Quantitative Assessment of the Reliability of Water Treatment Plant as an Example of Anthropotechnical System

Abstract

The anthropotechnical system is a system of human–technical object–environment. In addition to the reliability of technical objects, the reliability of humans is also important in the proper functioning of such a system. The water supply system is an example of an anthropotechnical system. The operator of such a system is responsible, among other things, for the proper functioning of the water treatment plant, in which the production of water takes place. The operator’s task is to control individual subsystems and technological objects in order to supply water of the right quality, in the right quantity, at the required pressure to the consumer at any time, without making an error. The work shows the reliability of the anthropotechnical system assessment using the example of a water treatment plant located in south-eastern Poland. The single-parameter method and the reliability index were used to analyze the reliability. The reliability of the tested object was analyzed for the technical and anthropotechnical variant. The results indicate that after taking into account the influence of the operator on the reliability of the tested system, a decrease in the reliability index of 11% is observed. In order to minimize the negative influence of human factor on the reliability of the anthropotechnical system, it is recommended to duplicate the system operator, which allows for increasing the level of reliability of the anthropotechnical system.

1. Introduction

The reliability of the anthropotechnical system, i.e., the system of human–technical object–environment, is equally influenced by the reliability of technical objects and the reliability of humans [1]. Humans can affect the anthropotechnical system in two ways—as its operator or as its user. Data available in the literature indicate that the human factor has an influence on the occurrence of over 90% of accidents in the nuclear industry, over 80% of accidents in the chemical and fuel industry, over 75% of maritime accidents, over 70% of aviation accidents and over 75% of failures in water supply systems (WSS) [2,3,4,5,6,7].

The WSS requires human activity in the form of an operator who uses technical objects and affects the environment. This is an example of an anthropotechnical system [8]. The WSS operator is responsible for controlling and managing the operation of the water supply system, its subsystems and technological processes in order to maintain the continuity of its operation while maintaining requirements for the reliability and safety of its operation [8]. The reliability of the WSS operation is based on the ability to supply water of the right quality, in the right quantity, at the required pressure at any time. Within the WSS, humans also play the role of a system user, being a water consumer whose task is to use the system in accordance with its intended purpose and to follow safety rules in order to protect themselves and the system from the effects of possible failures [9].

WSS is also classified as critical infrastructure system; therefore, the priority should be to manage the reliability and security of the system so that the system is highly resilient to adverse events [10]. Taking into account the reliability theory at the design stage and at the operation stage is one of the ways to ensure the security of water supplies. From the point of view of reliability theory, when designing a WSS, it is necessary to formulate criteria for reliable operation of facilities, select appropriate technical solutions, determine the method and scope of facility backup, select an appropriate water source to ensure the required amount of water and secure the method of power supply. The criteria for the reliable operation of facilities should take into account the interests of water recipients and water supply companies, including determining the effects of unreliable operation of the system—both material and social effects. The elements used to build the system should be of the best possible quality and characterized by high reliability. The simplest way to ensure an appropriate level of reliability is to use surplus facilities and devices through redundancy. Different types of redundancies are used: strength, parametric, structural, functional and informational. Redundancy may concern individual elements or the entire system or subsystem in general. During operation, from the point of view of reliability theory, an optimal strategy for repairs and technical inspections of objects should be developed and a rational stock of spare parts should be established [11]. The choice of the strategy for repairs and technical inspections should be preceded by an analysis of the repair costs and losses caused by component failures, thanks to which it is possible to determine the minimum operating costs as the sum of the cost of spare parts, repair costs and component failure costs [9,11]. In turn, a rational stock of spare parts should include often-used components and the components with the lowest reliability. An important aspect in maintaining operational reliability is also maintaining operational documentation that allows for reliability analyses, including technical and economic analyses or prognostic analyses in terms of failure occurrence in the entire system [11]. Increasing the reliability level of the WSS generates increased operating costs; therefore, when designing and operating the system, the water companies should strive to maintain the required level of reliability or higher while minimizing the costs of obtaining it, and ensuring appropriate technical and hydraulic parameters of the WSS operation [9,11].

The reliability of water supply systems has been the subject of many research papers. Over the years, these studies have evolved, from conceptual studies [9,12,13,14] through case studies [15,16,17] to proposals of different approaches to assess the reliability of water supply systems [18,19,20,21,22]. The researches took into account various elements of water supply systems, such as the reliability of water sources [23,24,25], the reliability of water supply networks [15,17,21] or the reliability of water treatment plants [12,16]. In the work [16], the reliability of an example WTP was tested, for which the reliability index was determined. However, the influence of the human factor was not taken into account in these works. There are very few studies that consider the operator’s influence on the reliability of the WSS. In the works [6,7] the identification of human errors accompanying WSS failures was presented. In the works [8,26,27] a theoretical framework for the analysis of the reliability of the WSS operator was proposed. In the work [28], the use of Fault Tree Analysis was proposed to assess the reliability of the WSS operator. In the work [29] a method for assessing the safety of the WSS, taking into account the human factor, was presented. In the work [30], the value of the probability of making errors by the WTP operator was determined using the fuzzy-Bayes CREAM (Cognitive Reliability and Error Analysis Method). These studies form the basis for the development of a method for assessing the reliability of the WSS taking into account the influence of the human factor, which was identified as an interesting research gap.

