1 Introduction

The transformers represent one of essential devices in the power energy sector and in distribution network as well as in many industrial household applications. Transformers transmit the energy form generation stations to distribution stations through the transmission lines with high efficiency based on characterization of equivalent circuit parameters and the related losses [1]. Therefore, the analysis of power systems and realistic operation of transformer needs for accurate transformer model. Many attentions have been investigated to exhibit the transformer parameters with minimizing its losses, minimizing the operational cost, and improving its performance. transformer’s unknown parameters provide nonlinear model due to its frequency dependence which is making that solution of transformer parameters task to optimality is a true challenge [2].

Estimating transformer parameters became a huge and necessary task for the best transformer design to achieve necessary standards and requirements [3, 4]. Moreover, the state of a transformer's functioning, such as steady or transient situations, has an impact on the estimation of any unknown parameters [5, 6]. Several techniques can be used to estimate these parameters, including: the well-known tests such as the no-load and short-circuit tests [7, 8], physical sizing of transformer [9], manufacturer’s data [10,11,12], and under various load information [7]. Primarily, the analytical techniques have been employed to quickly assess the transformer's physical sizing using finite element analysis.

Cast iron dry transformer parameters are extracted using logical method and compared with the same obtained from both finite element method and actual data [13]. Recently, metaheuristic optimization methods (MOMs) have been seen a boom in a variety range of optimization tasks, where they do not require the convexity/continuity of objective functions (OFs) and no reliance on the gradient information. Accordingly, several MOMs have been presented extensively in wide fields of optimization due to their capability in solving more complicated the optimization tasks as they do not require the gradient information of the OF and start with a population of initial guess [7]. Some of MOMs include the salp swarm [14, 15], Harris hawks [16], crow search [17, 18], barnacles mating [19], quantum sine cosine [20], artificial ecosystem (AEO) [21], and atom search [22] optimizers.

On the other hand, many attentions have been presented in the literature to estimate the optimal parameters of transformer models as well as electric motors, storage units, and fuel cells [23,24,25,26]. The accuracy of the optimization algorithms is tested by comparing the extracted parameter values against the actual ones [27,28,29,30,31,32,33,34]. For instance, particle swarm optimization (PSO) has been introduced to estimate transformer parameters with evaluating some physical dimensions such as loss parameters, winding inductance, and capacitance [6], where both single-phase (1Ph) and three-phase (3Ph) power transformer parameters were extracted using the load testing data that were acquired. Forensic-Based optimization [1] has been proposed to estimate the parameters for the 1Ph transformer (1PhT).

Adding to the above-mentioned, the slime mold optimizer has been used to estimate the parameters of 1Ph and 3Ph transformers, where the obtained results have been compared with other optimizers [35]. Bacterial Foraging [36] has been presented to extract the 1PhT parameters based on the data driven from load testing. Chaotic Optimization [7] has been investigated to estimate the equivalent circuit parameters of 1PhT. Also, Manta Ray Foraging Optimization and its chaotic version have been presented to identify the parameters of 1PhT, where no-load losses have been incorporated into the OF [3]. Moreover, coyote optimizer, Jellyfish search optimizer, and machine learning approach have been introduced to extract parameters of 3Ph and 1PhTs [37,38,39], respectively. Also, the transformer parameters have been evaluated using multi-objective evolutionary optimization and verified by contrasting the outcomes with actual measures and behavior [14]. Also, distribution transformer parameters have been extracted at frequencies between 1 kHz and 1 MHz using a simple black-box approach that uses an optimization technique and transfer functions estimated from recorded voltage ratios [40]. On the other hand, the parameters of 1PhTs have been identified with reference to saturation and inrush current using artificial hummingbird optimizer (AHO) [41] and Nelder–Mead optimization [42] and using inrush measurements [43]. Also, the nonlinear magnetizing characteristics of 1PhTs have been determined using minimum information [44]. Voltage and current measures have been taken into consideration to estimate 1PhTs parameters using hurricane optimization algorithm [45], crow search algorithm [46], and Black-Hole optimization [47].

Despite the substantial amount of recently developed algorithms in this field, there may arise a question here as why researchers are still interested in developing new optimization algorithms or improved variants. This query can be answered by the means of the No Free Lunch (NFL) theorem [48] as it logically affirms that no one can suggest a solution algorithm for dealing with all optimization issues. This implies that the success of an algorithm in addressing a particular set of issues does not ensure that it can also solve all optimization issues with different natures. In other words, regardless of the higher performance on a subset of optimization issues, all optimization methods perform similarly on average when considering all optimization problems. The NFL theorem enables researchers to develop novel optimization algorithms or enhance/modify the existing ones for dealing with subsets of problems in various domains.

This work's major contributions include (i) to accurately determine the best 1PhT’s unknown parameters using a novel promising metaheuristic optimizer namely dwarf mongoose optimization algorithm (DMOA) [50] and (ii) to analyze both steady-state and aging performances of transformers. The DMOA simulates the dwarf mongoose behavior and skills during the foraging behavior for its food. The performance of the DMOA has been assessed by applying it on three issues: 19 classical functions with complex natures, the IEEE CEC 2020 benchmark functions, and 12 engineering design problems [49]. Then, an improved DMOA (IDMOA) is presented to characterize the parameters of the 1PhT. Comparing the DMOA’s performance to other approaches, it has been offered a competitive and advanced one. However, as a recent optimizer, its performance lacks for balancing the exploitation and the exploration abilities when dealing with complicated optimization landscapes, including multimodal nature and high dimensional situations. Due to this, local optimum trapping may occur, and the search process may be degraded. Motivated by the issues, it can be mentioned that an improved variant of the DMOA is attempted to produce high-quality solutions with mitigating the premature convergence dilemma. This work proved that IDMOA can obtain the lowest SAEs among the others. Also, the efficiency of the IDMOA has been confirmed by applying the extracted parameters in calculating the hotspot temperatures and aging of a transformer and assured that the per unit life of the transformer is obtained as per the IEEE Std C57.91 rules [50].

