Mathematical Formulation of Management Targets

Abstract

In this Chapter, the main management targets of maritime grids are comprehensively reviewed and several important targets in this book are described in detail, including the (1) navigation task: the ship should arrive at the destination via the scheduled route within a pre-given time-period; (2) energy consumption: achieving the pre-given management task by consuming the minimum energy, including the navigation tasks of ships and the logistic tasks of ports; (3) gas emission and energy efficiency: reducing the gas emission and improving the energy efficiency while accomplishing the pre-given management task; (4) reliability under multiple failures: improving the ability to continuously supply critical loads during multiple failures; (5) Lifecycle cost: extending the lifetime of auxiliary equipment and reducing the corresponding lifecycle cost; and (6) quality of service: improving the satisfaction degree of customers, such as the heating/cooling service in the cruise ship, the cold-ironing service for the berthed-in ship, on-time rates of the cargo delivery. Conventionally, most of the above targets are vital operating indexes in the maritime industry. However, under the new background of extensive electrification, the above targets become energy-related and therefore could be controlled and optimized via optimization-based energy management.

3.1 Overview of the Management Tasks

Generally, maritime grids are designed to fulfil different missions, and during the missions, different management tasks should be achieved, which can be mainly classified as five types: (1) thermodynamic tasks; (2) environmental tasks; (3) economic tasks; (4) logistic tasks; (5) service tasks. Five types are grouped in Table 3.1.

Table 3.1 Management tasks and articles

Chapter 1 has clarified the focus of this book: the long-term energy management of maritime grids, therefore the thermodynamic tasks are beyond the scope. In this chapter, six tasks are selected for their deep relationsip with the maritime grids.

3.2 Navigation Tasks

3.2.1 Typical Cases

As we all know, the ocean area covers more than 70 percent of our planet. In the following Fig. 3.1, we can see the main maritime shipping routes have connected the whole world. With the help of this meshed route grid and the main junctions, the bulk cargos, containers, and passengers are freely traveling by ships. Therefore, the primary management tasks for the maritime grids are the navigation tasks, which require the ships to arrive at the destination on time.

Fig. 3.1

Main maritime shipping routes, reprinted from [15], open access

There are many different types of navigation tasks, and in this section, three representative cases are to show the navigation tasks for ships: (1) ferry routes for a short two-way trips; (2) cruise routes for a long-distance traveling; (3) cargo/container ship route for inland/oversea trading.

3.2.1.1 Ferry Route

In many rivers, lakes, or straits, it is neither economic nor environmental to build bridges above the water, and in those areas, ferries can usually act as one of the main transport vehicles. In the following Fig. 3.2, two ferry routes are shown.

Fig. 3.2

Two cases of ferry routes

Figure 3.2a shows the ferry routes between Singapore to Batam. The current ferries connect two ports in Singapore with five ports in Batam. There is a combination of 100 ferry-crossings each day across seven ferry routes, and seven routes are operated by four ferry companies, including Sindo Ferry, Horizon Fast Ferry, Batam Fast Ferry, and Majestic Fast Ferry, with the shortest crossing taking around 50 min (HarbourFront Centre to Sekupang) [16].

Figure 3.2b shows the ferry route between the banks of the Yangtze River in Chongqing, China, which connects the downtown of Chongqing “Chaotianmen Square” with the Nanbin Road. This route is a famous tourism route in China which has very nice urban views of the downtown and therefore it is not suitable to build a bridge. This ferry line has been operated for more than 30 years and the entire voyage consumes about 30 min.

Besides the above two cases, ferries are widely used in many other places in the world. Especially in north Europe, ferries can convey both the passengers and cars to pass many strait georges which are not suitable to build bridges.

Generally, the voyage of ferries is usually much shorter than other ships like the cruises or cargo ships, and the ferry routes are often located near cities or towns. Therefore the ferries are the pioneers of electrified ships for environmental concerns. The practical cases include the first all-electric ferry “ampere” in Denmark, which is navigated only on batteries and can provide services with ZERO emission, and other cruise ships in Norled company.

3.2.1.2 Cruise Route

Cruise ships are mostly used for commercial purposes. Different from the short trips of ferries, cruise ships need to navigate for weeks with thousands of passengers and staff. Before the widespread of airlines, cruise ships were the only way for inter-continent traveling. Nowadays, cruise routes are mostly for tourism. Figure 3.3 gives two examples of cruise routes.