The aim of the work was to perform a quantitative assessment of the reliability of the anthropotechnical system belonging to critical infrastructure. The assessment was conducted for a case study of Water Treatment Plant using a single-parameter reliability assessment based on the reliability index. Taking into account the anthropotechnical nature of the water supply system during the reliability assessment allows for better representation of real conditions. Previous studies in this area focused only on the assessment of the technical reliability of water supply systems.

2. Materials and Methods

2.1. Reliability Indicators

The parametric method was used to quantitatively assess the reliability of the anthropotechnical system. Based on the reliability indicator values for individual system components, the reliability indicator for the entire system can be determined. Reliability of systems can be quantitatively described using reliability indicators that characterize the properties of the system and its elements as well as random processes related to its functioning. The basic indicators used to assess reliability include the following [11]:

• Mean failure-free operating time T_p [d]—expected value of the random variable T’_p, defining the system (or element) operating time between two successive failures. This can be determined using Formula (1) or it can be calculated based on operating data with Formula (2) [11]:

$$T_p=E\left({T\prime }_p\right)={\int }_0^{\infty }tf\left(t\right)dt$$

(1)

$$T_p={\displaystyle \frac{1}{k+z}}·\left(\sum _{i=1}^k{t}_{pi}+z·t\right)$$

(2)

where:

-

E(T’_p)—expected value of a random variable T’_p;

-

t—observation time [d];

-

f(t)—probability density of a random variable T’_p;

-

k—number of periods of failing objects operation;

-

z—number of periods of non-destructive objects operation;

-

t_pi—value of the i-th operating time of the failing objects.

• Average renewal time T_o [d]—expected value of the random variable T’_o defining the renewal time. This can be calculated using Formula (3), or it can be calculated based on the operating data with Formula (4) [11]:

$$T_o=E\left({T\prime }_o\right)={\int }_0^{\infty }t{f}_o\left(t\right)dt$$

(3)

$$T_o={\displaystyle \frac{1}{n_o}}·\sum _{i=1}^{n_o}{t}_{oi}$$

(4)

where:

-

E(T’_o)—expected value of a random variable T’_o;

-

f_o(t)—probability density of a random variable T’_o;

-

n_o—number of repairs in the analyzed time period;

-

t_oi—duration of i-th repair.

• Failure intensity index λ(t)—determines the number of failures per unit of time [1/d]. It is determined using Formula (5), or it can be calculated based on operating data with Formulas (6) and (7) [11]:

$$\lambda (t)={\displaystyle \frac{dE\left({T^{\prime }}_p\right)}{dt}}$$

(5)

$$\lambda (t)={\displaystyle \frac{n(t,t+∆t)}{N·∆t}}$$

(6)

$$\lambda ={\displaystyle \frac{1}{T_p}}$$

(7)

where:

-

T_p—the average value of the operating time between successive failures is equal to E(T’_p) [d];

-

n(t,t + Δt)—total number of failures in a time interval Δt;

-

N—number of tested elements;

-

Δt—observation time, [d].

• Renewal intensity index µ(t)—determines the number of failures removed per unit of time. It can be determined using Formula (8), or it can be calculated based on operating data with Formulas (9) and (10) [11]:

$$\mu (t)={\displaystyle \frac{f_o\left(t\right)}{1-P_o\left(t\right)}}$$

(8)

$$\mu (t)={\displaystyle \frac{n(t,t+∆t)}{N·∆t}}$$

(9)

$$\mu ={\displaystyle \frac{1}{T_o}}$$

(10)

where:

-

P_o(t)—probability of renewal of the system;

-

f_o(t)—probability density of a random variable T’_o;

-

T_o—the average value of the system renewal equals to E(T’_o) [d];

-

n(t,t + Δt)—the number of all elements whose renewal was completed in the time interval t + Δt;

-

N—number of tested elements;

-

Δt—observation time, [d].

• Reliability index K_g—determines the probability that the system will be in an efficient state in a given time interval. It can be determined using Formula (11), or it can be calculated based on operating data with Formula (12) [11]:

$$K_g={\displaystyle \frac{\mu }{\mu +\lambda }}$$

(11)

$$K_g={\displaystyle \frac{T_p}{T_p+T_o}}$$

(12)

where:

-

µ—renewal intensity index [1/d];

-

λ—failure intensity index [1/d];

-

T_p—mean failure-free operating time [d];

-

T_o—mean time to repair [d].

