1 Introduction

The Smart Grid implementation requires effort from several sectors, including governments, industry, education, and end users, among others. The traditional electrical grid modernization involves the implementation and development of advanced technology. Because of this modernization, the electrical power stages of generation, transmission, distribution, and consumption have increased their functionality and participation. The integration of communication and information technologies in Smart Grids enables bidirectional communication between stakeholders and the main grid. Similarly, the Smart Grid promotes the high penetration of renewable resources in the main grid through distributed generation installation. Thus, according to [1], the main functional benefits acquired by the modernization of the grid are reliability, security, and efficiency.

In [2], the National Institute of Standards and Technology (NIST) formulate the Smart Grid in seven domains; namely, Customers, Markets, Service Providers, Operations, Bulk Generation, Transmission and Distribution. These domains communicate with each other to achieve the same objectives. Hence, the implementation of Smart Grid requires an integration of several technologies, new standards and a new engineering workforce with multidisciplinary skills that render it capable of solving problems in all the fields related to Smart Grid implementation.

However, to develop and implement these new technologies is difficult when the engineers hired have a traditional formation and are not aware of the specific problems brought about by technology integration. Meanwhile, established firms dominate their sectors with well-known technological models. This demonstrates that the multidisciplinary skills needed by employees in engineering firms have not been adequately taught in the universities [3]. This is translating to lack of specialization in engineering fields [4].

Another urgency that needs attention is that we currently live in a world where products and production are always changing. Demand forces shorter and more rapid innovation cycles, pushing engineers to develop and comprehend different disciplines, to apply fundamental skills in information and communications technology and to interact with people from other careers and cultures [5] in order to deal with the complex challenges posed by the world.

Recognizing these problems, some universities started to develop educational models that connect the engineering classes and laboratories with real-world firms that give students true-life challenges to solve, helping them to develop needed competencies. The strengthening relationships among universities and industries [6] are allowing collaborative efforts to define specific areas of innovation in educational methodologies. Benefited by these partnerships with industry, engineering colleges can implement industry inputs to the design and teaching of entrepreneurship programs [3] for the new generations of engineers.

Nevertheless, before a different educational approach is implemented, it is necessary to evaluate the current conditions of the students to learn if a new educational model would improve their motivation and learning.

In [7], a Smart Grid Education and Workforce Training Center are proposed as solutions to the lack of specialized engineers. In their document, they mention the importance of developing new educational training courses, including the training programs for power generation.

Several universities around the world have proposed new courses and specialization programs related to Smart grid, as well [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. According to [28], the training program should provide multidisciplinary skills for the formation of new engineers and include the fields of knowledge related to Smart Grid technology and its implementation.

The proposal of this work is described in the next sections: Sect. 2 presents a Smart Grid class with a laboratory; it discusses the laboratory, the syllabus, and the current approach from other educational institutions. In Sect. 3, this article discusses the measurement of the students’ perception of class through the use of signal detection theory (SDT) and the implementations of fuzzy logic, type 1 (FLT1) and fuzzy logic, type 2 (FLT2). Finally, a case study of a Smart Grid class in Tecnologico de Monterrey is presented with its evaluation results in Sect. 4. The general discussion and conclusions of this work are presented in Sects. 5 and 6, respectively.

2 Learning perception for a Smart Grid class

2.1 Current class methodologies

Regarding the education of electrical engineering (EE) undergraduate students, there have been educational methods that establish new ways of imparting classes to create a complete education and give their students the ability to solve more complex problems than the ones presented to them using the traditional concepts. This has come about as a response to low retention rates and the students’ lack of knowledge about their disciplines and the usefulness of their courses to the real world. [29, 30].

With the new technologies, policies and resources used in the electrical field, it is necessary that universities modify the way that the generation, transmission, distribution and consumption of energy is taught, as well as the use of new technologies in the industry, such as cyber-security protocols for user-data management [31] and the use of renewable sources [32]. One way this can be done is by creating different course methodologies specified in three different levels: a fundamental level for understanding the general concepts related to energy and power; an applied one to develop the practical skills required in the industry; and an advanced level for the students who look for a specialized degree in the Smart Grid technology [33].

According to [34], electrical engineering laboratories and their courses need to plan and design activities fit for both introductory and advanced topics to guarantee that every student can be accommodated regardless of the semester they are in. Another essential aspect mentioned is that every laboratory must have enough commercially affordable equipment (with low energy consumption ratings) to make sure every student can work with the most updated technologies. Also, the new class teaching methodologies must instruct their students not only about the engineering part of the Smart Grid but also must cover socioeconomic and environmental aspects [35] and the international standardization [36] that regulates the Smart Grid operations throughout the world.

In recent years, there has been more frequent emphasis that students must have the ability to experience teleoperation and real-time simulations even far from the physical laboratory and to practice with the equipment through multiple experiments where they are able to measure, control and modify the different parameters that determine how electrical machines or power systems work [29, 37,38,39]. However, there are some challenges that universities need to overcome to create a proper interactive e-learning methodology for engineering programs [40].

