Stochastic Frontier-Based Analysis of Energy Efficiency in Russian Open-Pit Mining Enterprises

Abstract

This article is devoted to the study of the possibilities for improvAzing the quality of energy management systems adopted at open-pit mining enterprises in the Russian Federation. The main idea of the work is to apply stochastic boundary value analysis methods using the production function for individual and integral estimates of the performance of energy-consuming objects when performing various types of technological work. It is shown that mining enterprises are experiencing problems in the field of rational energy consumption due to the lack of strictly formalized ways to determine the frontiers of the efficiency value of the parameter of specific energy consumption (SEC). A justification is given for the need to apply stochastic frontier analysis (SFA) methods and use the Cobb–Douglas production function to account for the nonlinearity and stochasticity of the operating conditions of energy-consuming mining objects. The results of a statistical analysis of the data on the operation of EKG-10 excavators at operating enterprises in Siberia are presented, as well as an assessment of their energy efficiency using the adopted approach based on planning the target value of SEC. The results of computational experiments on constructing an energy efficiency model using the SFA/Cobb–Douglas function for various data segmentation options are presented. Computational experiments have been conducted to compare variants based on the Cobb–Douglas production function and translog function with semi-normal and exponential distribution forms for the same data set. A comparative assessment is given of the approaches to the complex analysis of activities adopted at enterprises and proposed in this study, characterizing potential hidden energy losses in the range from 4.53% to 20.73%.

1. Introduction

It is known that the efficiency of any enterprise is closely related to the rational use of energy resources [1,2], which is especially typical for such production facilities as mining [3]. Therefore, various energy efficiency management tools are being actively implemented at enterprises in the Russian Federation today, ranging from individual means of technical regulation of equipment energy consumption [4,5] to the integration of centralized strategic energy management in accordance with the ISO 50001 series of international standards [6,7]. However, in particular, the application of standards in many mining enterprises does not differ in consistency and the sufficient level of integration into the processes of technological work management [6,8]. Moreover, the lack of explicit, well-founded, and unified approaches to assessing energy efficiency indicators in the industry, as well as the lack of tools to adequately interpret the dynamics of the functioning of energy-consuming objects (ECO) of enterprises, suggests that problems persist in the field of rational resource consumption and energy management in general.

As a rule, the specific energy consumption parameter is used as a traditional controller indicator of efficiency at enterprises—the ratio of the consumed volume of energy resources to the volume of technological work performed on the production of products (rock mass) [9,10]. Obviously, the use of the SEC parameter to analyze the energy efficiency of a manufacturing enterprise and its individual energy-consuming objects is explained by ease of interpretation and the inherent characteristics of the production process. Minimization of this indicator can be used as one of the criteria of the procedure for managing technological work in the context of rational energy consumption. It should be borne in mind that this parameter still has natural measurement ranges due to the technical and technological features of the ongoing processes and the operational characteristics of energy-consuming objects, such as dump trucks, excavators, etc. This means that, for individual ECO and processes, there are limits to the maximum permissible and efficient values of specific energy consumption, which must be calculated by taking into account individual empirical or statistical characteristics. Based on the individually calculated target value of the SEC, the results of production activities can be determined to a valid extent, characterizing the amount of savings or overspending of energy resources, and, in the case of a dynamic analysis of the work of the ECO, appropriate operational management measures can be taken.

However, to date, in several open-pit mining enterprises under consideration, the determination of efficient specific energy consumption is carried out based on intuitive expert assessments by specialists of enterprises who do not use (although, perhaps, subjectively consider) in their analysis the calculated statistical patterns of behavior of ECO parameters under various conditions of technological processes. Moreover, this parameter is used only when retrospectively analyzing the results of activities for individual time intervals (shift, month, and year), which, in general, does not allow any operational actions to be taken to manage work for rational energy consumption [6,10,11]. The final analysis of the work carried out based on comparing the planned (target) value of specific energy consumption with the actual indicator obtained, in the absence of a formal way to determine the statistically significant frontier of the efficient SEC, may not only be a little informative, but also hide high economic losses.

In this regard, the purpose of this work is to develop and substantiate an approach to improve the quality of energy efficiency assessment of processes of industrial enterprises for open-pit mining through the use of econometric methods based on stochastic frontier analysis (SFA) models [12,13,14]. This work is based on a statistical analysis of data on the operation of EKG-10 electric mining excavators obtained from operating enterprises in the Russian Federation and is a logical continuation of our early work in this field [15]. In particular, earlier we noted a number of problems and proposed some solutions, both in terms of the organization of energy management systems at Russian enterprises as a whole, and individual problems with obtaining production data (manual input, a large number of abnormal values). The main idea of this work is to validate the hypothesis on the presence of hidden energy losses with the current use of the SEC value to determine the limits of resource efficiency. To test the hypothesis, statistically sound and proven SFA production functions were used [16,17].

2. Application of Stochastic Boundary Analysis to Assess Energy Efficiency

2.1. Traditional Approaches to the Analysis of Energy Efficiency in Mining Enterprises

First of all, it should be noted that the technological processes of open-pit mining have a characteristic specificity, which consists of:

• The nonlinearity of production processes, the continuous dynamics of which depend on a large number of technical–technological, mining–geological, and weather–climatic factors, with a complex, and in some cases stochastic, nature of occurrence [10,11], which is difficult to formalize strictly;
• Heterogeneity of the main controlled objects and their parameters, which are mining and transport equipment. The differences between the objects are expressed both in terms and modes of operation and in the initial passport characteristics for similar objects of different series, brands, and manufacturers [18];
• Multicriteria of integral indicators of the efficiency of production activities [10,11,18], which primarily include various kinds of technological criteria, such as, for example:
  • Maximizing the volume of products produced (minerals and rock in general);
  • Minimizing the downtime of technological equipment (mainly dump trucks and excavators);
  • Minimizing the consumption of energy (and material) resources;
  • Etc.

It is assumed that the achievement of all performance indicators should be carried out simultaneously, so that the construction of production plans and the development of control actions involve the use of complex multi-criteria optimization methods, which, obviously, are computationally expensive procedures. Moreover, when managing open-pit mining operations, it is necessary to consider a number of requirements for safety, reliability, and quality in general, which further complicates ways to ensure integrated production efficiency. Thus, management procedures, as a rule, are decomposed into separate tasks aimed at analyzing and optimizing the activities of specific technological objects and production processes in various decision-making time intervals.

Traditional approaches to the management of individual objects and processes of open-pit mining, in a generalized form, include [10]:

• Formation of an annual production plan for the volume of mineral extraction in accordance with the characteristics of the deposit and the accepted technologies of uncovering;
• Discretization of the plan for separate time intervals (quarters, months, and shifts) with subsequent forecasting and planning of production volumes and resource consumption for individual technological objects for each specified time interval;
• Direct execution of planned tasks, continuous monitoring and analysis of performance results to take measures to eliminate discrepancies or recalculate planned indicators.