2.2. Reliability Structures

In the parametric method, calculations are performed based on the reliability scheme of the tested system, in which the system elements are presented as links representing technical connections between individual objects and devices. Each link is assigned a numerical value of a selected reliability indicator, e.g., the K_g reliability index [11]. The reliability of the entire system is determined by calculating the reliability indicator successively for individual structures. The following reliability structures can be distinguished [11]:

• Serial structure—used if the failure of any element causes the failure of the entire system. The K_g reliability index for the serial structure is equal to the product of the reliability indexes of the individual system elements. It is calculated using Formula (13):

$$K_g=\prod _{i=1}^n{K}_{gi}$$

(13)

where:

-

K_gi—reliability index of the i-th element of the system [-];

-

n—number of elements.

• Threshold structure—used if failure to “k” of “m” homogeneous system elements causes failure to the entire system. It is called threshold structure of the “m-k of m” type. For the threshold structure, the reliability index K_g is calculated according to Formula (14):

$$\sum _{k=0}^{k_p}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{k}}\right)·{\left(K_{g0}\right)}^{m-k}·{\left(K_{p0}\right)}^k$$

(14)

where:

-

k—number of damaged elements;

-

k_p—permissible number of damaged elements;

-

m—number of all elements;

-

K_g0—single element reliability index [-];

-

K_p0—single element unreliability index [-].

• Parallel structure—used if the failure of the entire system occurs as a result of the failure of all system elements at the same time. The K_g reliability index of the parallel structure is described by Formula (15):

$$K_g=1-\prod _{i=1}^n\left(1-K_{gi}\right)$$

(15)

where:

-

K_gi—reliability index of the i-th element of the system [-];

-

n—number of elements.

2.3. Anthropotechnical Reliability Structures and Indicators

The anthropotechnical system is a system of operator, technical object and the environment. When adapting the parametric method to the analysis of the reliability of the anthropotechnical system, a human (system operator) should be treated as an element of the system. They should be included as an additional link in the reliability scheme. This link represents the probability that the operator will perform their work without any mistakes. It is described using the reliability index K_g, similarly to other elements of the system. The operator and the technical object form an anthropotechnical pair with a serial structure [31], where the failure of the entire system occurs as a result of failure to the technical object or operator error. Figure 1 shows the serial structure for the operator and the technical object.

According to the definition of the reliability index for technical objects, it is the probability that the object will be in a state of efficiency in a given time interval. The state of efficiency of a technical object is understood as the state in which the object performs its task [11]. Similarly, the definition of the operator reliability index was formulated as the probability that the operator will perform their work while being in a state of efficiency, without making an error in the given environmental conditions and a given time interval. Knowing the probability of an operator making an error (HEP), the operator reliability index can be determined according to Formula (16):

$$K_g=1-HEP$$

(16)

where:

-

HEP—human error probability.

In order to know the HEP value, a human reality assessment should be performed. There are many studies available in the literature on various technical systems, in which the probability of human error was determined using various HRA methods [30,31,32,33,34,35,36,37,38,39]. Based on the results of these studies, it is possible to determine the reliability index for the operator of various technical systems. In relation to the water treatment plant operator, the HEP values were determined in [30]. In this work, the fuzzy-Bayes CREAM was used to determine the HEP value. The research was conducted on a group of 40 WTP operators who were subjected to a survey. In the study, they assessed the impact of 9 Common Performance Conditions, e.g., working conditions, availability of procedures and plans or qualifications and training on their work process. Then, a Bayesian network model was created that took into account the interdependencies between these factors and the resultant HEP value for each of the tested operators was determined.

2.4. Reliability Requirements for Water Supply Systems

A water supply system is reliable if it fulfils its tasks, i.e., supplies the consumer with water of the appropriate quality, in the appropriate quantity, under the required pressure, at any time, at an acceptable price [9,11]. The effects of the unreliable operation of the water supply system related to the failure to meet any of the above-mentioned criteria may be of the following nature [11]:

-

Technical and economic—they mainly concern industrial water recipients, for whom the lack of water supply means interruptions in production or reduced efficiency; deterioration of product quality; damage to technical equipment, causing financial losses;

-

Health and sanitary—these mainly concern households and public utility buildings. Lack of or interruptions in water supply pose a threat to maintaining appropriate sanitary conditions in the place where people live. Inadequate water quality can also be a source of epidemics;

-

Psychological and social—lack of or long-term interruptions in water supply affect the quality of people’s lives, causing social tensions that can lead to riots. This aspect is particularly observed during crisis situations;

-

Catastrophic—these may be related to the effects mentioned above, but their scale is very large, posing a threat to human health and life.

Maintaining the reliable operation of a water supply system is burdened with an increase in the operating costs of the system. The reliability of the system should be at such a level that the benefits resulting from its maintenance compensate for the costs it causes. For this purpose, it seems appropriate to define reliability criteria for water supply systems, taking into account the effects of unreliable operation of the water supply system.

