Integrated Stochastic Approach for Instantaneous Energy Performance Analysis of Thermal Energy Systems

Abstract

To ascertain energy availability and system performance, a comprehensive understanding of the systems’ degradation profile and impact on overall plant reliability is imperative. The current study presents an integrated Failure Mode and Effects Analysis (FMEA)–Markovian algorithm for reliability-based instantaneous energy performance prediction for thermal energy systems. The FMEA methodology is utilized to identify and categorize the various failure modes of the gas turbines, establishing a reliability pattern that informs overall system performance. Meanwhile, the Markovian algorithm discretizes the system into states based on its operational energy performance envelope. The algorithm predicts instantaneous energy performance according to upper and lower bounds criteria. This integrated methodology has been subjected to testing in three case studies, yielding results that demonstrate improved reliability and instantaneous energy performance prediction during system degradation. It was observed that after 14 years of operation, the likelihood of major failures increases to 79.6%, 88.7%, and 82.8%, with corresponding decreases in system performance reliability of 10.1%, 4.5%, and 7.8% for the Afam, Ibom, and Sapele gas turbine plants, respectively. Furthermore, the percentage of instantaneous mean power performance relative to the rated capacity is 37.9%, 35.1%, and 46.3% for the three gas turbine plants. These results indicate that the Sapele thermal power plant performs better relative to its rated capacity. Overall, this integrated methodology serves as a valuable tool for monitoring gas turbine engine health and predicting energy performance under varying operating conditions.

1. Introduction

The trend of the increasing use of technology in the energy and power industries has influenced the development of high-performance gas turbine systems [1]. The intent of which is to maintain and/or increase efficiency and performance, which subsequently induces additional fuel cost and maintainability challenges. Most importantly, the material response and survivability of gas turbine power plants under high-temperature regimes have not been comprehensively understood. In such a harsh (high temperature) operating environment, material degradation/deterioration plays a critical role in the plant components′ reliability and performance for the period under consideration. While certain criteria can be considered during the initial design phase of the infrastructure, it is essential to acknowledge that the system’s performance and responses can exhibit significant diversity throughout the operational cycle of a power plant. Specifically, in the demanding context of a gas turbine power plant, it becomes imperative to accurately capture and account for the dynamic variations and variability in material composition within the harsh operational environment. This meticulous consideration is vital for conducting a comprehensive and robust analysis of subsystem performance and reliability [2,3]. Also, complexity and dependency exist among critical subsystems that result in common cause failure or shock-induced failures among subsystems. The reviewed research literature shows that there is a limited understanding of the correlation of the gas turbine system’s performance with the subsystem’s reliability state [4]. This could be a result of the complex interdependencies and stochastic nature of the degradation-induced failure characteristic of the subsystems. There is a need to characterize the system’s performance in terms of mechanical degradation for reliability-based integrity management and energy performance analysis.

The current study presents an approach that integrates the FMEA with the Markovian algorithm for reliability-based energy performance analysis of gas turbine plants under degradation. The FMEA identified and classified the failure factors in the gas turbine plants. The identified failure modes are then mapped to define the performance state of the plants to indicate the failure propagation during operations. The performance states are structured and evaluated based on the initial state and transition probability matrix. The Markovian approach analyzes the effects of the state transition on the overall gas turbine system performance. The integrated approach is tested on gas turbine plants in tropical rainforests in Africa. The approach provides a better performance analysis, which is able to clearly deduce instantaneous energy output profiles for the systems under degradation. The application of the proposed approach provides condition monitoring tools for sustainable energy performance and optimum integrity management of the asset.

2. Overview of Gas Turbine Systems Analysis

Gas turbine power systems are subjected to a high-temperature operating environment. The harsh operating environments could create multiple failure tendencies and complexity, making it difficult to predict the system’s performance. The occurring failure modes/scenarios could have an economic and developmental impact with considerable financial losses. This mostly results in a loss of energy production and high downtime due to system breakdown. The functionality of a system can be described by its reliability. Several studies have been conducted to understand the performance of gas turbine systems and energy generation in the open-source literature [2,3,4,5,6,7,8,9]. Several different techniques have been proposed for reliability-based system analysis. These can be grouped into qualitative, quantitative, dynamic quantitative, and stochastic models and shall be examined in the following sections.

2.1. Reliability Approach for Gas Turbine Systems

A reliability analysis can be performed for different gas turbine systems and components, such as mechanical, electronic, electrical, and software components, at various stages of the engineering process. This is vital from the design stage to the operational phase of the gas turbine project. The FMEA approach captures various causes of failure by containing the common cause factors in a systemic structure, allowing for a reliability prediction [10]. Also, the different failure causes, events, hazards, faults, effects, and consequences associated with the infrastructure are identified. This shows the system’s structural interactions and performance functions, considering external factors, such as those experienced in high-temperature energy systems. The use of sheet-based, table-based, and diagram-based qualitative reliability methods to characterize system health states has been presented in the literature [10,11,12,13,14].

