFA) for efficient operation of hybrid AC and multi-terminal direct current (MTDC) power grids. The multi-objective framework aims at achieving economical, technical and environmental benefits by minimising the total production fuel costs, minimising the transmission power losses and minimising the environmental emissions in the AC/MTDC transmission systems. The MRFA imitates three separate independent foraging organisations of the manta rays. It is updated incorporating an additional Pareto archive to preserve the nondominated solutions. A dynamic adaptation of the fitness feature is employed by iteratively varying the form of the employed fitness function. Furthermore, a fuzzy decision-making technique is activated to finally pick the appropriate operating point of the AC/MTDC power grids. The proposed technique is compared with other reported algorithms in the literatures. The applications are conducted on three test systems. These systems are IEEE 30-bus, IEEE 57-bus test power systems in addition to real part of the Egyptian grid at West Delta region. Numerical results demonstrate that the proposed MO-MRFA has great effectiveness and robustness indices over the others. Nevertheless, the proposed MOMRFA is successfully extracting several Pareto solutions that meet the techno-economic requirements with accepted environmental concerns.


INTRODUCTION
Nowadays, the international community directs for efficient production and utilization of electrical energy in power systems. and attractive performance compared with the high voltage AC systems especially for long submarine transmission distances above 60 km and that networks of capacity greater than 500 MW [1,2]. Two options of HVDC technologies are found as currentsource converter HVDC (CSC-HVDC) and voltage-source converter HVDC (VSC-HVDC). The VSC-HVDC is the recent and widest spread for its techno-economic advantages [3,4]. The objective of the optimal power flow (OPF) is to specify the most suitable values of the control variables that involve generator output power, generator voltage, VAR operating outputs, transformer tap settings, FACTS variables, and other newly attached devices to reduce pre-specified fitness functions while meeting the operational system limits. OPF is searching  for the optimal balance of economic and environmental efficiency of electrical networks. It looks for the optimal settings of the system control variables for effective operating points. This optimum state is distinctive of objective functions such as minimising the costs of generation, decreasing the grid losses, increasing the loadability of the transmission lines, minimising potential pollutants, ameliorating the voltage quality, and providing better security whilst retaining various equality and inequalities constraints. [5,6]. The combination between HVDC systems and AC power networks modifies the control and operational techniques of electrical grids. Traditional operational procedures must be updated and adjusted to this advanced aspect of interconnected MTHVDC power grids. These technologies will impact on several optimization problems in the field of power and energy

FIGURE 3
The configuration of the modified system of IEEE 30-bus test system [28] systems like energy hub problem [7][8][9], scheduling of electrical consumption in multi-chiller systems [10], electricity load forecasting [11], assessment of renewable energy sources [12], reactive power dispatch [13,14], state estimation [15], economic load dispatch [16], and OPF [17] etc. Due to the vital role of the OPF in operating the power networks, its extension plans, and the correlated economic studies, it needs technical and performance improvements to manage the hybrid power grids. The OPF problem in the embedded AC/MTHVDC power grids is characterised with more non-convex, non-linear, multimodal, and complex problems as [18]. Numerous researchers discussed the OPF issue for either AC power systems [19,20] or DC grids [21,22]. Recently, the mathematical OPF model for hybrid AC-MTHVDC systems has received an excessive interest of interest in literature [23−27]. In all these reports, the evaluation of VSC-HVDC systems was limited to considering only a two-terminal design. The addition of meshed DC systems to AC power grids raises the difficulty of their operation and management study [23,28]. Some researchers disregard the model of some AC/MTHVDC components to simplify the analysis, like ignoring DC load flow equations [29]. Even though VSC-MTHVDC technologies were used in the load flow estimation of hybrid AC/DC networks in many prior studies as stated above, but limited literature implementations include VSC-MTDC devices in the optimization sense. In Ref. [4], the hybrid AC/DC networks has FIGURE 6 TAEE minimisation convergence rates of MRFA for case 3   been presented [23][24][25][26][27][28][29][30], which presented several simplifications to deal with the considered problem. Secondly, the influence of the reactive power compensators and tap settings that are related to the transformers existed in AC power grid have been neglected, whereas the methods used are dependent on the orig-inal starting point with certain simplifications that decrease the precision.
On the level of optimization algorithms, there are a huge number of articles that are described their applications in the AC electrical systems for solving the OPF problem that is divided  into single and multi-objective categories [31,32]. For solving the OPF problem incorporating the unified power flow controller (UPFC) in [33], an adaptive grasshopper optimizer (AGO) has been proposed by emerging the Levy flight concept into the standard grasshopper optimizer. In [34], a modified fractional order PSO has been presented to minimise  the power losses as a single objective target. Also, modified wind driven versions have been applied for solving the OPF

