Nutritional and Antioxidant Values of the Black Plum (*Vitex doniana*)

*Koba Fatou Traore, Kisselmina Youssouf Kone, Amédée Pascal Ahi, Doudjo Soro, Nogbou Emmanuel Assidjo and Marianne Sindic*

## **Abstract**

This study was conducted to first determine the nutritional potential and the antioxidant activity of black plum fruit pulp and peel. For these characterizations, classic methods were used. The results indicated high fibre and ash contents of black plum from all localities studied, ranging, respectively, from 34.79 ± 0.07–39.83 ± 1.85% and 4.91 ± 0.45–5.91 ± 0.41% for pulp, and 46.38 ± 0.09–50.21 ± 1.07% and 4.16 ± 0.81–4.28 ± 0.20% for peel. The mineral analysis revealed that Black plum pulp and peel are high in potassium (1863.00 ± 1.4–2584.55 ± 3.54 mg/100 g dry weight [DW]) and calcium (355.30 ± 2.52–389.52 ± 3.54 mg/100 g DW). Both the peel and pulp are characterised by a good essential amino acids profile of the protein. The total polyphenol, flavonoid and anthocyanin contents of pulp and peel ranged from 202.51 ± 4.19 to 463.45 ± 6.85 mg gallic acid equivalent (GAE)/100 g of Dry Weight (DW), 75.71 ± 1.03 to 145.55 ± 1.03 mg quercetin equivalent (QE)/100 g DW, and from 1.91 ± 0.08 to 8.28 ± 0.83 mg cyanidin 3-O-β-D-glucoside equivalent (C3GE)/100 g DW respectively. Thus, these fruits constitute a good source of important nutrients for health.

**Keywords:** Nutritional values, antioxidant, black plum, valorization

## **1. Introduction**

Fruits and vegetables have always been considered as essential sources of micronutrients and dietary fibre for the body to function properly. Moreover, regular fruit consumption is recommended for disease prevention and health benefits due to the nutrient composition of fruits, which includes vitamins, minerals, fibre and bioactive compounds [1]. In Côte d'Ivoire, these nutrients are generally delivered by some cultivated tropical fruit, including mangoes, papaya, citrus fruit, as well as imported temperate climate fruit, such as apple, pear and grape [2]. These tropical fruits are often not readily accessible to the populations beyond production time, especially in the developing countries due to their high cost or scarcity in the local market [3, 4]. However, these shortages correspond to the production stage of some wild fruit-bearing species, underutilised because of the limited knowledge

regarding their nutritional values. Assessing the quantitative and qualitative value of this fruit (including peel, pulp and seed) could reveal novel food sources with potential health-protective properties.

Among these wild plant resources, black plum (*Vitex doniana*) is an abundant and widespread species in tropical countries, such as Côte d'Ivoire [5–8]. In Côte d'Ivoire, it is found, for example, in Pakobo, Ahiérémou and Zougoussi, located in the centre of the country [9], Seguela in the west–north, Korhogo and Tingrela in the north [10], and Bondoukou in the east–north region [11]. Various parts of the plant, including the leaves and bark, have been used for many decades by some rural populations to treat diabetes, ulcer, diarrhoea, dysentery, asthma, insomnia and various other illnesses. The fruit of this species is an ellipsoid drupe of about 2.5–3.0 cm in height and 2.0–2.5 cm in diameter, and rich in sugar [12, 13]. The unripe fruit is green but turn brown–black when ripe, and are eaten raw [9, 14]. Nutritional analyses indicate that black plum is a good source of vitamins, such as calcium, iron, magnesium, phosphorus and vitamin C [15–18]. In addition, the pulp provides useful quantities of flavonoids with strong antioxidant properties [19].

Interestingly, no comparative study between the nutritional composition of black plum pulp and peel has yet been carried out. This study sought to show the nutritional potential and antioxidant properties of black plum peel and pulp, by analysing the physicochemical and nutritional properties in order to contribute to the valorization of this fruit, produced in large quantities, even in some remote areas of Côte d'Ivoire, particularly in the regions of Ferke, Tiebissou and Yamoussoukro.

## **2. Materials and methods he entirety**

## **2.1 Sample preparation**

The ripe black plum were harvested in three Côte d'Ivoire localities: Ferkéssédougou (Ferke), Tiébissou and Yamoussoukro. The fresh fruit was washed with tap and distilled water, sequentially. After drying the fruit with a paper towel, the pulp and peel were separated by hand using a stainless-steel knife, dried in an oven (Memmert U30-Gemini BV, GmbH, Germany) at 50°C for 24 h, then milled using a miller (ZBK220077–88 LW74d(B) A, China. Milled pulp and peel were packaged in polyethylene bag using a vacuum packager (NG 10121 MULTIVAC, Belgium) and frozen at −19°C until analysis. All tests were realised in triplicate.

## **2.2 Chemical and biochemical analyses**

## *2.2.1 Ash, protein, fat, carbohydrate and dietary fibre contents*

Ash, crude protein, fat and total carbohydrate were determined by standard methods AOAC [20]. Dietary fibre was analysed by AOAC 991.43. [21], and total carbohydrates by the protocol reported by Bertrand and Thomas [22]. The energy value (kcal/100 g) was calculated by multiplying the protein, fat and carbohydrate contents (g) by factors of 4, 9 and 4 kcal/g, respectively.

