An Assessment of the Non-Repeatability of a Diesel Engine Cycle-by-Cycle Operation Under Variable Load and Speed Conditions

Abstract

The non-repeatability of the internal combustion engine’s cycle-by-cycle (CCN-R) operation directly affects pollutant emissions, fuel consumption, and energy efficiency. Reducing this non-repeatability is an important part of efforts to improve the environmental performance of power units. Cycle variability analysis allows the identification of engine operating areas that promote unstable combustion and increased emissions of harmful exhaust components. The aim of the study was to quantitatively assess the cycle-to-cycle non-repeatability COV of selected operating parameters of the Perkins 1104D-E44TA diesel engine. The analyses covered the maximum cylinder pressure (p_max), the mean indicated pressure (IMEP), and the crankshaft rotation angle corresponding to the occurrence of maximum pressure (α). The measurements were carried out on an engine dynamometer at 25 operating points, covering speeds 1000–2200 r./min and load torques 200–400 N × m, recording 500 consecutive operating cycles at each point. The results showed that the most stable engine operation occurred at medium rotational speeds and moderate loads, where COV_pmax values did not exceed 0.5% and COV_IMEP values were lower than 1.0%. Increased pmax non-repeatability (up to 2.10%) and very high α angle variability (up to 100–140%) were observed at high rotational speeds and high loads. Only in the case of COV_IMEP was a significant reduction in repeatability observed compared to idling. The results obtained from cycle-by-cycle non-repeatability analyses can ultimately, after being supplemented with exhaust gas composition testing, be used as tools to support engine control optimization in order to reduce pollutant emissions and improve combustion efficiency.

1. Introduction

The global trend towards electrification of transport is only present in selected countries, such as Norway [1]. According to [2], combustion engines are still the basis of heavy transport, construction, agriculture, and energy. Exxon’s forecasts for 2040 indicate that the share of diesel engines (CI) in global energy consumption will increase to 50%. This is a serious problem, as oil production, which is remaining at the same level or even declining, will not be able to meet the growing demand for fossil fuels, especially diesel fuel (DF). For this reason, alternative fuels to DF are still being sought. Estimates [3] confirm a 15% increase in the share of alternative fuels in the total fuel pool for engines by 2040. This is not an optimistic forecast, as it will only cover part of the global increase in energy demand [4]. This indicates insufficient action in the area of sustainable development and waste management [5]. On the other hand, estimates by the International Energy Agency (IEA) clearly indicate that the transport sector accounts for approximately 24–27% of global CO₂ emissions related to energy use, a significant part of which comes from compression ignition engines [6]. At the same time, emission standards are being systematically tightened (Euro 6/7, Stage V, Tier 2), covering not only average emission values but also their stability over time [7]. The non-repeatability of the combustion process leads to temporary increases in HC, CO, and PM emissions, which significantly affect the effectiveness of exhaust gas treatment systems [8,9]. Studies presented in [10,11] show that under conditions of increased cycle-to-cycle non-repeatability, HC emissions can increase by as much as (20–40)% and CO by (15–30)%, especially at low loads. For this reason, cycle-to-cycle non-repeatability (CCN-R) [12], which in some studies is referred to as cycle-to-cycle variations (CCV) [7], becomes a significant problem not only in terms of energy, but also in terms of operation and the environment.

In general terms, an internal combustion engine is a thermodynamic system in which the chemical energy of fuel is converted into mechanical work through a sequence of repetitive cycles/operating cycles. Under ideal conditions, each cycle is identical, but in real conditions, due to the imperfections of components and processes, significant differences between successive combustion cycles are observed [8,10,13]. It turns out that the CC-NR phenomenon occurs in both spark ignition and compression ignition engines. It is mainly intensified by: rotational speed, load, EGR rate, injection strategy resulting from ECU settings and adaptation, and the type of fuel used [14,15,16]. The coefficient of variation in mean effective pressure COV_IMEP is most often used for the quantitative assessment of N-RCCN-R/CCV. In the literature [8,9,17], the following COV_IMEP ranges are assumed to indicate the stability of an internal combustion engine:

• <2%—stable operation;
• 2–5%—normal operation, limits of perceptible instability;
• 5–10%—noticeable instability of operation, vibrations, decrease η_i;
• >10%—instability of operation, noticeable torque variation, significant decrease η_i and increased vibrations, occurrence of so-called misfires.

The CCN-R phenomenon is the result of the overlap of stochastic and deterministic phenomena. The main causes of this phenomenon are turbulence fluctuations in the cylinder, heterogeneity of the fuel–air mixture, variability of ignition/auto-ignition conditions, and local differences in temperature and mass of residual exhaust gases, including those from the EGR system [18]. Experimental studies have shown that even under steady operating conditions, the maximum combustion pressure p_max can vary by 5–15% between cycles, while the CA50 angle can vary by ±2–5 deg [19,20,21]. This means that a certain proportion of cycles/revolutions deviate from stable conditions, resulting in a decrease in the average indicated efficiency.

The energy effects of CCN-R are primarily a decrease in the efficiency of converting the chemical energy of the fuel into mechanical work. An increase in COV_IMEP from around 2% to 6–8% can lead to a decrease in η_i by 3–7%, an increase in BSFC by 4–10%, and an increase in energy losses in the form of heat and unburned fuel [10,22]. These effects are particularly pronounced in engines operating under lean combustion conditions and with a high EGR ratio exceeding 20–25%, where a sharp increase in N-RCC is observed [23,24].

From an economic point of view, CCN-R leads to increased fuel consumption and higher operating costs. The literature shows that cycle-to-cycle non-repeatability can increase average fuel consumption and reduce engine energy efficiency, which has a direct impact on operating costs [25]. Variable combustion conditions contribute to torque fluctuations and dynamic crankshaft vibrations, which in turn lead to higher dynamic loads and accelerated mechanical wear [26]. Furthermore, simulation and experimental studies indicate that high amplitudes of cyclic irregularities have a significant impact on predicted fuel consumption and emissions, and measurements based on average cycles may significantly underestimate this impact under real-world conditions [27].