One of the first attempts to determine the required level of reliability, described by the reliability index, was presented in [11].

Table 1. Reliability indicators for water supply systems [11].

Water Supply System Category		Coverage of Total Water Demand [%]	C [1/year]	T0 [h]	Kg
I	Large (>50,000 recipients)	0–70	≤0.02	≤24	≥0.9999453
		71–99	≤2	≤24	≥0.9945206
		100	≤3	≤24	≥0.9917809
II	Average (500–50,000 recipients)	0–70	≤0.2	≤24	≥0.9994542
		71–99	≤3	≤24	≥0.9917809
		100	≤6	≤24	≥0.9835617
III	Small (<500 recipients)	0–70	≤1	≤24	≥0.9972603
		71–99	≤6	≤24	≥0.9835617
		100	≤12	≤24	≥0.9671233

presents the values of the required level of the WSS reliability index. They were calculated based on the parameter of the frequency of water supply system failures C and the average renewal time T₀. The base point was the assumption that the permissible average duration of a total lack of water supply to a household should not be longer than 24 h with a frequency of no more than once a year. When we are dealing with a total interruption in water supply in the entire WSS, these values should be interpreted in relation to the water supply subsystem (i.e., water intakes, pumping stations, treatment systems and tanks). These values have been proposed for over 30 years and require updating to current standards due to the development of new technologies in the field of water supply or the emergence of new threats to the reliability and safety of WSS operation.

2.5. Research Object

The case study for the reliability assessment of the anthropotechnical system was made on the example of the Water Treatment Plant (WTP) located in south-eastern Poland. The analyzed WTP supplies water to 200,000 recipients. The designed water production capacity is 84,000 m³/d. Raw water is taken from a local river using a bank-chamber intake. The water treatment plant consists of two independent technological lines A and B. They are based on the same water treatment processes: initial water ozonation; coagulation; filtration through a sand bed (station A) and anthracite-sand bed (station B); indirect ozonation; filtration through a carbon bed; UV disinfection; chlorine compound (chlorine gas and chlorine dioxide) disinfection; and, depending on need, correction of the water pH. The treated water is collected in water tanks on the WTP and then pumped into the water supply network. The length of the water supply network is 1140 km. The system includes 41 water pumping stations and 19 network tanks, with a total capacity of 53,000 m³. Figure 2 shows a conceptual technological scheme of the WTP.

3. Results

In the first stage, WTP reliability analysis was carried out using the single-parameter method and the K_g reliability index. Based on the information obtained from the water supply company and the technological scheme, a reliability scheme was prepared, as shown in Figure 3.

Table 2. Reliability indicators K_g of the system elements.

Symbol	Object	Kg
SS	Suction strainer	0.999
IW	Intake window	0.9992
CC	Contact chamber	0.9970
P1	1° pump	0.9681
MQ	Quick mixer	0.9891
MT	Static mixer	0.998
MS	Slow mixer	0.9729
ST	Sediment settler	0.9986
RF	Rapid filter	0.9846
TT	Technological water tank	0.9910
TP	Technological pump	0.9890
OG	Ozone generator	0.9980
CF	Carbon filter	0.9980
UV	UV lamp	0.9870
Cl	Chlorinator	0.9985
TW	Treated water tank	0.9892
P2	2° pump (high pressure)	0.9905

presents the values of reliability index for the system elements (based on data provided by the water supply company).

Based on the collected data, the reliability index for the studied WTP was determined. The results of calculations are presented in

Table 3. Calculations of reliability index K_g of the WTP as a technical system.

Object	Kg (For Single Element)	Reliability Structure Type	Kg (For Reliability Structure)	Object	Kg (For Single Element)	Reliability Structure Type	Kg (For Reliability Structure)
TECH LINE A				TECH LINE B
MQ	0.9891	parallel	0.9999	MT	0.998	threshold 2 of 4	0.992
MS	0.9729	series	0.9715	MS	0.9729	threshold 3 of 6	0.9999
ST	0.9986			ST	0.9986	threshold 4 of 6	0.9999
MS + ST	0.9715	threshold 6 of 9	0.9999	RF	0.9846	threshold 3 of 4	0.9989
RF	0.9846	threshold 6 of 8	0.9998	RF + RF	0.9989	pararell	0.9999
TT	0.991	-	0.991	TT	0.991	-	0.991
TP	0.989	threshold 1 of 3	0.9999	TP	0.989	threshold 1 of 3	0.9999
TECH LINE A (total)	-	series	0.9905	TECH LINE B (total)	-	series	0.9827
Object		Kg (For Single Element)		Reliability Structure Type		Kg (For Reliability Structure)
SS		0.999		series		0.9982
IW		0.9992
SS + IW		0.9982		threshold 2 of 4		0.9999
CC		0.997		threshold 1 of 2		0.9999
P1		0.9681		threshold 1 of 3		0.9999
TECH LINE A + TECH LINE B		-		parallel		0.9998
OG		0.998		threshold 1 of 2		0.9999
CC		0.997		threshold 1 of 2		0.9999
CC + CC		0.9999		parallel		0.9999
CF		0.998		threshold 4 of 6		0.9999
UV		0.987		threshold 5 of 6		0.9976
UV + UV		0.9976		parallel		0.9999
Cl		0.9985		threshold 1 of 2		0.9999
TW		0.9892		parallel		0.9998
P2		0.9905		threshold 1 of 3		0.9999
TW + P2 + P2		-		series		0.9996
P2		0.9905		threshold 2 of 4		0.9999
TW + P2 + P2		-		series		0.9997
TW + P2 + P2 + TW + P2		-		parallel		0.9999
WTP (total)		-		series		0.9988