## *2.2.2 Minerals analysis*

Dried pulp and peel (0.5 g in each case) of Black plum were transferred to digestion flasks containing 5 mL of HCl/HNO3 (1:3). After 2 h of slowly boiling, the mixture was cooled at room temperature, then 50 mL of distilled water was added

## *Nutritional and Antioxidant Values of the Black Plum (*Vitex doniana*) DOI: http://dx.doi.org/10.5772/intechopen.99129*

and followed by filtration. The cooled solution was placed in flasks and transferred to the atomic absorption spectrophotometer (Analyst 200, Perkin Elmer, Waltham, MA, USA) for mineral determination. The phosphorous was measured spectrophotometrically. Standard calibration curves constructed for each element were used for direct quantification.

## *2.2.3 Amino acid analysis*

The black plum pulp and peel amino acids composition was determined based on a previous method [23]. Duplicate samples were hydrolysed by transferring around 50 mg of sample weighed accurately in a 15 mL flask in which 5 mL of HCl (6.0 N) was added. The flask was closed under vacuum, nitrogen-purged and digested at 110°C for 24 h. Sulphur-containing amino acids were determined by using performic acid. The amino acids were analysed using a Biochrom 20+ amino acid analyser, Cambridge, United. Kingdom).

## *2.2.4 Phenols extraction*

Phenols were extracted by the method of Soro et al. [19]. Dried V. doniana pulp and peel samples (1 g) were homogenised in 10 mL solution of 80% methanol and 2% formic acid, using an Ultra TurraxT25 basic homogeniser (Heldoph Instruments D-91126, Schabach, Germany) at room temperature. The homogenate was sonicated for 30 min in a Bandelin electronic RK 541 H sonicator (Heinrichstrasse 3–4 D-12207, Germany) and then centrifuged at 9400 × g for 25 min in a DBS centrifuger (PCB 1500, Italy). The supernatant was collected and the precipitate extracted again with 10 mL of 80% methanol, under the conditions previously described. The two supernatants were mixed and filtered using Whatman filter paper No.1. The final methanolic extract was stored at 25°C to be used in determination of total phenols, flavonoids and anthocyanin contents.

## *<IP_ADDRESS> Determination of total phenols*

Total phenolic compounds (TPC) determination was performed as described by Gao et al. [24] Phenolic extract (100 μL) was mixed with 0.2 mL Folin–Ciocalteu reagent (Sigma), 2 mL of H2O and 1 mL of 15% Na2CO3 and the absorbance measured at 765 nm in a spectrophotometer (Thermo Scientific™ 75003631, ThermoFisher Scientific SAS, Strasbourg, France) after 2 h incubation at room temperature. Gallic acid was used for the calibration curve with a concentration range from 0 to 200 mg/L. Total phenols were expressed as mg gallic acid equivalent (GAE)/100 g DW.

## *<IP_ADDRESS> Determination of total flavonoids*

Total flavonoid (TF) contents were determined according method used by Meda et al. [25], but slightly modified. A volume of 0.5 mL of sample methanolic extract was diluted in 0.5 mL of distilled water. Then, 0.5 mL of aluminium chloride 10% (w/v) and the same volume of 1 M sodium acetate was added. Finally, 2 mL of distilled water was added and absorption reading at 415 nm was taken after 30 min against a blank sample consisting of a 4 mL methanolic extract without aluminium chloride. Quercetin was used for the calibration curve with a concentration range from 0 to 3.125 mg/mL. Results were expressed as mg quercetin equivalent (QE)/100 g DW.

## **2.3 Statistical analysis**

All data were analysed using ANOVA based on Tukey's HSD multiple comparison test at p < 0.05. Software RStudio version 1.2.1335 2009–2019 was used, and triple analyses were performed. The results were presented as mean ± standard deviation.

## **3. Results and discussion**

The nutritional composition of black plum pulp and peel harvested in three localities of Côte d'Ivoire (Ferke, Tiébissou and Yamoussoukro) are given in **Figures 1**–**3**. Values are compared on a dry weight (DW) basis, except for moisture content.

## **Figure 1.**

*Comparison between the nutritional composition of pulp and peel of Vitex doniana fruit harvested in Ferke, Tiebissou and Yamoussoukro.*

## **Figure 2.**

*Comparison between pulp and peel amino acid contents of black plum harvested in Ferke, Tiebissou and Yamoussoukro.*

*Nutritional and Antioxidant Values of the Black Plum (*Vitex doniana*) DOI: http://dx.doi.org/10.5772/intechopen.99129*

## **Figure 3.**

*Comparison between pulp and peel micromineral contents of Vitex doniana fruit harvested in Ferke, Tiebissou and Yamoussoukro.*

## **3.1 Ash content**

There was no significant difference between pulp and peel of fruit among the localities, but the ash contents of Ferke and Tiebissou fruit pulps were significantly higher (p < 0.05) than those of Yamoussoukro (**Figure 1**).