Fig. 3.3

Two cases of cruise routes

Figure 3.3a shows a cruise route from Jakarta-Singapore-Penang, which is operated by the “Genting Dream” since 2016 [17]. The “Genting Dream” weights 151,300 dwt, and has 335 meters length, which can accommodate 4000 passengers and 2000 staff. The entire voyage lasts three days.

Figure 3.3b shows a cruise route from Shanghai-Naha-Kagoshima-Shanghai, which is operated by the “Norwegian Joy” since 2017 [18]. The “Norwegian Joy” weights 167,725 dwt, and has 333 meters length, which can accommodate 3800 passengers and 1800 staff. The entire voyage lasts six days.

In other places of the world, such as the Baltic Sea, the North Sea, the Caribbean Sea, and the Mediterranean Sea, there exist many types of cruise routes, and with the demand explosion of tourism, traveling by cruise ships will be more popular in the future.

3.2.1.3 Cargo/Container Ship Route

Nowadays, most of the oversea trading and a certain part of inland trading are based on maritime transportation, i.e., the cargo/container ships. Figure 3.4 shows a cargo/container ship route from Dalian, China to Aden, Yemen.

Fig. 3.4

A case of cargo-container ship route, reprinted from [19], with permission from Elsevier

The total navigation time in Fig. 3.4 from Dalian to Aden takes 20 days. The oil tanker sails four times annually. Typical schedules are, the ship sets sail at 8:00 am on January 1st, April 1st, July 1st, and October 1st from Dalian, and returns on January 25th, April 25th, July 25th, and October 25th respectively from Aden [19].

3.2.2 Mathematical Model

Three main types of navigation routes have been described above. There exist many modeling methods for navigation tasks. In the following, a general complete model will be described in the first place, and a simplified model is then depicted in detail.

3.2.2.1 Time-Space Network Modeling

Generally, the decision variables during different navigation routes include: (1) the calls for the ports, i.e., choosing which port to berth in; (2) the navigation speed during each time-period; and (3) the total navigation time, i.e., determining the total navigation time to meet the requirements of customers. With the decision variables above, the mathematical model of navigation tasks can be shown by the following time-space network as Fig. 3.5, which is also shown in Refs. [20,21,22] as the navigation routing problems.

Fig. 3.5

Time-space network modeling of the navigation tasks

Assuming ${\mathcal{T}}$ is the total navigation time-period and is divided into $N_{{\left| {\mathcal{S}} \right|}}$ time-intervals. Then the navigation task is modeled in a directed graph ${\mathcal{G}} = (\bigcup\nolimits_{(t = 1)}^{{(N_{|} S|)}} {} S_{t} ,A_{v} ),$ where $S_{t}$ is the navigation point set (ports) which can be selected in the t-th time interval, and $A_{v}$ is the arc set which connects two concessive time-intervals, i.e., t and t + 1. Each element is denoted as $a = \left( {f,t} \right) \in A_{v} = \left( {s_{t}^{f} ,s_{t + 1}^{t} |\forall s_{t}^{f} \in S_{t} ,\forall s_{t + 1}^{t} \in S_{t + 1} ,t \in \left[ {1,N_{{\left| {\mathcal{S}} \right|}} - 1} \right]} \right)$. For the t-th time-interval, the ship can choose an arc $a = \left( {s_{t}^{f} ,s_{t + 1}^{t} } \right)$ as the navigation route and $x_{t}^{a} = 1$ correspondingly, and in other cases, $x_{t}^{a} = 0$. The distance of $(s_{t}^{f} ,s_{t + 1}^{t} )$ is denoted as $l_{a}$. The cruising speed is denoted as $v_{t}^{c}$ in each navigation route. Then the navigation model can be shown as follows.

$$ v_{t - 1,t}^{\hbox{min} } \le v_{t}^{c} \le v_{t - 1,t}^{\hbox{min} } ,\,\,t \in \left[ {T_{t - 1} ,\,T_{t} } \right] $$

(3.1)

$$ T_{t} = T_{t - 1} + T_{t}^{Na} , t \in N_{{\left| {\mathcal{S}} \right|}} $$

(3.2)

$$ T_{t}^{Na} = \Delta t_{t} = \frac{{\sum x_{t}^{a} \cdot l_{a} }}{{v_{t}^{c} }}, t \in N_{{\left| {\mathcal{S}} \right|}} $$