FIGURE 11
VSCs capability curve and their optimal operating points for the modified 30-bus system for minimising the power losses [35] and improving the voltage profile with coordinated UPFC [36]. In [37], a hybridised model between the dragonfly algorithm (DA) and the PSO has been presented for handling the OPF considering thyristorcontrolled series compensator (TCSC). In [38], a hybrid optimization algorithm based on moth swarm and gravitational search algorithms has been introduced for the OPF considering wind power. In [39], a modified multi-objective pigeon optimizer was applied for the OPF problem. In [40], the adaptive multiple teams' perturbation-guiding Jaya (AMTPG-Jaya) optimization algorithm has been utilized for the OPF problem employing several populations where different movement pathways have been executed for each population. In [41], an adaptive differential evolution (ADE) has been carried out to the OPF problem, where the crossover and scale rates have been dynamically changed through the search space. In addition, a dynamic reduction strategy to the population size have been activated but it is applied only on a small system of IEEE 30test system. On the other side, little metaheuristic optimizers have been presented to handle the OPF problem for hybrid AC/DC grids despite the significant advancements in the field of artificial intelligence optimization. In [42], the PSO algorithm has been devoted for solving multi-objective form of the OPF problem in AC/DC grids with ignoring the economic factor of generation cost reduction. In [43], equilibrium optimizer has been carried out for treating the multi-objective OPF model in AC/DC grids, but the objectives were roughly transformed into a single objective model using the weighting factors. In [44], genetic optimization algorithm (GA) was addressed the secure model of the OPF in order to minimise total power losses as just a single goal. Manta ray foraging algorithm (MRFA) is a recently optimizer that is developed to tackle real valued optimization tasks. It was motivated by the intelligent and special tactics of the manta rays to be executed in their foraging search about their food [45]. In this reference, its efficiency is evaluated compared to other new optimization techniques using many benchmark and industrial design models. In [46], a serious application of the MRFA has been performed for handling the OPF problem.
This paper modifies the MRFA to obtain the optimal operation of the AC/MTHVDC power grids with a multi-objective function. It is upgraded as a multi-objective manta ray foraging algorithm (MO-MRFA) by incorporating an additional Pareto archive to preserve the non-dominated solutions. The appli-

FIGURE 12
The configuration of the modified IEEE 57-bus system

FIGURE 13
Convergence rates of the proposed MRFA for TFC and TPL minimisation of the modified IEEE 57-bus system cations are carried out on two standard tests systems namely the modified IEEE 30-bus, the modified IEEE 57-bus test power systems and a one practical network from the Egyptian Network at the West Delta Region. To prove the proposed MO-MRFA capability for such purposes, the obtained simulation results are compared with some modern algorithms such  as: grey wolf optimiser (GWO) [47]; particle swarm optimizer (PSO); salp swarm algorithm (SSA) [48]; multi verse optimizer (MVO) [49]; dragonfly optimization algorithm (DA) [50]; crow search optimization algorithm (CSOA) [51,52]; bat optimization algorithm (BAT) [53,54]; marine predators optimization algorithm (MPO) [55]. The main contributions of this study are summarized as follow: • New improved MO-MRFA is developed for solving the OPF problem via finding the optimal coordinated controls (OCC) of the AC/MTHVDC electrical grids. • The operation procedure is formulated as multi-dimension operation framework that respects the technical economical operational requirements of hybrid grids. • It seeks to minimise the overall cost of generation, their produced emissions, and the overall losses on AC, MTHVDC power lines and VSC stations.  The rest of the sections of this paper are organized as follows: Section 2 lays out the concept of the optimum operation of the hybrid networks. Section 3 demonstrates the proposed method based on MO-MRFA. Section 4 records simulation findings for three AC/MTHVDC hybrid networks. Section 5 concludes the findings.