This current paper suggests an improved variant of the DMOA, named as IDMOA to decide the unidentified parameters of the EPT model. IDMOA operates with a population of solutions to effectively explore the search space. In this context, the Nelder–Mead Simplex (NMS) method is integrated to efficiently promote the neighborhood searching and enhance the exploitation ability. At initial stage, the OF is adapted for the transformer electrical equivalent extraction parameters in terms of the sum of absolute errors (SAEs) among numerous factors from nameplate data of transformers. Moreover, the corresponding operating inequality restrictions are incorporated with the SAEs goal. The proposed IDMOA is applied on two different cases with transformer ratings as 4 kVA and 15 kVA, respectively. Moreover, the cropped outcomes of IDMOA are compared with other recent challenging optimizers. The performances of DMOA and IDMOA are compared with other recognized competitors including AEO, equilibrium optimizer (EO), gradient-based optimizer (GBO), gray wolf optimizer (GWO), memory-based hybrid dragonfly algorithm (MHDA), and other well-known optimizers from the literature. To assure the exactness of the proposed IDMOA optimizer, the transformer hotspot temperature and its effects on the transformer aging according to IEEE Std C57.91 [50] are investigated using the parameters stemmed from the optimizer and from the practical (actual data). Transformer with 15 kVA is used for the comparisons which proofed the effectiveness of the IDMOA.

The remainder parts of this work are presented as follows: Part 1 declares the mathematical model of the 1PhT. Part 2 expresses the transformer model based on optimization prospective. In part 3, the procedures of the DMOA and IDMOA algorithms are demonstrated in detail. Part 4 shows the simulation results of the DMOA and IDMOA algorithms regarding the 1PhT parameters. Also, the steady state is assessed by the proposed algorithm in Part 5. The extracted parameters by the IDMOA are utilized to assess the transformer aging and its hotspot temperature in Part 6. Finally, the conclusions with extension of further research are highlighted in Part 7.

2 Single-phase transformer mathematical modeling

The 1PhT can be represented referred to its primary side as revealed in Fig. 1. The primary winding resistance and reactance, core loss resistance and leakage reactance are defined by \({R}_{P}\,{ \mathrm{and}\,\,X}_{P }, {R}_{c}\) and\({X}_{m}\), respectively. The secondary side load, winding resistance and reactance denoted to primary side are called \({Z}_{L}^{^{\prime}},\) \({R}_{s}^{^{\prime}}\) and \({X}_{s}^{^{\prime}},\) respectively [41, 51, 52].

Fig. 1
figure 1

Equivalent Transformer circuit

Full size image

The transformer three unknown variables pairs \({R}_{P}\,{ and\, X}_{P }, {R}_{c}\) and \({X}_{m}\,\mathrm{and}\,{R}_{s}^{\mathrm{^{\prime}}}\), and \({X}_{s}^{\mathrm{^{\prime}}}\) can be composed as \({Z}_{P}\), \({Z}_{m}\), and \({Z}_{s}^{^{\prime}}\) which are calculated as following:

$${Z}_{p}={R}_{p}+j{X}_{p}$$
(1)
$${Z}_{m}=({R}_{c}\times j{X}_{m})/{(R}_{c}+j{X}_{m})$$
(2)
$${Z}_{s}^{^{\prime}} ={R}_{s}^{\mathrm{^{\prime}}}+\mathrm{j}{X}_{s}^{\mathrm{^{\prime}}}$$
(3)

If the transformer is supplied with input voltage \(\left({V}_{P}\right)\) to deliver output voltage \(\left({V}_{s}^{^{\prime}}\right)\) connected to the load \(\left({Z}_{L}^{^{\prime}}\right),\) then, the equivalent transformer impedance\(\left({Z}_{\mathrm{eq}}^{^{\prime}}\right),\) primary current \(\left({I}_{p }\right),\) load current \(\left({I}_{s}^{\mathrm{^{\prime}}}\right),\) input power\(\left({P}_{i}\right),\) outpower\(\left({P}_{o}\right),\) transformer efficiency \(\left( \eta \right),\) and voltage regulation \(\left({V}_{\mathrm{reg}}\right)\) are estimated as follows [41]:

$${Z}_{\mathrm{eq}}^{^{\prime}}={Z}_{P}+\left({Z}_{m}.[{Z}_{L}^{^{\prime}}+{Z}_{s}^{^{\prime}}]\right)/\left({Z}_{m}+{Z}_{L}^{^{\prime}}+{Z}_{s}^{^{\prime}}\right)$$
(4)
$${I}_{p }={V}_{p }/{Z}_{\mathrm{eq}}^{^{\prime}}$$
(5)
$${I}_{s}^{\mathrm{^{\prime}}}=\{\left({Z}_{m}\right)/\left({Z}_{m}+{Z}_{L}^{^{\prime}}+{Z}_{s}^{^{\prime}}\right)\}. {(V}_{p }/{Z}_{\mathrm{eq}}^{^{\prime}})$$
(6)
$${V}_{s}^{^{\prime}}={Z}_{s}^{^{\prime}} . {I}_{s}^{\mathrm{^{\prime}}}$$
(7)
$${P}_{i}=\mathrm{real}\left[{V}_{p} .{I}_{p}^{*}\right]$$
(8)
$$P_{o} = {\text{real}}\left[ {V_{s}^{\prime } .I_{s}^{{\prime {\text{*}}}} } \right]$$
(9)
$$\eta \% = P_{o}/P_{i} \times 100$$
(10)

