Keywords

1 Introduction

The International Maritime Organization (IMO) report pointed out that the carbon emission of the shipping industry in 2007 was nearly 1 billion tons. If the shipping industry does not control the carbon emission, the carbon emission of the shipping industry will double, accounting for 18% of the total global carbon emission in 2050. Due to the proposal of the “double carbon” goal, energy conservation and emission reduction are less and less valued by all walks of life, and the shipping industry is no exception. Jiangsu Province responded positively and continued to promote the adjustment of industrial and energy structure to realize the synergistic effect of pollution and carbon reduction. The realization of carbon emission reduction of shipping industry needs to start with reducing ship energy consumption.

The researches on ship energy consumption optimization at home and abroad are mainly carried out in the following four directions: first, improve energy efficiency through the discovery and application of new energy; Second, the improvement of ship propulsion system to improve energy efficiency (Liang 2020; Yuan et al. 2017); Third, ship operation management and route scheduling to reduce energy consumption (Zhou 2020; Zhang 2020); Fourth, improve energy efficiency and reduce ship energy consumption through speed optimization (Li et al. 2020; Chang and Wang 2014; Kim et al. 2012). Ship speed optimization has the advantages of no need to modify the ship structure, convenient operation and less cost. There are many studies on ship speed optimization to reduce ship energy consumption. According to the measured data, Yuan et al. (2017) made a statistical analysis on the environmental factors of ship navigation. The data analysis shows that by optimizing the ship speed, the energy consumption efficiency of ships under different navigation conditions can be significantly improved. Sun (2019) studied the marine propulsion system and established the fuel consumption model of the marine diesel engine. The operation route of ship navigation is segmented, and the optimization of ship speed is completed by using genetic algorithm. However, there are few studies on the energy efficiency optimization of inland ships, and there is a relative lack of research on the energy consumption optimization of canal ships.

There are many factors affecting ship energy consumption, which are the main factors to be considered in ship energy consumption optimization. Through Spearman correlation analysis, Fan et al. (2017) proved that water flow and channel depth are the main factors affecting ship energy consumption. The energy efficiency of inland ships is significantly affected by the navigation environment including wind speed and direction, water depth and speed, as found in Yan et al. (2018). Therefore, this paper selects the channel water depth as the main influencing factor, establishes the mathematical model of ship fuel consumption, establishes the speed optimization algorithm, and calculates the economic speed corresponding to the water depth of the characteristic channel of the north Jiangsu Grand Canal under the guidance of the lowest ship energy consumption. The research results can be used to guide the navigation of ships in the northern Jiangsu canal section, and have important practical significance for reducing ship fuel consumption and realizing the “double carbon” standard.

2 Model Establishment

2.1 Ship Energy Consumption Model

2.1.1 Analysis of Ship Navigation Resistance

The basic resistance of the ship refers to the resistance of the water to the underwater part of the hull when the naked ship travels in calm water and deep water without accessories. The basic resistance of ship flow includes friction resistance and residual resistance, and the calculation formula is shown in formula (1).

$${R}_{0}={R}_{f}+{R}_{r}$$
(1)

where, \({R}_{0}\) is the basic resistance of ship water flow (N); \({R}_{f}\) is the friction resistance (N); \({R}_{r}\) is differential pressure resistance, also known as residual resistance (N).

The basic flow resistance of inland ships can be calculated according to the formula of basic flow resistance of zvankov self-propelled ships, as found in Li (2002), as shown in formula (2).

$${R}_{0}=\left(0.17{SV}^{1.83}+{C}_{r}{C}_{B}{A}_{M}{V}^{17+4Fr}\right)\cdot g$$
(2)

where,

$$S=Ld\left[2+1.37\left({C}_{B}-0.274\right)\frac{B}{d}\right]$$
(3)
$${C}_{B}=\frac{\Delta }{LBd}$$
(4)
$${A}_{M}={C}_{M}Bd$$
(5)
$${F}_{n}=V/\sqrt{gL}$$
(6)

where, \(S\) is the wet surface area of the hull, which can be approximately estimated according to formula (3), (m2); V is the speed of the ship to water (m/s); \({C}_{B}\) is a square coefficient, which can be calculated according to formula (4); \({A}_{M}\) is the water immersion area of the middle cross section, which can be calculated according to formula (5), (m2); \({F}_{n}\) is Froude number; \({C}_{r}\) is the residual resistance coefficient; \(L\) is the waterline length of the ship (m); \(B\) is the width of the ship (m); \(d\) Is the draft of the ship (m); \({C}_{M}\) is the mid cross section coefficient.