The idea of renovating university laboratories has been a new objective for different institutes around the world. The Princess Sumaya University for Technology (PSUT) in Jordan installed four laboratories to teach power electronics operations and their use in multiple power systems, including electrical generation machines used in the industry, especially ones that use solar or wind energy [41].

Some universities have not changed their laboratories physically but have opted instead to use simulations and virtual practices. The University of Minnesota (UMN) updated their power electronics curriculum to train future engineers in aspects of installing and monitoring a city that uses solar energy, maintenance of a wind generator and even the use of smart meters in current homes [42].

2.2 Smart Grid class

In recent years, the interest in Smart Grid implementation around the world has been growing exponentially due to commitments to reduce electrical energy consumption from fossil resources and switching to electrical energy generated by renewable resources. The Smart Grid implementation requires the integration of information and communications technologies that allow the compilation and processing of data in real time. Additionally, the electrical grid transformation integrates new actors and scenarios which make electrical energy generation safer, more reliable, and sustainable. One example is the transferring from centralized supply to decentralized supply, making electrical energy transmission predictable with a flexible and unidirectional flow.

Because of these transformations, high-quality electrical systems that deliver economic, efficient, and safe supplies of energy are possible. The European Commission has been working since 2011 on a normative called M/490 [43]. Its main objective is the updating of electrical grid technology to achieve interoperability with other systems. In order for this type of progress to occur, the formation, training, and specialization of engineers [28] are necessary both in the universities and in the workforce in all fields related to Smart Grid technology and development. This specialization requires that interested persons develop multidisciplinary skills that will equip them for modeling of applications; interoperability and interconnection among models; development of data models that address protection, integrity and privacy; communication networks and management of information systems; standards and protocols of communication; integration of renewable resources; control and power electronics; transmission and distribution systems; and SCADA protection systems.

Following the new educational model adopted by Tecnologico de Monterrey, Tec 21 [44], the syllabus for the Smart Grid class presented in this work intends to provide an integral formation to the university’s undergraduate engineering students. The objective of this syllabus is to improve the student’s competitiveness in the electrical field by developing their disciplinary and transversal competencies, such as innovative entrepreneurship, social intelligence, civic and ethical commitment, logical thinking, communication, and digital transformation. This program implemented in Tecnologico de Monterrey with undergraduate mechanical engineering students considers the engineering, environmental, economic, and political aspects related to the use of the new Smart Grid technology. The structure of the class was designed based on various courses implemented in multiple universities around the world (see Table 1). The final planning for the Smart Grid class included the following topics:

Table 1 Program analysis from different Smart Grid related classes
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  1. 1.

    Introduction and fundamentals of electric and power systems.

  2. 2.

    Power systems.

  3. 3.

    Control and operation of power systems.

  4. 4.

    Renewable generation systems.

  5. 5.

    Integration of renewable resources to the electrical grid.

  6. 6.

    Quality administration of the electrical grid.

  7. 7.

    Advanced measurement technologies.

  8. 8.

    Information and communication technology.

  9. 9.

    Normative, security, and privacy aspects of data.

  10. 10.

    Practical case study using the Smart Grid laboratory.

The purpose of giving these topics to the students is for them to understand that what makes the electrical grid “smart” is the combination of electrical and communications infrastructures to create a network that integrates the actions of the transmitters of energy and the end users in a two-way flow of communication. Also, it is intended for them to understand that energy is generated, transmitted, monitored, and consumed in a sustainable, economical, and secure manner. [47].

2.2.1 Introduction and fundamentals

The first block of the course concentrates on the explanation of the frequency-domain analysis, specifically, with phasors that represent the amplitude and phase of a sinusoidal signal of the form:

$$ V = V_{m} e^{j\theta } , $$

where Vm represents the amplitude and θ the phase. By knowing this expression, the students are capable of using the vectorized representation to calculate multiple impedances.

They were also taught how a Phasor Measurement Unit (PMU) works, according to the IEEE specifications, to produce synchronized phasor estimates of voltage and/or current at a high-sampling rate to determine the health of the electrical distribution system. By collecting data from multiple PMUs, the students could generate a global view of the behavior of the different interconnected modules. This knowledge was used further in the course for fault detection, increasing power quality, and preventing power outages. The general structure of a PMU is shown in Fig. 1.

Fig. 1
figure 1

Phasor measurement unit (PMU)

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Once the idea of the phasor was comprehended, the students were introduced to the different kinds of power (instantaneous, average, apparent and complex); what RMS values are and, lastly, how to evaluate the behavior of multiple-polyphase circuits, such as the single-phase, three-wire system; and the different connections between Y and ∆.