For instance, Katsavounis et al. [15]. and Scheu et al. [16] proposed the sheet-based approach checklist to determine and identify failure-influencing factors in the design, operation, and maintenance of energy power plants. The checklist provides a set of questions for reliability engineers and system operators for hazard identification to gather the facts on critical contributing parameters. Lazakis and Kougioumtzoglou [17] applied Hazard Identification (HAZID) and Failure Mode Effect Analysis (FMEA), as well as the Bayesian Network, in their failure analysis studies for an energy system; the probability of failure, as well as the annual cost with or without failure, was clearly analyzed. Similarly, Sharma et al. [18] proposed the use of FMEA for failure mode analysis of an aircraft gas turbine system. The FMEA was applied to grade the severity of all potential failure modes and for design improvement. The authors demonstrated the application of the FMEA on the rotor support system and identified poor welds, seals, or connections; chemical impurities in metal; voids, cracks, thin spots, or coatings; dirt or contamination on surfaces or in materials; incorrect positioning of the part; and design errors are common causes of early failure in gas turbine rotor support systems.

Several quantitative tools have been proposed to capture the likely failure modes and their effects on the mechanical systems’ reliability/performance [19,20,21,22,23]. Kappas [23] presents a review that focuses on the use of probabilistic methods for reliability analysis of gas turbine components failures. The author carried out a statistical failure analysis based on the component’s failure data over time by using a probability distribution. This is used to accurately predict the probability of occurrence of the given failure modes/components of the gas turbine systems.

A further methodology that experimentally and deductively represents the root cause of failures in the gas turbine system has been demonstrated [3,9,21,24]. Kolagar et al. [3] analyzed turbine failure due to the degradation of the blade. Bai [25] proposed the hybrid analytic hierarchy process (AHP) and fault tree technique for the reliability analysis of gas turbine components. This hybrid methodology captures the complexity of the fault based on the AHP criteria framework. The AHP establishes a pairwise comparison matrix for the criticality of the subsystems as it affects the functionality of the overall plant. The hierarchy structure is mapped into the fault tree for a quantitative analysis. However, the models could be limited for higher order complexity and with associated high degree of uncertainty. The stochastic nature of the failure characteristics of gas turbine components has not been fully studied yet.

2.2. A Stochastic Approach for Gas Turbine Systems’ Reliability

The failure characteristics of critical gas turbine systems could be space- and time-dependent. This could be a result of the causative failure factors mechanism. For instance, the corrosion mechanism, which is a failure-induced factor, presents an unstable phenomenon that could be difficult to predict. Also, the failure state of the subsystems can be represented using a multi-stage approach in which a soft failure (partial failure or pseudo-operational state) is presented [26]. In recent times, Jiang et al. [26] have demonstrated the Markov process’s applicability for the prediction of the reliability of gas turbine subsystems. The Markov process is a state–space methodology that represents the failure rate as the rate of the transition of one component from a functioning state to a degraded state and to a failed state for three-state systems. The authors further integrate the Markov process with reliability importance indices for component priority determination. This was evaluated based on Birnbaum’s measurement indices. The approach shows a capacity for the stochastic modelling of the gas turbine systems; however, the correlation of the characterized state with power output performance was not considered. Limited studies exist in the open-source literature that have characterized the performance output with the reliability state of the gas turbine mechanical subsystems. There is a need to quantitatively correlate the mechanical systems’ reliability with the plant’s performance output, considering interdependencies among gas turbine subsystems.

The current study presents the development of an integrated FMEA–Markovian approach for instantaneous energy performance predictions for gas turbine plants under degradation.

3. Methodology

This section presents the developmental steps used to create an integrated energy-based stochastic model for the prediction of a gas turbine power plant’s reliability and energy performance, as shown in Figure 1. The approach’s development begins with defining the plant’s systems and their subsystem performance and identifying the systems’ likely failure modes. The following steps briefly describe the model development process.

Figure 1. An algorithm for the reliability-based energy performance analysis of a gas turbine plant.

Step 1: The plant under study is defined based on its system and subsystem performance states, functionality, and health state based on degradation data for elements such as fuel systems, generator systems, combustion systems, shafting systems, cooling systems, etc. This also includes identifying the likely failure modes of the systems and their subsystems using FMEA and developing a logical relationship among the failure-causative parameters.

Step 2: The various plant performance dataset is collected. This includes, but is not limited to, failure data, repair/maintenance data, reliability, and manufacturer’s data. The data are used to estimate the failure rate of the various subsystems statistically, as well as the mean time between failures, mean time to failure, and mean time to repair. When one considers multi-state system (MSS) evolution in the space of states during a system operation period T, the following random variables may be of interest:

Time to failure, ${\mathit{T}}_{\mathit{f}}$ is the time from the beginning of the system’s life up to the instant when the system first enters the subset of unacceptable states.

The time between failures, ${\mathit{T}}_{\mathit{b}}$ is the time between two consecutive transitions from the subset of acceptable states to the subset of unacceptable states.

Number of failures, ${\mathit{N}}_{\mathit{T}}$, is the number of times the system enters the subset of unacceptable states during the time interval [0, T].

Equations (1)–(3) show the mathematical relations for the highlighted parameters’ prediction.

$$\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n} \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e} \mathrm{t}\mathrm{o} \mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e} \left(MTTF\right)={\lambda }^{-1}$$

(1)

For multi-state systems, the MTTF can also be the cumulative time during which the process remains in each state where j > i, such that

$${MTTF}_i=\sum _{j=i+1}^k{\displaystyle \frac{1}{{\lambda }_{j,j-1}}}$$

(2)

$$\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n} \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e} \mathrm{t}\mathrm{o} \mathrm{r}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r} \left(MTTR\right)={\mu }^{-1}$$

(3)

where λ and μ are the failure rate and repair rate, respectively.