At the same time, it should be noted that, first of all, enterprises are interested in maximizing the volume of rock mass, and de facto, management procedures, including operational regulation of ECO operating modes, are mainly aimed at optimizing this indicator. In this regard, the resulting efficiency of the production activities of enterprises largely depends on the quality of execution of planned (shift) tasks and measures for operational monitoring, analysis, and management of the progress of technological work. To ensure such measures, today, ECOs are equipped with specialized on-board systems, including a set of sensors, computing and transmitting devices that record information about various production and operational indicators of the equipment. The typical structure of on-board systems, as a rule, includes satellite navigation devices, electric and fuel flow meters, and various devices that determine the dynamics of changes in the position of individual parts of equipment and the load caused by interaction with the rock mass during mining and transport operations [5,18]. The information received from individual ECOs is aggregated in centralized control rooms and used both in the processes of operational management of the progress of production work and as statistical information for the in-depth analysis and planning of integrated activities of enterprises.

However, with the potential availability of a large amount of technological information, currently at the enterprises under consideration, the analysis of the energy efficiency of individual ECOs and production processes remains a problem in the field of rational energy consumption. Despite the adoption of the energy management strategy, the procedures for determining the efficiency of resource consumption and subsequent analysis of the results of production activities are characterized by a certain voluntarism. Mainly, this problem is caused by the accepted methods of calculating an established normative indicator, namely specific energy consumption, used both to evaluate the activities of individual ECO and for an integrated assessment of processes and the enterprise as a whole.

The existing energy management processes can be described in a general way as follows:

Responsible persons from among the personnel of enterprises, based on their own experience, carry out an expert assessment of the “optimal” value of specific energy consumption in the form of a target (planned) threshold. Such an assessment, as a rule, involves considering empirical observations of the monthly total consumption of each type of energy resource (electricity, diesel fuel, etc.) at the enterprise (1). The estimated monthly consumption volumes of energy resources are evenly distributed for each ECO instance (excavators, dump trucks, etc.), in accordance with the type of energy consumed (2). Further, in accordance with the production plan (evenly distributed planned values of the volumes of extracted and transported rock), a planned (target) value is set for each ECO, and the value of specific energy consumption is in the form of the ratio of the estimated consumption of energy resources to the estimated planned value of the volume of work (3).

This procedure can be written in the form of the following formulas:

$${\widehat{E}}_{m,k}\cong {\displaystyle \frac{{\sum }_{g=1}^G{\sum }_{n=1}^N{\overline{E}}_{g,n}}{G_{m,k}}},$$

(1)

$${\widehat{E}}_{n_m}={\displaystyle \frac{{\widehat{E}}_{m,k}}{N}},$$

(2)

$${\widehat{SEC}}_{n_m}={\displaystyle \frac{{\widehat{E}}_{n_m}}{{\widehat{V}}_{n_m}}},$$

(3)

where ${\widehat{E}}_{m,k}$ is an empirical–heuristic average estimate of the estimated volume of the $k$-th energy resource for the $m$-th ($m=1,M$) discrete time interval (month); ${\overline{E}}_n$ is the average observed value of the volume of the consumed energy resource by the $n$-th energy-consuming object ($n=1,N$); $g=1,G$ is the number of observations of the same time interval (month) that fell into the experience of the responsible person; ${\widehat{E}}_{n_m}$ is a heuristic average estimate of the amount of energy resources expected to be consumed by the nth ECO in the $m$-th time interval; ${\widehat{V}}_{n_m}$ is the planned value of the volume of work (mining or transportation of rock) by the $n$-th ECO in the $m$-th time interval; and ${\widehat{SEC}}_{n_m}$ is the planned (target) value of specific energy consumption for the $n$-th ECO in the $m$-th time interval.

The obtained individual estimate of the planned (target) value of specific energy consumption is set for each smaller discrete step included in the $m$-th interval, which means uniform time intervals in the form of shifts (12 h). Based on the obtained specific energy consumption targets, further monitoring and analysis of the shift and monthly results of the ECO activities are carried out, so that:

$$\left\{\begin{array}{c}{SEC}_{n_m}\le {\widehat{SEC}}_{n_m}\to \left(efficiently\right) and {∆E^{+}}_{n_m}=E_{n_m}-{\widehat{E}}_{n_m}\\ {SEC}_{n_m}>{\widehat{SEC}}_{n_m}\to \left(inefficiently\right) and {∆E^{-}}_{n_m}=E_{n_m}-{\widehat{E}}_{n_m}\end{array}\right.,$$

(4)

where ${SEC}_{n_m}$ is the actual value of specific energy consumption calculated on the basis of data on the volume of work performed and energy resources consumed obtained using on-board ECO systems; $E_{n_m}$ is the actual value of the volume of energy resources consumed by the $n$-th ECO in the $m$-th time interval; and ${∆E^{+}}_{n_m}$—the amount of resources saved; ${∆E^{-}}_{n_m}$—the amount of resources overspent.

It is obvious that this approach to assessing energy efficiency indicators in monitoring and analyzing the results of ECO activities has a number of disadvantages, namely:

• The calculation of primary planned indicators of energy resources is carried out using estimates based primarily on the experience of the decision maker. This approach cannot be called sufficiently objective, leveling the risks of human error, and is guaranteed to cover a representative and statistically significant number of observations;
• Using a uniform averaging of values, even for the same type of ECO, is an invalid approach, at least not taking into account the specifics of the service life and operating conditions of a particular ECO instance;
• In the process of calculating planned and actual consumption indicators of energy resources, the specifics of the types of technological work for which estimates of efficient specific energy consumption may differ significantly are not taken into account. Thus, the performance of two completely similar ECO types of work, such as, for example, the excavation of minerals and the excavation of waste rock, with equal values of the “extracted” rock mass, will consume different amounts of energy resources due to the peculiarities of the mining and geological conditions;
• A linear dependence of the amount of work performed on the amount of resources spent is assumed, which is more intuitive than a well-founded and statistically supported hypothesis.

Thus, the limit of the value of the planned specific energy consumption is set as an auxiliary, but small, informative parameter, which does not really explain the level of efficiency or inefficiency of ECO activities in certain situations. In fact, technological downtime of equipment, overfulfilments or deficiencies in the volume of work performed, and the specifics of individual ECO and technological processes that arise for unforeseen reasons are not considered. The use of this parameter, in the context of monitoring and operational management of the progress of production processes in order to offset the discrepancy between the actual values and the planned ones, is not provided. It is also necessary to consider separately the high noise level of the initial data obtained from on-board systems and the presence of a large number of abnormal values described in our previous work [15], which may affect real estimates of energy efficiency. Based on the above, it can be concluded that the existing approach to energy management adopted at enterprises of the Russian Federation currently requires significant changes and the use of methods to achieve greater objectivity, validity, and accuracy in determining key energy efficiency indicators.

2.2. Stochastic Frontier Analysis and Cobb–Douglas Production Function

Based on Farrell’s seminal works [19,20], various parametric and non-parametric methods for measuring efficiency have emerged. Notably, stochastic frontier analysis (SFA) models and data envelopment analysis (DEA) models have proven to be particularly useful in evaluating the efficiency of production units. SFA was introduced independently by Aigner et al. [21] and Meeusen and van der Broek [22], while the DEA approach was pioneered by Charnes et al. [23]. Subsequently, both methodologies have undergone substantial development and have attained considerable popularity.

The primary benefit of nonparametric DEA is its ability to operate without the need for a pre-specification of the functional form of the production function. However, all deviations from the frontier occurred due to variability in the measurement, which the DEA considers as an inefficiency. Considering the inherently stochastic nature of the production processes in the mining industry, an SFA model is more suitable for measuring the efficiency than a deterministic DEA approach.