These results contrast sharply with those of previous authors, notably, Soro et al. [19], who obtained average ash contents of 1.4% and 3.53%, respectively, for the fruit pulp of V. doniana harvested in V-Baoulé and Nord of Côte d'Ivoire. Comparatively higher ash contents of 11.50% were reported by Vunchi et al. [18], whereas Agbede and Ibitoye [16] stated levels (5.1%) similar to the current work. In addition, compared with the pulp from fruit of the same genus, the ash content found here is higher than that of baobab (4.5 ± 0.2%; Magdi, 2004) and avocado (2.1 ± 0.6%; [26]).

This ash content variation within the same fruit and between the same species may be due to the water content and soil properties [27]. The ash content of food reflects its total mineral content, and so Black plum would be a valuable source of minerals.

## **3.2 Protein content**

Pulp and peel protein contents of Black plum ranged from 2.11 ± 0.15 to 2.28 ± 0.16% (**Figure 1**). A markedly higher protein content (2.61 ± 0.07%) occurred in the peel of fruit from Yamoussoukro than Ferke and Tiebissou, respectively, of 2.35 ± 0.10 and 2.40 ± 0.13%, which had more protein in their pulps than peels (**Figure 1**). In comparison to these data, higher protein values (3.04%) have been recorded for Black plum in Uganda by Acipa et al. [15], and in the work by Jacob et al. [28] for *Strychnos spinosa* (8.72 ± 0.02%), *Diospyros mespiliformis* (6.99 ± 0.02%) and *Dialium guineense* (5.23 ± 0.01%). The fruit pulps studied have a low protein content, as per most fruits [26].  From these results, Black plum is not a useful source of lipids.

## **3.4 Total carbohydrates content**

Both the pulp and peel studied in this work, are low in carbohydrate. Yamoussoukro pulp samples were the highest in carbohydrate compared with the other samples (**Figure 1**), while fruit from Ferke showed intermediate carbohydrate contents (7.94 ± 0.36%). Nonetheless, both values were lower than those found in the pulp of Black plum from Burkina Faso (19.68–20.25%; [33]) and Nigeria (28.40 ± 1.06%; [16]), as well as the pulp of some other fruits, including mango (11.9 ± 0.41% DW) but close to the fruit of *Adansonia digitata* (8.25 ± 0.03– 9.25 ± 0.21%; [33]). The age of the tree, and environmental factors, such as climate and soil, could be the basis of the observed differences because Ferke is located in a dry savanna zone, with a very hot and dry climate (Sudanese climate) and ferralitic soils [34, 35], while Yamoussoukro and Tiebissou, located in mesophilous and savanna wooded areas, respectively, have a Baoulean climate, both with moderately desaturated ferralitic soils [36].

Among the localities, Ferke (6.49 ± 0.00%) corresponded to fruit peel with the highest total carbohydrate content, while an average of 4.36% carbohydrate occurred in the peel of fruit harvested in Tiebissou and Yamoussoukro. Comparatively richer sources of carbohydrate are the peels of mango Raspuri (28.20 ± 0.60% DW), mango Badami (20.80 ± 0.20% DW; [37]) and tomato (32.16 ± 1.11% DW; [38]). However, fruit pulp contained the most carbohydrates whatever the locality (**Figure 1**). The high carbohydrate content of Ferke fruit peel would be linked to the climate, as this region is located in the northern part of Côte d'Ivoire where a generally high temperature occurs. Given the Black plum are low in carbohydrate, it can be concluded that regular consumption of this fruit would be beneficial for people with obesity, who are most often subject to a diet restricted in carbohydrates.

## **3.5 Dietary fibre**

This study showed that dietary fibre is a main component in the pulp (34.79 ± 0.07–39.83 ± 1.85%) and peel (46.38 ± 0.09–50.21 ± 1.07%) of Black plum harvested in the three localities of Côte d'Ivoire. The dietary fibre levels in the pulp of Black plum are high when considering the amounts present in the pulp of white guava (3.50 ± 0.01%; [39]) and Deglet-Nour and Allig date varieties (14.4 ± 1.12% and 18, 0.45%; [40]. Moreover, the values obtained in this work approximate those recorded by Grigelmo-Miguel and Martín-Belloso [41] for the three orange varieties, Navel (35.4 ± 1.4%), Salustiana (35.9 ± 0.5%) and Valencia Late (36.9 ± 0.3%), which are well-recognised as sources of dietary fibre.

In addition, black plum peel displayed a much higher level of dietary fibre than some other fruit peels too, especially avocado (43.9 ± 2.7%), pineapple (16.3 ± 2.5%) and papaya (16.6 ± 2.2%) [42]. Interestingly, the peel fibre contents were significantly higher for the fruit harvested in Yamoussoukro (46.38 ± 0.09–50.21 ± 1.07%)

*Nutritional and Antioxidant Values of the Black Plum (*Vitex doniana*) DOI: http://dx.doi.org/10.5772/intechopen.99129*

than Ferke and Tiebissou (**Figure 1**). This trend could be explained as a fruit defence against heat stress. Ferke and Tiebissou fruit pulp results agree well with results found by Lamghari et al. [43], Gorinstein et al. [44] and Morais et al. [26]. Similar to the present study, these authors also obtained significantly higher (p < 0.05) dietary fibre values in the studied peels than pulps. It is well known that diets high in dietary fibre are associated with the prevention and treatment of various diseases, such as diverticular and coronary heart disease, colon cancer and diabetes, besides contributing to weight loss in individuals with obesity [45, 46].