Also, when the transformer is loaded, its temperature will increase due to its power loss [53]. If the transformer loading is increased more than its nameplate data, it will be subjected to overheating, shorten its life and may lead to insulation fail and risks. IEEE Std C57.91 [50] and IEEE C57.12.00 [54] formulated the relation between transformer temperature, loading and aging as per the following equations:

$${\text{Perunit}}\,{\text{Life}} = Xe^{{\left( {Y/\left[ {\theta _{{{\text{HS}}}} + 273} \right]} \right)}}$$
(11)
$${\text{Perunit}}\,{\text{Life}} = 9.8x10^{{ - 18}} e^{{\left( {1500/\left[ {\theta _{{{\text{HS}}}} + 273} \right]} \right)}}$$
(12)
$${\text{AAF}} = e^{{\left( {\{ 1500/383\} - \{ 1500/\left[ {\theta _{{{\text{HS}}}} + 273} \right]\} } \right)}}$$
(13)
$$\theta _{{{\text{HS}}}} = \theta _{{{\text{AM}}}} + \Delta \theta _{{{\text{TO}}}} + \Delta \theta _{{{\text{HS}}}}$$
(14)
$$\Delta \theta _{{{\text{TO}}}} = \left\{ {\Delta \theta _{{{\text{TO}} - U}} - \Delta \theta _{{{\text{TO}} - t}} } \right\}\left[ {1 - e^{{1/\tau _{{{\text{TO}}}} }} } \right] + \Delta \theta _{{{\text{TO}} - t}}$$
(15)
$$\Delta \theta _{{{\text{TO}} - t}} = \Delta \theta _{{{\text{TO}} - R}} \left[ {\frac{{\left( {K_{t}^{2} .R + 1} \right)}}{{\left( {R + 1} \right)}}} \right]^{n}$$
(16)
$$\Delta \theta _{{{\text{TO}} - U}} = \Delta \theta _{{{\text{TO}} - R}} \left[ {\frac{{\left( {K_{U}^{2} .R + 1} \right)}}{{\left( {R + 1} \right)}}} \right]^{n}$$
(17)
$$\tau _{{{\text{TO}},R}} = \frac{{C\Delta \theta _{{{\text{TO}} - R}} }}{{P_{{T,R}} }}{\text{for}}\,{\text{n}} = 1$$
(18)
$$\tau _{{{\text{TO}}}} = \tau _{{{\text{TO}},R}} \left\{ {\frac{{\left( {\frac{{\Delta \theta _{{{\text{TO}} - U}} }}{{\Delta \theta _{{{\text{TO}} - R}} }}} \right) - \left( {\frac{{\Delta \theta _{{{\text{TO}} - t}} }}{{\Delta \theta _{{{\text{TO}} - R}} }}} \right)}}{{\left( {\frac{{\Delta \theta _{{{\text{TO}} - U}} }}{{\Delta \theta _{{{\text{TO}} - R}} }}} \right)^{{\frac{1}{n}}} - \left( {\frac{{\Delta \theta _{{{\text{TO}} - i}} }}{{\Delta \theta _{{{\text{TO}} - R}} }}} \right)^{{\frac{1}{n}}} }}} \right\}{\text{for}}\,\,{\text{n}}\,\,{\text{ < }}\,1$$
(19)
$$C=\left\{\begin{array}{c}0.06*{W}_{\mathrm{CL}} + 0.04*{W}_{\mathrm{TF}} + 1.33* {\mathrm{VOL}}_{O} For\,ONAN\,Transformers \\ 0.06*{W}_{\mathrm{CL}} + 0.06*{W}_{\mathrm{TF}} + 1.93* {\mathrm{VOL}}_{O} For\,OFAF,Transformers\end{array}\right.$$
(20)

where \(\theta _{H}\) is the winding hottest-spot temperature, X is a constant (\(9.8 x {10}^{-18})\) as per IEEE, Y is a constant (\(1500)\) as per IEEE, \(\theta _{{{\text{AM}}}}\) defines the ambient temperature (AT), \(\Delta \theta _{{{\text{TO}}}}\) denotes top-oil rise (TOR) over AT in °C, \(\Delta \theta _{{{\text{HS}}}}\) defines winding hottest-spot rise over top-oil temperature, \(\Delta \theta _{{{\text{TO}} - U}}\) is the ultimate TOR over AT in °C, \(\Delta \theta _{{{\text{TO}} - t}}\) denotes the initial TOR over AT in °C, \(t\) is the duration of changed load in hours, \(\tau\) defines the thermal time constant for the transformer (accounting for the new load, and accounting for the specific temperature differential between the ultimate oil rise and the initial oil rise), \(\Delta \theta _{{{\text{TO}} - R}}\) is TOR over AT at full load current (FLC) (determined during test report), \({K}_{t}\) is ratio of load of interest to rated load, R is Ratio of load loss at rated load to no-load loss, and n is empirically derived exponent used to calculate the variations of \(\Delta \theta _{{{\text{TO}}}}\) with changes in load.