Therefore, within the framework of this work, the application of the stochastic frontier analysis model and the Cobb–Douglas production function is proposed in order to explore the possibilities of improving the quality of energy management systems. The main idea of the work is to substantiate the boundaries of efficient energy consumption, taking into account the following hypotheses and considerations:

• The use of traditional indicators for monitoring and managing energy consumption processes in the form of key interrelated parameters—the volume of work performed and the energy resources consumed—in order to preserve the simplicity of interpretation of the procedure for specialists of mining enterprises, as well as to ensure the possibility of obtaining analysis results based on representative historical and currently produced data;
• The nonlinearity and stochasticity of the behavior of such parameters are due to:
  • Natural patterns inherent in the production processes of open-pit mining;
  • Technical and technological capabilities of ECO functioning in various operating modes;
  • Hidden (unobservable) or difficult-to-formalize factors that affect the efficiency of production activities.

Thus, the task of the study was to select a model that would most accurately describe the relationship between the amount of energy consumed and the amount of work performed in relation to the ECO and the technological process under consideration, would take into account the nonlinearity of the relationship of parameters, and would also allow for and explain the presence of random errors in the statistical data.

So, based on the formulation of the research problem, one of the possible ways to solve it is the use of stochastic frontier analysis methods based on the Cobb–Douglas production function model. This group of econometric methods has proven itself well in solving many similar problems [12,13,14] and is intended to substantiate empirical patterns in the behavior of parameters of economic and production systems [16,17]. The general idea of such methods is the formation of a nonlinear model that supports the presence of various kinds of errors, where key performance indicators are considered as a dependent variable, and factors influencing the results of the system’s activities act as independent parameters. In general, to account for the impact of a possible stochastic error and identify the frontiers of efficiency and non-efficiency of resource costs in obtaining products, an SFA approach with a function of the following type can be used:

$$\mathrm{l}\mathrm{n}\left(y_i\right)=\mathit{ln}f\left(x_i;\beta \right)+v_i-u_i,$$

(5)

where $y_i$ is the volume of products produced by the $i$-th point in time; $f\left(x_i;\beta \right)$ is a function of the influence of factors (resources spent) $x_i$ with elasticity coefficients $\mathsf{\beta }$; $v_i~N(0, {\sigma }_v^2)$—stochastic error; $u_i~N^{+}(0,{\sigma }_u^2)$—inefficiency error. The random variables $v_i$ and $u_i$ are independent of each other and of the factors $x_i$.

The type of function $f\left(x_i;\beta \right)$ in SFA models can be selected according to the different conditions of the original problem, data, and the subject area under consideration. The choice of a specific function, ultimately, can provide the most accurate description of the behavior of the sampling parameters. In the context of this study, we do not aim to find the most accurate SFA model because, first, we compare it with the traditional linear approach using the SEC parameter. In this regard, we consider the most common as the basic function—the production function or the Cobb–Douglas utility function. This function is designed to describe the dependence of production volume on the cost of labor and capital in the system, which in the most general form, and, in particular, for the case considered in this study, can be represented as a dependence of the volume of output depending upon the amount of resources spent. The classical model of the Cobb–Douglas production function is a function of the following type:

$$Q=A\ast L^{\alpha }\ast K^{\beta },$$

(6)

where $Q$ is the volume of production; $A$ is the technological scale factor; $L$ is the volume of labor costs; $K$ is the volume of capital costs; and $\alpha$ and $\beta$ are the elasticity coefficients for labor and capital, respectively.

Given that, in our case, only one independent factor is used for the SFA model, namely energy consumption, the stochastic production function is written as:

$$y_i=A\ast {x_i}^{\beta }\ast exp\left(v_i\right)\ast \mathrm{e}\mathrm{x}\mathrm{p}\left({-u}_i\right),$$

(7)

In this case, $y_i$ means the volume of work performed $V_{n_m}$ (3) for the mining (excavation) or transportation of rock, and as a factor $x_i$, the volume of spent energy resources $E_{n_m}$ (3) is considered exclusively. Then, $v_i$ is defined as a random deviation of the volume of spent energy resources, explained by stochastic or non-formalized factors of the production process, and $u_i$ is a deviation in energy consumption, explained by the inefficient performance of technological work.

Also, an SFA model using the translog function can be considered as one of the possible options, which, as a rule, provides a more accurate description of the data under study and, in particular, in the case of one independent parameter. This model can be written as follows:

$$\ln \left(y_i\right)=\ln A+{\beta }_{x_i}\ln \left(x_i\right)+{\displaystyle \frac{1}{2}}{\beta }_{{x^2}_i}{\ln }^2\left(x_i\right)+v_i-u_i,$$

(8)

In this study, the translog function will be used for additional final confirmation of the main hypotheses regarding the nonlinearity and stochasticity of the behavior of energy efficiency parameters, as well as an illustrative comparative example of evaluating the accuracy of the basic model with the Cobb–Douglas function.

To calculate the parameters of the energy efficiency function, it is necessary to use the logarithmic likelihood function [22,23,24], which, in general, can be represented as:

$$l\left(\beta , {\sigma }^2, \lambda \right)=K\mathit{ln}\sqrt{{\displaystyle \frac{2}{\pi }}}+K\mathit{ln}{\sigma }^{-1}+\sum _{k=1}^K\mathit{ln}\left(1-\Phi ({\epsilon }_i\lambda {\sigma }^{-1}\right))-{\displaystyle \frac{1}{2{\sigma }^2}}\sum _{k=1}^K{{\epsilon }_i}^2,$$

(9)

where $\beta$ is the coefficient of elasticity of energy consumption; ${\sigma }^2$ is the total variance of the energy consumption error; $\lambda$ is the parameter of the error density function; $k=1,K$ is the observation number; $\Phi$ is the distribution density function of the standard normal distribution; and ${\epsilon }_i=ln\left({y_i}^k\right)-\mathrm{l}\mathrm{n}f\left({x_i}^k, \beta \right)$ is the error of the predicted value of the volume of work performed $V_{n_m}$ (3) of the amount of energy consumed $E_{n_m}$ (3). It should also be noted that the choice of the distribution form can also affect the accuracy of the model and, as a result, the reliability of the results obtained. But, as we noted earlier, in the framework of this study, we do not aim to find the most accurate model, due to the initial hypotheses put forward, which require primary confirmation.

The parameters $\left(\beta , {\sigma }^2, \lambda \right)$ of the logarithmic likelihood function make it possible to fit the model using known empirical data. The maximization of the likelihood function is performed using one of the standard optimization algorithms, such as, for example, the BFGS algorithm (Broyden–Fletcher–Goldfarb–Shanno algorithm).

The total variance of the ${\sigma }^2$ error is the sum of the variances of the stochastic error ${\sigma }_v^2$ and the inefficiency error ${\sigma }_u^2$. The stochastic error has a normal distribution and has a random effect on the dependent variable—the amount of work that cannot be explained by any obvious reasons for energy consumption. While the inefficiency error has a semi-normal distribution, it does not correlate with the stochastic error and indicates the immediate inefficiency of the processes. At the same time, an increase in the inefficiency value $u_i$ reduces the expected value of the dependent variable, and the parameter $\lambda$, calculated as:

$$\lambda ={\displaystyle \frac{{\sigma }_u}{{\sigma }_v}},$$

(10)

It shows the ratio of the inefficiency error to the stochastic error, so that its largest value determines the impact of inefficiency on the consumption of resources for the work performed.