VSCs modelling in AC/MTHVDC power grid
The configuration of AC/MTHVDC power grid is illustrated in Figure 1. This configuration has three VSCs interconnected by three DC lines (R dc ). Each VSC represents a controlled voltage source (Vc i ) that is linked to AC grid buses through transformer and phase reactor with equivalent impedance of (R ik + jX ik ). Vs k refers to the AC grid bus voltage. Therefore, the MVA (S ki ) injected into the VSCs of the AC grid is issued by: where, the injected active and reactive powers are denoted as P and Q, respectively. N ac and N VSC indicate the AC buses number that are connected to DC buses and VSCs, respectively. I ki is the related injected current that can be estimated as: From these two equations, both active and reactive powers (Pc i + jQc i ) at each VSC and the related AC side (Ps k + jQs k ) are calculated as: VSCs capability curve and their optimal operating points for WDPN test system (6) where, the DC line conductance and susceptance are denoted by G and B, respectively; θ ki indicates the phase angle deviation between VSC and the attached AC side.

OCC problem of hybrid AC/MTHVDC power transmission systems with environmental pollution minimisation
The OCC problem represents the joint optimization of multiobjective functions without exposing the regarded the hybrid power grids limitations. It is formulated in a multi-objective model to reduce the total fuel costs (TFC), the total power losses (TPL) in VSC stations, AC, and MTHVDC power grids, and the total atmospheric environmental emissions (TAEE) associated with the power stations.

Control variables of the OCC problem
For AC/MTHVDC power grids, the independent/control variables are two types. Firstly, traditional AC power grid controls for generator outputs of the active power (Pg 1 , Pg 2 , …… Pg Ng ), generator voltages (Vg 1 , Vg 2 , ……., Vg Ng ), transformer tap settings (Tap 1 , Tap 2 , ……, Tap Nt ), and reactive power injection of VAR sources (Qc 1 , Qc 2 , ……, Qc Nq ). N g , N t , and N q are the total number of generators, on-load tap transformers, and VAR devices, respectively [56,57]. Second, advanced controls of VSC systems in which four types of controlled strategies as [58]: Strategy 1: V dc -Q c constant control; Strategy 2: V dc -V c constant control Strategy 3: P dc -Q c constant control; Strategy 4: P dc -V c constant control; The first control strategy preserves fixed DC side voltage that generates fixed reactive power on the AC side. The second strategy enables fixed voltage on both AC and DC side buses. The third strategy offers fixed active power to be transferred on DC transmission line and fixed reactive power on the AC grid. The fourth strategy gives fixed active power transmitted to DC line and fixed voltage at the AC bus.

Dependent variables of the OCC problem
Likewise, these variables can be listed on the AC and MTHVDC systems. Upon on AC system, they are usually bus load voltage magnitudes (VL 1 , …, VL NPQ ), VAR outputs from the Ng generators are Qg 1 , Qg 2 , …, Qg Ng . The transmission line flows in NF lines are: SF 1 , …, SF NF . NPQ, and NF are, denoted to the total number of load buses, and system lines, respectively. For MTHVDC system, the voltage at DC buses and the MW flow though their lines are considered as dependent variables.