(3.3)

$$ T_{t}^{min} \le T_{t} \le T_{t}^{max} , t \in N_{{\left| {\mathcal{S}} \right|}} $$

(3.4)

$$ \mathop \sum \limits_{{s_{t}^{f} \in \delta^{ + } \left( k \right)}} x_{t}^{a} - \mathop \sum \limits_{{s_{t + 1}^{t} \in \delta^{ - } \left( k \right)}} x_{t}^{a} = b_{k} ,\forall k \in \bigcup\limits_{t = 1}^{{N_{{\left| {\mathcal{S}} \right|}} }} {} S_{t} $$

(3.5)

where $v_{t - 1,t}^{min}$, $v_{t - 1,t}^{max}$ are the minimum and maximum navigation speed during time-period $t \in \left[ {T_{t - 1} ,T_{t} } \right]$; $T_{t}^{Na}$ is the consumed time of the navigation between $\left( {s_{t - 1}^{f} ,s_{t}^{t} } \right)$; $T_{t}^{min} , T_{t}^{max}$ are the minimum and maximum consumed time when arriving at the port; $\delta^{ + } \left( k \right)$ (resp. $\delta^{ - } \left( k \right)$) denotes the set of arcs with the tail (resp. head) k; and $b_{{s_{t}^{f} }} = 1$, $b_{{s_{t + 1}^{t} }} = - 1$ and $b_{k} = 0$ for other cases.

Equation (3.1) represents the navigation speed should be within the upper and lower limits when navigation; Eq. (3.2) calculates the total consumed navigation time; Eq. (3.3) calculates the navigation time between two ports; Eq. (3.4) limits the total navigation time; Eq. (3.5) ensures the connectivity of the navigation scheme.

3.2.2.2 Simplified Modeling Method

In most cases, the navigation route is pre-determined and there is no need to re-schedule the route, and in those scenarios, the only action for determining an energy dispatch scheme is to adjust the navigation speed, and the above model in Sect. 3.2.2.1 can be simplified as Fig. 3.6. This simplification has been utilized in many research works [9, 10, 14, 23, 24].

Fig. 3.6

Simplified navigation task model with a fixed route, reprinted from [9], with permission from IEEE

Illustrated in Fig. 3.6, the voyage is divided into several time-intervals, and the duration of each time-interval is denoted as $\Delta t$. In each interval, the cruising speeds should be within the upper and lower bounds. Those time-intervals can be classified into two categories: (1) when the ship berths in the port (berthed intervals, denoted as $T_{b}$); (2) when the ship cruises within speed bounds (cruising intervals, denoted as $T_{c}$). During most time of $T_{c}$, the ship cruises around its nominal speed, while during the time-intervals right approaching the port (partial-speed interval, denoted as $T_{p}$), it cruises at a slower speed. The relation between $T_{b}$, $T_{c}$ is $T = T_{b} + T_{c}$, while the entire voyage horizon is $\left| {\mathcal{K}} \right|T$, and $\left| {\mathcal{K}} \right|$ is the number of ports.

As indicated in Fig. 3.6, the voyage distance between two consecutive time-intervals, i.e. t-th and t-1-th time-interval, is the accumulation of cruising speed with voyage duration $\Delta t$, which is represented as (3.6). Other constraints are shown in Eqs. (3.7)–(3.9).

$$ Dist_{t} = Dist_{t - 1} + v_{t}^{c} \cdot \Delta t $$

(3.6)

$$ \left( {1 - \delta_{D,k}^{max} } \right) \cdot Dist_{k}^{R} \le Dist_{t} \le \left( {1 + \delta_{D,k}^{max} } \right) \cdot Dist_{k}^{R} ,t \in T_{p} ,t \ne \left| {\mathcal{K}} \right|T $$

(3.7)

$$ Dist_{{\left| {\mathcal{K}} \right|}}^{R} \le Dist_{{\left| {\mathcal{K}} \right|T}} \le \left( {1 + \delta_{{D,\left| {\mathcal{K}} \right|}}^{max} } \right) \cdot Dist_{{\left| {\mathcal{K}} \right|}}^{R} $$