Transformers with different cooling modes will have different n values, which approximate effects of change in resistance with changing load.\({\tau }_{\mathrm{TO},R}:\) The top-oil time constant at rated kVA, \({\tau }_{\mathrm{TO}}\) : The top-oil time constant, \({P}_{T,R}\) : Total power loss at rated load, \(C\) : Thermal capacity of the power transformer (watt-hour per °C) depends on the cooling system, \({W}_{\mathrm{CL}}:\) weight of core and coils, \({W}_{\mathrm{TF}} :\) weight of tank and fittings, \({\mathrm{VOL}}_{O}\) : Oil volume in gallons of oil, \(\mathrm{ONAN}\) : Oil Natural Air Natural, and\(\mathrm{OFAF}\) : Oil Forced Air Forced

3 Extraction of the optimum transformer parameters

The main goal from extracting optimum transformer parameters is to achieve the best practical operation performance of the transformer. This can be achieved by the minimization of the sum absolute errors between the practical (SAE) and calculated transformer parameters. The OF is adapted in (21) as following:

$${\text{OF}} = {\text{minimize}}\left( {\left| {I_{p} - I_{{p - {\text{act}}}} } \right| + \left| {I_{S}^{\prime } - I_{S-act}^{\prime }} \right| + \left| {V_{S}^{\prime } - V_{S-act}^{\prime } } \right| + \left| {\eta -\eta_{{{\text{act}}}} } \right|} \right)$$
(21)

where \({I}_{p-\mathrm{act}}\), \({I}_{s-\mathrm{act}}^{\mathrm{^{\prime}}}\), \({V}_{s-\mathrm{act}}^{^{\prime}}\), \(\eta _{{{\text{act}}}}\) are the real values of the primary drawn current, output delivered current, output voltage, and transformer efficiency, respectively. The determination of the best values of the six unidentified transformer parameters (\({R}_{p}\), \({R}_{s}^{\mathrm{^{\prime}}}\), \({X}_{p}\), \({X}_{s}^{\mathrm{^{\prime}}}, {R}_{c}\), \({X}_{m}\) ) requires defining upper and lower limits collected from practical side for the six parameters as per the following (22):

$$\left\{\begin{array}{c}{{R}_{p-\mathrm{min}}\le R}_{p}\le {R}_{p-\mathrm{max}}\\ {R}_{s-\mathrm{min}}^{\mathrm{^{\prime}}}\le {R}_{s }^{\mathrm{^{\prime}}}\le {R}_{s-\mathrm{max}}^{\mathrm{^{\prime}}}\\ {{X}_{p-\mathrm{min}}\le X}_{p}\le {X}_{p-\mathrm{max}}\\ {X}_{s-\mathrm{min}}^{\mathrm{^{\prime}}}\le {X}_{s }^{\mathrm{^{\prime}}}\le {X}_{s-\mathrm{max}}^{\mathrm{^{\prime}}}\\ {{R}_{c-\mathrm{min}}\le R}_{c}\le {R}_{c-\mathrm{max}} \\ {{X}_{m-\mathrm{min}}\le X}_{m}\le {X}_{m-\mathrm{max}}\end{array}\right.$$
(22)

where the min/max pairs values of the six unidentified parameters (\({R}_{p}\) , \({R}_{s}^{\mathrm{^{\prime}}}\) , \({X}_{p}\) , \({X}_{s}^{\mathrm{^{\prime}}}, {R}_{c}\) , \({X}_{m})\) are, respectively, symbolled as (\({R}_{p-\mathrm{min}}\) and \({R}_{p-\mathrm{max}}),\) (\({R}_{s-\mathrm{min}}^{\mathrm{^{\prime}}}\) and \({R}_{s-\mathrm{max}}^{\mathrm{^{\prime}}}\) ), (\({X}_{p-\mathrm{min}}\) and\({X}_{p-\mathrm{max}}\) ), (\({X}_{s-\mathrm{min}}^{\mathrm{^{\prime}}}\) and \({X}_{s-\mathrm{max}}^{\mathrm{^{\prime}}}\) ), (\({R}_{c-\mathrm{min}}\) and \({R}_{c-\mathrm{max}}\) ), and (\({X}_{m-\mathrm{min}}\) and \({X}_{m-\mathrm{max}}\) ).

4 The proposed methodology

This section exhibits the basics of the DMOA and NMS method as well as the proposed integrated version that named as IDMOA methodology.

4.1 The basics of DMOA algorithm

In this section, the mathematical aspect of DMOA is presented [49]. The DMOA was proposed based on dwarf mongoose behavior while searching for food source. Mongooses offer impressive behaviors during foraging process as they split the population into three social classes, alpha (α) category, scout category, and babysitters. In terms of optimization of view, the DMOA starts its optimization searching by initializing a set of solutions within the search space as follows:

$${X}_{ij}={\mathrm{LB}}_{j}+\mathrm{rand}.\left({\mathrm{LB}}_{j}-{\mathrm{UB}}_{j}\right), i\in N, j\in D$$
(23)

where \(\mathrm{rand}\) stands for a random number produced from [0,1] according to uniform distribution. \({\mathrm{LB}}_{j}\) and \({\mathrm{UB}}_{j}\) define the limits of the \(j\mathrm{th}\) dimension search space. \(N\) and \(D\) denote the population size and total number of dimensions. After the initialization, each group of the DMOA performs its own dramatic behavior to capture the food, where the following subsections introduce the details regarding these groups

4.1.1 α-category

Once the swarm of mongooses is initialized, each solution is evaluated based on the fitness of function. In this context, the probability value regarding each solution is determined and then, it is used to select the α-female as follows:

$$\alpha =\frac{{\mathrm{fit}}_{i}}{{\sum }_{i=1}^{N}{\mathrm{fit}}_{i}}$$
(24)

The total number of mongooses within the α-group is considered as \(N-bs,\) where \(bs\) defines the number of babysitters. Peep stands for the vocalization the α-female that retains the family on path.