2.2.2 Power systems

The power systems topic started with the explanation of how the main electrical machinery is comprised; topics included generators, transformers, power lines, loads, transmission lines, protective devices, and the differences between the synchronous and asynchronous machines. Once the general behavior was explained, the course taught the students multiple mechanical concepts, such as angular position, velocity and acceleration, torque, and the relationship of work and power under Newton’s law of rotation.

Other concepts, such as the magnetic field and electromechanical energy conversion, gave the students proper knowledge of the basic functionalities of the most commonly used machinery in the electrical field. From these topics, the students were taught about different control methodologies that are implemented to make sure the best performance of these machines is obtained by evaluating what is the optimal output for the system.

2.2.3 Renewable generation systems

This part of the course focused on explaining the functionalities of different renewable generation systems, specifically, photovoltaic, wind, and hydroelectric ones. For every one of these systems, the general characteristics and operation modes were explained to the students, giving them a practical approach to each system to understand the limits, pros, and cons of all.

In the case of the hydroelectric plant, the classification of the power plants was taught, indicating how big each type should be, how much water it needs to process and in which locations they are built. Also, the general power flow from the reservoir to the generator was explained (see Fig. 2), teaching the students about the four main losses that occur during this process; namely, the penstock losses (friction and turbulence), hydro-losses (turbulence and viscosity), turbine losses and generator losses.

Fig. 2
figure 2

Power flow of a hydroelectric plant

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For the wind energy section, the students were taught about the basic structure of a wind turbine, as well as the basics of aerodynamics, the power generation system functionalities and the different types of turbines.

Once those basic concepts were covered, some of the economic factors that are involved in the creation of wind-energy farms were taught for the students to know the difficulties and planning involved in the construction of a plant capable of producing a certain amount of energy.

The teaching of photovoltaic energy included the overall characteristics and operation modes of all the components of the panel, from the different cell types used to the different arrangements of cells and panels to generate energy. The equivalent circuit was reviewed with the students for them to understand that although it could be seen that photovoltaic energy is sophisticated, it can be analyzed as a simple RC circuit. Finally, the different behaviors of photovoltaic systems when they are grid-connected versus being in stand-alone modes were explained.

2.2.4 Integration of renewable resources

This part of the course focused on how renewable energy systems have been integrated into the main grid around the world. Students were taught about the different policies that have been made to increase the percentage of renewable energy produced locally and worldwide. They were taught the different aspects that are needed to be considered for the integration of these systems; i.e., the technical aspects regarding the installation of the systems, the regulatory and market considerations of energy storage, and how centralized and distributed generation works when these renewable energy systems are implemented.

2.2.5 Quality administration in the electrical grid

In the quality administration section, all the students were taught about the proper protection and monitoring of the transmission and distribution lines of the electrical grid through a SCADA (Supervisory Control And Data Acquisition) system. With this topic, the professor taught the importance of real-time visualizations during the monitoring and, also, the different monitoring capabilities these systems have had through the years to ensure that the power system grid conditions were being adequately observed.

Regarding the use of the SCADA system, students were taught how it is currently being used as a distribution management system (DMS) for fault-location applications, for capacitor-and-voltage-regulators control, for load allocation and unbalanced load flow analyses [47], among other uses.

2.2.6 Advanced measurement technologies

This section of the course is a continuation of the SCADA system operation description, where its uses for fault detection and service restoration purposes were described. During this part of the course, the student would observe that most of the faults that occur on distribution and transmission lines are linked to the insulation of one or more phases. They also analyzed how smart switching and large-scale energy storage could reduce power outages while maintaining the quality of the energy that is being transmitted or stored.

2.2.7 Information and communication technology

Due to the enormous technological implementations that are being integrated into the Smart Grid, new communication and data-manipulation techniques are required to accurately control the generation, transmission, distribution, and consumption of electrical energy. By comprehending this topic, the students were capable of using communication protocols for multiple purposes; for example, the use of real-time transfer of electrical measurements for protecting relays, automating the exchange of information in the feeder, bus bar and substation levels of the Smart Grid, and generating accurate visibility of transmission system power flows across multiple power connectors [47].

2.2.8 Normative, security and privacy of data

One of the last topics taught to the students was the explanation of the most important standards and protocols used to regulate the behavior of the Smart Grid. After these sessions, students were familiarized with multiple protocols, like the Smart Grid communications protocol IEC61850, a protocol that provides communication solutions for substations, control centers, hydro-electric power plants, distributed energy resources and wind farms [47]. Other protocols were the DNP3 and IEC60870-5 (also known as the SCADA protocols), two standards that monitor and control field equipment measurement [47]; the IEEE C37.118 standard that defines the exchange of synchronized phasor measurements [47], among others. Due to the real-world cases of illegal access into electrical power systems, the topic of cybersecurity was also given to the students to teach them about the main security risks that have been documented through the past years. Finally, the use of advanced metering infrastructure (AMI) and demand-side management (DMI) was taught [47] to help protect the information of both industry and the end users.