For the multi-state elements, the system and sub-systems are discretized into states that characterize the degradation process of the system. This represents the state of soft/minor failures as the elements transit from an operational to a degradable state. For any element in the system designated $\mathit{j}$, there is ${\mathit{k}}_{\mathit{j}}$ different states with corresponding performance rates based on the energy or exergy analysis of the plant. This is represented by ${\mathit{g}}_{\mathit{j}}=\left\{{\mathit{g}}_{\mathit{j}1},\dots \dots {\mathit{g}}_{\mathit{j}{\mathit{k}}_{\mathit{j}}}\right\}$ based on the Markovian stochastic process. However, the current state of the element $\mathit{j}$ and the values of the element of the performance rate (net power output) ${\mathit{G}}_{\mathit{j}}$(t) at any instant time $\mathit{t}$ are randomly distributed. The performance rate of the element ${\mathit{G}}_{\mathit{j}}$(t) can take values from ${\mathit{g}}_{\mathit{i}}:{\mathit{G}}_{\mathit{j}}\left(\mathit{t}\right)\mathit{ϵ}{\mathit{g}}_{\mathit{j}}$. When mapping the system performance to the degradation state of the subsystems, the element evolution in the state space is used as a performance degradation criterion to characterize the stochastic process $\left\{\mathit{G}\left(\mathit{t}\right)|\mathit{t}\ge 0\right\}$. The intensity of the state transition from state $\mathit{i}$ to state $\mathit{i}-\mathbf{1}$ is ${\mathit{\lambda }}_{\mathit{i},\mathit{i}+1},\mathit{i}=\mathbf{2},\dots \dots .,\mathit{k}.$ But, for repairable systems, the repair rate (${\mathit{\mu }}_{\mathit{j}\mathbf{,}\mathit{j}\mathbf{+}\mathbf{1}})$ is factored into the transition process considering minor or major repairs via a restoration profile, as shown in Figure 2.

Figure 2. State transition diagram for the gas turbine systems (Adopted from Nitonye et al. [27]).

Step 4: To evaluate the transition probability matrix, the prior state of the mechanical system and subsystems are defined. This defines the initial state of the system at the start of operation. It is represented by Equation (4). The generator matrix describes the state transition intensity (failure and repair rates) for the system elements under consideration. Equation (5) is based on the Kolmogorov set of differential equations for a system element with three state spaces to evaluate the transition across the states. Equation (6) shows the transition probability matrix.

$$\pi =\left(\begin{array}{c}{\pi }_1\\ {\pi }_2\\ \begin{array}{c}:\\ \begin{array}{c}:\\ {\pi }_N\end{array}\end{array}\end{array}\right)$$

(4)

${\pi }^{T}A={\left(\right({\pi }^{T}A)}_1\dots \dots ,\left({\pi }^{T}A\right)N),$ with elements, ${\left({\pi }^{T}A\right)}_j=\sum _{i=1}^N{\pi }_i{a}_{ij}$

$$\begin{array}{l}{\displaystyle \frac{d{P}_3\left(t\right)}{dt}}=-\left({\lambda }_{\mathrm{3,2}}+{\lambda }_{\mathrm{3,1}}\right)P_3\left(t\right)+{\mu }_{\mathrm{2,3}}{P}_2\left(t\right)+{\mu }_{\mathrm{1,3}}{P}_1\left(t\right)\\ {\displaystyle \frac{d{P}_2\left(t\right)}{dt}}=\left({\lambda }_{\mathrm{3,2}}\right)P_3\left(t\right)-\left({\lambda }_{\mathrm{2,1}}+{\mu }_{\mathrm{2,3}}\right)P_2\left(t\right)+{\mu }_{\mathrm{1,2}}{P}_1\left(t\right) \\ {\displaystyle \frac{d{P}_1\left(t\right)}{dt}}=\left({\lambda }_{\mathrm{3,1}}\right)P_3\left(t\right)+{\lambda }_{\mathrm{2,1}}{P}_2\left(t\right)-\left({\mu }_{\mathrm{1,2}}+{\mu }_{\mathrm{1,3}}\right)P_1\left(t\right) \end{array}$$

(5)

where $\pi$ is the prior probability distribution of the system states given as $\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$ when the system is fully operational. $P_1,\dots \dots ., P_3$ are the transition probabilities for the four-state element. Initially, the system had $P_3\left(0\right)=1, P_2\left(0\right)=P_1\left(0\right)=0$. Table 1 shows the transition profile based on the three states (operational, minor failure, and major failure) in the system’s performance.

Table 1. State transition relationship between the turbine components.

$t$	$t+∆t$
	O	FMinor	FMajor
O	$e^{-\left({\lambda }_{\mathrm{3,2}}+{\lambda }_{\mathrm{3,1}}\right)∆t}$	${\displaystyle \frac{{\lambda }_{\mathrm{3,2}}}{{\lambda }_{\mathrm{3,2}}+{\lambda }_{\mathrm{3,1}}}}\left(1-e^{-\left(\left({\lambda }_{\mathrm{3,2}}+{\lambda }_{\mathrm{3,1}}\right)∆t\right)}\right)$	${\displaystyle \frac{{\lambda }_{\mathrm{3,1}}}{{\lambda }_{\mathrm{3,2}}+{\lambda }_{\mathrm{3,1}}}}\left(1-e^{-\left(\left({\lambda }_{\mathrm{3,2}}+{\lambda }_{\mathrm{3,1}}\right)∆t\right)}\right)$
FMinor	${\displaystyle \frac{{\mu }_{\mathrm{2,3}}}{{\lambda }_{\mathrm{2,1}}+{\mu }_{\mathrm{2,3}}}}\left(1-e^{-\left(\left({\lambda }_{\mathrm{2,1}}+{\mu }_{\mathrm{2,3}}\right)∆t\right)}\right)$	$e^{-{(\lambda }_{\mathrm{2,1}}+{\mu }_{\mathrm{2,3}})\Delta t}$	${\displaystyle \frac{{\lambda }_{\mathrm{2,1}}}{{\lambda }_{\mathrm{2,1}}+{\mu }_{\mathrm{2,3}}}}\left(1-e^{-\left(\left({\lambda }_{\mathrm{2,1}}+{\mu }_{\mathrm{2,3}}\right)∆t\right)}\right)$
FMajor	0	0	1