(3.8)

$$ \left\{ {\begin{array}{*{20}c} {\left( {1 - \delta_{v}^{max} } \right)v^{n} \le v_{t}^{c} \le \left( {1 + \delta_{v}^{max} } \right)v^{n} } & {\forall t \in T_{c} } \\ {\eta_{p} \left( {1 - \delta_{v}^{max} } \right)v^{n} \le v_{t}^{c} \le \eta_{p} \left( {1 + \delta_{v}^{max} } \right)v^{n} } & {\forall t \in T_{p} } \\ {v_{t}^{c} = 0} & {\forall t \in T_{b} } \\ \end{array} } \right. $$

(3.9)

where $Dist_{t}$ is the traveling distance at $t$-th time interval; $\delta_{D,k}^{max}$ is the maximum tolerance for traveling distance deviation; $\delta_{v}^{max}$ is the range of navigation speed; $\eta_{p}$ is speed ratio when berthing out. This model is under the assumption of a fixed navigation route, which is suitable for energy dispatch analysis in many practical cases.

3.3 Energy Consumption

There are many energy sources in the maritime grids, such as diesel engine/generators (DGs), fuel cells (FCs), energy storage systems (ESSs), renewable energy generation, and so on. To achieve better environmental benefits, the minimization of energy consumption is an important management task.

3.3.1 Diesel Engines/Generators

DG acts as the main energy source for most of the commercial ships and the auxiliary energy sources of ports. DG can scale from several kilowatt to tens of megawatt in different application scenarios. Generally, DGs can be classified as three main types by their rotating speed: (1) slow-speed two-stroke DG; (2) medium-speed four-stroke DG; and (3) high-speed four-stroke DG. A general case of diesel engines in ship is shown as Fig. 3.7a [25].

Fig. 3.7

Typical structure of diesel engines [25]

The main differences between the above types of DGs are the rotating speed. The slow-speed two-stroke diesel engines are typically defined as the one with its rotating speed less than 400 rpm. The rotating speed of the medium-speed four-stroke diesel engines usually is limited within 400~1400 rpm, and the high-speed four-stroke diesel engine has more than 1400 rpm. In addition, the slow-speed two-stroke diesel engine only has two strokes in a full operation cycle, which leads to greater ability to export power than the other two types, meanwhile the size and capacity are also larger.

Currently, the efficiency of the slow-speed two-stroke diesel engine can achieve 52%, compared with 42% for common land-based vehicles. Typical specific fuel-oil consumption (SFOC) curves of different diesel engines are shown in Fig. 3.8.

Fig. 3.8

Typical specific fuel-oil consumption (SFOC) curves

From Fig. 3.8, different diesel engines have their highest efficiency at 60~80% of rated power, and the energy consumptions under different power levels can be modeled as quadratic polynomial equations, shown as follows.

$$ FC^{DG} = \hbar^{2} \cdot \left( {r_{DG} } \right)^{2} + \hbar^{1} \cdot r_{DG} + \hbar^{0} $$

(3.10)

where $FC^{DG}$ is the fuel consumption of the diesel engine; $r_{DG}$ is the loading factor of the diesel engines, which is defined as $r_{DG} = {\raise0.7ex\hbox{${P_{DG}^{t} }$} \!\mathord{\left/ {\vphantom {{P_{DG}^{t} } {P_{DG}^{R} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P_{DG}^{R} }$}}$, and $P_{DG}^{t}$, $P_{DG}^{R}$ are the current power and rated power of the diesel engine; $\hbar^{2} , \hbar^{1} ,\hbar^{0}$ are the coefficients, which can be derived from the experimental curves like Fig. 3.8.

In conventional cases, the slow-speed two-stroke diesel engines are mostly used as the primary energy sources in traditional ships for propulsion. The main reason for this wide usage is the ability to coordinate with various types of propellers. In traditional ships, the propulsion system is directly connected with the main diesel engine, shown as Fig. 3.7b. As we all known, the speed of propeller is low, only around 100 rpm. The low-speed diesel engines can therefore well accommodate various types of propellers. But for the medium and high-speed diesel engines, an extra speed reduction transmission system should be installed and brings 3~5% energy loss. However, in AESs, the propulsion system has no necessity to directly connect with the diesel engines, then the medium and high-speed diesel engines can be served as the main energy sources with convenience. In port-side applications, diesel engines usually act as the prime movers of power generators, and are the power backups for emergency usages, or sharing the power demand in peak hours.