2.2.9 Smart grid

In the last part of the course, the students were asked to develop a research project that joins all the information reviewed during the semester to simulate a complete Smart Grid, based on renewable and non-renewable energy plants, multiple distribution lines, and variable end-user conditions. While they investigated this, the teacher explained to them the future tendencies foreseen for the Smart Grid, including both technical and regulatory aspects. At the end of the course, every student needed to deliver a report about the functionalities of their simulations and their conclusions about the use of new technologies to improve the generation, distribution, and consumption of electrical energy.

2.3 Smart Grid laboratory

The class was distributed along the sixteen weeks that covered the semester to teach between one and two theoretical topics and one practical session in the laboratory, as shown in Table 2. Considering that for one week, the group had two classes, the students had a total of 32 sessions where they could practice and learn about the different aspects related to the Smart Grid technology. Every two weeks, the students needed to present an investigation and the results related to the practical activity worked during the past sessions. They were evaluated for their research and problem-solving abilities. As seen in Table 2, there is a week with an activity called I Week, which refers to an Innovation Week, where the students attended practical sessions in local industry instead of taking a class at school.

Table 2 Smart Grid laboratory practical sessions distribution
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The laboratory equipment that the students had at their disposal was comprised of different modules that helped simulate the different stages of electrical energy (see Fig. 3). Energy generation was analyzed through the simulation of conventional methods, such as a coal plant. The transmission of energy through the use of renewable resources, such as solar, wind or water, was analyzed with a particular module that simulates the transmission lines (with resistance included to evaluate energy loss and other phenomena that occur in real-life situations). Distribution and consumption were simulated with modules that represented the resistive, capacitive, and inductive loads of end users (whether they were single persons or a small community). All these steps could be regulated and measured through modules that show the voltage, current, and power being consumed, transmitted or generated; changes could be made in real-time according to the control and communication within a SCADA system. The equipment that students had at their disposal during their classes is shown in Table 3.

Fig. 3
figure 3

Smart Grid laboratory equipment

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Table 3 Smart Grid laboratory modules and number of pieces [48]
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3 Perception measurement

Even though universities know that their students need to develop knowledge and abilities that allow them to solve the current world’s problems, there is no general methodology that explains how to assure that their classes reach that goal [49]. Therefore, educators have difficulty creating educational approaches and assessment methods for developing competencies in their students and evaluating their understanding of the global, societal, economic, and environmental (GSEE) contexts [50] of today’s global problems.

Depending on the nature of the student (passive or active), the definition of a good/bad class will be different, which also makes it more challenging to implement a global teaching/learning method to ensure an optimal educational model.

Passive students look for a quantitative increase of knowledge through memorizing, while active students look for learning methods that acquire knowledge through procedures that are used in practical situations as a way to help them to understand their reality [49]. The proper design of a class must consider both perceptions to include all students in the learning process.

One of the most widely-used methods to evaluate if the class teaching methodology is properly educating undergraduate students is the use of surveys, from which the institution learns whether or not the students consider that the material, the quality of the explanations, the duration of the sessions, have been beneficial in their formation [37, 38, 51]. However, the responses given by students are not evaluated according to the perception of the professor, making it difficult to determine if the student answers were given according to their real opinions or if they were answering in a way that the professor and the institution would feel assured that the class methodology was successful. To properly evaluate this, signal detection theory (SDT) can be used to reduce the uncertainty of the student’s answers.

3.1 Signal detection theory

A method used for analyzing the effectiveness of the surveys and design scenarios mentioned before is signal detection theory (SDT), which shows that the detection of a stimulus (signal) or a signal–noise (non-positive stimulus) depends on its intensity and the psychological state of the individual [6].

Despite detecting different stimuli in people, such values are classified in binary categories, which gives only four possible responses, as shown in Table 4, where the participants’ responses (decisions) are compared with the presence or absence of the evaluated signal.

Table 4 Signal detection theory possible responses [52]
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The idea of using SDT is to evaluate how effectively the subject recognizes or responds to the presence of certain stimuli (this will be denoted as a signal S), while also observing how frequently the subject responds to the absence of it (denoted as noise N). A Yes response given to a stimuli is called a Hit (H), while a Yes response to noise is called a False Alarm (FA); a No response to a stimuli is called a Miss (M), while a No response to noise is a Correct Rejection (CR) [52].

The design of SDT calls for presenting to the subject a series of trials, where S is present multiple times but not always in N. The observer needs to determine in which case it is present and when it is not through a simple yes–no response. Both S and N follow a normal Gaussian distribution (as seen in Fig. 4); areas of overlapping by them implies the presence of S inside N [52]. According to [52], SDT works under certain assumptions that help in evaluating both the distance between signal and noise and the user’s strategy. The first assumption is that the responses are given based on the intensity of a hidden variable, which makes the user answer, “Yes,” when it is larger than a predetermined threshold. Second, such hidden value follows a normal distribution for both the signal and the noise. Third, the signal is added to the noise, meaning that both have the same shape.