where O is the operational state, F^Minor is the minor failure state, and F^Major is the major failure state.

$$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x} = A = \left\{a_{ij}\right\}=\left[\begin{array}{cccc}{P}_{11}& P_{12}& \dots \dots & P_{1m}\\ P_{21}& P_{22}& \cdots \cdots & P_{2m}\\ ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮\\ P_{m1}& P_{m2}& \cdots & P_{mm}\end{array}\right]$$

(6)

Step 5: The evaluated transition probability from step 4 is used as an input to predict the system’s element reliability state. Based on this result, the system reliability can be predicted using Equation (7) [28]. This can also be mapped to the possible performance level of the gas turbine plant. For instance, assume the possible performance level of the plant based on the power output as $\left\{g_k^m,g_k^p,g_k^o,\right\}$, the system element means that an instantaneous performance occurs at time $t,$ and $\left(E_t\right)$ can be predicted using Equation (8).

$$\mathrm{T}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{r}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} R_i\left(t\right)=1-P_1\left(t\right)=\sum _{j=i+1}^k{P}_j\left(t\right)$$

(7)

$$E_t=\sum _{k=1}^3{g}_k{P}_k\left(t\right)=g_k^o{P}_3\left(t\right)+g_k^p{P}_2\left(t\right)+g_k^m{P}_1\left(t\right)$$

(8)

Therefore, in a different instance, the system’s reliability based on the energy output performance could be predicted and used to develop a profile for integrity-related decision-making. Three energy performance criteria are formulated for the analysis, as shown below. Upper and lower bounds for the energy demand can be generated based on the $g_k$ values to develop a mean instantaneous energy generation profile using Equation (8). The approach of mapping the thermal plant system’s health with performance using Markovian process modelling has been demonstrated by Lisnianski et al. [29] and is based on the operational performances of the selected plants.

For a fully operational state: $80 \mathrm{MW} < g_k^o \ge 160 \mathrm{MW}$.

For a partial/minor failure or loss state: $20 \mathrm{MW} < g_k^p \ge 80 \mathrm{MW}$.

For a major failure: $0 \mathrm{MW} < g_k^m \ge 20 \mathrm{MW}$.

4. Methodology Implementation

The developed Markovian stochastic process was tested on three gas turbine power plants operating in the tropical rainforest. The first plant is an ALSTOM GT-I3E2 manufactured by ALSTOM, Saint-Quen-sur-Seine, France, and has an installed capacity of 180 MW (base load 160 MW). The ALSTOM GT-13E2 operates on a simple Brayton cycle. The main mechanical components include a five-stage turbine, a 21-stage axial compressor mounted on the same shaft, and a single annular ring coiled counter flow combustion chamber placed in between the compressor and the turbine. The second plant is a GE-Frame 9/PG9161 (E), manufactured by General Electric, Louisville, KY, USA, with three (3) units and a total installed capacity of 191 MW (unit 1 with 38 MW, unit 2 with 38 MW, and unit 3 having 115 MW). In the period referred to in this study, only phase one and unit 3 (GTG-3) with 115 MW installed capacity were operational, exporting an average of 2400 MWH per day to the national grid. The GE-Frame 9/PG9161 (E) is a single-shaft gas turbine operating on a simple Brayton cycle with a three-stage axial flow turbine, a seventeen-stage axial flow compressor mounted on the same shaft, and a can-annular type combustion chamber. The third plant has four (4) Gas Turbine Generating Units of 112.5 MW capacity each, totaling 450 MW. It is a GE Frame 9E (GE FRAME PG9171E) manufactured by General Electric, Louisville, KY, USA, a simple-cycle operation plant with provision for future conversion into a combined-cycle plant. The GT is a single shaft, three bearing heavy duty industrial unit with a 17–stage axial flow compressor, can-annular type combustor and a 3-3-stage axial flow turbine. The environmental conditions show that the relative humidity varies between a minimum of 70% and a maximum of 90%, and power plants mostly operate at average temperature of 28 °C in the Niger Delta, although most power plants installed in the Delta are designed for 15 °C ISO conditions. Field operational and components’ failure and maintenance data were collated from log sheets kept in the control rooms of the plants and direct measurements from the human–machine inter-phase (HMI) from the various GT plants. Ten years’ worth of data were collected from the plants for analysis. The key information and parameters include inlet (ambient) and exit temperatures and the pressures of the different components of the gas turbine engine (compressor, combustor, and turbine), air, fuel, and exhaust gas flow rates, and overall operational management strategy. The specific fuel consumption, work output, power output, and pressure ratio were calculated using standard thermodynamic relations. These data were used as the input data, and the analysis results (output) are shown in Section 5.