To obtain an inverse (predicted) estimate of inefficiency $\widehat{u^k}$ (6) for the $k$-th observation, the following formulas are used based on error variances. ${ϵ}^k$ is the error value of the likelihood function and the optimized value $\lambda$:

$$\widehat{u^k}=E\left(u^k|{ϵ}^k\right)={\displaystyle \frac{{\sigma }_u^2}{{\sigma }^2}}\left({ϵ}^k+{\displaystyle \frac{{\sigma }_v^2}{{\sigma }_u^2}}\lambda \right),$$

(11)

Then, use the method of truncated performance ratings (${TE}^k=[0, 1]$), allowing for determining the actual evaluation of the efficiency of energy consumption in the performed volume of work, where ${TE}^k=0$ indicates the absence of efficiency and ${TE}^k=1$ reports the maximum efficiency:

$${TE}^k={\displaystyle \frac{{y_i}^k}{f\left({x_i}^k, \beta \right)\cdot \mathit{exp}\left(v^k\right)}}\ast \mathit{exp}\left(-\widehat{u^k}\right),$$

(12)

Thus, based on the obtained value of ${TE}^k$, for each of the known observations, it is possible to determine the randomness in the observed dependence of variables, the efficiency or inefficiency of the achieved result, and the subsequent calculation of the corresponding values of overspending or saving energy resources. At the same time, depending on the amount of sampling of the observation step, the application of this model can be implemented both for the resulting analysis of shift or monthly ECO activities, and as an operational dynamic assessment when monitoring the progress of work.

In addition, in order to test the hypotheses about the affiliation of the observed parameters to other forms of distribution, in this work, we redefined the likelihood function. Thus, acting by analogy with [24,25], the original form of the function (9) for the normal/semi-normal form of the distribution was transformed into the normal/exponential:

$$l\left(\beta , {\sigma }^2, \lambda \right)=-K\mathit{ln}\left({\displaystyle \frac{\lambda \sigma }{\sqrt{1+{\lambda }^2}}}\right)+{\displaystyle \frac{K}{2{\lambda }^2}}+\sum _{i=1}^{K}ln\Phi \left(-{\displaystyle \frac{{\epsilon }_i\sqrt{1+{\lambda }^2}}{\sigma }}-{\displaystyle \frac{1}{\lambda }}\right)+{\displaystyle \frac{\sqrt{1+{\lambda }^2}{\sum }_{i=1}^K{\epsilon }_i}{\lambda \sigma }},$$

(13)

So, as a result, the evaluation was carried out for the following types of functions:

• The SFA/Cobb–Douglas function with the normal/semi-normal form of the parameter distribution in the likelihood function;
• The SFA/Cobb–Douglas function with the normal/exponential form of the parameter distribution in the likelihood function;
• The SFA/Translog function with a normal/semi-normal distribution of parameters in the likelihood function;
• The SFA/Translog function with a normal/exponential distribution of parameters in the likelihood function.

Figure 1 illustrates the application of the SFA function for analyzing the results of ECO activities in the performance of production processes of mining enterprises proposed in this study. The basic idea is:

• In the primary exclusion of abnormal values (red dots) to offset the influence of extreme data;
• Calculating the function and obtaining the efficiency frontier using estimates of ${TE}^k$;
• An integral assessment of the distribution of the general population relative to the efficiency frontier, calculating the volumes of overspent energy resources ${∆E^{-}}_{\Sigma }$ and saved energy resources ${∆E^{+}}_{\Sigma }$;
• Comparing the estimates obtained with the results of the SEC application.

3. Results

3.1. Description of the Analysis of the Source Data and Their Characteristics

To test hypotheses about the incorrectness of the current approach to energy management, data obtained from operating enterprises were used. As an illustrative example for this study, data on the same type of energy-consuming objects, namely electric excavators EKG-10, were selected. It should also be noted that, in order to preserve the uniformity of the data in the illustrative example, ECOs were selected based on the principle of uniformity, not only of the passport technical characteristics but also the proximity of the service life, technical–technological, and mining–geological conditions of operation. Thus, data were selected from one of the coal mines in Siberia for three EKG-10 excavators operating at the same work site in 2021–2023, including such information as types of technological work performed, time parameters of work (system date and time, estimated time of completion of each type of work in minutes), hourly and shift estimates of the volume of work performed (m³) and the amount of electricity consumed (kWh), and planned (target) values of specific energy consumption.

The generated array of technological information consisted of 14,006 aggregates of observations of 12 h shifts, which obviously exceeded the expected value. The initial analysis showed a large number of duplicate records obtained as a result of assigning the same data to different members of the ECO crews, as well as a number of records that were subsequently identified as abnormal—primarily those where the presence of completed volumes of work was recorded in the absence of electricity consumption. Based on the results of data preprocessing and cleaning, a general set of 4598 unique shift records was formed, which were later used in the construction of the SFA/Cobb–Douglas production function.

Because one of the hypotheses considered in this study was the assumption of differences in the behavior of energy management parameters for different types of technological work, the resulting data set was evaluated in terms of the distribution of records according to relevant information. Thus,

Table 1. Distribution of records by type of technological work.

Type of Work (Name)	Number of Entries (Shifts)	Percentage of the Total Sample
Technological work 1	15	<1%
Technological work 2	122	2%
Technological work 3	1081	19%
Technological work 4	3395	60%
Technological work 5	999	18%

shows the general distribution of records for three excavators by five types of work performed before excluding abnormal data, while for Technological Work 1, a statistically insignificant (less than 1%) number of records was observed, as a result of which, this set was excluded from further analysis.

To simplify the perception of the research materials, the names of the works were depersonalized. However, in order to understand the reasons for excluding certain records during further analysis, it is necessary to give some comments. So, the general meaning of technological work for the ECO under consideration is a cyclic procedure for excavating rock by scooping, moving the boom with a bucket, and emptying the bucket, which, under certain conditions, can be considered as identical processes. However, it is necessary to keep in mind the fact that the aggregates of shift data obtained from the enterprise’s information base do not actually have complete uniformity by type of technological work. In one 12 h shift, up to three different types of work can be performed by one ECO, but the shift is assigned the value of only the type of work with the highest time value. When analyzing the results of shift and monthly ECO activities, the company does not consider hourly detailing of the types of technological work performed, and the assessment of energy efficiency for the ECO is determined in a general, integral form. At the same time, in reality, the types of work under consideration are associated with fundamentally different technical–technological and mining–geological conditions of operation. It is known that Technological works 1–4 are directly related to the excavation of rock. However, the actual conditions of performance have a number of differences that may affect energy costs, namely:

• Differences in the physical–geological and geometric properties of the excavated rock: empty rock mass (Technological work 1 and 3) and minerals (Technological work 2 and 4);
• Differences in intensity (frequency of repetition of operations per unit of time) and actual movement of the excavator (static or dynamic location) and its individual mechanisms (emptying the bucket into the truck body or into the dump).