Fig. 4
figure 4

SDT signal and noise functions

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Considering all of this, the behavior of the SDT model works, as illustrated in Fig. 4, where the Hit Rate (HR) represented the proportion of signal 1 (Yes) when the subject also responded 1, and the False Alarm Rate (FAR) represents the proportion of signal 0 (No) to which the subject responded 1 [6]. Both values are calculated by Eqs. (1) and (2), respectively.

$$ \textit{HR} = P\left( {\hbox{``}1{\hbox{''}}|1} \right) $$
(1)
$$ \textit{FAR} = P\left( {\hbox{``}0{\hbox{''}}|1} \right) $$
(2)

As seen in Fig. 4, the amount of H and FA depends on the position of the criterion value, generating a high amount of HR and FAR when the criterion is low and vice versa [53]. For this reason, the sensitivity of the measures is calculated with the distance dI between both rates, using Eq. (3). The position of the criterion corresponds to a response bias β expressed with Eq. (4), where C = 1[Z(HR) + Z(FAR)], and it corresponds to the criterion between the two distributions [52, 53].

$$ d^{j} = Z\left( \textit{HR} \right) - Z\left( \textit{FAR} \right) $$
(3)
$$ \beta = e^{\ln \beta } = e^{{\ln (d^{t} * C)}} = e^{{Z\left( \textit{FAR} \right)^{2} - Z\left( \textit{HR} \right)^{2} /2}} $$
(4)

The relationship between both H and FA is called a receiver operating characteristic (ROC), which helps measure the accuracy of the distance dI. It can be observed comparing an upper and lower criterion between HR and FAR, although it is easier if both of them are evaluated by implementing a z transformation to both of them [53]. An example of both ROC visualizations is shown in Fig. 5.

Fig. 5
figure 5

ROC a probabilistic and bz coordinates visualization [53]

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3.2 Fuzzy detection theory

By applying fuzzy logic to SDT (FDT), it is possible to set the signal values between 0 and 1, allowing more response options, based on different stages of the judgments of confidence (e.g., yes-sure, yes-not sure, no-not sure, no-sure) [54]. This results that the response given by the subject remains fuzzy, i.e., without forcing it into a crisp value.

Because of the fuzzy nature of the method, every result will have a degree of belonging in H, M, FA, and CR. So, to evaluate the distance dI and the strategy of every subject, it is necessary to evaluate the four possible outcomes. The four steps to properly analyze the data by using FDT are described below [31].

First, the mapping functions for the signal and noise values will have a membership value between the range of [0,1]; then, by using Eqs. (5a), (5b), (5c) and (5d) [54], the membership values for every hit (H), miss (M), false alarm (FA) and correct rejection (CR) are calculated, respectively. After n observations, the ratio for H, M, FA, and CR is calculated by (5e), (5f), (5g), (5h) [54], respectively. Once all the ratios were obtained, it is possible to calculate the sensitivity index dI and likelihood ratio β. In the case of dI, Eq. (3) is still used; however, β will be calculated by Eq. (5i), which uses the ordinate value of HR and FAR, described with Eqs. (5j) and (5k) [6, 54], respectively.

$$ H = { \hbox{min} }\left( {s,r} \right) $$
(5a)
$$ M = { \hbox{max} }\left( {s - r,0} \right) $$
(5b)
$$ \textit{FA }= { \hbox{max} }\left( {r - s,0} \right) $$
(5c)
$$ \textit{CR} = { \hbox{min} }\left( {1 - s,1 - r} \right) $$
(5d)
$$ \textit{HR} = \frac{{\sum \left( {H_{i} } \right)}}{{\sum \left( {s_{i} } \right)}} $$
(5e)
$$ \textit{MR} = \frac{{\sum \left( {M_{i} } \right)}}{{\sum \left( {s_{i} } \right)}} $$
(5f)
$$ \textit{FAR} = \frac{{\sum \left( {\textit{FA}_{i} } \right)}}{{\sum \left( {1 - s_{i} } \right)}} $$
(5g)
$$ \textit{CRR} = \frac{{\sum \left( {\textit{CR}_{i} } \right)}}{{\sum \left( {1 - s_{i} } \right)}} $$
(5h)
$$ \beta = \frac{{Y\left( {HR} \right)}}{{Y\left( \textit{FAR} \right)}} $$
(5i)
$$ Y\left( \textit{HR} \right) = \frac{1}{{\sqrt {2\pi } }}exp\frac{{ - Z(\textit{HR})^{2} }}{2} $$
(5j)
$$ Y\left( \textit{FAR} \right) = \frac{1}{{\sqrt {2\pi } }}exp\frac{{ - Z(\textit{FAR})^{2} }}{2} $$
(5k)