5. Results and Discussion

The research objective was to develop a reliable methodology for reliability-based system performance prediction for energy generation over time, given the health state of the plant subsystems.

5.1. Failure Modes Analysis of Plants’ Performances

A failure assessment technique was adopted to assess the plants’ functionality in view of the design performance. The Failure Mode and Effect Analysis (FMEA) explored the system subsystem performance, their failure characteristics, and the common causes of failure in the overall turbine plant for critical decision-making. The FMEA was applied to grade the severity of all potential failure modes and for design improvement. The results of the FMEA on the compressor, combustion, rotor support system, turbine, and generator are shown in Table 2.

Table 2. Principal events in gas turbine power plant performance failure.

Main Events	Codes	Intermediate Events	Codes	Basic Events	Codes
Electric generator system failure	T1	Generator trip fault	A2	Damage to generator casing	A11
Starting/governing system failure	K	Generator damage	A1	Generator Rotor bowing	A12
Failure of compressor system	E	Generator electrical fault	A3	Damage to generator guide way	A13
Failure of air-inlet system	C	Operational fault due to human error	A4	Loss of field to the main exciter	T31
Failure of turbine system	I	Failure of relay system	A5	Short circuit fault	T32
Failure of combustion system	G	Failure due to damaged fuse	A6	Loss of a.c supply to excitation	T33
		Failure to generate output voltage	T2	Coupling shaft stress	T61
		Generator excitation system failure	T3	Loose connectivity	T62
		Turbine failure to reach desired speed	T4	Pump motor insulation failure	C71
		Turbine blade failure due to Improper pressure	T5	Motor failure due to high current	C72
		Loose coupling of shaft	T6	No power supply at the motor terminal	C73
		Improper burning of fuel	T7	Motor overload	C91
		Low/high pressure of combustor air inlet	T8	Motor overheats due to cooling failure	C92
		Turning gear failure	K1	Motor failure due to wrong connection	C93
		Start-up system failure	K2	Failure of cooling system of the lube system	E51
		Gearbox system failure	K3	Faulty pressure gauge	E61
		Torque converter system	K4	Failure of heating system	E62
		Lubrication system failure	E5	Failure of temperature sensor	E63
		Loss of water supply from reservoir	E6	Ballooning of torque converter	K41
		Vane system turbine failure	I1	Clutch of the torque converter	K42
		Turbine radial bearing failure	I2	Earth fault-induced failure	A31
		Turbine cylinder failure	I3	Copper dusting occur in the generator	A32
		Turbine exhaust system failure	I4	Generator fault due to reverse voltage	A33
		Turbine cylinder exhaust system failure	I5	Generator failure due to inter turn fault	A34
		Exhaust connection system failure	I6	Generator failure due to short circuit fault	A35
		Combustion system wear and tear	G1	Too early operation from human operator	A41
		Igniter system failure	G2	Too late operation from human operator	A42
		Combustion cooling system failure	G3	Out sequence operation by human operator	A43
		Damaged basket of shell	G4	Delay in open the relay contact	A51
		Failure of evaporator cooler	C1	Relay contacts freeze	A52
		Failure of water supply system	C3	Relay contacts bounce	A53
		Pipe section rupture	C4	Relay coils burn out	A54
		Faulty pump system	C5	Fuse open on low current	A61
		Valves failed to open on demand	C6	Fuse fails to open on high current	A62
		Damaged pump motor	C7	Fuse partially opened	A63
		Rupture seal house	C8	Vane’s damage	I11
		Motor Overheating	C9	Pins of vane system damage	I12
		Compressor blade failure	E1	Broken blade rings of turbine cylinder system	I31
		Journal bearing failure	E2	Broken blade of the turbine system	I33
		Compressor thrust bearing failure	E3	Failure of the exhaust collector of the exhaust system	I41
		Journal bearing casing failure	E4	Failure of the pads of the radial bearing	I21
				Failure of thermocouples in the radial bearing system	I22, I52
				Damage of the shell of the radial bearing system	I23
				Damage of the static seal of the exhaust system	I61
				Damage of the exhaust pipes	I62
				Damage casing of the exhaust cylinder	I51
				Flame tube failure	G11
				Shell cylinder failure	G12
				Igniter system spring failure	G21
				Igniter system piston failure	G22
				Failure of the igniter in the igniter system	G23
				Blockage of the cooling system vessels	G31
				Failed to open of the bypass valve of the cooling system	G32
				Cooling system controller failure	G33
				Damaged transition piece in the basket	G41
				Damaged burner in the basket	G42
				Humid badges burst	D1
				Water pipes damage	D2
				Pipes failure due to corrosion	D3
				Valve failure due to mechanical fault	D4
				Valve failure due to controller fault	D5
				Rupture of seal house due to high temperature	D9
				Seal house rupture due to dryness	D10
				Motor overload	D11
				Filter system clogged	D14
				Broken rings in the blade system	F1
				Broken inlet guide vane in the blade system	F2
				Vibration of shaft	F3
				Broken stationary blades in the casing system	F4
				Damaged casing in the cooling system	F5
				Rupture of the pad in the casing system	F6
				Broken housing of the thrust bearing	F7
				Broken shoes of the thrust bearing	F8
				Broken filler of thrust bearing	F9
				Pressure gauge failure	F11
				Failure of heating system	F12
				Vibration of gearbox bearing	K31
				Failure of gearbox self shifting clutch	K32
				Excessive hogging of the gearbox bearing	K33

To further assess the health state of the plants (i.e., the Afam gas turbine plant, Ibom gas turbine plant, and Sapele gas turbine plant, respectively), the mean time to failure, which is a reliability measurement parameter, was evaluated. The mean time to repair, characterized by the plant’s maintenance strategy, was evaluated, as shown in Table 3a–c, for the Afam gas turbine plant, Ibom gas turbine plant, and Sapele gas turbine plant, respectively. The failure rate and repair rate for the plant subsystems were predicted and used to model the formulated Markovian stochastic approach.