3.3.2 Fuel Cell

Generally, fuel cell is a power source technology like diesel engines, but fuel cell directly transforms the chemical energy of fuel into electricity, thus in operating characteristics, the fuel cell is similar to the energy storage. For illustration, the fuel cell structure and its integration into ships are shown as Fig. 3.9a and b, respectively.

Fig. 3.9

Illustration of fuel cell

Since no spinning parts and no combustion process, fuel cells combine the characteristics of diesel engines and energy storages, i.e., the ability to continuously ouput power like diesel engines and the high efficiency like energy storages. In addition to the advantages of installment space and scalable capacity, fuel cells are viewed as a promising alternative energy source for the maritime grids, especially for the ships. In this field, fuel cell based on polymer exchange membrane (PEM) is the most mature technology, and has been already applied in ship applications [26, 27]. Additionally, fuel cells using conventional hydrocarbon fuels also have gained great concerns, such as Molten Carbonate Fuel Cell (MCFC) and Solid Oxide Fuel Cell (SOFC). Figure 3.10 gives the power characteristics of a methanol fuel cell [28].

Fig. 3.10

Power characteristic curves of fuel cell [28]

From Fig. 3.10, we take the voltage curve when the fuel flow rate equals 12 mL/min as an example. The curve can be divided into phase I~III: (1) Phase I, electrochemical polarization zone; (2) Phase II, Ohm polarization zone; and (3) Phase III, concentration polarization zone. Phase I happens when the current is small, and in this phase, the electrochemical polarization effect enlarges the internal resistance of fuel cell. The voltage curve therefore has a deep drop. Then in phase II, the internal resistance is kept as a constant and the voltage characteristic follows Ohm’s law. At last, in phase III, the concentration polarization effect dominates the process and further enlarges the internal resistance, meanwhile, the voltage suffers a deep drop. From the above Fig. 3.10, we can also find that the change of fuel flow rate has a significant effect on Phase III but smaller effects on Phase I and II, respectively.

In the power curves, the power of fuel cell will firstly increase with the current, then the power becomes saturated, at last in phase III, the power suffers a dramatic drop. For a well-designed fuel cell, the Maximum-Power-Point-Tracking Method (MPPT) will keep the fuel cell in the state of maximum power [29]. The fitting of the maximum power in each curve with the fuel flow rate, shown as the brown curve in Fig. 3.10, can represent the energy consumption model of a fuel cell. The formulation is shown as follows.

$$ \begin{array}{*{20}c} {P^{FC} = g^{2} \cdot \left( {r_{FC} } \right)^{2} + g^{1} \cdot r_{FC} + g^{0} } \\ {FC^{FC} = p_{f} \cdot r_{FC} \cdot \Delta t} \\ \end{array} $$

(3.11)

where $P^{FC}$ is the power of fuel cell; $r_{FC}$ is the fuel flow rate of fuel cell; $g^{2} , g^{1} , g^{0}$ are coefficients; $FC^{FC}$ is the fuel consumption of fuel cell; $p_{f}$ is the unit price of fuel; $\Delta t$ is the length of time period.

3.3.3 Energy Storage

Generally, the operation of maritime grids includes the grid-connected and islanded modes, and most of maritime grids need to work in the shifting between two modes. For example, when the ship berths in a port and connects on cold-ironing equipment, the ship operates in grid-connected mode, and when the ship berths out, it works in islanded mode. For other ocean platforms, i.e., drilling platforms, offshore wind farms, they may work in different modes by cases. For example, when the offshore wind farms are connected with the main-land power system, they work in grid-connected mode and when they are connected with the islands, they work in islanded mode.

To keep the reliability in the above two modes, ESS is an important component for all types of maritime grids to act as an energy/power buffer between the generation-side and demand-side. In long-term timescale, an important application of ESS in maritime grids is to shave the peak load, which is shown in Fig. 3.11. When in peak load, ESS can discharge to share the power demand and in valley load, the ESS can charge and the load demand increases. Then the peak load can be viewed as “shaving” to other time periods and the maximum power demand can be reduced. This “load shaving” ability is very important to the maritime grids since most of the ships or ports don’t have much power reserve, and the proper operation of ESS is essential to the reliability, security and stability of maritime grids.