Multi-objective OCC problem
The mathematical representation of the OCC problem is expressed as follows: Subject to: g(x,y) = 0 ( 8 ) h(x,y) ≤ 0 ( 9 ) where, O is the considered vector of M objectives; x is the independent variables; y is the dependent variables. In this research, the OCC problem is formulated with multiobjective concerns for minimising the overall fuel costs and environmental emissions from generation stations. In addition, transmission losses are considered. The cost of fuel generation is modelled on the basis of the loading effect of the valve point, which is practically followed by several ripples in the power grid. As a consequence, the TFC is modelled as [59]:  The primary source of air contaminants in power plants is the fossil fuelled generation stations are where nitrogen oxides (NOx), sulphur oxides (SOx), and second-carbon dioxide (CO 2 ) are released. The TAEE in ton/hr is represented as a sum of the quadratic exponential equation, as in [59]: where γ i , β i , α i , ξ i , and λ i are the regarding coefficients of the emitted atmospheric pollutants. Another objective may be considered to minimise the total losses (Total loss ) in the AC/MTCDC grids which are the combination of the power losses in the AC transmission grid (PL AC ) as in [56], the combination of power losses in DC transmission grid (PL DC ), and the VSC losses (PL VSC ) as: In its simplified formula, the quadratic relationship of VSC losses can be expressed with the VSC current injected (Ic i ) [60] as: where φ 1 , φ 2 , and φ 3 are the loss coefficients related to each VSC.

Constraints of the OCC problem for AC/MTDC power transmission systems
As equality constraints, the AC grid power balance equations are usually stated as: (17) where, N b is the total number of buses; PL and QL are, respectively the active and reactive demands; G ij and B ij are the mutual conductance and susceptance between buses i and j, respectively. Also, the DC grid power flow constraints must be considered as: where, P dc,k is the MW power injected into the bus (k). In addition, the operating variables for AC power grids must be met under the respective restrictions as described in [56,57]: Comparably, there are inequality restrictions linked to the MTDC grid as following [18]: However, the capability curve of every VSC can be preserved by enforcing the following restriction [18]: is indeed the circle centre of VSC PQ curve which has its diameter is d. "max" and "min" superscripts signify the maximum and minimum values of the respective variables.

MRFA
MRFA is a new computational optimizer [45]. It is driven by the intelligent and special tactics of the manta rays in the foraging search. It imitates three different independent foraging organizations. The first is the foraging chain, in which some of them forage in a limited cooperative manner, as the manta rays are arranged in an ordered line to encircle the largest amount of plankton in their gills. The cyclone forging is the second strategy, where several manta rays are joined together in a spiral formation to create a consolidated spiralling peak. Which pushes the water into the surface and draws the plankton within their mouths. The latter technique is for somersault. The individu-als are looking for the location of the planktons and swimming towards them. They signify the individual targets that are searching for the minimum fitness location of the plankton. The MRFA identifies the number of the population (P N ) of the manta rays and the full number of iterations (M It ). Then, each D-dimensional individual (Y) is originally generated as: where, Y m,n is the individual of every manta ray (m); n indicates each control variable; r corresponds to a random value and it belongs to the range [0, 1]. The control variables limits are indicated by the superscripts 'min' and 'max'.
In the first technique, the foraging chain, Equation (32) is used to upgrade each individual as: where, Y * and Y indicate, respectively, the updated and old manta ray positions. Y B is the plankton food which is represented via the best position that achieves the highest fitness or concentration. σ refers to a weight coefficient that changes from one iteration to the next: In the second technique, iterations are split evenly into two sections. The first reflects on enhancing the MRFA exploration. As a result, each manta ray position is modified by using Equation (34), in the first bisection of the iterations, as: where, Y R is a formed individual randomly within the bounds recognized as follows: The adaptive weighting factor, β, is modified as follow: .sin(2ßr 1 ) where, Itr is the present iteration, and r 1 is an arbitrarily number that is uniformly distributed in the range [0, 1]. The second bisection of the iterations relies on strengthening the MRFA exploitation. In this regard, each position is modified as: In the third technique, somersault foraging, each individual is adjusted to the best extracted location as follows: 3 .Y m,n )∀m ∈ P N and n ∈ D (38) where somersault factor, Sf = 2, regulates the somersault domains of the manta rays; the random numbers r 2 and r 3 are uniformly distributed in the range [0,1].