Thus, a shift analysis of the ECO activities without taking into account the hourly detail of the types of technological work performed may show irrelevant results, hide energy losses, or seriously underestimate the expected energy consumption. Figure 2 shows the graphs of the shift distribution of data on the volume of work performed and the amount of electricity consumed. As can be seen from the graphs, abnormal behavior of the parameters is observed for Technological work 5, with a high distribution density at zero values and a large number of peak deviations for one of the parameters. This behavior of the parameters is explained by the fact that the concept of “other work” is defined under Technological work 5, which de facto represents a large list of non-typical tasks related to the performance of work outside the regulations of the main production processes (repairs, emergency work, etc.). Considering the above, evaluating the energy efficiency or non-efficiency of Technological work 5 did not seem to be correct, and the set of relevant data was excluded for further analysis.

When visually evaluating Technological works 2, 3, and 4, it can be noted that, in general, there is a similar tendency in the behavior of the parameters but differing in absolute values. Thus, a general characteristic can be called a close to linear increase in the volume of work performed with an increase in electricity consumption, but upon subsequent detailed consideration, it will be shown that this dependence has a more nonlinear character, in which a significant increase in resource consumption does not bring a linear increase in the volume of work performed.

It should also be noted that there is the presence of abnormal behavior in the parameters with a large number of peaks and near zero marks in other types of work, which is often due not to natural features of production processes but to cases of manual rewriting of the initial data and a number of other factors, which we also reported earlier [15]. Figure 3 shows heat maps of the data distribution for Technological works 2, 3, and 4, which reflect in red the points that were later excluded as anomalies by using the well-known “Isolation Forest” algorithm [26,27].

It should be recognized that the exclusion of data in the resulting analysis of ECO activities is an incorrect approach because, in this case, it is impossible to calculate the actual volumes of extracted rock and spent energy resources. It should be noted that the exclusion of extreme values, of course, can affect the assessment of the efficiency of a particular ECO, but it allows you to more accurately determine the frontiers of efficient energy consumption for subsequent comparative analysis of the results of production processes.

Figure 4 shows an example of the distribution of data on individual excavators before excluding the abnormal values.

The graphs also show many extreme points that are poorly explained from the point of view of production processes. The density of point distribution for Excavator 1 and Excavator 3 is relatively high, while the graph for Excavator 2 shows a pronounced heteroscedasticity of the data. In general, the behavior of the parameters for individual ECOs also has a noticeable general trend that is close to linear.

Table 2. Statistical characteristics of the initial parameters of the analysis.

Parameter	T. work 2		T. work 3		T. work 4		Excavator 1		Excavator 2		Excavator 3
	Perf. Work	Power Cons.	Perf. Work	Power Cons.	Perf. Work	Power Cons.	Perf. Work	Power Cons.	Perf. Work	Power Cons.	Perf. Work	Power Cons.
Mean	1475.15	2655.21	9191.12	4001.18	952.023	3904.84	835.59	2791.7	8360.112	3844.83	713.0	3618.0
Min	120.012	494.0	0.594	45.03	0.008	86.0	0.072	2.0	0.594	43.25	0.008	86.0
Max	3160.84	4747.99	27,101.4	12,330.0	2482.78	19,050.0	2270.58	19,050.0	27,101.42	12,330.0	2482.78	14,572.36
Median	1581.11	2643.63	9580.77	4236.01	939.61	3837.41	890.8	3238.91	8896.24	4035.63	709.68	3858.94
St. dev.	682.29	982.06	3948.38	1307.02	387.79	1279.18	517.76	1810.65	4429.05	1360.63	411.03	1770.02

shows the statistical characteristics of the initial data for individual technological works and for individual ECO.

For a general assessment of the data quality after cleaning from abnormal values and primary confirmation of the hypothesis about the nonlinearity of the behavior of the parameters, calculations of a number of statistical characteristics were performed, as shown in

Table 3. Statistical characteristics of the sample before and after the exclusion of anomalies.

Parameter	Clearing the Data	T. work 2	T. work 3	T. work 4	Excavator 1	Excavator 2	Excavator 3
Correlation	Before	0.3780	0.7337	0.8838	0.5719	0.7297	0.4614
	After	0.4867	0.8155	0.8738	-	-	-
R2	Before	0.1429	0.5383	0.7812	0.3271	0.5325	0.1732
	After	0.2369	0.6651	0.7635	-	-	-
Adjusted R2	Before	0.1421	0.5377	0.7739	0.3263	0.5319	0.1724
	After	0.2367	0.6648	0.7614	-	-	-
Condition number	Before	9782.6	18,960.0	32,720.1	10,713.0	19,187.0	225.84
	After	224.299	189.2	163.557	-	-	-
RMSE	Before	1367.25	885.7	483.212	1385.01	899.82	1519.01
	After	1468.02	1151.0	679.516	-	-	-
MAE	Before	1049.12	723.1814	433.380	782.175	730.906	806.921
	After	1255.65	974.0366	561.897	-	-	-
MAPE	Before	37.3552	34.7959	13.8363	19.9734	34.3212	23.6337
	After	42.9992	31.4390	24.5309	-	-	-

.

In accordance with the results shown in Table 3, we can conclude the following:

• The general significance of statistical characteristics, both for technological work and for excavators, allows us to talk about the existing difference in the behavior of parameters, which confirms the hypothesis that it is necessary to consider the ECO and the processes separately when analyzing the results of production processes. At the same time, the difference in the behavior of parameters for excavators, taking into account their initial identity, is difficult to explain, but may be due to the influence of the crews’ experience, weather conditions, or other reasons that are difficult to formalize without using additional information;
• After excluding anomalies for Technological works 2, 3, and 4, there is a noticeable improvement in the statistical parameters. Thus, we can note an increase in correlation and coefficient of determination, as well as a decrease in conditionality, which allows us to talk about an overall increase in the representativeness of the sample;
• The increase in errors with the growth of other statistical indicators suggests confirmation of the hypothesis of a nonlinear dependence of the volume of work performed and energy resources consumed, which is consistent with the chosen approach to building a model using stochastic frontier analysis methods.

3.2. Comparison of SEC and SFA Production Functions

To further confirm the hypothesis about the incorrectness of the results obtained within the framework of the traditional energy management strategy, the planned and actual indicators of specific energy consumption were restored, as well as the results of their comparative analysis in accordance with the procedures adopted at enterprises (1)–(4).

Figure 5 shows a visualization of the shift distribution of the actual values of specific energy consumption (blue dots) relative to the planned value (red line) for each of the three excavators during 2021–2023. It is believed that values below the line of the planned (target) value of specific energy consumption are efficient and indicate energy savings, while values above the line are inefficient and, accordingly, report energy losses (4). Based on the graph, it can be seen that the vast majority of shifts for Excavators 1 and 3 are located in the area of efficiency, while for Excavator 2, the distribution of shifts, on the contrary, is in the area of inefficiency.

Table 4. Estimates of specific energy consumption values by year.