Fig. 3.11

Effects of energy storage in peak load shaving

The energy output of ESS is actually from other time-period, and the energy consumption of ESS is for the energy losses in the charging and discharging process, which is shown as follows.

$$ E_{t}^{B} = \left\{ {\begin{array}{*{20}c} {E_{t - 1}^{B} - P_{t - 1}^{ESS} \cdot \eta_{ch} \cdot \Delta t} & {\forall t \in {\mathcal{T}}\backslash 1,P_{t - 1}^{ESS} < 0} \\ {E_{t - 1}^{B} - \frac{{P_{t - 1}^{ESS} }}{{\eta_{dis} }} \cdot \Delta t} & {\forall t \in {\mathcal{T}}\backslash 1,P_{t - 1}^{ESS} \ge 0} \\ \end{array} } \right. $$

(3.12)

where $E_{t}^{B}$ is the energy stored in the t-th time period; $P_{t}^{ESS}$ is the power of ESS; $\Delta t$ is the length of time-period; $\eta_{ch} , \eta_{dis}$ are the charging/discharging efficiency of ESS.

3.3.4 Renewable Energy Generation

Renewable energy generation has been viewed as the solution to global fossil fuel depletion. Similar in maritime grids, renewable energy generation has also been gradually integrated. In Chap. 1, we have described many practical cases of renewable integration into ships. Here we give several cases to illustrate the development of renewable energy generation in ports.

The first case is from the Valencia port, Spain. This port plans to construct a breakwater dam and install tidal energy generation on it, shown as Fig. 3.12. The total capacity of the tidal energy generation can be 2.5 MW. A more detailed plan is proposed by the Houston port [13], which is shown in Fig. 3.13.

Fig. 3.12

Integration of tidal energy generation [30]

Fig. 3.13

Future development of Houston port, reprinted from [13], with permission from Elsevier

In Fig. 3.13, Spilman’s island (area 6) is planned for the photovoltaic (PV) integration, and the PV power can be used to share the power demand of Houston port, such as cold-ironing, various port cranes and electric transportation. In the future, renewable energy generation will play an even more significant role in maritime grids.

The renewable energy generation harvests different types of energy and transforms them into electricity. Its integration will reduce the usage of fossil fuel, which can be viewed as “negative fuel consumption”. Its model can be shown as follow.

$$ FC^{RE} = - {\raise0.7ex\hbox{${E^{RE} }$} \!\mathord{\left/ {\vphantom {{E^{RE} } {\eta_{FC} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\eta_{FC} }$}} $$

(3.13)

where $FC^{RE}$ is the fuel consumption reduction by renewable energy generation; $E^{RE}$ is the total energy generated by the renewable energy generation; $\eta_{FC}$ is the average efficiency of fossil fuel to electricity.

3.3.5 Main Grid

As above, the main grid is also an important energy source for maritime grids, especially for the ports. When the ship berths in a port and connects to the cold-ironing equipment, the main energy source also becomes the main grid. Generally, maritime grids will purchase electricity from the main grid according to the negotiated price, and the energy amount is measured at the substation. The model is shown as follows.

$$ FC_{MG} = p_{t} \cdot E_{MG} $$

(3.14)

where $FC_{MG}$ is the price paid for electricity purchase; $p_{t}$ is the electricity price in t-th time period; $E_{MG}$ is the purchased electricity amount.

3.4 Gas Emission

As we all know, the great concern for gas emission in the maritime industry is the main motivation of maritime electrification, including the electrification of ships and ports. In the last decade, various energy regulations have progressively stimulated the innovations and targeted technology of all components that influencing the system performance from their design phases. As two main representatives of maritime grids, the management tasks of gas emission for the ships and ports are described as follows.

3.4.1 Gas Emission from Ships

3.4.1.1 Greenhouse Emission and Energy Efficiency Indexes

Currently, the plans of Energy Efficiency Design Index (EEDI) and the Energy Efficiency Operational Index (EEOI) are one part of the IMO’s strategies to control the greenhouse emission from ships, which have two main roles: (1) providing a benchmark for comparing the energy efficiency of vessels; (2) setting a minimum required efficiency level for different ship types, size segments or cargo volumes.