Based on the results observed, black plum is a potentially good dietary fibre source and could contribute towards the prevention and treatment of several degenerative diseases. Therefore, these fruits could be included in some food formulations for possible fibre enrichment.

## **3.6 Amino acids**

Similarly to most fruits, such as apple [47], medlar [48], strawberry [49] and tomato [50], the main amino acids present in Black plum pulp were glutamic acid and aspartic acid, which had averages of 0.27 mg/100 g DW (**Table 1**). Conversely, the peels were particularly high in threonine and proline, with mean contents of 0.29 ± 0.01 and 0.27 ± 0.01 mg/100 g DW, respectively. Furthermore, the percentages of essential amino acids in the protein of black plum pulp and peel compare well with standard protein, according to the World Health Organisation (WHO; **Table 2**). In pulp protein, only two amino acids or amino acid pairs (methionine/ cysteine and lysine) had scores below 100%, of a few percentage points only or around 50% (lysine). However, only isoleucine (47.82–80.42%), lysine (41.60– 51.62%) and valine (30.80–41.43%) percentages characterised the peel protein. In addition to Yamoussoukro fruit, Ferke and Tiebissou fruit pulps contained higher threonine, histidine and methionine amounts than their peels (**Figure 2**). The peels were rather rich in valine and leucine. Despite the low protein content, black plum protein has a good amino acids profile.

## **3.7 Mineral contents**

Regarding the macrominerals, all samples contained high amounts of potassium (K), followed by calcium (Ca), magnesium (Mg), phosphorus (P) and sodium (Na). In addition, the main micro-minerals were iron (Fe), followed by manganese (Mn) and Zinc (Zn). The concentration ranges of K, Ca, Mg, P and Na in Black plum, were respectively, 1863.00 ± 1.41–2283 ± 1.41, 355.30 ± 2.52– 389.52 ± 3.54, 221.00 ± 1.41–255.5 ± 2.12, 118.00 ± 0.71–179.00 ± 2.83 and 38, 30 ± 7.77–53.33 ± 1.53 mg/100 g DW for the pulp, and 2155.50 ± 4.95–2584.55 ± 3.54, 301.50 ± 2.12–325.50 ± 0.00, 199.67 ± 1.15–250.00 ± 2.83, 105.50 ± 2.12–142.00 ± 1.41 and 39.70 ± 5.69–62.31 ± 2.08 mg/100 g DW for the peel (**Table 3**).

The pulp and peel of fruit collected in Tiébissou and Yamoussoukro exhibited the highest levels of K, while the lowest K levels (2155.50 ± 4.95 and 1863.00 ± 1.41 mg/100 g DW, respectively) occurred in those from Ferke. The K content of the fruit pulp of V. doniana in this work is higher than that obtained by Ladeji and Okoye [51] of 127.2 mg/100 g DW, and Vunchi et al. [18] of 15.70 ± 0.26 mg/100 g DW. In addition, the observed Black plum pulp and peel K contents are higher than those in banana (382 ± 15 and 337 ± 7 mg/100 g DW; [52]) and mango (185 ± 11 and 444 ± 13 mg/100 g DW; [53]. The pulps Ca concentrations were significantly similar (p > 0.05) among the localities, whereas Ca concentration was highest in Yamoussoukro fruit peel (389.52 ± 3.54 mg/100 g DW), and lowest in Tiebissou fruit peel (355.30 ± 2. 52 mg/100 g DW). In addition, except for Tiebissou

## **Table 1.**

*Amino acids contents (mg/100 g DW) of pulp and peel of black plum harvested in Ferke, Tiebissou and Yamoussoukro.*

## *Nutritional and Antioxidant Values of the Black Plum (*Vitex doniana*) DOI: http://dx.doi.org/10.5772/intechopen.99129*


**4. Conclusions**

This research paper has opened a new Li-ion battery modeling research direction in the HEV BMS applications field by performing several investigations on ARX and ANFIS alternative accurate battery models with a high impact on improving the battery SOC estimators' accuracy and their robustness, design, and real-time implementation in MATLAB and Simulink environments.

The effectiveness of the modeling and SOC estimation strategies is demonstrated through an extensive number of simulations in a MATLAB R2021b software environment. The preliminary simulation results are encouraging, and extensive investigations will be done in future work to extend the applications area. The performance analysis from the last section reveals that ANFIS battery models overpass the second-order linear ARX polynomial battery model in terms of SOC and terminal voltage accuracy and by their capability and suitability to simplify the battery model structure and build robust and accurate SOC Li-ion battery estimators with a high terminal voltage prediction accuracy. The AEKF SOC estimator accuracy based on combined ANFIS model structure is also very accurate compared with AEKF SOC estimator based on ARX dynamic part model with the SOC absolute value lower than 1%, better than the usual 2% SOC value reported in the literature field. Both alternative models are based only on the measurement input-output data set collected by a data acquisition (DAQ) system incorporated in the BMS of HEVs. Besides, the battery SOC and output voltage signals' accuracy is not affected by noise as long as the AEKF SOC estimator is very robust.