Table 3. System performance and energy generation loss due to plant health state ((a)—Afam Gas Turbine Plant, (b)—Ibom Gas Turbine Plant, and (c)—Sapele Gas Turbine Plant).

Likely Failure State	Cumulative Mean Time to Failure (hours)	Rate of Failure	Cumulative Mean Time to Repair (h)	Rate of Repair
(a) System performance and energy generation loss due to plant health state (Afam Gas Turbine Plant)
Axial shaft bearing leakage	622.98	1.61 × 10−3	200.9	4.98 × 10−3
System disturbance/low system voltage/grid failure	4499.7	2.22 × 10−4	398.9	2.51 × 10−3
Fan problem	483.45	2.07 × 10−3	287.8	3.47 × 10−3
Bus bar differential protection failure	3980.9	2.51 × 10−4	729.5	1.37 × 10−3
Medium pressure turbine differential expansion failure	90.8	1.10 × 10−2	25.3	3.95 × 10−2
Transformer leakage/GT oil leakage	105.98	9.44 × 10−3	65.6	1.52 × 10−2
Generator breaker failure	6789.9	1.47 × 10−4	698.8	1.43 × 10−3
Generator pole slippage	567.8	1.76 × 10−3	209.3	4.78 × 10−3
Relay system failure	386	2.59 × 10−3	117.1	8.54 × 10−3
Pump motor system failure	433	2.31 × 10−3	89.9	1.11 × 10−2
Blade system failure	98.9	1.01 × 10−2	76.7	1.30 × 10−2
Vane system failure	105.99	9.43 × 10−3	55.8	1.79 × 10−2
Thrust bearing failure	144.87	6.90 × 10−3	98.6	1.01 × 10−2
Radial bearing failure	981.3	1.02 × 10−3	288.7	3.46 × 10−3
Combustion system failure	166.9	5.99 × 10−3	67.3	1.49 × 10−2
Governing system failure	188	5.32 × 10−3	88.2	1.13 × 10−2
Turning gear failure	78.9	1.27 × 10−2	47.4	2.11 × 10−2
Seal house ruptured	487.2	2.05 × 10−3	209.7	4.77 × 10−3
Rotor bowing failure	312.8	3.20 × 10−3	112.5	8.89 × 10−3
Mechanical fault failure	478.6	2.09 × 10−3	101.4	9.86 × 10−3
Vibration induced failure	201.8	4.96 × 10−3	89.7	1.11 × 10−2
Gearbox system failure	812.8	1.23 × 10−3	398.1	2.51 × 10−3
Control system failure	109.87	9.10 × 10−3	67.8	1.47 × 10−2
Turbine system failure	81.98	1.22 × 10−2	109.6	9.12 × 10−3
Torque system failure	321.8	3.11 × 10−3	99.8	1.00 × 10−2
Igniter system failure	657.78	1.52 × 10−3	110.9	9.02 × 10−3
(b) System performance and energy generation loss due plant health state (Ibom Gas Turbine Plant)
Axial shaft bearing leakage	589.7	1.70 × 10−3	109.8	9.11 × 10−3
System disturbance/low system voltage/grid failure	2199.9	4.55 × 10−4	483.2	2.07 × 10−3
Fan problem	398.8	2.51 × 10−3	190.4	5.25 × 10−3
Bus bar differential protection failure	1778.1	5.62 × 10−4	478.9	2.09 × 10−3
Medium pressure turbine differential expansion failure	100.4	9.96 × 10−3	144.8	6.91 × 10−3
Transformer leakage/GT oil leakage	39.6	2.53 × 10−2	12.6	7.94 × 10−2
Generator breaker failure	4987.1	2.01 × 10−4	289.7	3.45 × 10−3
Generator pole slippage	345.3	2.90 × 10−3	118.7	8.42 × 10−3
Relay system failure	502.2	1.99 × 10−3	386.2	2.59 × 10−3
Pump motor system failure	298.9	3.35 × 10−3	113.1	8.84 × 10−3
Blade system failure	64.2	1.56 × 10−2	12.5	8.00 × 10−2
Vane system failure	211.8	4.72 × 10−3	56.7	1.76 × 10−2
Thrust bearing failure	238.9	4.19 × 10−3	102.4	9.77 × 10−3
Radial bearing failure	1011.9	9.88 × 10−4	597.8	1.67 × 10−3
Combustion system failure	189.6	5.27 × 10−3	45.6	2.19 × 10−2
Governing system failure	119.8	8.35 × 10−3	87.7	1.14 × 10−2
Turning gear failure	88.9	1.12 × 10−2	31.5	3.17 × 10−2
Seal house ruptured	507.1	1.97 × 10−3	105.6	9.47 × 10−3
Rotor bowing failure	402	2.49 × 10−3	209.2	4.78 × 10−3
Mechanical fault failure	339.9	2.94 × 10−3	89.6	1.12 × 10−2
Vibration induced failure	288	3.47 × 10−3	66.8	1.50 × 10−2
Gearbox system failure	908	1.10 × 10−3	206.2	4.85 × 10−3
Control system failure	118.9	8.41 × 10−3	47.5	2.11 × 10−2
Turbine system failure	67.9	1.47 × 10−2	109.7	9.12 × 10−3
Torque system failure	409.8	2.44 × 10−3	211.1	4.74 × 10−3
Igniter system failure	779.7	1.28 × 10−3	219.4	4.56 × 10−3
(c) System performance and energy generation loss due to plant health state (Sapele Gas Turbine Plant)
Axial shaft bearing leakage	543.8	1.84 × 10−3	112.1	8.92 × 10−3
System disturbance/low system voltage/grid failure	3887.3	2.57 × 10−4	897.5	1.11 × 10−3
Fan problem	354	2.82 × 10−3	89.7	1.11 × 10−2
Bus bar differential protection failure	4002.9	2.50 × 10−4	172.2	5.81 × 10−3
Medium pressure turbine differential expansion failure	97.9	1.02 × 10−2	56.3	1.78 × 10−2
Transformer leakage/GT oil leakage	105.98	9.44 × 10−3	57.1	1.75 × 10−2
Generator breaker failure	5986.7	1.67 × 10−4	456.8	2.19 × 10−3
Generator pole slippage	433.5	2.31 × 10−3	109.8	9.11 × 10−3
Relay system failure	408.4	2.45 × 10−3	99.8	1.00 × 10−2
Pump motor system failure	309.7	3.23 × 10−3	202.6	4.94 × 10−3
Blade system failure	79.9	1.25 × 10−2	89.9	1.11 × 10−2
Vane system failure	110.7	9.03 × 10−3	68.7	1.46 × 10−2
Thrust bearing failure	209.5	4.77 × 10−3	101.2	9.88 × 10−3
Radial bearing failure	1011.4	9.89 × 10−4	209.7	4.77 × 10−3
Combustion system failure	204.8	4.88 × 10−3	67.8	1.47 × 10−2
Governing system failure	108	9.26 × 10−3	77.8	1.29 × 10−2
Turning gear failure	104.8	9.54 × 10−3	56.2	1.78 × 10−2
Seal house ruptured	508.7	1.97 × 10−3	201.2	4.97 × 10−3
Rotor bowing failure	411.4	2.43 × 10−3	78.9	1.27 × 10−2
Mechanical fault failure	399.8	2.50 × 10−3	103.4	9.67 × 10−3
Vibration induced failure	199.2	5.02 × 10−3	56.7	1.76 × 10−2
Gearbox system failure	598.2	1.67 × 10−3	287.5	3.48 × 10−3
Control system failure	249.1	4.01 × 10−3	97.5	1.03 × 10−2
Turbine system failure	69.9	1.43 × 10−2	87.9	1.14 × 10−2
Torque system failure	246.5	4.06 × 10−3	109.8	9.11 × 10−3
Igniter system failure	789.8	1.27 × 10−3	243.9	4.10 × 10−3