The EEDI plan was first announced at the 62nd session of IMO’s Marine Environment Protection Committee (MEPC 62) with the adoption of amendments to MARPOL Annex VI, IMO [31]. After that, four important guidelines from IMO were enforced in the MEPC 63 in 2012 [32,33,34,35]. However, EEDI is used to measure greenhouse emission in the design phase for new ships. To implement this index for ships which have already been in service, EEOI is proposed as an amendment of EEDI in 2013 [36]. The general simplified formulas of EEDI and EEOI are shown as follows.

$$ EEDI = \frac{{\left( {Engine\,power} \right) \cdot SFC \cdot CF}}{DWT \cdot speed} $$

(3.15)

$$ EEOI = \frac{{\left( {Engine\,power} \right) \cdot SFC \cdot CF}}{{\left( {Cargo\,weight} \right) \cdot speed}} $$

(3.16)

where $SFC$ is the specific fuel consumption of engine (g/kW); $CF$ is the conversion factor of unit fuel to greenhouse emission; $DWT$ is the deadweight of the ship; $speed$ is the navigation speed of ship. The difference between $EEDI$ and $EEOI$ is the deadweight to replace the cargo weight. Both the EEDI and EEOI can measure the greenhouse emission per unit transportation task.

From the above definitions, the ship which has higher energy efficiency will have lower values of EEOI and EEDI. A detailed description of EEDI and the meaning of each parts are shown as Fig. 3.14.

Fig. 3.14

The EEDI calculation formula

IMO also sets many reference lines for various ship types, and each type of ship should attain smaller EEDI than the reference line. The reference lines for some ship types are shown in Table 3.2.

Table 3.2 Energy sources of different port-side equipment (data from [37])

IMO also sets many reduction targets in different time-period, i.e., (1) phase 0, 2013–2015; (2) phase 1, 2015–2020; (3) phase 2, 2020–2025; (4) phase 3, 2025 and later. Phase 1 requires a 10% reduction in the reference lines compared with phase 0, and phase 2 requires a 15~20% reduction in the reference lines compared with phase 0, and phase 3 requires a 30% reduction in the reference lines compared with phase 0. As we can see, the regulation for the greenhouse emission from ships will be even stricter in the future, and gradually becomes the primary management task for maritime grids.

3.4.1.2 NOx and SOx Emission from Ships

The concerns for both the NOx and SOx emission and some corresponding regulations are depicted in Sect. 1.4. Figure 3.15a and b respectively gives the typical emission characteristic of diesel engines for NOx and SOx [38].

Fig. 3.15

(data from [38])

NOx and SOx emission characteristics

From Fig. 3.15, the unit emission of both NOx and SOx will fall at first and then stabilize when the loading factor increased. In this sense, quadratic models can be formulated to represent the NOx and SOx emission from ships.

$$ GE^{Ni} = g_{2}^{Ni} \cdot \left( {r_{DG} } \right)^{2} + g_{1}^{Ni} \cdot r_{DG} + g_{0}^{Ni} $$

(3.17)

$$ GE^{Su} = g_{2}^{Su} \cdot \left( {r_{DG} } \right)^{2} + g_{1}^{Su} \cdot r_{DG} + g_{0}^{Su} $$

(3.18)

where $GE^{Ni} , GE^{Su}$ are the NOx emission and SOx emission; $g_{2}^{Ni} , g_{1}^{Ni} ,g_{0}^{Ni}$ are the coefficients for NOx emission; $g_{2}^{Su} , g_{1}^{Su} ,g_{0}^{Su}$ are the coefficients for SOx emission; $r_{DG}$ is the loading factor of diesel engine.

3.4.2 Gas Emission from Ports

Generally, the gas emission from ports can be from three aspects, (1) maritime operation, including the approaching, hoteling and berthing-out of ships; (2) yard operation, including the operation of port-side logistic equipment, such as quay cranes, transferring vehicles and gantries; (3) generated hinterland logistic system operation, including the railways or land-based transportation system to transfer the cargo from the ports to the inland.

On the other side, the gas emission from ports has diversified types. Figures 3.16 and 3.17 respectively gives the breakdowns of different gas emissions in Taranto port [39] and Los Angel port [40].