*Investigations of Using an Intelligent ANFIS Modeling Approach for a Li-Ion Battery… DOI: http://dx.doi.org/10.5772/intechopen.105529*


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characterization with XPS of the surface of polymer microparticles loaded with

**9**

*India*

Sanjeev R. Inamdar

*Department of Physics,*

*Laser Spectroscopy Programme,*

*Karnatak University, Dharwad*

**Rotational Dynamics of Nonpolar and Dipolar**

**Molecules in Polar and Binary Solvent Mixtures**

The absorption of photons by a molecule leads to its excitation. An electronically excited molecule can lose its energy by emission of ultraviolet, visible, infrared radiation or by collision with the surrounding matter. Luminescence is thus the emission of photons from excited electronic energy levels of molecules. The energy difference between the initial and the final electronic states is emitted as fluorescence or phosphorescence (Lakowicz, 2006). Fluorescence is a spin-allowed radiative transition between two states of the same multiplicity (e.g., S1 → S0) whereas; phosphorescence is a spin-forbidden radiative transition

The mechanisms by which electronically excited molecules relax to ground state are given by the Jablonski diagram as shown in Fig. 1. The absorption of a photon takes a molecule from ground state (singlet state, S0) to either first excited state (singlet state, S1) or second

Fig. 1. Jablonski diagram of transitions among various electronic energy levels

→S0).

**1. Introduction**

between two states of different multiplicity (e.g., T1


*Edited by Ling Zhang*

Advances in forest management will enhance the sustainable development of human society, and should be focused on. Under the context of global change, soil nutrients, especially nitrogen, should be carefully managed and monitored in plantations experiencing intensive nitrogen input, and forests with exotic plant invasion disturbance, considering its substantial contribution to global nitrous oxide. One negative effect of global change could be loss of biodiversity, which could be maintained by forest management. In addition, advanced technologies should also be developed to prevent fire in forests considering its increased frequency. Importantly, policies and technologies should also be developed for advanced forest management, such as deep learning in plant disease prevention, and quantitative strategic planning matrix in management of forest conservation.

ISBN 978-1-83968-306-0

Advances in Forest Management under Global Change

Published in London, UK © 2020 IntechOpen © Artiom Vallat / Unsplash


**4. Total Ignorance in Measurements**

Let us here consider a simple example to show how, in measurement procedures, the situation called total ignorance by Shafer is very often present.

A calibrator provides a reference voltage of 24 V, and some multimeters of the same typology (4 1 2 Leader 856) are employed to measure this voltage. The instrument data sheet provides the measuring accuracy as ± % of reading ± number of digits, and the value of each digit is given by the resolution in the considered range. According to the data sheet, Table 7 provides the resolution and the measuring accuracy in the different ranges. For the measurand *Vx* = 24 V, which is the reference voltage in the proposed example, the range is 30 V and therefore, according to the specifications, the measurement accuracy is ±(0.05% *Vx* + 2 mV) = ±0.014 V.

**Table 7.** The multimeter data sheet.

Two different measurement procedures are considered:

1. All multimeters are employed to measure the reference voltage.

2. Only one multimeter is employed to measure the reference voltage.

Figure 2 shows, with the orange line the reference voltage and with the pink crosses the value measured by 10 different multimeters. Since different instruments are employed, it is likely to happen that the measured values fall around the reference value. In this situation, it could be possible to apply a probabilistic approach by considering the following: the mean of the measured values; an uncertainty interval around the evaluated mean, and a pdf over this interval (but only if a high number of different instruments are employed).

**Figure 2.** Up: values measured by different multimeters. Down: values measured by a single multimeter.

This situation represents very well the calibration procedure that is performed by the instruments' manufacturer to provide the accuracy interval, which reflects the behavior of all instruments of the same typology. However, this situation is very seldom met in practice, because generally only one instrument is available and employed. Under this more common situation, when only one multimeter is employed, the value measured by the multimeter will be shifted with respect to the reference value. Moreover, if different measurements were taken, this would not help to better estimate the reference value since all measured values would be shifted more or less the same amount with respect to the reference value, as shown by the green circles in Figure 2. In fact, all measured values are taken, in this case, by the same instrument and, therefore, are affected by the same systematic error, even if a small variation can be observed, due to the presence of also random phenomena.

In this last case, even if the mean of the measured values is taken, no better estimate of the measurand can be obtained. Additionally, even if an interval is built, according to the dispersion of the measured values, this interval would not contain the value of the measurand. Therefore, to provide a good uncertainty interval, it is necessary to refer to the accuracy interval provided by the data sheet. The data sheet does not provide any pdf associated with this interval, and therefore, no pdf can be assigned to the obtained interval.