. Table 3 is divided into three sectors. The upper right and left sectors contain calculations of reliability index for individual structures of technological lines A and B. The lower sector of the table contains calculations for the remaining WTP objects. The final result of the calculations is the reliability index K_g for the WTP object as a whole.

According to the requirements presented in Table 1, for large water supply systems with 100% degree of coverage of water demand, the required reliability index K_g value for the water supply system is 0.9917809. Therefore, the tested WTP meets the reliability requirements (WTP (total) K_g = 0.9988). This result refers to the operation of the cooperating subsystems of water intake, water treatment subsystem, water pumping subsystem and water storage subsystem, which together constitute the water supply subsystem.

The above analysis covers only the technical aspects of the WTP operation, based on the operational data of individual objects and devices such as failure-free operation time T_p, renewal time T₀, failure intensity λ or renewal intensity μ. As we know, WTP is an example of an anthropotechnical system, where a human plays an equal role to technical devices, being the operator of these objects. In an anthropotechnical system, not only can technical objects and devices be unreliable, but also the operator. In the next step, a reliability analysis was carried out, taking into account the human as an element of the anthropotechnical system. For this purpose, the reliability scheme was modified by adding elements representing the work of the operator, controlling individual subsystems in the system, such as the intake subsystem, the treatment subsystem, the pumping subsystem and the water storage subsystem. The modified reliability scheme is shown in Figure 4.

Table 4. Reliability index values for water treatment plant operator (based on [30]).

Symbol	Object	Kg
		Min	Average	Max
IO	Water Intake Process Operator	0.9287	0.9864	0.9993
OO	Water Treatment—Ozonation Process Operator	0.9493	0.9863	0.9994
CO	Water Treatment—Coagulation Process Operator	0.9433	0.9868	0.9995
FO	Water Treatment—Filtration Process Operator	0.9287	0.9853	0.9993
DO	Water Treatment—Disinfection Process Operator	0.9423	0.9879	0.9994
PO	Water Pumping Operator	0.9254	0.9863	0.9992
SO	Water Storage Operator	0.9357	0.9861	0.9995

presents the values of the reliability index for WTP operators divided into individual subsystems. These values were determined based on the results of the work [30], in which the human reliability analysis was performed for group of WTP operators. The results of this work present the probability of making an error by system operator while controlling individual subsystems, based on which reliability indexes were calculated.

The calculations of the reliability index for the anthropotechnical system (in the form of operator—WTP) are presented in

Table 5. Calculations of reliability index K_g of WTP as anthropotechnical system.