In the context of reducing greenhouse gas emissions, alternative fuels are becoming increasingly important, particularly FAME, HVO, and various types of waste fuels, including pyrolytic fuels [28,29,30,31]. Due to its high CN value 70–90, low aromatic content, and repeatability of ignitions, HVO has a (20–40)% lower COV_IMEP value than diesel fuel, with a 1–2 deg decrease in COV_CA50 [29,32,33,34,35]. Waste fuels, on the other hand, due to their higher viscosity and variability in composition, often lead to an increase in COV_IMEP by 30–60%, especially at low loads and high EGR rates [24,36,37]. Despite the favorable CO₂ emissions balance in the LCA analysis, their impact on combustion stability and engine durability is not yet fully understood.

Despite the extensive literature on cycle-to-cycle non-repeatability, there is still a lack of comprehensive studies covering a wide range of speeds and loads simultaneously. Most of the work to date has been limited to selected engine operating points. This has made it difficult to relate the results to actual engine operating conditions, especially over a wider range of external engine characteristics.

With this in mind, the aim of the study was to determine the cycle-by-cycle unevenness of a diesel engine over a wide range of rotational speeds and loads. To achieve this goal, an engine with more than one cylinder and an engine dynamometer were used. It was recognized that tests on one-cylinder laboratory engines offer great cognitive opportunities, but do not reflect actual multi-cylinder engines, in which the operation of one cylinder affects the others. This clearly indicates the innovative nature of the study, in which cycle-by-cycle non-repeatability will be determined in relation to a number of operating parameters. Importantly, the tests will be conducted on a real engine, rather than on a laboratory one-cylinder engine, as is often the case, which also increases their applicability. The scientific contribution of this study will be maps of the distributions of the maximum pressure in the cylinder, the average indicated pressure, and the angle at which the maximum pressure occurs in the cylinder. Importantly, the results will be related to idling.

2. Materials and Methods

2.1. Subjects of the Research

The object of the study was a Perkins 1104D-E44TA (Perkins Engines Company Limited, global locations) four-cylinder diesel engine with a displacement of 4.4 L, equipped with a common-rail fuel supply system, a turbocharger, and an intake air cooler. The engine complied with the European EU Stage II/IIIA and American EPA Tier 2/Tier 3 emission standards for machines intended for off-road use. The basic technical data of the test object is presented in

Table 1. Basic technical parameters of the Perkins 1104D-E44TA (Perkins Engines Company Limited, global locations) engine [38].

Parameter	Description, Value
engine type	CI, TDI, with intercooler
number of cylinders/valves	4/16
diameter × piston stroke	105 mm × 127 mm
displacement	4400 cm3
compression ratio	16.2:1
max. power/rotational speed	90.1 kW/2200 r./min
max. torque/rotational speed	514 N × m/1400 r./min
combustion chamber	ω type
max. boost pressure	1.8 bar
fuel system	high-pressure, 2-phase common-rail
number of openings in the injector	6
max. fuel pressure	1800 bar
crank radius	63.5 mm
connecting rod length	219.1 mm

.

The tested engine was fueled with diesel fuel with a 5% addition of bio-components, sourced from the ORLEN network. The CN number of the fuel was min. 51 according to PN-EN ISO 5165 [39].

2.2. Test Stand

An Automex engine dynamometer was used to conduct the tests (Figure 1). The test engine, 2, and brake, 3, were mounted on the frame, 1, using an auxiliary frame and vibration isolators. The engine was connected to the brake using a shaft, 4, equipped with couplings and compensating joints. The engine was supplied with fuel, 5, and air, 6, and exhaust gases, 7, were removed and cooled, 8. The test stand was equipped with a number of additional sensors monitoring the operation of the system, which were not directly related to the test cycle. These included: sensors for the temperature of the engine and brake coolant, oil, intake air, and exhaust gases; oil, fuel, and boost pressure; an exhaust gas composition analyzer; and air and fuel flow meters. The ventilation system controlled the air exchange in the room to ensure constant engine operating conditions.

An Elektromex EMX—200/6000M (Elektromex Centrum, Bychlew, Poland) eddy current brake was used to apply the load to the tested engine. Its basic technical data is presented in

Table 2. Basic technical data of the EMX—200/6000M (Elektromex Centrum, Bychlew, Poland) eddy current brake [40].

Parameter	Value
max. power consumption	200 kW
max. speed	6000 r./min
max. torque	700 N × m
coolant requirement	6 m3/h

.

To analyze the cycle-by-cycle non-repeatability, it was necessary to measure the cylinder pressure, angular position, and rotational speed of the crankshaft, as well as the engine load. To achieve this goal, the equipment used is presented in Figure 2.

The basic technical data of the testing equipment is presented in

Table 3. Basic technical data of the AVL GH13P pressure sensor [41].

Parameter	Value
measuring range	(0–250) bar
nominal sensitivity	15 pC/bar
sampling frequency	every 0.1 deg (~150 kHz at 1500 r./min)
linearity	</= ±0.3% FSO
operating temp. range	to 400 °C

,

Table 4. Basic technical data of the AVL 365C optical angle encoder [42].

Parameter	Value
max. rotational speed	20,000 r./min
overload capacity	ok. 400 g
permissible operating temp. (electronics)	−40 °C to 70 °C
permissible operating temp. (mechanical/optical)	−40 °C to 120 °C
type of analysis	rotating, swiveling
output resolution I	3600, 1800, 720, …, 36 pulses/rotation
output resolution II	720, …, 36 pulses/rotation

and

Table 5. Basic technical data of the Keli type DEE force transducer [43].

Parameter	Value
rated capacities	300 kg
sensitivity	2 ± 0.003 mV/V
TC zero	±0.002% F.S. 10 °C
TRC span	±0.002% F.S. 10 °C
operating temp.	−30 to +70 °C
max. safe overland	120% F.S.