Year	Indicator	Excavator 1	Excavator 2	Excavator 3
2021	$\widehat{\overline{SEC}}$	0.688; [0.580; 0.830]
	$\overline{SEC}$	0.298	1.923	0.208
	$∆\overline{SEC}$	0.390	−1.235	0.480
2022	$\widehat{\overline{SEC}}$	0.688; [0.580; 0.830]
	$\overline{SEC}$	0.299	2.137	0.174
	$∆\overline{SEC}$	0.389	−1.449	0.514
2023	$\widehat{\overline{SEC}}$	0.688; [0.580; 0.830]
	$\overline{SEC}$	0.308	2.166	0.214
	$∆\overline{SEC}$	0.380	−1.478	0.474

shows annual estimates of the average value of planned (target) specific energy consumption—$\widehat{\overline{SEC}}$, average monthly estimates of actual specific energy consumption—$\overline{SEC}$, and an integral estimate of the deviation of the planned from the actual for the year $∆\overline{SEC}$.

Table 5. Integrated analysis of ECO performance using SEC.

Indicator	Excavator 1	Excavator 2	Excavator 3
Efficient shifts (%)	99.9	11	100
Inefficient shifts (%)	0.1	89	0
${∆E^{+}}_{\Sigma }$ (kWh)	2,288,950.83	79,890.83	3,599,419.88
${∆E^{-}}_{\Sigma }$ (kWh)	−26.75	−6,897,425.81	0
${∆E^{total}}_{\Sigma }$ (kWh)	2,288,924.08	−6,817,534.98	3,599,419.88

shows integral estimates for the entire observation period, reflecting the percentages of efficient and inefficient shifts, considering the use of the planned SEC value and the total shift volumes of energy resources saved ${∆E^{+}}_{\Sigma }$ and overspent ${∆E^{-}}_{\Sigma }$, as well as the total final estimate of energy consumption over 3 years ${∆E^{total}}_{\Sigma }$.

In accordance with the estimates received, it is necessary to give several comments:

• As can be seen from Figure 5 and Table 4, the value of the planned (target) specific energy consumption, although it has some “seasonal” (monthly) variation, but the actual indicators of specific energy consumption have a significant difference both in comparison with the excavators and in comparison with the total value of the target specific energy consumption;
• The 100% distribution of shifts in the efficiency zone for Excavators 1 and 3 causes some skepticism about the adequacy and reliability of the approach to evaluating the ECO activities using a single SEC value that does not take into account the individual characteristics of the equipment and the technological processes performed;
• An integral analysis of the results of the ECO activities in terms of absolute electricity costs, in turn, shows an overspending of resources of 929,191.02 kWh, which could serve as an argument in favor of the relevance of the current approach to energy efficiency assessment. However, taking into account the details of the indicators, this fact also raises doubts about the reliability.

Based on the approach proposed in this study, in order to confirm the initial hypotheses and study the applicability of the SFA model to describe energy efficiency, further work was conducted as follows. The Cobb–Douglas production function was used as a basic function, which was used to perform step-by-step calculations of performance indicators when grouping data according to the principles of division only into separate types of technological work; division only into separate ECO; and division into separate types of work for an individual ECO. Using the calculated parameters of the models, the adequacy of the proposed data separation and the overall improvement in the descriptive ability of the nonlinear relationship of energy efficiency indicators were determined. In accordance with the results obtained, the grouping of data by individual types of work for individual ECO was used as the final sample. For this sample, additional calculations were performed based on the application of the translog function in order to compare the results obtained, generalize the accuracy of the models, and finally, confirm the hypothesis of the non-representativeness of the linear approach using SEC.

The results of the construction of the production function for certain types of technological work are shown in Figure 6 and in

Table 6. Function parameters for certain types of work.

Indicator	Technological work 2	Technological work 3	Technological work 4
$A$	6.64700	3.37386	3.63817
$\beta$	0.27625	0.56212	0.60975
${\sigma }^2$	0.15561	0.07124	0.06869
${\sigma }_v^2$	0.01541	0.01023	0.01550
${\sigma }_u^2$	0.14019	0.06101	0.05318
$\lambda$	3.01584	2.44155	1.85218

.

The functions constructed for an individual ECO without separation by type of technological work are shown in Figure 7 and

Table 7. Function parameters for individual ECO without separation by type of work.

Indicator	Excavator 1	Excavator 2	Excavator 3
$A$	5.10514	5.72823	6.21953
$\beta$	0.46602	0.30853	0.35063
${\sigma }^2$	0.06663	0.13047	0.09099
${\sigma }_v^2$	0.01202	0.03271	0.00094
${\sigma }_u^2$	0.05462	0.09776	0.08159
$\lambda$	2.13194	1.72882	2.94636

.

The results of the construction of the production functions for individual ECOs for certain types of technological work are shown in Figure 8 and Figure 9 and in

Table 8. Function parameters for individual ECOs for individual types of work.

Indicator	Excavator 1/T. work 4				Excavator 2/T. work 2				Excavator 2/T. work 3				Excavator 3/T. work 4
	Cobb–Douglas		Translog		Cobb–Douglas		Translog		Cobb–Douglas		Translog		Cobb–Douglas		Translog
	Half.	Exp.	Half.	Exp.	Half.	Exp.	Half.	Exp.	Half.	Exp.	Half.	Exp.	Half.	Exp.	Half.	Exp.
$A$	5.59968	4.67584	6.9939	3.56010	3.56055	3.45517	5.74882	6.55709	5.34312	2.78065	5.3158	−0.91385	6.36854	6.17319	5.8536	6.83016
$\beta \left({\beta }_{x_i}\right)$	0.39314	0.52189	−0.12653	0.87481	0.61976	0.61731	−0.02283	−0.30076	0.34924	0.62477	0.16025	1.46731	0.32829	0.34949	0.46242	0.13999
${\beta }_{{x^2}_i}$	-	-	0.09241	−0.05700	-	-	0.09301	0.13417	-	-	0.04203	−0.09959	-	-	0.01688	0.03318
${\sigma }^2$	0.07232	0.05028	0.07341	0.04006	0.06636	0.03735	0.06065	0.03559	0.14980	0.07354	0.10625	0.04265	0.09454	0.07014	0.09433	0.07062
${\sigma }_v^2$	0.02215	0.02972	0.07341	0.0242	0.01614	0.03378	0.01829	0.03336	0.04539	0.04607	0.01508	0.04254	0.01041	0.01497	0.00898	0.01462
${\sigma }_u^2$	0.05017	0.02056	0.06205	0.01585	0.05021	0.00357	0.04236	0.00223	0.10441	0.02746	0.09117	0.00011	0.08413	0.05517	0.08535	0.056
$\lambda$	1.50488	0.83173	2.33722	0.80931	1.76361	0.32496	1.52206	0.25847	1.51669	0.77207	2.64622	0.05074	2.84327	1.91986	3.08222	1.95737

.

Table 9. Statistical characteristics of the parameters.

Characteristic	Excavator 1/T. work4		Excavator 2/T. work 2		Excavator 2/T. work 3		Excavator 3/T. work 4
	Perf. Work	Power Consump.	Perf. Work	Power Consump.	Perf. Work	Power Consump.	Perf. Work	Power Consump.
Mean	1070.966	3540.445	1494.712	2709.593	9312.152	4050.914	834.353	4240.85
Min	0.384	351.483	225.816	988.0	100.65	55.679	17.584	760.943
Max	2243.768	7620.135	3072.174	4747.993	21,354.304	7121.677	2090.056	7571.443
Median	1072.104	3556.512	1593.534	2643.625	9628.29	4266.315	810.052	4256.0
St. dev.	386.948	903.597	641.328	936.876	3784.671	1224.032	338.161	1041.451

shows the statistical characteristics of the parameters used for the final calculations of the construction of the efficiency function for individual ECOs and individual types of work.