Proposed MO-MRFA for OCC in AC/MTHVDC grids
In the MRFA described in the previous subsection, new individuals are produced on the chain foraging strategy of Equation (32), the equations plan for the cyclone Equations (35) and (37) or somersault for Equation (38). Mainly, MRFA is very receptive to the best location of plankton food. Consequently, the suggested MO-MRFA makes use of this attribute by developing a dynamic fitness function adaptation dependent on varying the fitness form of each iteration The weighting factor, ω i , is correlated with each goal (i). As a result, the fitness function (O) can be expressed as follows: Depending on the following two equations, Equations (40,41), the called fitness function is modified accordingly in each run resulting in high variability in individuals with a high concentration with an adaptive normalized object. The normalization process, which is involved with the difference of weights, leads in the equal treatment of the different goals.
In order to build Pareto individuals, the suggested MO-MRFA is created, adding an external repository to maintain non-dominated individuals, and then Pareto dominance contrast is used to upgrade the repository. In each iteration, modified entities are compared to current Pareto individuals in the repository to remove the dominated solutions. As a result, it is revised and, if it is met, the exclusion technique is implemented by eliminating any of the Pareto solutions in the most populated areas using roulette wheel selection [61].
For handling the considered OCC problem, the power balance equality constraints in AC grid (Equations (16) and (17)) and in MTHVDC grid (Equation (18)) are assured by sequential AC/DC load flow method [46]. For the operating constraints of the control variables, they are beginning within their bounds and if either of them is exceeded during iterations, they are randomly reconstructed within the following acceptable range. The operating limits of dependent variables in the AC/ MTHVDC power systems are expanded by the quadratic penalties in the objective functions regarded. A solution that leads to some breach of the restrictions on this entire basis of dependent variables could not be chosen in the sequent iteration. Thus, in terms of the objectives accomplished, each new position is contrasted to its counterpart in the previous iteration. The position of the manta ray is changed if the current position is not dominated by the previous one. This approach retains variety and increases the consistency of the solution. Therefore, a set of optimal Pareto solutions is then generated and preserved. In order to acquire the compromise point for operating the system, a fuzzy decision-making tool [56] can be applied where the membership function (μ i ) can be assigned for each objective as follows: Then, a distinguished solution is then derived using a fuzzybased mechanism that obtains optimum membership (μ q ) as: where, q, m and n relate to each compromise individual corresponding to each step of W; the number of objectives and the number of compromise individuals, respectively. Figure 2 displays the flowchart of the suggested MO-MRFA to operate optimally the AC-HVDC power grids. As has been shown, the key advantage of the proposed strategy relative to the others is its balanced and enhanced exploitation/exploration characteristics. Every iteration was split into two successive sections by the proposed MRFA. As seen in the Figure. 2, "Run sequential AC/DC load flow" and "Check the limitations of the dependent variables" appear twice. The first part triggers the strategy of chain and cyclone, while the second part triggers the strategy of somersault. In both sections, the fitness function must be tested by running the power flow and testing the corresponding constraints.

SIMULATION RESULTS
The

Application results for the IEEE 30-bus system
Originally, the first test system, IEEE 30-bus test system, has of 6 generators, 41 lines, 30 buses, 4 on-load tap-changers and 9 capacitive shunts. As seen in Figure 3, the modified system has two MTHVDC grids [28]. The data of buses, power lines, and the limits of VAR sources are referred to [57]. The lower and upper bounds for load-bus voltage, tap-changer limits and generator voltages, are, respectively, 0.95 and 1.05 p.u. respectively. In the first MTHVDC grid, the VSCs control mode are VSC 1 is V dc -Q c mode, VSCs 2 and 3 are in P dc -V c mode. In the second MTHVDC grid, VSC 4 is V dc -Q c mode whilst VSCs 5 and 6 are in P dc -V c mode. The VSCs converting power are taken 100 MVA where the lower and upper voltages of the DC buses and VSCs are, respectively, 0.9 and 1.1 p.u. The shunt VAR injections are restricted by 5 MVA. Cost, emission, and losses coefficients of VSC stations are tabulated in Tables 1, 2 and 3, respectively.
Six cases are discussed to cover both single and multiobjectives frameworks as:

Results of single objective OCC optimization of the modified 30-bus AC/MTHVDC system
In this subsection, the MRFA for optimum operation of the hybrid AC/MTHVDC grid is implemented for the updated IEEE 30-bus test system with a single TFC, TPL and TAEE minimisation target, respectively for cases 1-3. Tables 4-6 show the optimal settings of the OCC control variables and the related effects are shown in respectively, for the MRFA and several competitive techniques, PSO, GWO, MVO, SSA, CSOA, DA, MPO, and BAT.