Every mongoose sleeps in the early sleeping mound that is set to \(\varnothing.\) The candidate food position is produced by using the following expression:

$${X}_{i+1}={X}_{i}+{\mathrm{ph}}_{i}.\mathrm{peep}$$
(25)

where \({ph}_{i}\) defines a uniform random number within the [− 1, 1], where the sleeping mound (sm) is updated after every iteration as follows:

$${\mathrm{sm}}_{i}=\frac{{\mathrm{fit}}_{i+1}-{\mathrm{fit}}_{i}}{\mathrm{max}\left\{\left|{\mathrm{fit}}_{i+1},{\mathrm{fit}}_{i}\right|\right\}}$$
(26)

The average or mean value regarding the sleeping mound can be obtained as follows:

$$\varphi =\frac{{\sum }_{i=1}^{N}{\mathrm{sm}}_{i}}{N}$$
(27)

The algorithm proceeds toward the scouting stage, where the sleeping mound or subsequent food source is assessed once the babysitter group exchange criterion is fulfilled.

4.1.2 Scout category

The scouts search for the next sleeping mound as they do not visit the former sleeping mounds, which ensures the exploration ability. For the DMOA model, the scouting and foraging are done simultaneously as explained in [49]. This movement is modeled regarding the failure or success evaluation in attaining a new sleeping mound. In other words, the movement of the mongooses depends on their performances; meanwhile, if the family goes far enough foraging, they will explore a new sleeping mound. The scout mongoose is simulated by the following form:

$${X}_{i+1}=\left\{\begin{array}{cc}{X}_{i}-\mathrm{CF}*{\mathrm{ph}}_{i}*\mathrm{rand}*\left({X}_{i}-\overrightarrow{M}\right),& {\varphi }_{i+1}>{\varphi }_{i}\\ {X}_{i}+\mathrm{CF}*{\mathrm{ph}}_{i}*\mathrm{rand}*\left({X}_{i}-\overrightarrow{M}\right),& \mathrm{else}\end{array}\right.$$
(28)

where \(\mathrm{rand}\) stands for an arbitrary random number ranged from 0 to 1, \(\mathrm{CF}={(1-t/T)}^{(2(t/T))}\) defines collective-volitive parameter that balances the exploration and exploitation searches among the mongoose group, and it is decreased linearly with the growth of iterations. \(\overrightarrow{M}={\sum }_{i=1}^{N}\frac{{X}_{i}*{\mathrm{sm}}_{i}}{{X}_{i}}\) is the vector that realizes the mongoose’s movement to the new sleeping mound.

4.1.3 Babysitters’ category

Babysitters represent subsidiary members that sit with the youngsters, and they are rotated regularly to permit mother (the α-female) to drive the remainder members of the group while performing the daily foraging. The babysitter exchange parameter is utilized to reset the food source and scouting information formerly held by the family associates substituting them. The working frame for the DMOA is shown in Fig. 2.

Fig. 2
figure 2

Working frame for the DMOA

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4.2 Nelder–Mead simplex technique

The NMS approach was first developed to handle unconstrained minimization circumstances [55]. Numerous scientific issues and practical engineering applications [56, 57] have been solved using this unique technique, which relies on the information of the OF to be solved without the need of its derivatives. As an iterative algorithm, the NMS method might be used. For instance, NMS would begin from a simplex generated by the starting vertices when faced with a \(D\) -dimensional minimization function. In each iteration, those vertices that are inferior to freshly generated vertices can be replaced, which results in the fulfillment of a new simplex. The simplex would be focused and nearer the best solution with repeated iterations. The detail procedures of the NMS method can be presented as the following steps [56]:

Step1: The vertices are numbered and sorted in ascending order based on the fitness values by the following Equation, \(f\left({x}_{1}\right)\le f\left({x}_{2}\right)\le \dots \le f({x}_{D})\le f({x}_{D+1}).\)

Step2: Compute the reflection point \({\mathrm{x}}_{\mathrm{r}}\) and calculate the value of the OF by (29). Substitute \({x}_{D+1}\) by \({\mathrm{x}}_{\mathrm{r}}\) and perform step 6, if \(f\left({x}_{1}\right)\le f({x}_{r})\le f({x}_{D+1});\) if \(f\left({x}_{1}\right)>f({x}_{r}),\) then, carry out step 3; if \(f({x}_{D+1})\le f({x}_{r}),\) then, perform the step 4:

$${x}_{r}=\left(1+\alpha \right). \overline{x }-\alpha {x}_{D+1}$$
(29)

where \(\alpha\) represents the reflection factor, \(\overline{\mathrm{x} }\) defines the center point of vertices except \({\mathrm{x}}_{\mathrm{D}+1},\) that is achieved by (30):

$$\overline{x }={\sum }_{i=1}^{n}{x}_{i}/D$$
(30)

Step 3: Compute the expansion point \({x}_{\mathrm{e}}\) using (31) and obtain the corresponding OF\({\mathrm{f}(x}_{\mathrm{e }}).\) If\({f(x}_{e })\le {f(x}_{r })\) , then \({x}_{D+1}\) is substituted by \({x}_{e};\) else, \({x}_{D+1}\) is substituted by \({x}_{r}.\) Then, carry out step 6:

$${x}_{e}=\left(1-\beta \right). \overline{x }+\beta {x}_{r}$$
(31)

where \(\beta\) represents the expansion factor.