According to the analysis of natural environment data and field investigation, it can be seen that the water depth in most areas of the north Jiangsu section of the Beijing Hangzhou Grand Canal is relatively small, and the shallow water effect has a significant impact on the ship navigation resistance. The mathematical expression of ship navigation resistance considering the shallow water effect can be expressed by formula (7).

$${R}_{h}={K}_{h}{R}_{0}$$
(7)

where,

$${K}_{h}=1+\frac{0.0065{V}^{2}}{\left(h/d-1\right)\sqrt{d}}$$
(8)

where, \({K}_{h}\) is the conversion coefficient of shallow water navigation resistance, which can be calculated according to formula (8); \({R}_{h}\) is the resistance of shallow water channel (N); \(h\) is the channel water depth (m); \(V\) is the ship speed (m/s).

2.1.2 Propulsion Characteristics of Diesel Engine

The working characteristics of marine diesel engine include speed characteristics, load characteristics and propulsion characteristics. Propulsion characteristics refer to the relationship between the performance parameters and the propeller speed and power when the diesel engine drives the propeller, as found in Zhang and Chen (2017).

According to the derivation of mathematical formula, the output power \({P}_{e}\) of diesel engine is directly proportional to the third power of diesel engine speed \({n}_{e}\), as shown in formula (9). According to the multiple fitting results of measured data, as found in Wang (2014), it can be inferred that the diesel engine fuel consumption rate \({g}_{e}\) and engine speed \({n}_{e}\) can be fitted by quadratic polynomial, as shown in Eq. (10).

$${P}_{e}={f}_{1}\left({n}_{e}^{3}\right)$$
(9)
$${g}_{e}={f}_{2}\left({n}_{e}^{2},{n}_{e}\right)$$
(10)

2.1.3 Open Water Characteristics of Propeller

The open water characteristics of propeller are mainly reflected by dimensionless coefficients such as thrust coefficient \({K}_{T}\), torque coefficient \({K}_{Q}\) and propeller efficiency, and these coefficients are only related to advance coefficient \(J\), the expression is shown in (11).

$${K}_{T}=\frac{T}{\rho {n}_{p}^{2}{D}^{4}}=f\left(J\right)$$
(11)

where,

$$J=\frac{{h}_{p}}{D}=\frac{{V}_{p}}{{n}_{p}D}$$
(12)

where, \(T\) is the ship thrust (N); \(Q\) is the propeller torque (N \(\cdot \) m); \(\rho \) is the density of water (kg/m3); \({n}_{p}\) is the propeller speed (r/min); \(D\) is the propeller diameter (m); \(J\) is the advance coefficient, which is the ratio of process \({h}_{p}\) to propeller diameter \(D\); \({V}_{p}\) is the propeller speed (m/s).

The functional relationship between thrust coefficient, torque coefficient, propeller efficiency and advance coefficient \(J\) can be drawn into the open water characteristic curve of propeller. The propeller open water characteristic curve can be fitted by inquiring the propeller open water characteristic map according to the propeller model.

In the actual navigation process, there are some errors, so it is necessary to correct the calculation of propeller open water characteristics.

Due to the viscous action of water itself, the ship will produce the wake effect of following the ship’s hull during navigation, and the propeller advance speed \({V}_{p}\) will be less than the ship’s navigation speed \(V\). Equation (13) is the relationship between them and the wake coefficient \(\omega \). The relationship between the corrected propeller speed \({V}_{p}\) and the propeller speed coefficient \(J\) is shown in Eq. (14).

$${V}_{p}=\left(1-\omega \right)\cdot V$$
(13)
$$J=\frac{{V}_{p}}{{n}_{p}D}=\frac{\left(1-\omega \right)\cdot V}{{n}_{p}D}$$
(14)

