4.1 Calculation of carbon emissions
There are various methods to calculate the specific carbon emissions, and the common methods are the statistical method based on the cost of fuel consumption (CO 2 EMISSION ANALYSIS FOR CONTAINERSHIPS BASED ON SERVICE ACTIVITIES | Semantic Scholar, n.d.) and the cost per carbon tax based on the carbon emissions(Hansen et al., 2008). Because considering only the cost of carbon emissions is often not enough to achieve the goal of carbon emission reduction, this paper uses the total carbon emissions of the whole process stage from the time the vessel berths until it leaves port to minimize them. Because the carbon emission of a vessel depends largely on the amount of its fuel burned, according to Hughes(Hughes, 1996), the fuel consumption rate is related to the vessel speed index, and the study by Du(Du et al., 2011) et al. determines more detailed data on this basis, and this paper directly uses his data to set the relationship coefficients between fuel consumption rate and speed for different sizes of ships: the coefficient for small vessels is µ = 3.5, and The coefficient for medium-sized vessels is µ = 4, and the coefficient for large vessels is µ = 4.5.
Before a vessel can berth in a port and start loading and unloading, it needs to berth at the port anchorage and wait for the berth to be available before it can berth and start its work. Therefore, the carbon emission of the whole berthing process can be seen as two parts: one part is the carbon emission while waiting for berthing at the port anchorage; the second part is the carbon emission when sailing from the port anchorage to the berth at the best speed.
$${F}_{i,1}={\alpha }_{1}\times \left[{c}_{i,0}+{c}_{i,1}{\left(\frac{R}{{t}_{i,b}-{t}_{a1,i}}\right)}^{{\mu }_{i}}\right]\times \left({t}_{i,b}-{t}_{a1,i}\right)+{\alpha }_{1}\times {c}_{i,0}({t}_{a1,i}-{t}_{a0,i})$$
$$\begin{array}{c} ={\alpha }_{1}\times \left[{c}_{i,0}\left({t}_{i,b}-{t}_{a1,i}\right)+{c}_{i,1}{R}^{{\mu }_{i}}{\left({t}_{i,b}-{t}_{a1,i}\right)}^{1-{\mu }_{i}}+{c}_{i,0}({t}_{a1,i}-{t}_{a0,i})\right] \#\left(1\right)\end{array}$$
The front part of Eq. (1) indicates that the fuel consumption of vessel i is the product of fuel consumption rate and sailing time, while the total carbon emission of vessel i during the voyage near the port area is the carbon emission factor multiplied by the fuel consumption; the second half indicates the total carbon emission of the vessel while waiting at the port anchorage. Where \({c}_{i}\) is the energy consumption factor of the engine of the vessel, R is the average distance of the vessel from the port anchorage to the port berth, and the carbon emission factor \({\alpha }_{1}=\) 3,110 g⁄(kg-fuel) (carbon per kg of fuel consumed), with reference to the widely accepted COSCO 2008 standard(T. Wang et al., 2019; Du et al., 2011). For the same distance, the sailing speed is excessively fast, although the docking time will be reduced, it will lead to the increase of fuel consumption rate and increase of carbon emission; the sailing speed is extremely slow, which in turn will lead to the increase of sailing time, and also the total fuel and carbon emission will be increased. In this paper we refer to the conclusion and method of Wang's paper (T. Wang et al., 2019) to make the least fuel consumption by controlling the speed of the vessel at berthing stage, and set the optimal route speed for each type of vessel as \({s}_{i}^{*}\), in order to find the optimal sailing speed only need to take the first half of Eq. (1) about the total consumption of sailing to the derivative of the time difference \({t}_{i,b}-{t}_{a,i1}\), and calculate its extreme value that is the optimal speed of each vessel. The optimal speed is:
$$\begin{array}{c}{s}_{i}^{*}={\left(\frac{{c}_{i,0}}{{c}_{i,1}\left({\mu }_{i}-1\right)}\right)}^{\frac{1}{{\mu }_{i}}}\#\left(2\right)\end{array}$$
Informing the ships that need to berth about their optimal speed and keeping their speed near the optimal speed will not only reduce the energy consumption of the vessels, but also reduce the increase of carbon emissions. However, it is difficult to maintain this point in the actual process, and for the convenience of the experiment in the subsequent experiments to unify all vessels from the port docking area to the berthing operation time is fixed to 1.