. The accuracy and sensitivity of the testing equipment indicated therein ensured the required values for further analysis. The torque of the load was determined by multiplying the force sensor readings by the length of the arm on which it was mounted.

Automex test bench control software [44] was used to regulate the engine operating conditions (Figure 3a). It was possible to plan a custom test scenario with variable parameters defined by the user. On the other hand, AVL IndiCom Mobile^TM software [45] was used to record the parameters required for the cycle-by-cycle analysis of engine operation non-repeatability software (Figure 3b) was used to record the parameters required for the analysis of cycle-by-cycle non-repeatability engine operation. This made it possible to record the maximum cylinder pressures p_max, the mean indicated pressure IMEP, and the angle α at which p_max occurred in individual cycles. The AVL IndiCom Mobile^TM software allowed for the determination of statistical parameters and the assessment of non-repeatability.

The AVL IndiCom Mobile^TM software also enabled the recording of pressure traces in the cylinder (Figure 4a), which allowed for accurate visualization of differences throughout the entire operating cycles (Figure 4b). However, due to the large volume of output files with a sufficient number of operating cycles for analysis, this type of comparative assessment was abandoned. In the literature, non-repeatability is usually assessed on the basis of peak pressure values p_max in the cylinder, which was also adopted as a determinant in the analyses in this study.

2.3. Research Methods

The assessment of the non-repeatability of the combustion engine’s operation cycle-by-cycle was carried out according to the plan presented in (

Table 6. Experiment plan.

T N × m	400	5.	10.	15.	20.	25.
	350	4.	9.	14.	19.	24.
	300	3.	8.	13.	18.	23.
	250	2.	7.	12.	17.	22.
	200	1.	6.	11.	16.	21.
	idle	1000	1300	1600	1900	2200
n, r./min

).

The experiment plan was not dependent on the WHSC and WHTC homologation cycles, as these procedures are used to assess exhaust emissions rather than to analyze the variability of combustion parameters cycle by cycle. The rotational speeds were increased by 300 r./min starting from 1000 r./min, while the loads were increased by 50 N × m starting from 200 N × m, which allowed for a total of 25 measurement points. As part of the comparison, additional tests were carried out on the engine running at idle speed, without load.

By plotting the test points on the external characteristics of the engine (Figure 5), the test range was highlighted, covering a large part of the area under the maximum torque curve. When creating the test plan, care was taken not to approach the maximum values of the torque curve in order to avoid engine failure. It should be noted here that the torque curve mapped IMEP and boost pressure (curve shape).

At each of the 25 points in the test plan, 500 cycles were recorded using AVL IndiCom Mobile^TM (at a α resolution of 1 deg) for the following values:

• maximum cylinder pressure—p_max;
• mean indicated pressure—IMEP;
• crankshaft rotation angle at which p_max occurs—α.

In AVL IndiCom [45], the maximum cylinder pressure p_max was determined directly from the pressure trace as a function of the crankshaft angle for each individual cycle. The algorithm analyzed the signals step by step, applying preprocessing by limiting high-frequency noise, correcting the pressure sensor offset, and correcting the TDC position. Importantly, the algorithm limited the search p_max only to α close to TDC (several degrees before and several degrees after), eliminating pressure peaks during suction and exhaust from the identification. Preliminary angle α analyses showed that the algorithm contained in AVL IndiCom Mobile^TM takes into account a α step of 1 deg when determining cycle-by-cycle non-repeatability. The exact signal processing procedures in AVL IndiCom Mobile^TM are not precisely described in the manufacturer’s documentation.

• In general, IMEP is determined on the basis of the relationship [12].

$$\mathrm{IMEP}={\displaystyle \frac{W}{V_{sc}}}={\displaystyle \frac{1}{V_{sc}}}{\displaystyle \oint p\hspace{0.17em}}\mathrm{d}{V}_{sc}$$

(1)

• The non-repeatability of the cycle-by-cycle engine operation was evaluated using the coefficient of variation [47,48,49].

$$\mathrm{COV}=\left({\displaystyle \frac{\sigma }{M}}\right)\times 100\%$$

(2)

• The standard deviation was calculated.

$$\sigma =\sqrt{{\displaystyle \frac{1}{N-1}}{\displaystyle \sum _{i=1}^N{\left(x_i-M\right)}^2}}$$

(3)

• The average value.

$$M={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{x}_i}$$

(4)

• Based on Equation (1), COV was calculated for p_max, IMEP, and α.

3. Results

3.1. Analysis of the Non-Repeatability of Maximum Pressure in the Cylinder

The characteristic values from 500 engine cycles obtained during registration using AVL IndiCom software (AVL List GmbH, Graz, Austria) [45] were further processed statistically using Matlab 2025b software (The MathWorks, Inc., MA, USA) [50]. After importing the data, the scatter of points was analyzed, from which a histogram was created, followed by a box plot and a probability plot (Figure 6). A summary of the results of the statistical analysis relating to non-repeatability p_max is presented in

Table A1. Results of statistical analysis of p_max values.