5.2. State Probabilities Prediction Under Different Plants’ Performance

The predicted failure rate and the repair rate for the gas turbine were deduced from the failure and repair data and used to characterize the plant’s transition states for the prevailing operation condition. The mean time to failure is a reliability function that can be estimated given the failure rate for the period under consideration. The maintainability of the plant with respect to optimum energy generation and plant availability can be deducted based on the results for the three gas turbine plants. For the Afam Gas Turbine Plant, the transition rate, as shown in Figure 3, indicates the likely stochastic change in the plant behaviour with time to the point of total failure under an imperfect maintenance strategy. Similarly, Figure 4 and Figure 5 show the performance state transition for the Ibom gas turbine and Sapele turbine plants, respectively.

Figure 3. Turbine health state profile for a period of operation (Afam gas turbine plant).

Figure 4. Turbine health state profile for a period of operation (Ibom gas turbine plant).

Figure 5. Turbine health state profile for a period of operation (Sapele gas turbine plant).

The result of the prediction of the plant performance based on the estimated state probability for the operational, minor failure and major failure profiles are presented in Figure 3. Based on the existing maintenance strategy for the power plant, the predicted state transition for fourteen (14) years of operations gives a likely state probability of 79.6% for the major failure state for the Afam gas turbine plant. The major failure based on the FMEA can be attributed to the turbine blade damage, axial shaft bearing damage and rotor bearing failure. The complex failure mechanism of the gas turbine is due to interdependencies among the sub-systems. The power generation and investment sustainability are a function of the plant’s health state and availability for the period under consideration.

After fourteen (14) years of operation of the Ibom and Sapele gas turbine plants, the plant performance state indicates an 88.7% and 82.8% likelihood of a major failure, respectively, as shown in Figure 4 and Figure 5. The essence of this method is to predict the system state performance given the plant operating and maintenance strategy and how these health state probabilities will impact the overall energy generation with time.

The results indicate a progressing decline in the reliability of the gas turbine system, which reflects the effect of the degradation of the component over time. The health state profiles give a true representation of the state’s performance. To further monitor the plant performance, the reliability is estimated after fourteen (14) years of operation for the three power plants, respectively. The results show a 10.1%, 4.5%, and 7.8% decline in the system performance reliability for the Afam, Ibom and Sapele gas turbine plants, respectively. The key performance indicators for condition monitoring can be updated based on the result analysis to sustain system performance and energy efficiency. One such indicator is the application of a proactive integrity management strategy for optimizing system maintenance.