According to the relevant principles of hydrodynamics, the forward rotation of the propeller at the stern will reduce the stern pressure, resulting in the increase of the resistance of the ship during navigation. Therefore, the thrust of the propeller \(T\) should not only offset the navigation resistance of the ship \(R\), but also offset the resistance \(\Delta R\) increase caused by the rotation of the propeller. The drag increment \(\Delta R\) is numerically equal to the thrust derating \(\Delta T\). Thrust derating fraction \(t\) is the ratio of thrust derating \(\Delta T\) to ship thrust \(T\), which can be used to express the relationship between ship thrust \(T\) and ship resistance \(R\), as shown in Eq. (15).

$$t=\frac{T-R}{T}$$
(15)

2.1.4 Engine Speed and Ship Speed Relation

Equation (16) is the calculation of ship thrust, and Eq. (17) is the relationship between ship thrust and ship navigation resistance.

$$T={K}_{T}\rho {n}_{p}^{2}{D}^{4}$$
(16)
$$T=R/\left(1-t\right)$$
(17)

Substituting the ship navigation resistance \(R=f\left(V\right)\) and Eq. (16) into Eq. (17), and substituting the known thrust derating fraction \(t\), propeller thrust coefficient \({K}_{T}\), advance coefficient \(J\) and wake coefficient \(\omega \), Eq. (17) can be transformed into a univariate quadratic inequality of propeller speed \({n}_{p}\), and the expression of engine speed and speed can be obtained by using the root seeking formula.

2.1.5 Fuel Consumption Model of Diesel Engine

The fuel consumption of marine diesel engine in a voyage can be calculated according to Eq. (18).

$$W={g}_{e}{P}_{e}t=f\left(V\right)$$
(18)

where, \(W\) is the fuel consumption of a voyage (g); \({g}_{e}\) is the fuel consumption rate of diesel engine (\(g/ (kW\cdot h)\)); \({P}_{e}\) is the output power of diesel engine host (kW); \(t\) is the sailing time (h).

2.2 Speed Optimization Model

2.2.1 Genetic Algorithm

Genetic algorithm is an evolutionary algorithm based on the evolutionary law of “natural selection and survival of the fittest” in the process of biological evolution.

Because more friendly operation interface for non-professional personnel and graphical calculation results, this paper adopts the American Mathworks’s GADST genetic algorithm and direct search toolbox brought by MATLAB. The main function of GADST genetic algorithm is \(ga\). It is to find the optimal individual with the smallest fitness function. Functions can be called with the algorithm statements (19).

$$x=ga\left(@fitness, nvars,A,b,Aeq,beq,LB,UB,@nonlcon,options\right)$$
(19)
$$\left[x,fval,exuitflag,output,population,scores\right]=ga\left(\cdots \right)$$
(20)

where, the input parameters are \(fitness, nvars,A,b,Aeq,beq,LB,UB,nonlcon,options\); the output parameter is \(x,fval,exuitflag,output,population,scores\).

The main function \(ga\) of genetic algorithm can be expressed as Eq. (21). Equation (22) is constraint condition.

$$minf\left(x\right)\left(the\ number\ of\ x\ is\ nvars\right)$$
(21)
$$\left\{\begin{array}{c}Ax\le b\\ Aeqx=beq\\ LB\le x\le UB\\ nonlcon\end{array}\right.$$
(22)

2.2.2 Algorithm Language for Speed Optimization Model

The expression of total fuel consumption of N segments is shown in Eq. (23). In genetic algorithm, Eq. (23) is the fitness function.

$$W=\sum_{i=1}^{n}{W}_{i}=\sum_{i=1}^{n}f\left({V}_{i}\right)\left(i=\mathrm{1,2},\cdots n\right)$$
(23)

where, \(W\) is the fuel consumption of the ship during the whole voyage (g); \({W}_{i}\) is the fuel consumption of the ship in segment \(i\); \({V}_{i}\) is the speed of segment \(i\); \(n\) is the number of segments of the whole voyage.