Par./Seq.	1.	2.	3.	4.	5.	6.	7.	8.	9.	10.	11.	12.	13.
M, bar	71.767	77.947	84.976	92.269	94.292	66.505	71.771	76.484	81.037	93.623	79.507	85.803	92.616
Me, bar	71.731	77.966	84.995	92.268	94.270	66.501	71.777	76.475	81.045	93.830	79.517	85.794	92.608
max., bar	73.687	80.150	87.186	95.592	96.951	67.360	72.885	77.870	82.602	97.483	80.624	86.992	93.930
min., bar	69.776	75.618	82.631	89.476	91.398	65.572	70.340	75.157	79.308	88.032	78.272	84.968	91.650
Mo, bar	71.864	77.728	84.542	90.726	93.601	66.482	71.561	76.184	80.850	89.703	79.351	85.343	92.625
σ	0.603	0.796	0.754	0.926	1.001	0.313	0.367	0.441	0.492	1.970	0.396	0.350	0.360
σ2	0.363	0.634	0.569	0.857	1.003	0.098	0.135	0.194	0.242	3.883	0.157	0.123	0.130
κ	3.286	3.010	2.950	3.447	2.621	3.072	3.556	3.258	2.985	2.959	2.821	2.905	3.548
s	0.173	−0.106	−0.053	0.093	0.040	−0.153	0.015	0.109	−0.031	−0.648	0.086	0.285	0.349
COV, %	0.840	1.021	0.888	1.003	1.062	0.471	0.512	0.576	0.607	2.105	0.498	0.408	0.389
K&S	1	1	1	1	1	1	1	1	1	1	1	1	1
par./seq.	14.	15.	16.	17.	18.	19.	20.	21.	22.	23.	24.	25.	idle
M, bar	99.538	105.326	88.004	97.390	102.994	106.885	108.441	97.665	101.948	106.278	107.995	111.763	52.589
Me, bar	99.530	105.370	88.055	97.388	102.955	106.420	108.160	97.657	101.930	106.230	107.940	111.790	52.526
max., bar	100.740	107.510	89.588	98.848	104.720	109.990	112.860	99.073	103.560	108.410	109.680	113.470	54.206
min., bar	98.670	103.140	86.350	95.889	101.410	105.060	106.950	96.587	100.680	104.870	106.710	109.910	51.102
Mo, bar	99.570	105.260	87.518	97.227	102.850	106.080	107.820	97.657	101.960	106.100	107.790	111.860	52.569
σ	0.355	0.735	0.618	0.481	0.611	1.270	1.073	0.406	0.447	0.545	0.513	0.606	0.615
σ2	0.126	0.540	0.382	0.231	0.373	1.613	1.150	0.165	0.200	0.298	0.264	0.367	0.378
κ	3.042	2.829	2.687	3.139	2.641	2.475	6.993	3.305	3.210	3.465	3.125	3.085	2.207
s	0.228	−0.081	−0.273	0.072	0.232	0.869	1.965	0.281	0.207	0.434	0.340	−0.240	0.227
COV, %	0.357	0.698	0.702	0.493	0.593	1.188	0.989	0.416	0.438	0.513	0.475	0.542	1.170
K&S	1	1	1	1	1	1	1	1	1	1	1	1	1

. The table also contains the result of the assessment of the normality of the distribution p_max in relation to the theoretical distribution according to the K&S test [51] for a certain finite number of observations (in this case 500).

Analyzing the sample results for sequence 1 of the experimental plan, at 1000 r./min and 200 N × m (Figure 6a and Table A1), the values p_max range from 69.78 to 73.69 bar, with an average of M = 71.77 bar. No clear upward or downward trend was observed, suggesting process stability over time. A slight increase p_max may have been due to the engine control system adapting to the load conditions. The histogram (Figure 6b and Table A1) showed a standard deviation of σ = 0.603, which was a moderate spread. The skewness reached a slight right-sided value of s = 0.1727. The kurtosis κ = 3.286 indicated a slightly leptokurtic distribution, close to normal, for which κ = 3 [50,52]. The logical value 1 (Table A1) in the K&S test showed that the distribution of results was consistent with the normal distribution. The median shown in Figure 6c was M_e = 71.731 bar, and the interquartile range was 0.779 bar, which indicated a slight variability of results. The frame limits (Figure 6c) corresponded to the quartiles Q₁ = 72.1585 bar and Q₃ = 71.3795 bar, while the whiskers reached 70.22 and 73.29 bar, respectively. In sequence 1 of the experimental plan, seven outliers were identified that fell outside the typical range of observations. The quantile–quantile plot (Figure 6d) allowed us to assess the conformity of the empirical distribution p_max with the normal distribution; the slight deviations visible in the tails were due to the presence of right-skewness (s = 0.1727). Due to the extensive nature of the results, the numerical results for the remaining sequences of the experimental design are summarized in (Table A1).

When analyzing the overall non-repeatability p_max (

Table 7. Histograms of dispersion p_max for individual sequences of the experimental design.

n, r./min
T, N × m		1000	1300	1600	1900	2200
	400
	350
	300
	250
	200
idle

), differences in the shape of the distributions were observed. Most of them, especially those for lower rotational speeds 1000–1600 r./min and lower loads 200–300 N × m, took on a shape similar to normal, maintaining symmetry and concentration around the mean value. This was confirmed by low values of s (close to 0) and κ (oscillating around 3), as well as low coefficients of variation COV_pmax (below 0.6%) (Table A1). In these engine operating ranges, the combustion process was stable, with fewer anomalies resulting in less variation in maximum pressure. The most noticeable deviations from the normal distribution occurred at a speed of 1900 r./min and a load of 350 N × m. The histogram showed a clear right-sided trend with strong asymmetry, which was confirmed by s = 0.8686. The value of κ = 2.4747 was close to normal, with a slight indication of leptokurtic distribution [53,54]. The flattening of the distribution is due to the presence of two clusters of different sizes (multimodal distribution). The K&S test indicated a logical value of 1, although the distribution matched the left, larger cluster of data. Similar, though less pronounced, disturbances in the distribution were observed at 2200 r./min and 400 N × m, where the results were shifted to the right and more dispersed, indicating greater variability in the maximum pressure in the cylinder.

In the case of idling, an increased spread of values and asymmetry of the distribution were observed, and the shape of the distribution suggested noticeable instability of combustion at idle speed, which resulted from the lack of load that stabilizes engine operation and increased adaptation of the control module [8].

A summary of the non-repeatability COV_pmax for individual sequences of the experimental plan (rotational speeds and loads) is presented in Figure 7a. The lowest COV_pmax values (<0.5) were recorded in the range of medium loads 250–350 N × m and medium to higher rotational speeds 1600–2200 r./min. This indicates relatively stable combustion conditions, which may be the result of the required cylinder filling and the effective process of self-ignition and flame front propagation resulting from the required fuel dose [55,56]. Such engine operating parameters were characterized by high repeatability of combustion cycles and favorable fuel atomization in the combustion chamber. The highest level of non-repeatability, COVp_max = 2.10%, was recorded for the sequence 1300 r./min and 400 N × m. The reason for this could have been the instability of the supercharging, combined with the fuel injection angle correction declared by the engine ECU, which adversely affected the efficiency of the combustion process.