Further, following the accepted approaches to the integrated assessment of the results of the ECO, the efficiency indicators of individual excavators for the entire observation period were calculated. When calculating the values, the data for a 12 h shift were evaluated, considering their heterogeneity, by dividing by types of technological work performed into separate time intervals (equal to 1 h), for which appropriate estimates of the efficiency frontier were obtained, and the resulting estimate was summarized in an integral form for the shift as a whole.

Thus, based on

Table 10. Integrated analysis of ECO performance using SFA/Cobb–Douglas function.

Year	Indicator	Excavator 1		Excavator 2		Excavator 3
		Normal/Half-Normal	Normal/Exponential	Normal/Half-Normal	Normal/Exponential	Normal/Half-Normal	Normal/Exponential
2021	Efficient shifts (%)	11	19	20	26	8	16
	Inefficient shifts (%)	89	81	80	74	92	84
	${∆E^{+}}_{\Sigma }$ (kWh)	11,167.87	18,792.26	85,398.83	108,060.42	8752.44	21,118.13
	${∆E^{-}}_{\Sigma }$ (kWh)	−182,810.05	−142,915.45	−760,280.53	−677,081.55	−249,233.06	−207,968.17
2022	Efficient shifts (%)	9	14	7	11	11	18
	Inefficient shifts (%)	91	86	93	89	89	82
	${∆E^{+}}_{\Sigma }$ (kWh)	9663.7	15,627.98	31,654.38	47,777.82	12,504.83	25,685.75
	${∆E^{-}}_{\Sigma }$ (kWh)	−202,011.49	−162,651.12	−1,539,565.6	−1,369,247.47	−178,034.36	−141,153.37
2023	Efficient shifts (%)	12	18	19	22	7	18
	Inefficient shifts (%)	88	82	81	78	93	82
	${∆E^{+}}_{\Sigma }$ (kWh)	8927.71	13,430.45	45,453.15	55,730.87	4251.0	13,847.71
	${∆E^{-}}_{\Sigma }$ (kWh)	−119,810.28	−95,770.91	−492,245.64	−449,055.62	−145,364.34	−117,724.28
${∆E^{total}}_{\Sigma }$ (kWh)		−474,872.54	−353,486.79	−2,629,585.41	−2,283,815.53	−547,123.49	−406,194.23

, the integral efficiency of the ECO over the entire observation period can be estimated as an overspend of 3,651,581.44 kWh when using the Cobb–Douglas function in the normal/half-normal form of distribution or 3,043,496.55 kWh when using the Cobb–Douglas function in the normal/exponential form of distribution.

Based on

Table 11. Integrated analysis of ECO performance using SFA/Translog function.

Year	Indicator	Excavator 1		Excavator 2		Excavator 3
		Normal/Half-Normal	Normal/Exponential	Normal/Half-Normal	Normal/Exponential	Normal/Half-Normal	Normal/Exponential
2021	Efficient shifts (%)	11	31	24	48	8	16
	Inefficient shifts (%)	89	69	76	52	92	84
	${∆E^{+}}_{\Sigma }$ (kWh)	12,172.01	39,615.28	125,227.82	309,154.25	9046.26	21,026.06
	${∆E^{-}}_{\Sigma }$ (kWh)	−182,778.72	−114,627.65	−654,625.64	−393,546.19	−249,274.21	−207,989.93
2022	Efficient shifts (%)	9	23	11	29	11	18
	Inefficient shifts (%)	91	77	89	71	89	82
	${∆E^{+}}_{\Sigma }$ (kWh)	10,172.16	30,005.66	59,640.87	229,469.96	12,658.02	25,630.67
	${∆E^{-}}_{\Sigma }$ (kWh)	−201,591.46	−133,963.93	−1,353,868.16	−860,877.61	−177,898.35	−141,193.75
2023	Efficient shifts (%)	12	29	24	33	8	18
	Inefficient shifts (%)	88	71	76	67	92	82
	${∆E^{+}}_{\Sigma }$ (kWh)	9850.55	28,586.38	65,931.77	141,771.78	4595.4	13,738.65
	${∆E^{-}}_{\Sigma }$ (kWh)	−119,273.29	−76,559.17	−441,341.64	−310,689.53	−144,998.2	−117,775.61
${∆E^{total}}_{\Sigma }$ (kWh)		−471,448.75	−226,943.43	−2,199,034.98	−884,717.34	−545,871.08	−406,563.91

, the integral efficiency of the ECO over the entire observation period can be estimated as an overspend of 3,216,354.81 kWh when using the translog function in the normal/half-normal form of distribution or 1,518,224.68 kWh when using the translog function in the normal/exponential form of distribution.

Table 12. Likelihood function values for different functions and distribution forms.

Year	Function and Form of Distribution	Excavator 1/ T. work 4	Excavator 2/ T. work 2	Excavator 2/ T. work 3	Excavator 3/ T. work 4
2021	Cobb–Douglas N./Half-n.	−231.2549	−14.5597	−99.0131	−156.4347
	Translog N./Half-n.	−233.6542	−14.8547	−100.1777	−156.9429
	Cobb–Douglas N./Exp.	−202.6789	−12.6381	−94.3652	−133.6013
	Translog N./Exp.	−225.0399	−14.8243	−94.7005	−134.3139
2022	Cobb–Douglas N./Half-n.	−208.0916	−9.6791	−177.7632	−181.6162
	Translog N./Half-n.	−208.1397	−9.7402	−184.9627	−181.6197
	Cobb–Douglas N./Exp.	−196.8884	−9.5758	−158.5829	−168.2663
	Translog N./Exp.	−162.0102	−6.5179	−168.3036	−168.3122
2023	Cobb–Douglas N./Half-n.	−183.0509	−11.7807	−25.2428	−85.1495
	Translog N./Half-n.	−184.7398	−11.7839	−45.8184	−86.1039
	Cobb–Douglas N./Exp.	−181.4367	−9.3482	−34.8808	−84.6748
	Translog N./Exp.	−183.9125	−11.1101	−46.0355	−74.5607
Total	Cobb–Douglas N./Half-n.	−207.4658	−12.0065	−100.6730	−141.0668
	Translog N./Half-n.	−208.8446	−12.1263	−110.3196	−141.5555
	Cobb–Douglas N./Exp.	−193.6679	−10.5207	−95.9429	−128.8475
	SFA/Translog N./Exp.	−190.3209	−10.8174	−103.0132	−125.7289

shows the values of the likelihood function for different functions and forms of distribution by year for individual excavators for certain types of technological work.