Object	Kg (For Single Element)	Reliability Structure Type	Kg (For Reliability Structure)	Object	Kg (For Single Element)	Reliability Structure Type	Kg (For Reliability Structure)
TECH LINE A				TECH LINE B
CO	0.9868	-	0.9868	CO	0.9868	-	0.9868
MQ	0.9891	parallel	0.9999	MT	0.998	threshold 2 of 4	0.992
MS	0.9729	series	0.9715	MS	0.9729	threshold 3 of 6	0.9999
ST	0.9986			ST	0.9986	threshold 4 of 6	0.9999
MS + ST	0.9715	threshold 6 of 9	0.9999	CO + MT + MS + ST	-	series	0.9787
CO + MQ + MS + ST	-	series	0.9866	FO	0.9853	-	0.9853
FO	0.9853	-	0.9853	RF	0.9846	threshold 3 of 4	0.9989
RF	0.9846	threshold 6 of 8	0.9998	RF + RF	0.9989	parallel	0.9999
FO + RF	-	series	0.9851	OF + RF + RF	-	series	0.9852
SO	0.9861	-	0.9861	SO	0.9861	-	0.9861
TT	0.991	-	0.991	TT	0.991	-	0.991
SO + TT	-	series	0.9772	SO + TT	-	series	0.9772
PO	0.9863	-	0.9863	PO	0.9863	-	0.9863
TP	0.989	threshold 1 of 3	0.9999	TP	0.989	threshold 1 of 3	0.9999
PO + TP	-	series	0.9862	PO + TP	-	series	0.9862
TECH LINE A (total)	-	series	0.9366	TECH LINE B (total)		series	0.9292
Object		Kg (for Single Element)		Reliability Structure Type		Kg (for Reliability Structure)
IO		0.9864		-		0.9864
SS		0.999		series		0.9982
IW		0.9992
SS + IW		0.9982		threshold 2 of 4		0.9999
IO + SS + IW		-		series		0.9863
OO		0.9863		-		0.9863
CC		0.997		threshold 1 of 2		0.9999
OO + CC		-		series		0.9862
PO		0.9863		-		0.9863
P1		0.9681		threshold 1 of 3		0.9999
PO + P1		-		series		0.9862
TECH LINE A + TECH LINE B		-		parallel		0.9955
OO		0.9863		-		0.9863
OG		0.998		threshold 1 of 2		0.9999
CC		0.997		threshold 1 of 2		0.9999
CC + CC		0.9999		parallel		0.9999
OO + OG + CC + CC		-		series		0.9861
FO		0.9853		-		0.9853
CF		0.998		threshold 4 of 6		0.9999
FO + CF		-		series		0.9852
DO		0.9879		-		0.9879
UV		0.987		threshold 5 of 6		0.9976
UV + UV		0.9976		parallel		0.9999
Cl		0.9985		threshold 1 of 2		0.9999
DO + UV + UV + Cl		-		series		0.9877
SO		0.9861		-		0.9861
TW		0.9892		parallel		0.9998
TW + TW		0.9998		parallel		0.9999
SO + TW + TW		-		series		0.986
PO		0.9863		-		0.9863
P2		0.9905		threshold 1 of 3		0.9999
P2 + P2		0.9999		series		0.9998
P2		0.9905		threshold 2 of 4		0.9999
P2 + P2 + P2		-		parallel		0.9999
PO + P2 + P2 + P2		-		series		0.9862
WTP (total)		-		series		0.8910

. Table 5 is constructed in a similar way to Table 3. It additionally contains elements presenting the values of the operator reliability indexes.

The operator and the controlled object in the anthropotechnical system make a serial structure. With a large number of different objects, the number of serial structures of anthropotechnical pairs (operator–object) also increases, which reduces the reliability index of the entire tested system. In the presented case study, the reliability index of the tested WTP (total) after taking into account the influence of the system operator reached the value K_g = 0.8910. In relation to the required K_g values presented in Table 1, after taking into account the anthropotechnical nature of the system, the tested WTP does not meet the requirements (where required K_g = 0.9917809).

The presented case assumes the work of a single operator, which, as proven, creates real threats to the reliability and safety of water supply. One of the basic ways to increase reliability is to duplicate elements [11]. For the most important technological processes, operator decisions should be made based on the decision of at least two operators, which reduces the probability of human error. In such a situation, the operators create an anthropotechnical pair with the controlled object as a serial–parallel reliability structure. Figure 5 shows the reliability scheme for the anthropotechnical system, in which the WTP objects are controlled by a pair of operators.

Table 6. Calculations of reliability index K_g of WTP as anthropotechnical system (doubled operator).