When we have a pdf over a given support, it is possible to assign a confidence interval (or degree of belief) to any subintervals of the support. However, when no pdf is assigned and no knowledge is available to assign a specific pdf, it is not possible to associate any confidence interval (or degree of belief) to any subintervals of the support. We are, therefore, perfectly in the case of Shafer's total ignorance, where a degree of belief can be assigned to the support (or universal set), but no degree of belief can be assigned to the subintervals (to the subsets of the universal set).

It clearly follows that total ignorance is present in the measurement field. Since belief functions better represent total ignorance, it is worth exploring these functions and the theory of evidence to find an alternative, more general way to handle measurement uncertainty and measurement results. It is not the aim of this paper to provide all definitions and mathematical details, for which the readers are referred to the published literature [10–14]. The next section will, therefore, give only some introduction to come to the possibility distributions (PD) and the random-fuzzy variables (RFV).

#### **5. The Random-Fuzzy Variables**

In the previous sections, belief functions are introduced and it is shown how they can suitably represent the available knowledge, including total ignorance. It is interesting to observe that belief functions are a generalization of the probability functions and the necessity functions. In this respect, it is first necessary to know what a focal element is.

Let us first define the basic probability assignment function:

$$\begin{aligned} m: P(X) &\to [0, 1] \\ m(\mathcal{Q}) &= 0 \\ \sum\_{A \in P(X)} m(A) &= 1 \end{aligned} \tag{3}$$

where *X* is the universal set, *P*(*X*) is the power set of *X* and ∅ is the empty set. According to (3), *m*(*A*) represents the degree of belief that an element *x* belongs to set *A* (only to set *A* and not to its subsets).

Set *A* for which *m*(*A*) > 0 is called the focal elements of *X*. When the focal elements are singletons, then it can be proved [9–12] that belief functions are probability functions, and the theory of evidence enters in the particular case of the probability theory. This shows that the belief functions are, as wanted by Shafer, a generalization of the probability functions. However, it is also interesting to consider another particular case of belief functions, which are called necessity functions and are obtained when the focal elements are all nested, as shown in Figure 3.

**Figure 3.** Example of nested focal elements, for sets and for intervals.

The upper plot in Figure 3 clearly shows that, when sets are considered, all sets can be ordered in such a way that *A*1 ⊂ *A*2 ⊂ ... ⊂ *An* ≡ *X*. When, instead of sets, intervals are considered, the lower plot can be drawn, which still satisfies *A*1 ⊂ *A*2 ⊂ ... ⊂ *An* ≡ *X*. This case is very interesting from the metrological point of view because there could be an analogy between these nested intervals and the confidence intervals of a given pdf at different, increasing levels of confidence.

The necessity function is defined as follows:

$$\operatorname{Vec}\left(A\_{\bar{j}}\right) = \sum\_{k=1}^{\bar{j}} m(A\_k)$$

and represents the degree of belief that an element *x* belongs to set *A* and to all its subsets. When the belief functions are necessity functions, then the theory of evidence enters the particular case of the possibility theory.

In the same way that probability density functions are defined in probability, possibility distribution functions (PD) are defined in possibility as follows:

$$r: X \to [0, 1]$$

where:

$$\max(r(x)) = 1$$

when *x* ∈ *X*.

It can be proved [10,11] that the nested intervals of Figure 3, together with their corresponding necessity functions *Nec*- *Aj* , represent confidence intervals at specific levels of confidence, coverage probability, or degree of belief *Nec*- *Aj* . Therefore, remembering the GUM words that "*the ideal method for evaluating and expressing measurement uncertainty should be capable of readily providing such an interval, in particular, one with a coverage probability or level of confidence that corresponds in a realistic way to that required*" [1], it can be stated that the possibility theory, which provides all confidence intervals at all confidence levels, is perfectly GUM compliant.

If the intervals of Figure 3 are not overlapped with each other but are positioned at different vertical levels α, such as *αj* = 1 − *Nec*- *Aj* , then a fuzzy variable is obtained, as in the example in Figure 4.

**Figure 4.** Example of possibility distributions and confidence intervals.

The fuzzy variable is commonly defined by its membership function which is, from the strict mathematical point of view, a PD.

Since a fuzzy variable (a PD) represents confidence intervals at all levels of confidence, a fuzzy variable can be used to represent in a very immediate way the result of a measurement [10–12]. Moreover, since different kinds of uncertainty contributions may affect the measurement procedure, the best way to represent the result of a measurement is the use of a fuzzy variable of type 2 and, in particular, a random-fuzzy variable (RFV). An RFV provides two PDs and can, hence, represent separately the effects on the measurement result of the different contributions to uncertainty. An example of RFV is given in Figure 5, with the red and violet lines. In an RFV, the uncompensated systematic contributions are represented by the internal PD *rint*(*x*) (violet line), while the random contributions are represented by the random PD *rran*(*x*) (green line). The external PD *rext*(*x*) (red line) is obtained by the combination of the two PDs *rint*(*x*) and *rran*(*x*) [10–13].

**Figure 5.** Example of RFV.

Extending the considerations made for the fuzzy variables, it can be stated that the cuts *Xα* at levels α of the RFV are the confidence intervals associated to the measurement result at the confidence levels *Nec*(*Xα*) = 1 − *α* (as shown in Figure 6). In particular, the internal interval of each confidence interval is due to the effect on the measured value of the systematic contributions to uncertainty, while the external intervals are due to the effect of the random contributions.