Step 4: if \({\mathrm{f}(x}_{\mathrm{r }})\le {\mathrm{f}(x}_{\mathrm{D}+1 }),\) then obtain the external contraction point \({x}_{\mathrm{oc}}\) by (32) and assess this point using corresponding fitness value\({\mathrm{f}(x}_{\mathrm{oc}});\) else, the internal contraction point \({x}_{\mathrm{ic}}\) is computed using (33) with its corresponding objective value \({\mathrm{f}(x}_{\mathrm{ic}}):\)

$${x}_{\mathrm{oc}}=\left(1-\gamma \right). \overline{x }+\gamma {x}_{r}$$
(32)
$${x}_{\mathrm{ic}}=\left(1-\gamma \right). \overline{x }+\gamma . {x}_{D+1}$$
(33)

where \(\upgamma\) defines the contraction factor.

For the case that, the \({\mathrm{x}}_{\mathrm{oc}}\) is created; if \({\mathrm{f}(x}_{\mathrm{oc }})<{\mathrm{f}(x}_{\mathrm{r }}),\) then \({\mathrm{x}}_{\mathrm{D}+1}\) is substituted by \({\mathrm{x}}_{\mathrm{oc}},\) and carry out step 6; else, perform the step 5. Once, the \({\mathrm{x}}_{\mathrm{ic}}\) is obtained, if \({\mathrm{f}(x}_{\mathrm{ic }})<{\mathrm{f}(x}_{\mathrm{D}+1 }),\) then, substitute \({\mathrm{x}}_{\mathrm{D}+1}\) with \({x}_{\mathrm{ic}},\) and perform step 6; otherwise, run step 5.

Step 5: In accordance with (34), all vertices except for \({\mathrm{x}}_{1}\) are shrunk collectively to create a new simplex and advance to step 6:

$${V}_{i}=\left(1-\delta \right). {x}_{1}+\delta {x}_{i}. i\in D+1$$
(34)

where \(\delta\) denotes the shrinkage factor.

Step 6: Stop the search process if the cut-off condition is satisfied; otherwise, move on to the next iteration.

4.3 The proposed IDMOA

In this section, the primary motivation regarding the suggested IDMOA algorithm along with the stepwise explanation is presented in detail.

4.3.1 Motivation for enhancing the DMOA performance

As described earlier, the explorative and exploitative skills of DMOA are determined mainly by the α-group, scout group, and babysitters toward food sources. This searching behavior increases the diversity of solutions and strengthens the exploration pattern of DMOA to some degree of extent. However, dealing with multimodal optimization challenges with many valleys and peaks aspect degrades the exploration and exploitation skills of the algorithm and may lead to the trapping into local minima. Moreover, this trapping mode can be caused by the weaknesses of the exploitation capability of DMOA which hinders identification of the optimal solution. Therefore, to further strength the exploration and exploitation skills and ensure a suitable convergence precision, the NMS method is incorporated with the DMOA to ensure the accurateness of the optimal solution. The main feature of the NMS method is that it offers an effective optimization performance by tracking the position of the best solution, position of the worst solution, and the centroid of all solutions, which significantly aids in averting the weaknesses of the DMOA. In this regard, NMS is invoked to the DMOA to accomplish more refined local searching in the vicinity of the candidate feasible solutions while improving the exploitation trends. By this integration, the algorithm can avoid the inactive iterative, running without any improvement in the outcome.

4.3.2 Framework of the proposed IDMOA

In this work, NMS method is embedded into DMOA to improve the exploration and exploitation skills of the original DMOA version. In the initialization phase, the dwarf mongooses create a population of random positions (solutions) within the boundary limits, then, the DMOA-based algorithm invokes its social search groups, the α-group, scout group, and babysitters, to explore different regions within the search space. At the second phase, NMS method is invoked and located after the DMOA to carry out the exploitive feature whin neighborhood zones reached by DMOA. In this context, the best solution in the current population would be used to establish the primary simplex. Afterward, the simplex stages are continued for \(\mu\) iteration to update this solution.

The \(\mu\) is a crucial parameter that can cause the NMS techniques to be overemphasized if its value is too high. Contrarily, if its value is too low, this mechanism will not be able to fully utilize its local search capability. Its value is decided upon through repeated experiments as \(2d,\) and then, the search procedures of NMS method will be performed for \(\mu\) iterations and swapped back to the DMOA phase. The DMOA and NMS phases proceed through iterations until the stopping condition are satisfied. The pseudo code of the proposed IDMOA algorithm is demonstrated in Fig. 3.