Object	Kg (For Single Element)	Reliability Structure Type	Kg (For Reliability Structure)	Object	Kg (For Single Element)	Reliability Structure Type	Kg (For Reliability structure)
TECH LINE A				TECH LINE B
CO	0.9868	parallel	0.9998	CO	0.9868	parallel	0.9868
MQ	0.9891	parallel	0.9999	MT	0.998	threshold 2 of 4	0.992
MS	0.9729	series	0.9996	MS	0.9729	threshold 3 of 6	0.9999
ST	0.9986			ST	0.9986	threshold 4 of 6	0.9999
MS + ST	0.9715	threshold 6 of 9	0.9999	CO + MT + MS + ST	-	series	0.9787
CO + MQ + MS + ST	-	series	0.9866	FO	0.9853	parallel	0.9853
FO	0.9853	parallel	0.9998	RF	0.9846	threshold 3 of 4	0.9989
RF	0.9846	threshold 6 of 8	0.9998	RF + RF	0.9989	parallel	0.9999
FO + RF	-	series	0.9996	OF + RF + RF	-	series	0.9852
SO	0.9861	parallel	0.9998	SO	0.9861	parallel	0.9861
TT	0.991	-	0.991	TT	0.991	-	0.991
SO + TT	-	series	0.9908	SO + TT	-	series	0.9772
PO	0.9863	parallel	0.9998	PO	0.9863	parallel	0.9863
TP	0.989	threshold 1 of 3	0.9999	TP	0.989	threshold 1 of 3	0.9999
PO + TP	-	series	0.9997	PO + TP	-	series	0.9862
TECH LINE A (total)	-	series	0.9897	TECH LINE B (total)	-	series	0.9819
Object		Kg (For Single Element)		Reliability Structure Type		Kg (For Reliability Structure)
IO		0.9864		parallel		0.9998
SS		0.999		series		0.9982
IW		0.9992
SS + IW		0.9982		threshold 2 of 4		0.9999
IO + SS + IW		-		series		0.9997
OO		0.9863		parallel		0.9998
CC		0.997		threshold 1 of 2		0.9999
OO + CC		-		series		0.9997
PO		0.9863		parallel		0.9998
P1		0.9681		threshold 1 of 3		0.9999
PO + P1		-		series		0.9997
TECH LINE A + TECH LINE B		-		parallel		0.9998
OO		0.9863		parallel		0.9998
OG		0.998		threshold 1 of 2		0.9999
CC		0.997		threshold 1 of 2		0.9999
CC + CC		0.9999		parallel		0.9999
OO + OG + CC + CC		-		series		0.9996
FO		0.9853		parallel		0.9998
CF		0.998		threshold 4 of 6		0.9999
FO + CF		-		series		0.9997
DO		0.9879		parallel		0.9998
UV		0.987		threshold 5 of 6		0.9976
UV + UV		0.9976		parallel		0.9999
Cl		0.9985		threshold 1 of 2		0.9999
DO + UV + UV + Cl		-		series		0.9996
SO		0.9861		parallel		0.9998
TW		0.9892		parallel		0.9998
TW + TW		0.9998		parallel		0.9999
SO + TW + TW		-		series		0.9997
PO		0.9863		parallel		0.9998
P2		0.9905		threshold 1 of 3		0.9999
P2 + P2		0.9999		series		0.9998
P2		0.9905		threshold 2 of 4		0.9999
P2 + P2 + P2		-		parallel		0.9999
PO + P2 + P2 + P2		-		series		0.9997
WTP (total)		-		series		0.9972

shows the calculations of the reliability index for the WTP as the anthropotechnical system controlled by a pair of operators.

For two operators working in a parallel structure, it is assumed that for the operation of the system, it is enough for one of them not to make a mistake. In the presented case study, the WTP (total) reliability index after taking into account the influence of the pair of operators reached the value of K_g = 0.9972. In relation to the presented required values of K_g = 0.9917809 (Table 1), after doubling the operators of WTP technological processes, the object meets the reliability requirements.

Figure 6 presents a graphical interpretation of the obtained results of WTP reliability assessment using the K_g reliability index for the assessed variants: technical, anthropotechnical (single operator) and anthropotechnical (doubled operator).

For the obtained results, a sensitivity analysis was performed to investigate how changes in operator reliability affect the reliability of the entire facility. For this purpose, three simulations were performed, with the decrease in operator reliability index K_g by 1% (scenario 1), 5% (scenario 2) and 10% (scenario 3).

Table 7. Sensitivity analysis for WTP reliability index K_g.

Scenario	Decrease in Operator Reliability Index Kg	WTP Reliability Index Kg
		Single Operator	Doubled Operator
1	1%	0.8646	0.9944
2	5%	0.7640	0.9676
3	10%	0.6496	0.9024

presents the calculation results.

Decrease in operator reliability index causes a decrease in the value of the reliability index of the entire facility. In the case of the anthropotechnical variant with a single operator, a decrease in the operator reliability index K_g by 1% causes a decrease in the reliability index of the WTP by 2.64%; in the case of a decrease in the operator reliability index K_g by 5%, the reliability index of the WTP decreases by 12.70%, while a decrease in the operator reliability index K_g by 10% causes a decrease in the reliability index of the entire facility by as much as 24.14%. In the case of the anthropotechnical variant with doubled operator, the decrease in the operator reliability index has a smaller effect on the changes in the reliability index of the entire facility. A decrease in the K_g operator reliability index by 1% causes a decrease in the reliability index of the entire facility by 0.28%; in the case of a decrease in the K_g operator reliability index by 5%, the reliability index of the WTP decreases by 2.96%, while a decrease in the K_g operator reliability index by 10% causes a decrease in the reliability index of the entire facility by 9.48%.