In addition, the non-repeatability of individual sequences of the experimental plan was referred to idling, where the rotational speed was 800 r./min with no load (Figure 7b). In this case, the results at the test points were subtracted from the value obtained for idling, COV_pmax = 1.696%. A clear increase of 0.94% in non-repeatability p_max was recorded at 1300 r./min and 400 N × m. This was due to combustion instability under high load at speeds close to the maximum torque, where boost is precisely controlled by the feedback system [56]. At 1900 r./min and 350 N × m, COVp_max was similar (0.02%) to idling. At the other test points, COV_pmax had a lower value than for idling, which is consistent with [8,12]. The largest decrease in COV_pmax (0.81%) relative to idle speed was recorded for the point 1600 r./min and 350 N × m, which suggests a significant improvement in stability under these conditions. The general trend (Figure 7a) indicates increased COV_pmax values at low rotational speeds and high loads. This may be due to the extended cycle time and the difficulty in ensuring a homogeneous mixture composition under high load conditions [8].

In summary, changes in COV_pmax (Figure 7a,b) are a useful diagnostic tool for identifying areas of engine operation where increased combustion instability occurs. The results can be used as a basis for further optimization of engine operating parameters, ignition control strategies, and the fuel supply system, which will translate into improved efficiency and reliability of the power unit.

3.2. Analysis of the Non-Repeatability of the Indicated Mean Effective Pressure

Similarly to p_max, in order to illustrate IMEP statistical analyses, sequence 1 from the experimental plan (1000 r./min and 200 N × m) is presented as an example (Figure 8).

The IMEP values ranged from 6.31 to 6.78 bar, with an average of M = 6.5 bar. The lack of a clear upward or downward trend suggests that the process of generating the indicated pressure remains relatively stable over time (Figure 8a). The visible minor fluctuations may have been due to slight differences between cycles or disturbances. Individual extreme observations may indicate incidental disturbances.

The IMEP values with the normal distribution curve superimposed are shown in (Figure 8b). The standard deviation σ = 0.087 (

Table A2. Results of statistical analysis of IMEP values.

Par./Seq.	1.	2.	3.	4.	5.	6.	7.	8.	9.	10.	11.	12.	13.
M, bar	6.504	7.694	8.825	10.016	10.743	6.719	8.024	9.186	10.345	11.575	6.999	8.200	9.386
Me, bar	6.495	7.687	8.819	10.019	10.747	6.723	8.021	9.189	10.345	11.574	7.000	8.205	9.392
max., bar	6.780	8.035	9.155	10.306	10.960	6.924	8.240	9.405	10.584	11.987	7.320	8.513	9.649
min., bar	6.305	7.400	8.584	9.698	10.483	6.521	7.819	8.929	10.109	11.134	6.680	7.869	9.043
Mo, bar	6.550	7.529	8.780	10.017	10.673	6.653	8.004	9.180	10.296	11.616	6.798	8.110	9.226
σ	0.087	0.111	0.093	0.127	0.092	0.071	0.073	0.082	0.087	0.157	0.112	0.115	0.110
σ2	0.008	0.012	0.009	0.016	0.009	0.005	0.005	0.007	0.008	0.025	0.013	0.013	0.012
κ	2.409	2.724	2.906	2.577	2.381	2.934	3.119	3.165	2.722	2.534	2.618	2.615	2.683
s	0.209	0.137	0.174	−0.109	−0.208	−0.170	0.069	−0.193	−0.133	0.030	0.122	−0.188	−0.161
COV, %	1.337	1.444	1.059	1.271	0.859	1.049	0.912	0.895	0.844	1.357	1.606	1.402	1.167
K&S	1	1	1	1	1	1	1	1	1	1	1	1	1
par./seq.	14.	15.	16.	17.	18.	19.	20.	21.	22.	23.	24.	25.	idle
M, bar	10.565	12.161	7.490	8.725	9.904	11.003	12.242	7.571	8.906	10.311	11.510	12.657	1.275
Me, bar	10.570	12.160	7.483	8.711	9.893	10.995	12.233	7.580	8.911	10.318	11.517	12.654	1.268
max., bar	10.928	12.526	7.823	9.070	10.354	11.407	12.786	7.975	9.248	10.659	12.007	13.302	1.491
min., bar	10.144	11.823	7.151	8.441	9.629	10.693	11.925	7.177	8.501	9.944	10.991	12.045	1.073
Mo, bar	10.458	12.171	7.478	8.683	9.840	10.945	12.207	7.476	8.756	10.400	11.459	12.614	1.164
σ	0.159	0.138	0.127	0.115	0.117	0.121	0.122	0.134	0.139	0.134	0.158	0.212	0.088
σ2	0.025	0.019	0.016	0.013	0.014	0.015	0.015	0.018	0.019	0.018	0.025	0.045	0.008
κ	2.399	2.513	2.968	2.924	2.911	3.305	3.708	3.112	2.902	2.820	3.314	2.796	2.137
s	−0.191	0.010	0.071	0.365	0.278	0.488	0.613	−0.178	−0.297	−0.129	−0.073	−0.152	0.148
COV, %	1.502	1.131	1.691	1.321	1.185	1.099	0.998	1.775	1.559	1.302	1.377	1.677	6.925
K&S	1	1	1	1	1	1	1	1	1	1	1	1	1

) indicated a small spread of values. The skewness index s = 0.2085 suggested an almost symmetrical distribution, while the kurtosis at κ = 2.4087 indicated a platykurtic distribution, slightly deviating from the normal distribution. The shape of the histogram indicated compliance with the normal distribution, which was confirmed by the results of the K&S test. 1. The dispersion of results (Figure 8c) gave a median M_e = 6.495 bar, the interval between the lower quartile Q₁ = 6.4354 bar and the upper quartile Q₃ = 6.5727 bar formed an interquartile range of 0.1373 bar. This value indicated limited cyclical variability of the parameter. The whiskers of Figure 8c covered data 6.3048–6. 779 bar, with one outlier. The quantile–quantile plot (Figure 8d) showed only slight deviations at the ends of the distribution, which resulted from the presence of a slight right-skewness (s = 0.2085).