**Figure 6.** Example of RFV and some of the confidence intervals, which show the systematic effects and the random effects on the final measured result.

If RFVs can suitably represent measurement results, then it is important to understand how an RFV can be built and how two RFVs can be combined with each other, as will be briefly explained below; we refer the readers to the literature for more details [10–14].

## *5.1. RFV Construction*

To build an RFV, it is necessary to define the shape of the PDs *rint*(*x*) and *rran*(*x*), whose construction is different [10–14] since they represent different kinds of contributions.

As far as *rint*(*x*) is concerned, this PD represents the uncompensated systematic contributions to uncertainty. As shown in the example of the multimeter in the previous Section 4, generally, the only available knowledge is, in this case, the accuracy interval given by the manufacturer of the employed instrument in the data sheet. Therefore, the available knowledge can be represented by Shafer's total ignorance. As is also shown in Section 3, total ignorance is mathematically represented by the belief function [9–11]:

$$Bel(X) = 1$$

$$Bel(A) = 0 \quad \forall A \subset X$$

and by the rectangular PD, such as the one in violet line in Figure 5. It follows that *rint*(*x*) is rectangular in most situations, even if situations may exist that could lead to different shapes [10–14].

On the other hand, *rran*(*x*) must represent the random contributions to uncertainty and therefore, in most cases, a pdf is known or can be supposed. In this case, the corresponding PD can be easily obtained by applying the suitable probability–possibility transformation (different probability–possibility transformations are available in the literature to transform pdfs into PDs. The suitable transformation when PDs are used to represent measurement results is the maximally specific probability–possibility transformation, which preserves all confidence intervals and corresponding confidence levels) [10,15].

As an example, when the pdf is uniform, then the corresponding PD is triangular; when the pdf is triangular, then the corresponding PD is the orange one in Figure 7; when the pdf is Gaussian, then the corresponding PD is the blue one in Figure 7.

**Figure 7.** Example of PDs coming from given pdfs. Blue line: PD from a Gaussian pdf. Orange line: PD from a triangular pdf.

#### *5.2. RFV Combination*

When the measurement results are represented by RFVs and they must be combined, it is possible to take into account all the available metrological information about the nature of the contributions to be combined and the way these contributions combine in the specific measurement procedure. According to that, since PDs can be combined using many different mathematical operators, the most proper one can be chosen.

Without entering the details, for which the readers are referred to [10,16,17], it can be stated that the random contributions to uncertainty always compensate with each other during the combination, and therefore, an operator that simulates this typical probabilistic compensation should be chosen. On the other hand, the systematic contributions to uncertainty could compensate or not with each other during the combination, according to the specific contributions and the specific measurement procedure. Therefore, there should be the possibility to choose between a mathematical operator that simulates compensation and another one which does not compensate.

Let us first consider the evaluation of the joint PD, starting from two PDs. As an example, Figure 8 shows the results obtained by combining the same two PDs with the use of two different t-norms (for the definition of the mathematical t-norms, the readers are addressed to [15]): the min t-norm (on the left) and the Frank t-norm (on the right). In the upper plots, the two-dimensional joint PDs are shown, while in the lower plots, the corresponding α-cuts are shown. It can be easily seen how compensation applies when the Frank t-norm is employed, while no compensation applies when the min t-norm is employed.

**Figure 8.** Combination of uncorrelated contributions. The same initial PDs are considered on both the left and right figures. On the left, the min t-norm is applied; on the right the Frank t-norm is applied. In the upper plots, the joint possibility distributions are shown. In the lower plots, the corresponding α-cuts are shown.

Figure 8 refers to the combination of uncorrelated contribution. Without entering the details, the correlation can also be considered, as shown, as an example, in Figures 9 and 10.

**Figure 9.** Combination of correlated contributions when the min t-norm is applied. The same initial PDs as in Figure 8 are considered. **Right**: joint PD. **Left**: shape of its α-cuts.

**Figure 10.** Combination of correlated contributions when the Frank t-norm is applied. The same initial PDs as in Figure 8 are considered. **Left**: joint PD. **Right**: its α-cuts.

From Figures 9 and 10, it can be easily seen how correlation modifies the joint PDs and the corresponding α-cuts.

Once the joint PDs *rran*(*x*, *y*) and *rint*(*x*, *y*) are obtained, it is possible to evaluate the joint PD *rext*(*x*, *y*) and the final RFV (this is obtained by applying the famous Zadeh extension principle. The readers are referred to [9,10] for the details) [9,10,16,17].