The carbon emissions during the operation of the vessel in port can be calculated according to the carbon emission Eq. (3)(Hu et al., 2014). where the carbon emission factor\({ \alpha }_{2}=\) 683 g/kw-h, \(({\stackrel{\sim}{t}}_{i,d}-{t}_{i,b})\) is the total time of vessel operation in port, \({P}_{i}\) is the rated power of engine of vessel i, \({L}_{r}\) is the load factor, and \({E}_{i}\) the number of engine of vessel i.
$$\begin{array}{c}{F}_{i,2}={\alpha }_{2}\times \left[{P}_{i}\times {L}_{r}\times {E}_{i}\times \left({\stackrel{\sim}{t}}_{i,d}-{t}_{i,b}\right)\right]\#\left(3\right)\end{array}$$
The main carbon emission in the working phase of the port equipment is the energy consumption from the working process of QC and YT. When serving vessels for loading and unloading work, QC is the electric energy used, and YT is also used in this paper to consider electric energy. For QC service vessels, the general situation is to consider time-stopping QC scheduling, which means that QC will allocate a fixed quantity according to the estimated departure time of different vessels and will not make adjustments until this vessel is completed. Such a scheduling method will lead to a certain waste of resources due to the inefficient processing of QC and there will be time for QC and YT to wait for each other. Therefore, in this paper, we adopt a time-varying QC scheduling method to ensure that the number of QCs assigned to the same vessel can be different in successive time steps of the same vessel. In addition, we guarantee that all vessels will remain in the same position during processing. Also we only allow at most one QC to be transferred from the vessel being served to an adjacent vessel for each time step change, i.e., at most one QC number change per time step. Of course we assume that the transfer time of QC installation, movement, etc. is negligible. Since the number of QCs on each ship changes with time and the efficiency of each QC is the same, we also assume that the workload of each vessel is converted to the amount of work handled by the QC per unit of time. Then the statistical workload of all vessels working time QC together all processed is the total running time of QC. In this way the QC carbon emission of the whole loading and unloading process can be calculated as:
$$\begin{array}{c}{F}_{i,3}={\alpha }_{3}\times \omega \times {w}_{i}\#\left(4\right)\end{array}$$
where \(\omega\) is the energy consumption per hour of QC operation, as the consumption is electrical energy in kWh/hour; the carbon emission factor \({\alpha }_{3}\) of electricity consumption is related to the grid region, according to the regional grid benchmark emission factors of the National Development and Reform Commission on Climate Change(Eb, 2020), as shown in Table 2.
Table 2
Results of baseline emission factors for China regional grid for the 2019 emission reduction project
Network Name
|
\({\text{E}\text{F}}_{grid,OM Simple}\)
(t\({\text{C}\text{O}}_{2}/MWh\))
|
\({\text{E}\text{F}}_{grid,BM}\)
(t\({\text{C}\text{O}}_{2}/MWh\))
|
North Regional Power Grid
|
0.9419
|
0.4819
|
Northeast Regional Power Grid
|
1.0826
|
0.2399
|
East Regional Power Grid
|
0.7921
|
0.3870
|
Central Regional Power Grid
|
0.8587
|
0.2854
|
Northwest Regional Power Grid
|
0.8922
|
0.4407
|
Southern Regional Power Grid
|
0.8042
|
0.2135
|
Note: (1) OM in the table is the weighted average of marginal emission factors for electricity from 2015–2017; BM is the marginal emission factor for capacity as of 2017 statistics; (2) This result is calculated based on aggregated data of publicly available feed-in power plants |
For YT, its total consumption is related to the number of containers to be handled and the distance of transported containers to the corresponding yard. Because the yard area is in the form of two-dimensional plane coordinates and the mutual interference of YT transport routes is not considered, and it is assumed in this paper that the transport task of YT is the same in both cases, regardless of whether it is empty or loaded. So the transportation time of YT from berth to the corresponding area block of the yard is (5) equation: |
$$\begin{array}{c}{t}_{i,\text{Y}\text{T}}=\frac{\left|{x}_{i}-{x}_{s,i}\right|+{y}_{s,i}}{v} \forall i\in V\#\left(5\right)\end{array}$$
Equation (6) indicates the total energy consumption of vessel i in the port service process YT, because the empty load transport speed is the same, so the time is the same, while this paper does not consider the number of YT limit, so that there is no YT due to the number of insufficient waiting consumption; \({t}_{i,\text{Y}\text{T}}\) is the transport time of YT from the port berth to the corresponding yard, and YT can only transport a unit container each time.
$$\begin{array}{c}{F}_{i,4}={\alpha }_{3}\times \left[\left(\psi +{\psi }^{*}\right)\times {t}_{i,\text{Y}\text{T}}\times {w}_{i}\times {P}_{\text{Q}\text{C}}\right]\#\left(6\right)\end{array}$$
Thus adding up all the carbon emissions mentioned above, the total energy consumption of each vessel i during the entire loading and unloading service at the port can be calculated.