The upper and lower limit constraints of the speed optimization algorithm \(\left[LB,UB\right]\) are the size range of the speed \(V\), as shown in Eq. (24) and (25).

$${n}_{min}\le {n}_{e}\le {n}_{max}$$
(24)
$${V}_{min}\le {V}_{i}\le {V}_{max}$$
(25)

According to the above fitness value function expression and constraints, the genetic algorithm can be called with MATLAB statement. The algorithm is expressed as follows Eq. (26).

$$\left[x,fval\right]=ga\left(@fueloil\_total,n,\left[{}\right],\left[{}\right],\left[{}\right],\left[{}\right],LB,UB\right)$$
(26)

3 Ship Experimental Verification

3.1 Experimental Ship

In this study, a ship on the Beijing Hangzhou Grand Canal is selected as the research object, and its ship parameters are used to calculate the economic speed. The parameters of the ship are shown in Table 1.

Table 1. Ship related parameters.
Full size table

3.2 Fuel Consumption Model of Experimental Ship

The propulsion characteristic curve of the diesel engine is obtained from the operation manual of the Z6170ZLCZ diesel engine used by the research object ship. According to Eq. (9), the quadratic polynomial fitting of the diesel engine fuel consumption curve can obtain the mathematical expression of diesel engine fuel consumption and engine speed, as shown in Eq. (27). The curve is fitted with the third power function according to Eq. (10), and the mathematical expression of engine power is obtained, as shown in Eq. (28). The fitting curve is shown in Fig. 1.

$${g}_{e}=4.219\times {10}^{-4}\cdot {n}_{ei}^{2}-0.840\cdot {n}_{ei}+618.165$$
(27)
$${P}_{e}=2.947\times {10}^{-7}\cdot {n}_{ei}^{3}$$
(28)
Fig. 1.
figure 1

Z6170ZLCZ Engine propulsion characteristic fitting curve.

Full size image

According to the ship parameters and the previous calculation process, the mathematical model of fuel consumption of the research object ship on a certain leg can be obtained, as shown in Eq. (29).

$${W}_{i}=\left(4.219\times {10}^{-4}\cdot {n}_{ei}^{2}-0.840\cdot {n}_{ei}+618.165\right)\left(2.947\times {10}^{-7}\cdot {n}_{ei}^{3}\right)\frac{{L}_{i}}{{V}_{i}}$$
(29)

where, \({n}_{ei}\) is the engine speed of segment i; \({L}_{i}\) is the mileage of segment i.

3.3 Experimental Verification

In order to verify the mathematical model of ship fuel consumption established in this paper, an experimental route with a total length of 47.9 km is designed. During the experiment, the working conditions of the ship following experiment were obtained by consulting the crew, recording the display data of relevant instruments and real-time monitoring. The specific contents are shown in Table 2.

Table 2. Experimental conditions.
Full size table

The experimental conditions shown in Table 2 are substituted into the ship speed optimization model in this paper, and the optimal speed corresponding to the experimental conditions is calculated which is 6.96 km/h.

By substituting the empirical speed used in the experiment and the recommended speed calculated by the speed optimization model in this paper into the marine diesel engine fuel consumption model in this paper, the calculated fuel consumption consistent with the experimental conditions can be obtained. The calculated results and the real ship experimental results are listed in Table 3 for comparison and analysis. It can be seen that the error between the fuel consumption calculated by the fuel consumption model and the experimental fuel consumption is 1.26%, and the calculated fuel consumption corresponding to the recommended speed is 2.88% lower than the empirical speed.

Table 3. Model calculation and measured fuel consumption.
Full size table

4 Economic Speed Calculation and Result Discussion

4.1 Calculation of Economic Speed

It can be seen from Sect. 2.1 that the main environmental impact factor considered in this paper is water depth when calculating ship navigation resistance. Due to the interception of multi-level ship locks in the northern Jiangsu canal, the water level before the two ship locks has little change. Therefore, according to the division principle of one navigation section between the two ship locks, the northern Jiangsu section of the Beijing Hangzhou canal is divided into 11 sections, namely \({L}_{1}\), \({L}_{2}\), \(\cdots {L}_{11}\), and the mileage of each navigation section is shown in Table 4.

Taking the fuel consumption model of the whole voyage as the fitness function, the genetic algorithm is used to optimize the speed. According to the calculation flow in Sect. 2.4, the relationship between engine speed and speed of 11 segments is calculated according to different water depths of the segments, as shown in Eq. (28).

It is known that the engine speed of the research object ship during normal operation is controlled at 300–1200 (r/min). The upper and lower bound constraints of the speed optimization algorithm can be obtained through calculation, as shown in Eq. (29).