TFC Minimisation
As seen in the results obtained in Table 4, the TFC is decreased from $975.64 to $840.3 per hour with a reduction of 13.87 per cent relative to the original case using the MRFA, while the reduction ratio using the competitive algorithms from the lowest to the highest reduction compared with the initial case as: SSA, DA, GWO, BAT, MVO, CSOA, PSO and MPO are 9.24%, 12.01%, 12.42%, 12.56%, 12.96%, 12.99%, 13.26% and 13.75%, respectively.

TPL minimisation
Also, the suggested MRFA succeeds, as well, in meeting the minimum TPL, which is considered as the main objective function in the second case. Table 5 tabulated the settings of the control variables and the values of objective function obtained with the proposed MRFA compared with several competitive optimization algorithms. The TPLs are minimised from 8.57 MW with a percentage reduction of 28.12%. This reduction is the highest one compared with all reported methods in Table 5.

TAEE Minimisation
In case 3, it is attaining the minimum TAEE. By using the MRFA the TAEE is decreased to 0.203 ton/hr with a reduction of 16.11%. This value cannot be accomplished by other methods, as seen in Table 6.
A further success is observed in this table with losses reduction by 24.3%. The MRFA simulation results demonstrate outperforms over other algorithms with substantial achievement at each different objective function that achieves optimally operation of the hybrid AC/MTHVDC grid in cases 1-3. In addition, the bus voltages of the AC grids are substantially enhanced by the proposed MRFA as seen in Figure 4. The convergence rates for cases 1-3 are demonstrated in Figures 5 and 6 of the suggested MRFA for cases 1-3. From these statistics, the high capacity of the MRFA to find the minimal target considered is explained. Progress through iterations reflects the potential to find an optimal solution in an evolutionary manner.

Results of multi-objectives OCC of the modified 30-bus AC/MTHVDC system
In this subsection, the OCC in hybrid AC/MTHVDC systems is treated in a multi objective optimization form and solved by MO-MRFA for bi-and tri-objective functions. The outcomes of the control variables and the related results for the three multi-objective scenarios are seen in Table 7. Figures 7-9, respectively, demonstrate Pareto solutions for optimum operation of AC/MTHVDC with TFC and TAEE minimisation bi-objective functions (case 4), TFC and TPL minimisation biobjective functions (case 5) and TFC, TAEE and TPL minimisation tri-objective functions (case 6). The simulation findings that are obtained in these figures make it clear that the proposed MO-MRFA offers the best solution on the considered objectives. Compared to cases 1-3, the optimal compromises for TFC, TAEE and TPL are equivalent to the single goal values achieved.

Case 4
In this case, the available compromise options, between minimisation of TFC and TAEE minimisations, are obtained from TFCs of approximately 840 ($/hr) and TAEEs of approximately 0.44 (ton/hr) to TFCs of approximately 10,432 ($/hr) and TAEEs of approximately 0.209 (ton/hr) going through the optimum approach TFCs of 870.9 ($/hr) and TAEEs of 0.295 (ton/hr) as seen in Figure 7.

Case 5
In this case, a variety of compromise solutions are achieved between TFC and TPL minimisations beginning from TFC in the range 848-994 ($/hr) and TPL in the range 14.3-9.5 (MW). Figure 8 shows the best compromise solution is 873.8 ($/hr) and 11.73 (MW).

Case 6
In the same way, different effective solutions are acquired and well-distributed in Figure 9, including the optimum TFC compromise solution of 882.60 ($/hr), TAEE of 0.26490 (ton/hr) and TPL of 10.850 (MW). Figure 10 displays the bus voltages in the AC grid for the multi-objective cases and initial case by using proposed MO-MRFA, where all bus voltages are under the permissible limits. Figure 11 shows the VSCs capability curve and their optimal operating points of the modified 30-bus system for the previous 6-cases of single and multi-objective. From this figure, all the obtained operating points of the VSCs, with the proposed technique, are undoubtedly within their specified limits.