**4. Resilience metrics**

*Infrastructure Management and Construction*

*Electrical distribution grid SCADA system and tlc dependencies.*

station—BTS hereafter—of the telecommunication network). In turn, BTS are supplied by the energy provided by the SS of the EDN, thus configuring a dependency loop (no energy on a specific BTS, no telecontrol functionality of this BTS in favor of other SS of the network). In this work, we suppose that BTS do not have power backup, i.e., we will simulate the worst possible case. This implies that if a certain BTS depends on a certain substation SS that is in a damaged (or disconnected) state,

Each SS can be modeled as a finite state machine as shown in **Figure 3**. In normal conditions, the SS is in the initial "functioning" state. Starting from this state, the

• Failure state: when a failure in the SS occurs, transition 1 is activated. The SS

• Not functioning state: in case of a contingency, the protection devices of the grid will disconnect a number of secondary substations that will change their state from "functioning" to "not functioning." For example, when a SS moves into the failure state, all SS on the same line move into their not functioning state. A SS

that specific BTS will immediately stop functioning.

*Secondary substation (SS) finite state model.*

secondary substation (SS) can move into two different states:

remains in this state for the expected failure duration;

**38**

**Figure 3.**

**Figure 2.**

Let us assume to have an EDN characterized by its topology, with nodes *N* and links *L* corresponding to electrical stations and electrical lines, respectively. The function representing the *functioning state* for all the elements of the EDN is referred to as *F*:

$$F(\mathbf{N}, L, \mathbf{t}) = \mathbf{O} \forall \mathbf{t} \tag{1}$$

if all elements *N* and *L* are in a *functioning state* and all telecontrol functionalities are active. Let us now introduce a perturbation function *P* that can change the state of one EDN element from the *functioning state* to one of the other possible states. In such a case

$$P \colon F(N, L, t) \to F'(N, L, t) \tag{2}$$

where *F*′ (*N,L,t*) *>* **0** for *t* ∈ [*0,T*] and zero elsewhere. For the sake of simplicity, we will apply the perturbation *P* only to the electrical secondary station (referred to as SS). Time *T* represents the time when all elements have been repaired and the network comes back to its fully functional state *F(N, L, t) = 0*. A perturbation *P*, in principle, could affect one (or more) electrical station and bring it (or them) from the functioning state to the not functioning or the failure states.

The damage of a SS consequent to the introduction of *P* produces a sequence of perturbations on the network. These consist in the disconnection of other nodes along the line due to instantaneous opening of protection switches. The damaged nodes are replaced by power generators (PGs) to ensure electrical continuity to the node's customers. The damaged nodes will not be repaired in the time space of the simulation, but their function will be restored through the settlement of PGs. The disconnected nodes, in turn, are reconnected either through a telecontrol operation (if available) or by dispatching technical crews to provide manual reconnection. All such interventions require specific times, which are considered when defining a restoration sequence of interventions. The impact of the perturbation *P* on the EDN is measured using a key performance indicator (KPI) that is currently used by the Italian Energy Authority to estimate the level of service continuity of an EDN. Such KPI is expressed as the number of disconnected customers *ni* of the *i-th* EDN node times the duration *i* of its disconnection. Such a value is expressed in terms of *kilominutes* (i.e., 103 minutes). Thus, if the damage of the *i-th* SS of the network will result in the disconnection of *m* SS, each one for a time *j* ( *j =* **1**,*m*), the overall KPI outage metrics will be measured in terms of *i* that is defined as follows:

$$
\Gamma\_i = \sum\_{j=1}^{m} n\_j \,\,\tau\_j \tag{3}
$$

For a given perturbation *P*, the integral over the simulated time span of Eq. (3) represents the perturbed functional state of the grid defined in Eq. (2):

$$\int\_{\mathbf{0}}^{\mathbf{T}} \mathbf{F}^{'} \left(\mathbf{N}, \mathbf{L}, \mathbf{t}\right) d\mathbf{t} = \Gamma\_{i} \tag{4}$$

*i* , thus, represents the *impact* that the damage of an EDN element (the *i-th* node) can produce, by using an official KPI as a metric. The larger the value of *i* , the weaker the capability of the network to withstand the perturbation in terms of impacts produced on the EDN customers. In general the value of *i* depends on different factors (described in detail in Section 4) ranging from the topology of the network and the employed technologies to the efficiency of operator restoration procedures; therefore, it would not be inappropriate to correlate the value of *i* with the inverse of the resilience concept *R*. In other terms

$$\mathcal{R}^{-1} \propto \Gamma\_i \tag{5}$$

We can generalize the concept by checking the EDN behavior versus all possible perturbations. The overall operational network resilience will be thus associated with the inverse of the value of the integral of the distribution function of all the *i* values (*D*()) resulting from the failure of each one of the *N* nodes of the EDN (normalized with respect to the total number of nodes *N*):

$$\mathbf{R} \propto \frac{\mathbf{1}}{\epsilon \,\Gamma \,\mathrm{\,\,\mathrm{\,\,\mathrm{\,\mathrm{\,\mathrm{\,\mathrm{\,\mathrm{\,\mathrm{\mathrm{\,\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\cdot}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \,\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\,\mathrm{\mathrm{\text{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\cdot}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \,\mathbf} $$

The higher the impact, the lower is the resulting operational resilience of the EDN network.


**Author details**

Zachary Shea\*, William M. Singer and Bo Zhang School of Plant and Environmental Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**41**

*Soybean Production, Versatility, and Improvement DOI: http://dx.doi.org/10.5772/intechopen.91778*

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