Based on a collective analysis of histograms (

Table 8. Histograms of dispersion IMEP for individual sequences of the experimental design.

	n, r./min
T, N × m		1000	1300	1600	1900	2200
	400
	350
	300
	250
	200
idle

and Table A2), it was found that most distributions showed relatively good agreement with the normal distribution. They were symmetrical, well-fitted to the theoretical curve, and did not contain significant deviations or clear outliers. In many cases, the statistical parameter values immediately confirmed the stability and predictability of the engine’s operation. The idle histogram clearly deviated from the normal distribution. The IMEP values were asymmetrically distributed, with left-skewedness (s = 0.1476). A clear peak was noticeable, suggesting an excess of low values. For this distribution, κ = 2.137, which indicated a flattened leptokurtic distribution. The density curve did not fit the data distribution well, which indicated unstable engine operation under no load, resulting from irregular combustion. At a speed of 2200 r./min and a load of 400 N × m, the histogram was clearly asymmetrical. There was a negative skew (s = −0.1524), and the results took on a wider distribution than would be predicted by the normal model. The value κ = 2.7959 indicated a noticeably leptokurtic distribution. Similarly, at 2200 r./min and 350 N × m, there was a clear negative skew (−0.0729) and a slight sharpening of the peak of the distribution. κ = 3.3137, which indicated a platykurtic distribution. This suggested a greater dispersion of IMEP values and a lower frequency of values close to the mean. The standard deviation, equal to σ = 0.1584, confirmed the higher variability of the indicated pressure at high load torque. For a rotational speed of 1600 r./min and a load of 250 N × m, the histogram was slightly bimodal. Two local maxima were visible, suggesting that the results could have been due to changing combustion conditions. The negative skewness s = −0.1875 clearly indicated an asymmetric distribution with a longer tail on the left side. The value κ = 2.6154 also deviated slightly from the values typical for a normal distribution, which emphasized the complex nature of this measurement case. Overall, all IMEP histograms showed a better fit to the normal distribution at medium speeds and moderate load—the results were then more symmetrical and compact, indicating more stable combustion conditions. At low speeds, especially at idle, clear asymmetry and skewness were observed. In turn, at high speeds and high loads, the histograms become more stretched and skewed. An increase in torque favored a wider distribution, while moderate loads are associated with more regular engine operation [8,12].

By comparing COV_IMEP values (Figure 9a), it was shown that the engine operates most stably at a speed of 1300 r./min and load torques 250–350 N × m, where the COV_IMEP coefficient is <1.00. In turn, the highest level of non-repeatability (above 1.6) was observed for rotational speeds 1600–2200 r./min and a load torque of 2000 N × m, in one case at 2200 r./min and a load of 400 N × m. The variability of the COV_IMEP coefficient depending on the load and speed indicated that operation at low torque and high speed is associated with increased randomness of the combustion process. On the other hand, increasing the load resulted in an overall decrease in the non-repeatability coefficient, which suggested more favorable thermal conditions. However, all COV_IMEP values were below 2, which according to [8,9,17] means that the engine was running stably.

When evaluating the COV_IMEP differences for individual test points and idle speed, for which COV_IMEP = 6.9246% (Figure 9b), it was noted that the largest decreases 5.56–6.08% occur in the speed ranges 1000–1300 r./min and load torque 300–400 N × m. In turn, the smallest deviation (5.15%) was recorded under conditions of high rotational speed 2200 r./min and low load 200 N × m. Overall, at all test points, there was a decrease in engine irregularity of 5.1–6.1% compared to idling. The differences at individual test points of the experiment are consistent with the observations presented in [8].

The COV values for p_max and IMEP (Figure 7a and Figure 9a) indicated that while in the case of p_max the differences compared to idling could increase by up to 1% in two cases, in the case of IMEP, a decrease of more than 5% was observed at all analyzed points.

In summary, COV_IMEP values were higher than COV_pmax values, but showed a greater advantage over idling in all cases tested. This is important in the context of engine ECU calibration and the assessment of combustion efficiency under various load conditions.

3.3. Analysis of the Non-Repeatability of the Angle at Which Maximum Pressure Occurs in the Cylinder

Similarly to previous cases, sequence 1 from the experimental plan (1000 r./min and 200 N × m) was presented as an example of statistical analysis of the angle α. The values α oscillated around 9 and 10 deg, with visible repeating peaks (Figure 10a). The average value was approximately M = 9.64 deg, standard deviation, σ = 0.64 (

Table A3. Results of statistical analysis of α values.