For evaluating SDT through fuzzy signals, the conventional triangular and saturation functions are used to describe the membership of both signals and responses (see Fig. 6). By implementing these membership functions, it is possible to find the relationship between the crisp and fuzzy values of the answers, following the general description of their respective equations. Equation (6) for the triangular function, where α, β, and δ make reference to the left, middle and right points in the function, respectively; and Eq. (7) for the left saturation function, where now α and β are the lower and upper limits of the function [55].

$$ \mu \left( x \right) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {x \le \alpha } \hfill \\ {\frac{x - \alpha }{\beta - \alpha },} \hfill & {\alpha \le x \le \beta } \hfill \\ {\frac{\delta - x}{\delta - \beta },} \hfill & {\beta \le x \le \delta } \hfill \\ {0,} \hfill & {x \ge \delta } \hfill \\ \end{array} } \right. $$
(6)
$$ \mu \left( x \right) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {x < \alpha } \hfill \\ {\frac{x - \alpha }{\beta - \alpha },} \hfill & {\alpha \le x \le \beta } \hfill \\ {1,} \hfill & {x > \beta } \hfill \\ \end{array} } \right. $$
(7)
Fig. 6
figure 6

a Triangular and b left saturation membership functions

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3.3 Fuzzy type 2 signal detection (F2DT)

Unfortunately, there are times when information cannot be appropriately presented with just one membership function because the person is not capable of knowing an exact value for their answers (see Fig. 7). This means that some uncertainty needs to be considered when evaluating the responses of people. Fuzzy logic type 2, developed by Zadeh [56], supports these kinds of situations, showing a better performance than fuzzy logic type 1.

Fig. 7
figure 7

Crisp function, fuzzy function, and uncertainty presence in fuzzy systems

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Fuzzy type 2 sets are formed by three-dimensional membership functions described by the following equation

$$ \begin{aligned} \tilde{A} & = \left( {x,\mu_{{\tilde{A}}} \left( x \right)} \right)|x \in X \\ & = \left\{ {\mathop \int \limits_{x \in X}^{{}} \left[ {\mathop \int \limits_{{u \in J_{x}^{u} \subseteq \left[ {0,1} \right]}}^{{}} \frac{{f_{x} \left( u \right)}}{u}} \right]/x} \right\} \\ \end{aligned} $$
(8)

By using this representation, the fuzzy type 2 intervals can be visualized and analyzed as regular fuzzy type 1 structures [57], where the output will be the center of mass of a Footprint of Uncertainty (FOU) generated between an Upper Membership Function (UMF) and a Lower Membership Function (LMF), as seen in Fig. 8. For this work, all three membership functions used for the FDT will have both UMF and LMF to consider the uncertainty in the answers of the observers properly.

Fig. 8
figure 8

FOU region, UMF, and LMF of a fuzzy type 2 function

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Regarding the defuzzification method used for F2DT, the most commonly used is the Karnik–Mendel Iterative Procedure (KM) [58], an algorithm that finds the endpoint in the fuzzy type 2 interval approximating the value at the midpoint of it [59]. To do this, two centroids are found: one at the left part of the interval and another at the right, to finally obtain the center point between them by calculating the average between them.

Implementing the enhanced KM method published by Wu and Mendel [60], the procedure for obtaining the left and right centroids are obtained following the next steps. It is important to mention that the steps described here are the same for both centroids, but in order to reduce space usage, only the left centroid yl will be described in this paper.

First, the optimal initial switch for the centroid is expressed as

$$ y_{l} = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \underline{{x_{i} }} \left[ {\frac{{\overline{{w_{i} }} + \underline{{w_{i} }} }}{2}} \right]}}{{\mathop \sum \nolimits_{i = 1}^{N} \left[ {\frac{{\overline{{w_{i} }} + \underline{{w_{i} }} }}{2}} \right]}} $$
(9)

where wi is the value for the UMF of the respective xi, while \( \underline{{w_{i} }} \) is the value for the LMF. While iterating in this method, when yI= y, it will mean that the subsequent iterations are not providing an improvement to the last value of yl. The ending criteria will be when the switching condition kI= k occurs. Both yI and y are calculated using Eqs. (10) and (11), respectively.

$$ y' = \frac{{\mathop \sum \nolimits_{i = 1}^{k'} \underline{{x_{i} }} \overline{{w_{i} }} + \mathop \sum \nolimits_{i = k'}^{N} \underline{{w_{i} w_{i} }} }}{{\mathop \sum \nolimits_{i = 1}^{k'} \overline{{w_{i} }} + \mathop \sum \nolimits_{i = k' + 1}^{N} \underline{{w_{i} }} }} $$
(10)
$$ y = \frac{{\mathop \sum \nolimits_{i = 1}^{k} \underline{{x_{i} }} \overline{{w_{i} }} + \mathop \sum \nolimits_{i = k'}^{N} \underline{{w_{i} w_{i} }} }}{{\mathop \sum \nolimits_{i = 1}^{k} \overline{{w_{i} }} + \mathop \sum \nolimits_{i = k' + 1}^{N} \underline{{w_{i} }} }} $$
(11)

Once the initial value for yl is obtained, the algorithm runs as follows [59]:

1. The discourse values for x are sorted in ascending order, where i = 1, 2,…, N, assigning their corresponding weights wi.