Fig. 3
figure 3

Working frame for the proposed IDMOA

Full size image

5 Simulation and validations

The performance and accurateness of the proposed IDMOA in estimating the 1PhT's unknown parameters are evaluated through conducting to two case studies: 15 kVA and 4 kVA 1PhTs [1, 28]. For fair verifications, the results of the proposed IDMOA are compared with some well-known optimizers including AEO [21], EO [58], GBO [59], GWO [21], MHDA [60], and DMOA for further validations. In addition to that the results of IDMOA are compared against other related works from literature which are GA [27], PSO [27], Chaotic Optimization Approach (COA) [10], Imperialist Competitive Algorithm (ICA) [10], and GSA [10] (The reader is invited to see Sect. 5.4).

For fair stabilization, all the implemented competitors employ the same size of population of 50, maximum of iterations is equal to 200 with 20 independent runs. Besides, each of presented algorithm is implemented under MATLAB environment with AMD Ryzen 5 5600u with Radeon Graphics 2.30 GHz and 16 GB RAM. The suggested parameter configurations for each comparative algorithm are presented in Table 1 and are based on recommendations in the relevant literature. It is noteworthy that after conducting few experiments, the population sizes of presented algorithms are fixed at 50 [61].

Table 1 Configuration settings of the parameters for the competing algorithms
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5.1 Case 1: 15 kVA 1PhT

The suggested algorithm along with the comparative ones is investigated and applied to estimate the parameters of 15 kVA 1PhT which is associated with the following nameplate data: 15 kVA, 2400 V/240 V, one-phase, and 50 Hz. Moreover, the measured transformer characteristics regarding the voltages, currents, and efficiency at FLC are as follows: \({X}_{s-\mathrm{act}}^{\mathrm{^{\prime}}}\) = 2383.8 V, \({I}_{p-\mathrm{act}}\) = 6.20 A, \({I}_{s-\mathrm{act}}^{\mathrm{^{\prime}}}\) = 6.20 A, and \(\eta _{{{\text{act}}}}\) = 99.2% [10, 28].

The reported results for the transformer parameters and compared algorithms are tabulated in Table 2. In this sense, the representative OF by (21) is applied while taking the values referred to the transformer's primary side into account using short-circuit and standard Z-circuit tests.

Table 2 Extracted parameters using the IDMOA and compared methods for 15 kVA 1PhT
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Table 3 illustrates the results of the four items of the OF obtained by the proposed algorithm and the compared ones. Based on the recognized results, it can reveal that the proposed algorithm can provide errors sum remarkably close to zero, while the errors sum for the GBO and EO can provide competitive edge. For further assessment, the proposed IDMOA and the compared one’s convergence behavior are analyzed and compared as depicted in Fig. 4. It can be observed that the IDMOA starts to be stable with minor SAE changing at nearly fiftieth iteration with superior minimum SAE value equals to 0.0335124, while the GBO and EO start to be stable at 100 and 120 iterations, respectively. It can be established that IDMOA can provide faster converge rate pattern than the compared ones and affirm the high-quality outcome of the IDMOA.

Table 3 Error’s comparisons among the IDMOA and other competing ones for 15 kVA 1PhT
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Fig. 4
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Convergence rate for the IDMOA and compared ones for test case 1

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5.2 Case 2: 4 kVA, 250/125 V 1PhT

The performance of the proposed IDMOA is further evaluated on another type of the 1PhT to extract its associated parameters with the following nameplate data 4 kVA, 250/125 V, one-Phase, 50 Hz. The extracted parameters are compared with other optimizers as shown in Table 4. Moreover, the characteristics of the voltages, nominal load currents, and efficiency at FLC of this case are obtained and compared with other well-established algorithms as in Table 5. It can be realized that the sum of percentage error achieved by IDMOA is 1.12e−5 which is the lowest one over the other optimizers. In addition, the qualitive performance of the IDMOA and the other ones is assessed by depicting the convergence trends as in Fig. 5. In this sense, it is noted that the IDMOA reaches fixed SAE value at approximately the 50 iteration while the nearest behavior optimizers AEO, MHAD and GBO starts to be stable at 125, 180, and 200 iteration. This is to confirm that IDMOA has the faster convergence rate than other optimizers.

Table 4 Extracted parameters using the IDMOA and compared methods for 4 kVA transformer
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Table 5 Errors’ comparison for IDMOA and other methods for 4 kVA transformer at FLC
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Fig. 5
figure 5

Convergence curves for IDMOA and the other optimizers for case 2

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The reader can note that there are small deviations between the exact reported SAEs, and those circulated in Tables 3 and 5 due to the approximations as apparently 6 round digits are used.

5.3 Statistical assessment

In this subsection, the performance’ stability of the algorithm is assessed by conducting some statical measures which can affirm that the obtained solution not happened by chance. In this sense, each algorithm is performed for 20 independent runs with recording some of statistical descriptive tests which are the best value (Min.), average value (Mean), median value (Median), worst value (Max.) of the OF, and standard deviation (Std). The results for these measures are exhibited in Table 6 for the studied cases, where the better values are highlighted with bold format. Based on achieved results, it is noted that the IDMOA can provide progressive results over the compared ones for Case 1 and can provide competitive results with the AEO, EO, and GBO for the best value of the OF. However, in terms of mean value of the OF, the proposed IDMOA can provide superior results than the others. Finally, according to the statistical indices, the IDMOA has a promising competitive performance for the parameter extraction of the 1PhT.