Simulation results of the standard IEEE 57-bus system
The IEEE 57-bus system has 57 buses, 80 lines, 8 generators, 17 on-load tap changing transformers and 3 shunt capacitive sources. The data for buses, transmission lines, and the limits of VAR generations are referred to [63]. The lower and upper limits for the generator and load voltage are 0.94 and 1.06 p.u., respectively. The lower and upper limits values for load voltages and tap transformer are 0.95 and 1.1 p.u., respectively, where the reactive compensation is limited by 30 MVAr. This system is modified with five VSCs and four DC connected lines as shown in Figure 12. The VSCs are located at bus 26-29 and 52, respectively. These selected buses are not connected between generators units and are connected lines of high-power flow in the initial case. The DC lines are added between VSCs 1-2, 2-3, 3-4 and 4-5. VSC 1 is Vdc-Qc control mode whilst the other four VSCs are in Pdc-Vc control mode.
In this section, three cases are considered for this test system. For cases 1 and 2, the validity of the proposed technique is examined by comparing the results with the standard GWO, PSO, SSA, MVO, DA, CSOA and BAT. In case 3 the proposed technique is applied for TFC and TPL minimisation.
(A) TFC minimisation: Table 8 gives the results of optimal scheduling of the control variables and the corresponding simulation results of the modified IEEE 57-bus test system with single objective TFC minimisation using the proposed MRFA and several competitive algorithms. From the obtained results, the MRFA can minimise the TFC from 53,673.15 to 41,923.63 $/h with a reduction of 21.89%.
(B) TPL minimisation: Table 9 shows that the proposed technique succeeds in achieving the lowest power losses in case 2 compared with other competitive techniques. The total power losses are reduced by 67.68% using the proposed MRFA, which has the highest power loss reduction compared with all of the competitive algorithms as reported in Table 9. Figure 13 displays the convergence rates of the proposed MRFA for OCC of hybrid AC/HVDC of the 57-bus IEEE test system with single objective TFC and TPL minimisation, which depicts the fast convergence characteristic of the proposed technique.
(C) TFC and TPL bi-objective minimisation: Case 3 introduces the optimal solution of the OCC of hybrid AC/MTHVDC with bi-objective functions TFC and TPL minimisation to identify the capability of the proposed MRFA for searching optimal solutions and compromises the conflicted objectives as given in Table 10.
TFC can be minimised from 53,673.15 to 42,448.91 ($/hr) and power losses is decreased from 52.044 to 18.054 MW. Also, the proposed technique can maintain all constraints within their allowable limits. Figure 14 shows the AC grid voltage profile for the initial case and the three cases of TFC minimisation, TPL minimisation and bi-objective TFC and TPL minimisation. Figure 15 shows the VSCs capability curve and their optimal operating points of the modified 57-bus system. There are three cases that cover single and multi-objective frameworks. From this figure, it found that all the obtained operating points of the VSCs, with the proposed technique, are undoubtedly within their specified limits.