Table 4. Channel mileage and mean water depth of 11 segments of northern Jiangsu canal.
Full size table
$$\left\{\begin{array}{c}{n}_{e1}=141.21{V}_{1}+42.365\\ {n}_{e2}=140.65{V}_{2}+44.857\\ {n}_{e3}=146.23{V}_{3}+20.169\\ {n}_{e4}=122.03{V}_{4}+27.965\\ {n}_{e5}=145.82{V}_{5}+21.988\\ {n}_{e6}=154.18{V}_{6}+14.837\\ {n}_{e7}=121.82{V}_{7}+8.9210\\ {n}_{e8}=132.97{V}_{8}+78.922\\ {n}_{e9}=142.74{V}_{9}+35.611\\ {n}_{e10}=150.73{V}_{10}+0.327\\ {n}_{e11}=124.09{V}_{11}+38.694\end{array}\right.$$
(30)
$$\left[LB,UB\right]=\left[\begin{array}{c}1.825, 8.198\\ \mathrm{1.814,8.213}\\ \mathrm{1.914,8.068}\\ \mathrm{2.229,9.605}\\ \mathrm{1.901,8.079}\\ \mathrm{2.042,7.879}\\ \mathrm{2.389.9}.777\\ \mathrm{1.662,8.431}\\ \mathrm{1.852,8.157}\\ 1.\mathrm{988,7.959}\\ \mathrm{2.104,9.359}\end{array}\right]$$
(31)

According to the ship parameters and the above calculation process, the whole process fuel consumption mathematical model of the research object ship on the northern Jiangsu section of the Beijing Hangzhou Grand Canal can be obtained, as shown in Eq. (32). Taking the fuel consumption model of the whole voyage as the fitness function, the genetic algorithm is used to optimize the speed.

$$W=\sum_{i=1}^{11}{W}_{i}=\sum_{i=1}^{11}f\left({V}_{i}\right)\left(i=\mathrm{1,2},\cdots 11\right)$$
(32)

Then, aiming at minimizing the total fuel consumption of the ship’s voyage, the speed optimization algorithm is called to calculate the segmented economic speed.

4.2 Result and Discussion

According to the above calculating procedure, the optimized speed values based on the average water depth of each segment of the Beijing Hangzhou Grand Canal in northern Jiangsu are shown in Table 5.

Table 5. Segmented economic speed (unit: km/h).
Full size table

It can be seen from Table 5 that the optimal solution of segmented speed in the north Jiangsu section of the Beijing Hangzhou canal is in the range of 6–8 (km/h). The deeper the water depth in the section, the smaller the impact of water depth on ship navigation. Under the same working condition, the corresponding reduction of ship fuel consumption and the greater the optimal solution of speed; On the contrary, the shallower the water depth of the section, the more obvious the influence of the shallow water effect on the ship’s navigation resistance, and the larger the ship’s fuel consumption under the same working conditions, so the smaller the optimal solution of the speed is obtained.

5 Conclusions

Through the analysis of the natural conditions of the canal in Northern Jiangsu, this study established a ship fuel consumption model and speed optimization model suitable for the canal environment. Taking a ship in the north Jiangsu canal as an example, the model established in this paper was experimentally verified. It can be seen that the economic speed calculated by the model can reduce the ship energy consumption by about 2.88% compared with the empirical speed. At the same time, the economic speed of the northern Jiangsu canal section was calculated, and the corresponding economic speeds of 11 sections are obtained. The following conclusions are drawn: the economic speed range of the north Jiangsu section of the Beijing Hangzhou canal is 6–8 km/h; Under the same working conditions, the deeper the water depth of the leg, the smaller the impact of the water depth on the ship’s navigation, and the greater the calculated economy. These models and conclusions are helpful to the study of canal economic speed in the future and has reference significance for the canal to achieve low energy consumption and carbon emission reduction.

The current study only considered the influence of water depth and speed on ship energy consumption, ignoring the influence of canal width. At present, it mainly carried out theoretical research, so the applicability of the research has some limitations. More experiments will be carried out to verify the reliability of the ship model in the future. The model will also be used to calculate the ship energy consumption and economic navigation capital of the characteristic ship types of the northern Jiangsu canal. And the canal width will also be included in the study.