Par./Seq.	1.	2.	3.	4.	5.	6.	7.	8.	9.	10.	11.	12.	13.
M, deg	9.644	9.794	9.776	9.886	10.750	0.804	1.444	1.206	1.346	2.498	2.786	2.370	2.452
Me, deg	10.000	10.000	10.000	10.000	11.000	1.000	1.000	1.000	1.000	3.000	3.000	2.000	2.000
max., deg	11.000	11.000	11.000	11.000	12.000	2.000	3.000	3.000	3.000	4.000	4.000	4.000	4.000
min., deg	8.000	8.000	9.000	9.000	9.000	0.000	1.000	0.000	0.000	−1.000	2.000	2.000	2.000
Mo, deg	10.000	10.000	10.000	10.000	11.000	1.000	1.000	1.000	1.000	3.000	3.000	2.000	2.000
σ	0.644	0.610	0.484	0.483	0.569	0.720	0.544	0.456	0.666	0.764	0.530	0.492	0.502
σ2	0.414	0.372	0.234	0.234	0.324	0.519	0.296	0.208	0.443	0.583	0.281	0.242	0.252
κ	2.923	3.209	2.949	3.918	3.029	1.964	2.313	6.491	3.176	5.459	2.880	1.671	1.181
s	−0.186	−0.228	−0.492	−0.292	−0.216	0.311	0.673	1.999	0.454	−0.980	−0.164	0.640	0.240
COV, %	6.673	6.230	4.951	4.888	5.296	89.571	37.643	37.815	49.458	30.571	19.019	20.739	20.481
K&S	1	1	1	1	1	1	1	1	1	1	1	1	1
par./seq.	14.	15.	16.	17.	18.	19.	20.	21.	22.	23.	24.	25.	idle
M, deg	2.058	1.168	1.418	1.204	1.512	0.848	1.082	1.212	1.058	1.186	0.754	1.158	2.542
Me, deg	2.000	1.000	1.000	1.000	1.500	1.000	1.000	1.000	1.000	1.000	1.000	1.000	3.000
max., deg	3.000	3.000	3.000	3.000	3.000	3.000	3.000	3.000	3.000	3.000	3.000	4.000	4.000
min., deg	1.000	0.000	1.000	0.000	0.000	−1.000	−1.000	0.000	−1.000	−1.000	−1.000	−2.000	1.000
Mo, deg	2.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.000	2.000	3.000
σ	0.383	0.676	0.510	0.613	0.609	0.771	0.788	0.783	0.822	0.783	1.058	1.161	0.559
σ2	0.147	0.457	0.260	0.375	0.371	0.594	0.621	0.612	0.676	0.613	1.120	1.348	0.313
κ	6.490	4.241	1.699	2.788	2.639	3.212	3.391	2.436	2.618	3.365	2.394	2.700	2.402
s	0.575	0.837	0.514	0.010	0.114	0.504	−0.022	0.088	−0.367	0.214	0.218	−0.264	−0.510
COV, %	18.626	57.859	35.945	50.871	40.262	90.893	72.803	64.561	77.711	65.996	140.341	100.252	22.004
K&S	1	1	1	1	1	1	1	1	1	1	1	1	1

). The lack of a long-term trend confirmed the stability of the cyclic variability. The values α, despite being grouped into four areas, were distributed asymmetrically around the value of 10 deg (Figure 10b). The distribution was slightly right-skewed, practically normal. There was a concentration in two local maxima (Figure 10c). The kurtosis value κ was close to 3, which, despite the small spread of values, confirmed the conformity of the distribution with the normal distribution. The quartiles were Q₁ = 9 and Q₃ = 10, respectively, and the interquartile range was 1 deg. There were no outliers below 8 deg or above 11 deg, suggesting no incidental disturbances or cycles deviating from the typical course. The quantile-quantile plot (Figure 10d) showed consistency with the normal distribution, especially in the middle range.

When evaluating the histograms of the angle distribution α (

Table 9. Histograms of dispersion α for individual sequences of the experimental design.

	n, r./min
T, N × m		1000	1300	1600	1900	2200
	400
	350
	300
	250
	200
idle

) collectively, deviations from the normal distribution were observed in some cases, manifested by multimodality (more than one local maximum appears) and asymmetry. These features were particularly pronounced at low loads 200 N × m and extreme rotational speeds 1000 and 2200 r./min. Under these conditions, the distributions are irregular in shape, often flattened or asymmetrical. For example, at 200 N × m and 1000 r./min, there is a noticeable uneven distribution of data around the mean, resulting from the presence of two local maxima. The best compliance with the normal distribution was observed in cases with average loads 250–350 N × m and moderate rotational speeds 1300–1600 r./min. Under these conditions, the histograms had a single, well-defined peak, and the results were symmetrical about the mean.

For a load of 300 N × m and a rotational speed of 1300 r./min, the statistical parameters (Table A3) indicated high repeatability of self-ignition and stability of the combustion process. As the engine speed increased, a shift α closer to TDC was observed. For example, for a rotational speed of 1000 r./min, the average angle value is approximately 10 deg, while for 2200 r./min it is approximately 1 deg, which may indicate a relative increase in self-ignition delay in terms of the crankshaft rotation angle at faster piston movement. At the same time, an increase in the dispersion of values α was observed, the standard deviation increased from 0.64 for sequence 1 to 1.16 for sequence 25, and the shape of the histograms lost its regularity.

Based on comparisons of the studies conducted, a clear correlation was observed between the angle α and p_max. In summary, the most stable and repeatable combustion process occurs at medium load and moderate rotational speeds, where the distribution of the angle α best matched the normal distribution. Under extreme conditions (high rotational speeds or low loads), the randomness of occurrence α increased, which affected the pressure curve in the cylinder and the overall efficiency of the engine.

Comparing the COV_α non-repeatability results (Figure 11a), it was found that the lowest level occurred at low rotational speeds 1000–1300 r./min, medium and high loads 250–400 N × m, where the coefficient values ranged from 4.89 to 6.23%. This meant that in this range, combustion was most repeatable from cycle to cycle. An increase in rotational speed and load resulted in an increase in non-repeatability. This was particularly evident at a rotational speed of 2200 r./min and a load of 400 and 350 N × m, where COV_α = (100–140)%. This suggested that under high load and high rotational speed conditions, combustion becomes more unstable. This was the result of a shortened cycle time, increased air flow turbulence, and differences in thermal and pressure conditions between cycles. COV_α values for rotational speeds 1200–1900 r./min ranged from 18.63 to 50.87%. There were sequences with results deviating from this rule, such as at a rotational speed of 1300 r./min and a load of 200 N × m, where the expected non-repeatability should be approximately 30%. However, probably due to supercharging disturbances, COV_α = 89.57%. High COV_α values could, in general terms, result from the mathematical relationship describing it. When the values of the α were assessed relative to TDC (then α = 0 deg), COV_α could take on high values, regardless of the actual physical dispersion of the combustion process.

By calculating the difference between the individual sequences of the COV_α experiment plan and the value obtained for idling (22.00%), it was found that for 1000 r./min, a decrease in non-repeatability was observed under all load conditions, with difference values of 15.33–16.71% (Figure 11b).