Table 6 Statistical measures for the presented algorithms
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5.4 Assessments with other methods from the literature

In this subsection, the IDMOA was further with other competitors reported from the literature including PSO [27], GA [27], ICA [10], GSA [10], COA [10], and AHO [41] for Case 1 and PSO [1], FBI [1], JS [39], and AHO [41] for Case 2. The results of the original DMOA and the proposed IDMOA as well as the other advanced competitors are revealed in Table 7. In this sense, it can be observed that IDMOA exhibits progressive results compared to other the state-of-the-art emulators on the two cases.

Table 7 Comparison between the estimated parameters by the proposed IDMOA and that reported from the literature
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5.5 Statistical testing using the pairwise Wilcoxon tool

Nonparametric statistical tests represent an essential tool for the comparison investigation. In this sense, the widely used Wilcoxon signed-rank test [8] is adopted here to compare the results of two algorithms. Its fundamental concept is not only to keeping track of each algorithm's wins but also to rank the disparities among performances [8]. For each optimization task and every pair of methods, it was checked to see if there was a statistically significant difference between the median results. To conveniently obtain the median and the interquartile range (IQR) of each method, each method is conducted for 20 runs on each task. Then, the hypothesis's p-value is determined and is defined as follows:

  1. (1)

    \({H}_{0}\) : median (Method \(X\) ) = median (Method \(Y\) ).

  2. (2)

    \({H}_{1}\) : median (Method \(X\) ) \(\ne\) median (Method \(Y\) ).

To assess the outcome, the Wilcoxon signed-rank test is used, with a significance threshold of 0.05. \({H}_{1}\) is approved if the \(p\) -value \(\le 0.05\) since the medians of the two algorithms differ from one another and one of them is superior to the other in terms of the measure being utilized.

Otherwise, \({H}_{0}\) will be accepted, and the two techniques are incomparable if the \(p\) -value \(>0.05.\) Therefore, Table 8 exhibits the results of the \(p\) -value with the positive rank (\({R}^{+})\) and the opposite rank \(({R}^{-})\) of Wilcoxon’s test. Based on the p value, it can be noted that the IDMOA is incomparable with GBO for Case 1 and affirms the superior performance over the other optimizers (see Table 8).

Table 8 Results of Wilcoxon test
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6 Transformer aging assessment based on the proposed optimizer

As well known, the transformer parameters are vital in defining the no-load and load losses created in core and windings which can lead to transformer insulation degradation and shorten its expected nominal life. To verify the capability of the proposed IDMOA optimizer in extracting the truthful transformer parameters, its results have been applied to the IEEE Std C57.91 [50] for transformer aging, and the produced hotspot temperature equations previously mentioned in (12) to (21) and compare it with the same using actual parameters.

The initial parameters values are considered and/or collected from practical side as per the transformer name plate data for a certain manufacturer as following:

\(\theta _{{{\text{AM}}}}\) = 30 °C, \(\Delta \theta _{{{\text{TO}} - R}} = 50^{^\circ } {\text{C}}\) , \({K}_{t}=\) 0 to 1.2, \({K}_{U}=\) 1.2, R = 25, n = 1, and \({P}_{T,R}\) = 119.34 W for IDMOA optimizer and is equal to 120 W-based actual data.

The behavior of transformer hottest-spot temperature using the extracted parameters from IDMOA optimizer compared to the same created by actual values is revealed in Fig. 6. It can be noticed that the behaviors of the two cases are remarkably close specially at higher loading which is considered the higher danger operation condition. The percentage biased error between the two studied scenarios is shown in Fig. 7.

Fig. 6
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Loading effects on the transformer hottest-spot temperature

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Fig. 7
figure 7

Transformer hottest-spot temperature error

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It can be noticed that the error at 100% loading is about 0.007% while dropped to zero at 120% loading. Minor error appeared at low load conditions may be due to the measurements precision and can be condoned because of its minor effect on the hotspot temperature. Also, it can be observed from Fig. 6 (both actual and IDMOA curves are almost matched) that the hottest-spot temperature is less than 110 °C at FLCs as per IEEE Std C57.91 recommendations.

Transformer insulation life (aging) and transformer aging accelerator factor (AAF) behavior are plotted as in Figs. 8 and 9, respectively. It can be noted that the per unit life (aging) is nearly 1 at 110 °C, and the AAF is larger than 1 for winding hottest-spot temperatures greater than the reference temperature 110 °C and less than 1 for temperatures below 110 °C as indicated in Fig. 8.

Fig. 8
figure 8

Transformer insulation life (aging)

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Fig. 9
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Transformer AAF

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7 Conclusions

A novel attempt has been made in this work to develop an improved Dwarf Mongoose Optimizer (IDMOA)-based Nelder–Mead Simplex method to extract the parameters of power transformers exact equivalent model. In this context, the nameplate data are offered to investigate this goal through minimizing the sum of absolute errors across some chosen variables. Toward this goal, two test cases studies have been conducted to assess the performance of the proposed algorithm. further validations are affirmed by the comparisons some comparative methods. The minimum achieved values regarding the SAEs by the IDMOA are 3.3512e−2 and 1.12e−5 for 15 kVA and 4 kVA cases, respectively. In this sense, IDMOA can provide the lowest SAEs among the others. Finally, the effectiveness of the IDMOA has been verified by applying the obtained parameters in calculating the hotspot temperatures and aging of 15 kVA transformer which showed that the per unit life of the transformer is gained at less than 110 °C as per the IEEE Std C57.91 and C57.12.00 guidelines. The current IDMOA-based methodology can be extended for further assessments on larger capacities of power transformers (oil/dry) with two and tertiary windings.