Simulation results of the practical WDPN
The third AC/MTHVDC test system is the practical WDPN which consists of 52 buses, 108 lines and 8 generators [64,65]. It is modified with a MTHVDC grid with four VSCs and three DC lines as shown in Figure 16. The VSCs are located at bus 5, 36, 6, and 31, respectively. The DC lines are added between VSCs 1-2, 2-3 and 1-4. VSC 1 is Vdc-Qc control mode whilst the other three VSCs are in Pdc-Vc control mode.
The maximum and minimum values for the generator voltage are 1.06 and 0.94 p.u., respectively. The limits for the bus voltages are ±10%. For this test system, this section considers three cases. Two single objective case for TFC and TPL minimisation are introduced in cases 1 and 2, respectively. In case 3 the proposed technique is applied for simultaneous minimisation of TFC and TPL The proposed technique is examined with the standard for these two cases.
(A) TFC minimisation: Table 11 gives the simulation results of optimal values of the control variables and the corresponding results of the modified WDPN with single objective TFC minimisation using the MRFA and other competitive techniques. It is found that the proposed MRFA gives the best solution as the fuel cost as it is minimised from 25,164 to 23,095.5 $/hr with 8.22% reduction.
(B) TPL Minimisation: Also, the proposed MRFA succeeds in achieving the lowest power losses in Table 12. It has superior performance compared with other competitive techniques for TPL minimisation (case 2). The TPL is reduced using the proposed MRFA from 22.36 to 11.58 MW with 48.21% reduction. while its value is reduced by 44.95, 35.16, 39.36, 47.04, 43.76, 45.32, 37.83 and 47.69% using the GWO, PSO, SSA, MVO, DA, CSO, BAT and MPO, respectively. The convergence rates of the proposed MRFA for the previous two cases is displayed in Figure 17. This figure depicts the fast convergence characteristic and the capability of finding the optimal solution of the proposed technique.
(C) TFC and TPL bi-objective minimisation: The OCC of hybrid AC/MTHVDC with bi-objective functions TFC and TPL minimisation is introduced and the Pareto set solutions is given in Figure 18 while Table 13 tabulates the related optimal results. The capability of the proposed MO-MRFA is demonstrated for searching optimal solutions and compromises the conflicted objectives. In addition, a compromise solution is acquired with TFC of 23,490.56 ($/hr) and power losses of 22.814 MW. Besides that, all constraints are kept within their allowable limits for single and multi-objectives by using the improved MO-MRFA for voltage of AC grid as shown in Figure 19, and operating point of the VSCs, as shown in Figure 20

Statistical analysis of the compared algorithms
To evaluate the performance of the competitive algorithms, a set of statistical indices namely best (BE), average (AV), worst (WO) and standard deviation (STD) are evaluated and their results are reported in Table 14 for the modified 30-bus AC/MTHVDC and the modified WDPN test system. From this table, the proposed MRFA always acquires the competitive best indices of all statistical indices for minimising the fuel costs in case 1 that is applied for the modified IEEE 30-bus test system and the modified WDPN. It has a very small standard deviation for both systems that declares its higher robustness compared to other solution methods.

4.5
Computation complexity for the proposed MRFA Also, Table 15 shows the proposed MRFA computation complexity that is estimated using the concept of the big O notation [66,67]. The modified IEEE 57-bus system has the highest computational burden of O(660000) where the modified WDPN system is the lowest with O(360000). On the other side, the average computational time in minutes of the proposed MRFA is 10.0127 for the greatest computational complexity system of the modified IEEE 57-bus system, based on DELL Inspiron machine of model 2014. This time is suitable for the real time considered problem since the OPF is usually considered for the next 15 min. Also, this time can be much faster through parallel computing and more advanced computers or workstation machines that are usually existed and updated in practical call centres all over the world.

CONCLUSIONS
An improved multi-objective manta ray foraging algorithm (MO-MRFA) has been investigated for enhancing the optimal operation of hybrid AC/MTHVDC power grids. The considered problem aims at achieving technical and economic benefits considering the emissions concerns of the generation stations. Added to the previous aims, the problem has been formulated to involve the minimisation of the total of power losses at the VSC stations, AC, and the power MTHVDC transmission systems. The proposed optimisation algorithm mimics three distinct individual foraging organisations of the manta rays. It is integrated an external Pareto repository to conserve the non-dominated manta ray's positions. Furthermore, a fuzzy decision-making technique is activated to select the final candidate operating point of the hybrid AC/MTHVDC power grid. The applicability of the proposed MO-MRFA has been verified through different case studies on modified IEEE 30-bus, modified IEEE 57-bus and a practical part of the Egyptian system at the West Delta Region Power. Assessment of the proposed solution methodology is employed with significant improvements compared with several recent algorithms. Significant technical improvements are achieved with reduction up to about 70% in the power losses as well as economic achievements up to about 25%. The emission is benefited with reduction of 16%. The simulation results demonstrate high stability indices of the effectiveness and preponderance of the proposed algorithm for single and multi-objective frameworks. Also, the proposed procedure is benefited with well-diversified Pareto solutions. The best compromise operating solution is efficiently created to meet the operator specifications. Based on the effectiveness of the proposed algorithm, it is nominated for possible future applications such as combined heat and power dispatch, integration of renewable distributed generation and optimal operation of automated distribution networks.