4. Discussion

In the presented case study, for the analyzed WTP, an 10.78% decrease in the value of the K_g reliability index for the anthropotechnical system (K_g = 0.8910) was observed in relation to the K_g reliability index for the technical system (K_g = 0.9988). In the case of including a single operator in the assessment of the system reliability, the studied system did not meet the reliability requirements set for large water supply systems (K_g ≥ 0.9917809). In the case where a technical object or device is characterized by insufficient reliability, structural redundancy is used to increase it. It consists of ensuring a surplus of a given element in the system by adding it in a parallel structure to the existing object. In this way, a new element, the so-called redundant element, appears in the system, which increases the reliability of the system. In relation to the examined WTP, the unreliability of the system operator caused the reliability of the entire system to decrease, and it did not meet the reliability requirements. For this purpose, a single redundancy (doubling) of the system operator was made. The reliability analysis of the WTP for the anthropotechnical system with a duplicated operator showed that the reliability index increased to the level of K_g = 0.9972. Doubling the operator allowed the anthropotechnical system reliability index to increase by 10.62%, thanks to which the tested system met the reliability requirements for water supply systems. The sensitivity analysis performed showed that the system with doubled operators is also more resistant to changes in the values of the operator reliability index. This results from the reliability structure of the system of two operators, which create a parallel structure, characterized by higher reliability than the serial structure. The standard for managing the operation of the WTP should be the work of at least two operators in one shift. This will ensure that the system is maintained at the required level of reliability and ensure the safety of water supply, with relatively low costs of employing a second operator. One of the limitations of the proposed method is determining the reliability of the system operator. Human nature is complex, and it is difficult to model the operator’s reliability in the work process. Many factors influence human errors [3,4,6,33,40]. The most important of them include the following: unfavorable environmental conditions (lighting, vibrations, noise, microclimate, exposure to hazardous substances); improper ergonomics of the operator’s station—lack of resources to perform the task (tools, materials, documentation); improper work organization (complexity of tasks, lack of team cooperation, difficulties in communication and information flow); poor psychophysical condition of the operator (fatigue—physical or mental, related to the circadian rhythm of sleep and wakefulness; distraction—results from disorientation caused by excess stimuli; stress—occurs when the operator feels mental discomfort regarding the conditions or requirements related to the implementation of the task in a situation where they exceed their capabilities); low operator competences related to lack of knowledge and experience, resulting in a lack of understanding of the purpose of the task and the principles of its execution; and low motivation of operators (manifested by routine, feeling of pressure, lack of assertiveness and negligence). To determine the K_g operator reliability index, the results of the work [30] were used, in which the probability of making an error by WTP operator was determined using the Cognitive Reliability and Error Analysis Method (CREAM). This is based on the Contextual Control Model (COCOM), which describes the relationships between cognitive functions occurring in the human mind [1]. In simple terms, this model assumes that each human action results from a controlled process of using the possessed competences depending on the requirements of a given situation. Actions undertaken by humans are also intentional and reactive in nature in relation to the conditions accompanying the work process.

Another aspect that should be taken into account is the fact that the process of managing and operating WTP is currently highly automated. It therefore seems that an operator of such a system may be unnecessary. However, automatic systems, despite the use of a number of safeguards, may be unreliable and sensitive to contemporary threats, e.g., energy blackout, cyber terrorism. During a crisis situation with an unpredictable course or in the event of damage to automatic systems, it is the human (the WTP operator) who should be able to make operational decisions. Based on their experience, knowledge and familiarity with the specifics of a given WTP, they have a chance to restore it to a state of efficiency using manual control. Complete elimination of humans from the WTP operation process poses a potential risk to maintaining the reliability and safety of its operation.

An interesting direction for further research seems to be to conduct an analysis of the anthropotechnical reliability of WTP on a larger number of case studies, which will allow for the identification of the real human impact on the reliability of the functioning of WTP and water supply systems. This will allow for the formulation of new values of recommended reliability indexes for water supply systems, thanks to which it will be possible to take into account the anthropotechnical nature of the system already at the design stage. Designing systems with regard to reliability criteria is particularly important in the case of critical infrastructure systems such as the water supply system. It allows for a significant reduction in the negative effects of system failure on many levels: technical–economic, health–sanitary or psychological–social.

5. Conclusions

The quantitative assessment of the reliability of technical systems allows for precise determination of reliability indicators, based on which it is possible to manage system resources from the design stage, through the construction stage and the operation stage. The idea of designing a system with reliability requirements taken into account allows for achieving measurable benefits in later stages of operation. It allows one to avoid or minimize costs resulting from the general failure of the system. In anthropotechnical systems, in addition to the failure of technical objects, we also deal with the failure of the human—the operator of such a system. For this reason, omitting the influence of the human factor when assessing the reliability of the technical system is a mistake. A human in an anthropotechnical system may turn out to be the so-called weakest link.

This paper presents a method for assessing the reliability of WTP, which is also an example of an anthropotechnical system. The values of the reliability index of the tested system were determined for three variants: technical, anthropotechnical and anthropotechnical with a duplicated system operator. The analysis showed that the system operator has an impact on the reliability of the technical object—the case study of WTP showed a 10.78% decrease in system reliability, after taking into account the operator’s unreliability in the calculations. In order to increase the reliability of the anthropotechnical system, it is recommended that at least two system operators work simultaneously. This allows for an increase in system reliability by 10.62%. The water supply system belongs to the critical infrastructure, so maintaining an appropriate level of reliability and safety of its operation should be the primary goal of water supply companies. Reliability management should apply not only to technical objects but also to employees responsible for the maintenance and operation of the system, who have an equal impact on the reliability and safety of the water supply system.