4.2 Calculation of the total time to handle all vessels
In order to improve the motivation and sustainability of the port side, it is important to consider not only the minimization of carbon emissions, but also to improve the efficiency of terminal handling. So we consider the minimization of the total time for all vessels to finish handling. For the main cost of the port side QC operating costs, and penalties due to delays in loading and unloading, some will also consider the cost of labor costs, etc.(Eb, 2020), which are highly consistent with the minimization of the maximum completion time, i.e., the completion time is minimized while its cost is minimized. Therefore, a new objective function is added to minimize the total waiting time for completion of all vessels, which can be expressed as Eq. (7):
$$\begin{array}{c}Time=\sum _{{i}\in {V}}\left({\stackrel{\sim}{t}}_{i,d}-{t}_{a0,i}\right)\#\left(7\right)\end{array}$$
where \({\stackrel{\sim}{t}}_{i,d}\) is the actual departure moment of vessel i and\({t}_{a,i0}\) is the arrival moment of vessel i.
4.4 Establishment of formulae
Based on the above analysis, it can be calculated that the total carbon emissions of a vessel throughout the process B-QC-YAP from arrival to departure are roughly estimated as Eq. (9), taking into account the guaranteed efficiency on the part of the port operator, i.e. the minimization of the total handling time of all vessels (10), and the minimization of the distance from all berthing locations to their corresponding yard areas in order to ensure the coordination of the yard with the berths (11):
$$\begin{array}{c} min {{f}}_{1}=\sum _{{i}\in {V}}({F}_{i,1}+{F}_{i,2}+{F}_{i,3}+{F}_{i,4}) \#\left(9\right)\end{array}$$
$$\begin{array}{c} min {{f}}_{2}=\sum _{{i}\in {V}}\left({\stackrel{\sim}{t}}_{i,d}-{t}_{a0,i}\right) \#\left(10\right)\end{array}$$
$$\begin{array}{c} min {{f}}_{3}=\sum _{i\in V}\left|{x}_{i}-{x}_{s,i}\right|+{y}_{s,i} \#\left(11\right)\end{array}$$
subject to:
$$\begin{array}{c}{x}_{i}+{l}_{i}\le L \forall i\in V\#\left(12\right)\end{array}$$
$$\begin{array}{c}{t}_{a0,i}\le {t}_{a1,i}\le {t}_{i,b} \forall i\in V\#\left(13\right)\end{array}$$
$$\begin{array}{c}{x}_{i}+{l}_{i}\le {x}_{{i}^{{\prime }}}+M\left(1-{\theta }_{i{i}^{{\prime }}}\right) \forall i,{i}^{{\prime }}\in V ,i\ne {i}^{{\prime }}\#\left(14\right)\end{array}$$
$$\begin{array}{c}{\stackrel{\sim}{t}}_{i,d}\le {t}_{{i}^{{\prime }},b}+M\left(1-{\eta }_{i{i}^{{\prime }}}\right) \forall i,{i}^{{\prime }}\in V ,i\ne {i}^{{\prime }}\#\left(15\right)\end{array}$$
$$\begin{array}{c}{0\le \theta }_{i{i}^{{\prime }}}+{\theta }_{{i}^{{\prime }}i}\le 1 \forall i,{i}^{{\prime }}\in V ,i\ne {i}^{{\prime }}\#\left(16\right)\end{array}$$
$$\begin{array}{c}0\le {\eta }_{i{i}^{{\prime }}}+{\eta }_{{i}^{{\prime }}i}\le 1 \forall i,{i}^{{\prime }}\in V ,i\ne {i}^{{\prime }}\#\left(17\right)\end{array}$$
$$\begin{array}{c}\sum _{i}{\delta }_{it,j}\le 1 \forall i\in V,t\in T,j\in QC\#\left(18\right)\end{array}$$
\(\begin{array}{c}\sum _{j}{\delta }_{it,j}-\sum _{j}{\delta }_{i{t}^{{\prime }},j}\le 1+M\left(2-{\phi }_{it}-{\phi }_{i{t}^{{\prime }}}\right), \forall i\in V, \forall t {t}^{{\prime }}\in T t\ne {t}^{{\prime }} \#\left(19\right)\end{array}\) \(\begin{array}{c}\sum _{j}{\delta }_{it,j}-\sum _{j}{\delta }_{i{t}^{{\prime }},j}\ge -1-M\left(2-{\phi }_{it}-{\phi }_{i{t}^{{\prime }}}\right), \forall i\in V, \forall t {t}^{{\prime }}\in T t\ne {t}^{{\prime }} \#\left(20\right)\end{array}\)