The current study demonstrates the novel application of an integrated stochastic Markovian approach for the assessment of the system reliability and energy performance of gas turbine power plants. The results of the instantaneous energy generation prediction, given the state of the plants and the state probabilities, are detailed in the subsequent Section 5.3.

5.3. Instantaneous Energy Performance Under System Degradation

The time evolution of instantaneous power generation can be predicted by defining the stochastic process of the performance degradation of the gas turbine. Figure 6, Figure 7 and Figure 8 show the mean instantaneous energy performance for the Afam, Ibom, and Sapele gas turbine plants. The result in Figure 6 indicates a progressive decline in the plant power generation as a function of system deterioration and time. The stochastic performance process is space- and time-dependent, capturing the instability of the failure elements and propagating their effect on the overall power output for the period. The difference in the upper and lower bound performance ratings reduces as the operating time increases. For instance, at fourteen (14) years of operation, the Afam plant shows a 37.9% mean performance based on the rated capacity. This reflects a 62.06% decrease in the plant power generation capability in comparison to the design rating. As discussed earlier, the rate of the turbine system deterioration and poor maintenance strategy play a key role in the overall power output, as shown.

Figure 6. Mean instantaneous energy performance under system degradation state and failed maintenance strategy (Afam gas turbine plant).

Figure 7. Mean instantaneous energy performance under system degradation state and failed maintenance strategy (Ibom gas turbine plant).

Figure 8. Mean instantaneous energy performance under system state and failed maintenance strategy (Sapele gas turbine plant).

Similarly, the mean instantaneous power generation for the Ibom and Sapele gas turbine plants, as shown in Figure 7 and Figure 8, gives mean performances of 35.1% and 46.3% relative to their rated capacities. This result shows that the Sapele gas turbine performs better relative to its rated capacity. The state of the plant performance in this study is conditioned by the system’s health state and the poor maintenance strategy by the management of the plants. The results provide a well-informed performance analysis that could aid integrity managers and management in developing the best maintenance and performance improvement strategies to provide an optimum return on investment.

The results further show the amount of power loss based on the system’s degradation and poor integrity management. For investment assessment and overall return analysis, the energy generated per kW can be predicted in line with the net expected value. The proposed approach provides a performance monitoring tool for cost and energy managers to allow them to carry out proactive planning and adopt a sustainable maintenance strategy.

Figure 9 shows the result of the comparative assessment of plant mean instantaneous power generation under the same constraints. The result indicates that, after fourteen (14) years of operation, under the prevailing deterioration of the subsystems, the mean instantaneous power output will be 31.2 MW, 27.9 MW, and 29.3 MW for the Afam, Ibom, and Sapele gas turbine plants, respectively. The percentage drop in the plant’s performance compared with the design performance rating indicates the worst system failure scenario. It is important to note that every system management strategy that performs below average, as shown in the case study, should be reviewed. An updated analysis of the investment measurement could be carried out using the profile in Figure 9 to predict the expected return and identify areas of intervention. An improved, cost-effective management strategy is needed to sustain the plant’s overall performance and minimize energy loss.

Figure 9. Comparative mean instantaneous energy performance under system degradation state and failed maintenance strategy for the three gas turbines.

The stochastic Markovian process has demonstrated the capacity for a condition-based reliability performance for power generation and system availability prediction of gas turbine power plants. The current model offers an energy performance and well-informed integrity management tool for plant performance monitoring for optimum decision-making and energy availability. The proposed approach performs better than approaches proposed in previous studies [5,19,21,26].

6. Conclusions

Our research objective was to develop a robust and adaptive stochastic model for reliability-based energy system performance evaluation considering unstable degradation and a poor maintenance strategy. The study emphasizes the application of the Markovian process to characteristics of the failure causes’ elements’ state transition during the gas turbine power generation. Three gas turbine plants of different capacities were evaluated, considering the prevailing operation of the asset to develop the FMEA, fault propagation structure, and state transition probability, and the overall mean instantaneous power generation was predicted for a period of 20 years of operation. The following are the key findings that can be drawn from the current study.

• The proposed approach provides a holistic assessment of the failure mode of the gas turbine plant, considering the prevailing operating environment.
• The failure and repair rates, which defined the reliability and maintainability function, were reliably predicted from the failure and repair data. This was used to characterize the state transition of the failure causal elements for the three gas turbine plants, as shown.
• The analysis shows a 10.1%, 4.5%, and 7.8% decline in the system performance reliability for the Afam, Ibom, and Sapele gas turbine plants, respectively.
• The subsystem degradation impacted the mean instantaneous power output by different percentage drops with respect to the design performance rating.
• Under the prevailing deterioration of the subsystems and poor maintenance strategy, the plant’s percentage performance relative to the rated capacity is 37.9%, 35.1%, and 46.3% for the Afam, Ibom, and Sapele gas turbine plants in the 14th year of operation. This shows a different level of performance decline and its implication on the economic viability of the investment.
• The approach provides a performance monitoring tool for gas turbine engine health assessment and energy performance prediction for any given operating state of the asset. A proactive integrity management strategy could be inferred from the analysis of the results to support decision-making in the management options of the gas turbine power plants.

The proposed model provides a real-time capability to monitor the system’s health state and predict the instantaneous energy performance of the gas turbine under degradation. However, the developed approach can be further enhanced in future research by integrating the Markov Decision Process and dynamic modelling tool, which is able to capture the unstable degradation-induced failure characteristics and dependencies among failure causative factors for optimum condition monitoring.