4. Discussion

Based on the results shown in Figure 6, Figure 7, Figure 8 and Figure 9 and Table 6, Table 7 and Table 8, the following comments can be made:

• First of all, it should be noted that the application of the SFA models showed quite convincing results in terms of confirming the hypothesis of a nonlinear relationship between the studied parameters, which is characterized by the value of the elasticity coefficient;
• At the same time, there is a low impact of energy consumption on the amount of work performed, which, in general, is understandable and can be explained by the need to take into account other factors of production;
• It is noticeable that, in cases of more thorough separation of data by individual types of technological work and ECO, their heteroskedasticity, visually observed through a closer grouping of point clouds, significantly decreases, which also suggests the adequacy of the proposed approach;
• At the same time, the use of a “blind” method of clearing data from anomalies based on the “Isolation Forest” algorithm, although it helps to reduce the error spread, however, in some cases, it gives incorrect results in the form of excluding relevant data and including extreme outliers (for example, Figure 8c). In this regard, it is obviously necessary to apply other approaches to the search for anomalies, which can be implemented in the form of heuristic and statistical procedures that take into account the specifics of the nature of the data. In particular, in the future we plan to combine the previously proposed methods of anomaly detection with the current approach to energy efficiency assessment [15];
• Mainly, the application of the SFA model using the Cobb–Douglas production function in the construction of individual energy efficiency models shows clear differences in the implementation of certain types of technological work on the same type of excavators, expressed as differences in the calculated parameters. This fact allows us to talk about the potential applicability of considering the peculiarities of the functioning of individual ECOs and obtaining more adequate and reliable indicators of the energy efficiency of the implementation of production processes at the mining enterprises under consideration;
• Comparison of the translog function with the Cobb–Douglas function for different distribution forms, in general, gave the expected close results, confirming the initial hypotheses of the study. In all cases, the functions have a fairly close view, while the translog shows a slightly lower energy loss.

At the same time, it is quite difficult to draw convincing and unambiguous conclusions about which function and form of distribution are best suited. So, following Table 12, it can be noted that, in all cases, the half-normal distribution gave the value of the loglikelihood function lower than the exponential one, on the basis of which we can give the advantage to the last. In exponential form, the Cobb–Douglas function gives a slightly better result than the translog function for the log likelihood value. However, the difference in estimates in both cases is not high, and taking into account the value of observations in the data samples, as well as the initial higher accuracy, in our opinion, the translog function should be preferred.

The results obtained in accordance with Table 5, Table 10, Table 11 and

Table 13. Comparison of integrated energy efficiency assessment indicators for different models.

Indicator	SEC	SFA/Cobb–Douglas (Norm./Half.)	SFA/Cobb–Douglas (Norm./Exp.)	SFA/Translog (Norm./Half.)	SFA/Translog (Norm./Exp.)
Efficient shifts (%)	77	11	18	12	27
Inefficient shifts (%)	23	88	82	88	73
${∆E^{+}}_{\Sigma }$ (kWh)	5,968,261.54	217,773.93	320,071.37	309,294.87	838,998.69
${∆E^{-}}_{\Sigma }$ (kWh)	−6,897,452.56	−3,869,355.35	−3,363,567.95	−3,525,649.67	−2,357,223.37
${∆E^{total}}_{\Sigma }$ (kWh)	−929,191.02	−3,651,581.42	−3,043,496.58	−3,216,354.81	−1,518,224.67
Total energy loss (%)	7	27.73	23.11	24.42	11.53

show integral estimates of the analysis of production activities for the three excavators considered over the entire observation period in comparison with the results obtained for the same data using the target value of SEC. A detailed table with calculated values for each excavator for each type of work for individual years is provided in Appendix A.

Based on Table 12, it can be concluded that, by taking into account the proposed approach, the energy efficiency of production processes at the enterprises under consideration is significantly lower than expected. Thus, the difference in absolute values of the overexpenditure of electricity can range from 589,032.98 kWh to 2,722,389.98 kWh of hidden losses, which is equivalent to an additional 4.53–20.73% of total energy consumption for the entire period of observation for the considered technological work of 13,131,100.05 kWh. The results obtained are generally in line with expectations and correlate with the results in this area. Thus, the use of the SFA models to assess the effectiveness of production activities, as a rule, provides a better statistical justification of the frontiers. Obviously, the estimates obtained using the SEC contradict the general patterns and cannot be perceived as adequate in cases where the three-year efficiency of production activities is in the range of 99–100%. In this case, in our opinion, we were able to demonstrate this clearly, which will give an impetus to rethinking the situation.

The practical meaning and actual applicability of the proposed approach can be determined as follows. Calculating the boundaries of individual energy efficiency, the ECO can be implemented as a function of an automated energy management system [15]. The frontiers can be assessed on a regular basis, covering the entire range of available data for each ECO, or with the selection of optimal sliding windows to take into account the dynamics of technical and operational conditions. In particular, for the convenience of information perception by enterprise personnel, the actual efficiency limit according to the SFA can be transformed into the value of the SEC by comparing the planned amount of technological work with the corresponding effective amount of energy. If savings are achieved through the operational management of work based on the SFA estimates, the released energy and/or financial resources can be redirected to additional energy management activities.

Thus, some disadvantages of the proposed approach and the results obtained include:

• The absence of a large number of examples and intersections for all types of technological work performed for individual ECOs in order to obtain a more visual picture of comparing the results, better confirming the hypotheses of the study. This situation is explained by the initial need to obtain homogeneous data, in which many abnormal records were excluded. The presence of anomalies in general, including the presence of entire groups of non-routine technological work, requires additional research in order to find ways to work with them and take them into account in the integrated analysis of the activities of enterprises;
• The estimates obtained regarding the overspending of energy resources and the volume of hidden losses, in comparison with the accepted approach to assessing energy efficiency, in fact, do not allow us to say that these volumes could be saved. Since the approach proposed in this study also allows only a retrospective characterization of the results of activities, in order to take any operational measures to improve the energy efficiency of processes in the future, it is necessary to formalize the methods for the online monitoring of ECO and restructuring energy management systems using the SFA production function;
• The difficulty is in unambiguously summarizing the results and drawing universal conclusions, including for all mining enterprises, other types of energy-consuming objects (for example, dump trucks), and other time intervals (hour, day, month, year). In particular, due to the initial purpose of the comparison with the SEC, the dependence of work volumes on energy consumption alone was considered, which does not directly explain seasonal fluctuations and other non-linear factors of production. In the future, it is planned to take into account these gaps and expand the study using the proposed approach.

The following research directions are considered:

• First of all, additional in-depth research can be conducted to find and determine the most accurate and reliable SFA model. As the preliminary results of comparing different variants of the production function and the distribution form function have shown, we can obtain statistically significant differences in the assessment of potential electricity losses;
• Selection, justification, and verification of additional factors affecting the cost of production in the implementation of technological processes of mining industries;
• Formalization of methods for clearing data from abnormal values and ways of sampling records for operational (minutes, hours) monitoring of energy efficiency indicators of work performance;
• Research on other models and algorithms for evaluating energy efficiency based on DEA and SFA methods, including for other energy-consuming objects (dump trucks) and the complex relationship of ECO activities in the implementation of a full cycle of transport and technological work in mining.

5. Conclusions

As a brief conclusion, the following should be noted. The results obtained generally confirmed the initial hypothesis of the study and suggest that there are problems in the field of energy efficiency at the enterprises under consideration. Even though enterprises strive to improve energy consumption indicators, the implementation of individual measures does not have adequate and high-quality solutions. The approaches formulated in this paper for assessing the energy efficiency of individual objects and processes can serve as the necessary methodological and instrumental basis for restructuring production and improving integrated performance indicators. However, in order to use the proposed approach, further research and formalization of its application methods are required, taking into account the main guiding standards in the industry (such as ISO 50001:2018, etc.).