$$\begin{array}{c}\sum _{i}\sum _{j}{\delta }_{it,j}\le n \forall i\in V,t\in T,j\in QC\#\left(21\right)\end{array}$$
$$\begin{array}{c}\sum _{j}{\delta }_{it,j}\le M{\phi }_{it} \forall i\in V ,t\in T,j\in QC\#\left(22\right)\end{array}$$
$$\begin{array}{c}{n}_{i,min}\le {n}_{i,t}=\sum _{j}{\delta }_{it,j}\le {n}_{i,max} \forall i\in V\#\left(23\right)\end{array}$$
$$\begin{array}{c}\sum _{t}\sum _{j}{\delta }_{it,j}*{w}_{ij}\ge {w}_{i} \forall i\in V,t\in T \#\left(24\right)\end{array}$$
$$\begin{array}{c}{x}_{i}\ge 0 \forall i\in V\#\left(25\right)\end{array}$$
$$\begin{array}{c}{\theta }_{i{i}^{{\prime }}},{\theta }_{{i}^{{\prime }}i}, {\eta }_{i{i}^{{\prime }}},{\eta }_{{i}^{{\prime }}i}\in \left\{\text{0,1}\right\} \forall i,{i}^{{\prime }}\in V,i\ne {i}^{{\prime }}\#\left(26\right)\end{array}$$
$$\begin{array}{c}{\delta }_{it,j},{\delta }_{it,k}\in \left\{\text{0,1}\right\} \forall i\in V,j\in QC,k\in K,t\in T\#\left(27\right)\end{array}$$
The objective function (9) represents the total carbon emission of B-QC-YAP for all vessels during the whole period from arrival at port anchorage to departure, which includes the carbon emission of vessels, QC and YT; the objective function (10) represents the minimization of the total time from arrival to completion for all vessels; the objective function (11) represents the minimization of the distance from the berthing position of all vessels to their corresponding loading and unloading target yard area. Constraint (12) indicates that it is guaranteed that all ships must berth within the terminal boundary; constraint (13) indicates that the moment of departure of a vessel from the port anchorage must be later than the moment of arrival at the docking point, with the moment of berthing at the end; constraint (14) guarantees that if vessel i is located to the left of vessel \({i}^{{\prime }}\), the right position of vessel i must be smaller than the left position of vessel \({i}^{{\prime }}\); constraint (15) ensures that in the two-dimensional berthing time plane, if vessel i is located below ship \({i}^{{\prime }}\), then vessel \({i}^{{\prime }}\) must berth in the berth after vessel i leaves. Constraints (16) and (17) ensure that only one of \({\theta }_{i{i}^{{\prime }}}\) and \({\theta }_{{i}^{{\prime }}i}\) is equal to 1, and only one of \({\eta }_{i{i}^{{\prime }}}\) and \({\eta }_{{i}^{{\prime }}i}\) is equal to 1. Constraint (18) requires that any QC can only serve at most one vessel at any time. Constraint (19)(20) ensures that for any two consecutive time steps, the difference in the number of QCs assigned to the same vessel is no more than one, and that only one QC change is allowed for immediate change QC scheduling. Constraint (21) restricts that the total number of QCs assigned to a ship at any time step cannot exceed the available number. Constraint (22) specifies that no QC may be assigned to an unprocessed vessel. Constraint (23) limits the number of QCs assigned to each vessel. Constraint (24) specifies that the required processing volume per vessel must be met. Constraints (25)-(27) indicate the range of variables.