2. Establish the switching point k = round(N/2.4) and compute

\( a = \mathop \sum \limits_{i = 1}^{k} y_{i} \bar{\mu }_{{\tilde{B}}} \left( {y_{i} } \right) + \mathop \sum \limits_{i = k + 1}^{N} y_{i} \underline{\mu }_{{\tilde{B}}} \left( {y_{i} } \right) \)

\( b = \mathop \sum \limits_{i = 1}^{k} \bar{\mu }_{{\tilde{B}}} \left( {y_{i} } \right) + \mathop \sum \limits_{i = k + 1}^{N} \underline{\mu }_{{\tilde{B}}} \left( {y_{i} } \right) \)

\( y = \frac{a}{b} \)

3. Find the switching point kI∈ [1, N − 1], such that

\( y_{{k^{{\prime }} }} \le y \le y_{{k^{{\prime }} + 1}} \)

4. Check if kI= k. When that is true, the algorithm stops, and the left centroid is assigned cl= y; if false, continue.

5. Compute the equations

\( s = sign\left( {k^{\prime} - k} \right) \)

\( a^{\prime} = a + s\mathop \sum \limits_{{i = \hbox{min} \left( {k,k^{\prime}} \right) + 1}}^{{{ \hbox{max} }\left( {k,k^{\prime}} \right)}} y\left[ {\bar{\mu }_{{\tilde{B}}} \left( {y_{i} } \right) - \underline{\mu }_{{\tilde{B}}} \left( {y_{i} } \right)} \right] \)

\( b' = b + s\mathop \sum \limits_{{i = \hbox{min} \left( {k,k^{\prime}} \right) + 1}}^{{{ \hbox{max} }\left( {k,k^{\prime}} \right)}} \left[ {\bar{\mu }_{{\tilde{B}}} \left( {y_{i} } \right) - \underline{\mu }_{{\tilde{B}}} \left( {y_{i} } \right)} \right] \)

moreover, compute again

\( y' = \frac{a'}{b'} \)

6. Assign y = yI, a = aI, b = bI, k = kI and return to step 3.

Once both centroids are calculated, the midpoint is obtained through the following equation:

\( y = \frac{{c_{l} + c_{r} }}{2} \) (12)

4 Case of study

4.1 Design of test

Using the same methodology of analysis of FDT and a questionnaire created for measuring the perceptions of the students about their Smart Grid class, it was possible to determine if the teaching model was useful to develop real problem-solving classes and accomplish the class objective. With this in mind, the survey above was done based on The Engineer of 2020 survey [50, 61], and the particular concerns that the university, Tecnologico de Monterrey, had. This helped determine the student abilities and the perceptions the undergraduate students had regarding the Smart Grid theory and the experimentation provided in class.

The survey (see Table 5) was created with 30 questions, divided into four different sections, which measured the students’ perceptions about different parts related to the structure of the class they had. The first section covered questions 1 to 7 and measured the student’s development of abilities and skills. The second one was formed by questions 8–14; it measured their perceptions of the general structure any class would have. The third section evaluated questions 15–23 to determine the students’ perceptions about the current Smart Grid class they were coursing. Lastly, the fourth section, covered by questions 24–30, was the evaluation of the students’ perceptions about the class and its contents after finishing the course.

Table 5 Students perception survey
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Section one was based on some of the questions of [61], so, using the scale of Fig. 9, the possible answers described for each student how much the abilities were developed. However, the rest of the questions in the survey used a different scale (see Fig. 10) to measure how much the student agreed with the sentence. These questions were written considering the signal and noise presence mentioned in [52, 54] to correctly generate the FDT analysis of every student.

Fig. 9
figure 9

Survey section one answer scale

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Fig. 10
figure 10

Sections two, three, and four answer scale

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When using F2DT to evaluate the uncertainty in the observer’s answers, the width of such uncertainties (their corresponding FOU) varied according to the semester that each student was in. When a student was in one of the last semesters, his ideas regarding the advantages of the Smart Grid laboratory would be complete than the ones of a student who is only in their first third of the undergraduate program. Hence, the uncertainty width used in the F2DT was smaller for the students of the last semesters and broader for the ones who had completed less than half of their program. This relationship for the uncertainty width was calculated using the equation

$$ FOU_{width} = 1 - \left( {\frac{x}{9}} \right) $$
(13)

where x is the number semester of the nine semesters of the undergraduate program in which the student was enrolled when taking the class.

Besides the uncertainty width of each one of the student’s answers, the middle point β for the input membership function varied according to the signal of every question, which was determined by the professor of the class. This is because the expected response (hence the higher value for the answer) for every answer was different, depending on the professor’s opinion about what was the optimal answer. So, to change the position of β, the following equation was used:

$$ \beta = 7\left( {\frac{x}{100}} \right) $$
(14)

where x represents the percentage of how accurate were the responses concerning the professors’ opinions.