This meant that the engine ran more stably than in the no-load condition. In the range of average rotational speeds 1300–1900 r./min, the situation was more varied. There were both points with slight improvements, such as 3.38% at 1600 r./min and 350 N × m, and clear deteriorations, such as 118.34% at 2200 r./min and 350 N × m. A significant anomaly was found at a rotational speed of 1300 r./min and a load of 1300 N × m, where an increase in COV_α relative to idle speed of 8.51% was recorded, and this point was characterized by the greatest unrepeatability of COVp_max and COV_IMEP. This indicates insufficient injection angle adjustment, which affected the angle α.

In summary, a comparison of COV_α values at test points showed that the most stable engine operation occurs at low speeds and medium loads. An increase in rotational speed and load torque leads to a significant deterioration in repeatability, which can negatively affect the operating culture, efficiency, and durability of engine components.

4. Discussion

The cycle-by-cycle non-repeatability analysis allowed for a quantitative assessment of the stability of the Perkins 1104D-E44TA (Perkins Engines Company Limited, global locations) engine at 25 operating points covering speeds from 1000 to 2200 r./min and load torques from 200 to 400 N × m. The analysis covered the maximum cylinder pressure p_max, the mean indicated pressure IMEP, and the crankshaft rotation angle corresponding to the occurrence of maximum cylinder pressure α, using 500 consecutive operating cycles for each point.

The lowest values of the coefficient of variation in maximum cylinder pressure (COV_pmax) were obtained in the range of average rotational speeds 1600–2200 r./min and moderate loads 250–350 N × m, where COV_pmax did not exceed 0.5%. Under these conditions, the distributions p_max were characterized by low skewness s < 0.2 and kurtosis close to 3, which confirmed stable combustion. Increased p_max non-repeatability occurred at low rotational speeds and high loads. The highest value of COVp_max = 2.10% was recorded for 1300 r./min and 400 N × m, which can be associated with boost instability and dynamic injection angle corrections performed by the ECU system. Compared to idling COV_pmax = 1.696%, an increase in non-repeatability of 0.94% was observed at this point.

Analysis of the average indicated pressure showed lower IMEP sensitivity to local combustion fluctuations. For most test points, COV_IMEP values were lower than 1.0%, and the most stable engine operation was achieved at 1300 r./min and a load of 250–350 N × m, where COV_IMEP < 1.0%. The highest COV_IMEP values, exceeding 1.6%, were observed at high rotational speeds 1600–2200 r./min and low load, including for the point 2200 r./min and 400 N × m. Compared to idling, for which COV_IMEP was 6.92%, a significant decrease in IMEP non-repeatability in the range of 5.1–6.1% was obtained at all load points.

The angle of maximum pressure in the cylinder showed the greatest sensitivity to changes in operating conditions. The COV_α coefficient of variation values were lowest 4.89–6.23% for low rotational speeds of 1000–1300 r./min and medium and high loads of 250–400 N × m. As the rotational speed and load increased, there was a sharp increase in the non-repeatability of the angle α. For speeds of 2200 r./min and loads of 350–400 N × m, the COV_α values reached 100–140%, indicating a significant randomness in the occurrence of pmax. High COVα values could result from the mathematical relationship itself; when α was evaluated relative to TDC (α = 0 deg), COVα could take on high values regardless of the actual variability of the combustion process. Compared to idling (COV_α = 22.0%), for a speed of 1000 r./min, a decrease in non-repeatability of 15.33–16.71% was recorded at all load points. In turn, a significant deterioration in stability of +118.34% was observed for the point of 2200 r./min and 350 N × m. These results confirm a strong correlation between the variability of angle α, p_max, and IMEP, especially under high-load and high-speed conditions.

5. Conclusions

The tests conducted allowed for the assessment of cycle-by-cycle non-repeatability and stability of the Perkins 1104D-E44TA (Perkins Engines Company Limited, global locations) engine at 25 operating points covering speeds 1000–2200 r./min and loads 200–400 N × m, based on the analysis of p_max, IMEP, and the angle of maximum cylinder pressure α for 500 consecutive operating cycles. The following conclusions were drawn based on the tests conducted.

• The most stable engine operation was achieved in the range of medium rotational speeds 1600–2200 r./min and moderate loads 250–350 N × m, where COV_pmax values were lower than 0.5% and COV_IMEP did not exceed 1.0%.
• The maximum non-repeatability of the maximum cylinder pressure was COV_pmax = 2.10% and occurred at 1300 r./min and 400 N × m, which represented an increase of 0.94% compared to idling (COV_pmax = 1.696%).
• The indicated mean effective pressure IMEP was characterized by less variability than pmax; compared to idle speed (COV_IMEP = 6.92%), a decrease in IMEP non-repeatability in the range of 5.1–6.1% was obtained at all measurement points.
• The angle of maximum cylinder pressure was the most sensitive indicator of engine stability. The lowest COV_α values 4.89–6.23% were obtained at speeds 1000–1300 r./min and loads 250–400 N × m, while the highest values, reaching 100–140%, occurred at 2200 r./min and high loads. High COVα values could result from the mathematical relationship itself.
• An increase in engine speed caused a shift in the average angle of p_max occurrence towards TDC—from approximately 10 deg CA for 1000 r./min to approximately 1 deg CA for 2200 r./min—with a simultaneous increase in the spread of α values (0.64–1.16).
• A clear correlation was found between the non-repeatability of p_max, IMEP, and angle α, confirming the usefulness of their combined analysis as a diagnostic tool in assessing combustion stability and optimizing engine control strategies.

6. Future Work

Further research will focus on analyzing the impact of various conventional, alternative, and waste fuels on the cycle-to-cycle non-repeatability of combustion parameters, in particular p_max, IMEP, and the angle of maximum cylinder pressure α. It is planned to include fuels with different physicochemical properties, such as cetane number, viscosity, density, and fractional composition, including waste-derived fuels, in order to assess their impact on self-ignition stability, combustion behavior, and the potential for reducing pollutant emissions.