4.2 Analysis of results

The survey was applied to a group of twenty students in the electrical–mechanical engineering program who took the Smart Grid class in the last third of their final semesters. The first three sections of the survey were answered by the students at their mid-term period of the semester to make sure that they already had some opinion about their class. The fourth section of the survey was given to them after finishing the course to evaluate how useful the class structure had been for them.

All the answers were evaluated through SDT and the two versions of FDT (type 1 and type 2) to compare the results with the binary answers of the students and the results given after evaluating the uncertainties of their responses to the questions. For the first part of the analysis, the SDT and FDT type 1 analysis were made with the answers of every student, as seen in Table 6.

Table 6 Analysis of results for one student
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The general metrics obtained from all the students are shown in Table 7. To determine if the answers of the students to every question were Hits (H) or False Alarms (FA), the survey was answered by the course’s professor, who determined the scale that he perceived was correct for every question.

Table 7 General results using FDT
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Observing the results from the FDT analysis, we can observe that the distances dI between each student’s answer varies considerably from one to another; this is also observed in the bias β that every student had after evaluating their responses.

After the results using FDT were obtained, the students’ answers were evaluated with F2DT to evaluate how much the responses would vary when considering the uncertainty of the answers. The general results using this second method are shown in Table 8, where it can be observed that even though the distances dI of each of the students’ answers were bigger than using FDT; the bias β was considerably smaller than the ones produced by the fuzzy type 1 analysis.

Table 8 General results using F2DT
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In general, it can be observed that dI is higher when a bigger fuzzy analysis is used, and the minimum is reached when using the general SDT analysis. On the other hand, the bias β becomes smaller when using fuzzy logic types 1 and 2, being the minimum when using the latter. For this reason, it can be observed that the use of F2DT generates more accurate values for researching perception through surveys.

By observing these results, it can be observed that the students had a proper idea of how the Smart Grid technology is implemented in real-life situations. However, the methodology used in class indicates that more practical sessions are required for them to properly comprehend the behavior of all of the stages that are involved in the Smart Grid.

5 Discussion

Nowadays, the modernization of traditional electrical grids and their implementation requires technically specialized engineers. The Smart Grid paradigm includes several fields of knowledge, such as automatic controls, electrical machinery, power electronics, renewable energy, public policy, economics, and information and communication technologies. Thus, the formation of specialized engineers must provide multidisciplinary skills to tackle the challenges of Smart Grid implementation. To attain this objective, several courses have been proposed and developed in different universities around the world. However, some of the proposed courses would still be taught under the traditional methods of education. In this research, a course with hands-on experimentation in a Smart-grid Laboratory was proposed, developed, and implemented. This program focuses on a multidisciplinary formation. The course is designed with simulations and case studies as practices, which are carried out in the Smart-grid laboratory. This to approximate real-life scenarios in operations. In this way, the use of a Laboratory promotes the development of multidisciplinary competencies in undergraduate students, as measured by their perceptions of the class and the laboratory use.

6 Conclusions

This study presents the analysis of the students’ perceptions regarding the Smart Grid class and laboratory that was given in Tecnologico de Monterrey, using SDT, Fuzzy Signal Detection Type 1 (FDT) and Type 2 (F2DT) theories. This paper proposes the use of F2DT for generating an algorithm capable of considering the uncertainties in words and noise when answering a survey. The method proposed considers both the fuzzy ranges of fuzzy detection theory and the perceptions of every student regarding different aspects of the class teaching methodology. It compares their perceptions with the perceptions of their professor. This helps to measure the amount of student motivation generated when taking the Smart Grid class, while also measuring the stimulus signals for getting the correct answers.

The methodology of F2DT is useful for measuring the perception in surveys because it considers both the perception and the sensitivity conditions of the observers when answering the questions presented to them. The results of our research suggest that a strategy for changing the class methodology that is being used can be implemented, and it should consider what both the students and the professor consider to be the best way to learn the most relevant topics of a particular subject.

The use of SDT, FDT, and F2DT confirm that students have a clear sensitivity dI, which means that they are capable of understanding what the main objectives of the Smart Grid class and laboratory as well as the implications that the use of this technology has been. Also, it helps the professor in evaluating how the content and the practical activities are being presented to the group, allowing them to modify the class methodology to generate a better one, which improves the teaching/learning methodology even more for future generations of engineering students.

7 Future work

The next steps regarding this work will focus on creating new evaluation methodologies to assess the training of engineering students. Also, new theoretical concepts and case studies will be presented to reinforce the research and problem-solving competencies of the students, preparing them for the challenges of electrical grid modernization. Finally, the introduction of some remote laboratory sessions in the class syllabus will be added, allowing the students to work with the laboratory equipment in a virtual environment and exposing them to work on new projects that improve even more their technological abilities.