Fig. 3.16

(data from [39])

Gas emission breakdown of Taranto port, Spain

Fig. 3.17

Gas emission breakdown of Los Angeles port, US, reprinted from [40], open access

From Fig. 3.16, CO2 contributes to the majority of the total gas emission, and NOx, SOx, and particle mass (PM) are the other three main types of polluted gas emission. From Fig. 3.17, Ocean Going Vessels (OGV) are the main contributors for most of the gas emission, except the carbon monoxide (CO). Especially for the SOx emission, OGVs have contributed a 93.5% share. On the other hand, the cargo handling equipment is the highest contributor for CO emission, mainly for the incomplete combustion in the diesel engines of port cranes. At last, heavy-duty vehicles are also important CO2 contributors, as well as a major NOx contributor.

To measure the gas emission from ports, CO2, SOx and NOx are selected as the main representatives, and their calculations are similar and can be shown as Fig. 3.18.

Fig. 3.18

Calculation of gas emission from ports

Where $GE$ is the total gas emission, i.e., CO2, SOx, and NOx; $SFOC_{main} ,SFOC_{aux} ,$$SFOC_{eq}$ are the specific fuel oil consumptions (SFOCs) for the main engines, auxiliary engines, and the cargo handling equipment, respectively; $EL_{main} , EL_{aux} , EL_{eq}$ are the average loading factors for the main engines, auxiliary engines, and the cargo handling equipment, respectively; $EP_{main} ,EP_{aux} ,EP_{eq}$ are the capacities of the main engines, auxiliary engines, and the cargo handling equipment, respectively; $D$ is the radius of the emission control area (ECA); $V_{s}$ is the speed of ship when approaching the port. It should be noted that, Fig. 3.18 gives a general formula to calculate the gas emission of ports. The detailed model can be formulated after proposing the energy models of all the attached equipment.

3.5 Reliability Under Multiple Failures

During operation periods, maritime grids will face many types of failures, including equipment outages, short-circuit failures, and so on. In some severe scenarios, the failures may happen simultaneously and cause some serious consequences. To ensure the security of maritime grids, the reliability under multiple failures is an important management task. To simplify the modeling of multiple failures, only N-2 failures are considered in this book.

3.5.1 Multiple Failures in Ships

Different from the land-based maritime grids, such as ports, the ships are generally “islanded grids” when navigation. To ensure reliability, the ships are generally designed with two parallel buses. Some warships may even have four parallel buses to increase the survivity. Figure 3.19 gives a typical topology with two parallel buses, i.e., PB and SB. Each load, i.e., $P_{no,1} \sim P_{no,8} , P_{vs,1} \sim P_{vs,4}$, can receive electricity from two buses, which means any one-bus failure will not cause any loss of load. Currently, the topology as Fig. 3.19 with two parallel buses is a common design for commercial ships.

Fig. 3.19

Multiple failure types in ships, reprinted from [41], permission from IEEE

The multiple failures in the ship power system can be classified into two types: (1) the semi-island mode. This mode has coupled zones between the island parts. For example, in Fig. 3.19, when in semi-island mode, the $P_{no,2}$ can still receive electricity from ATG by SB via the switch $S_{S,2}$, $S_{p,2}$. (2) island mode. This mode has no coupling zones at all, and the total system is divided into two islands, which is shown as the red failures in Fig. 3.19. In the above two severe multiple failure types, the island mode is more serious than the semi-island mode. Some of the loads have to be cut off if necessary.

3.5.2 Multiple Failures in Ports

The port grids are similar to conventional land-based distribution networks, which will supply various types of service to the berthed-in ships. A typical case is shown in Fig. 3.20, which has an electrical network, a water network, and a heat network. Three networks are coupled together since the water pump is driven by electricity and the combined heat power (CHP) generator is the source of the heat network.

Fig. 3.20

A distribution network with multiple services (representing a port)

As we can see, the networks in Fig. 3.20 are all radial ones, and any failure of equipment or branch will cause the loss of load demand (electricity, water, or heat). Then the network should be reconfigured to restore service. For example, when W3 is in failure, W4, W5 will have no water supply, then W2 and W5 should be reconfigured, then the water supply of W3~W5 can be restored.

For multiple failures, there are two main types: (1) heterogeneous failures in different networks, such as one failure in electrical network and one failure in the water network; (2) homogeneous failures in the same network, such as two failures are both in the electrical network, or in water network. The latter failure mode has been well studied to improve network reliability [42, 43]. The former one involves different types of networks, which is not well studied at present.

3.5.3 Reliability Indexes

Reliability is the ability of the network to provide services in different operating conditions. Until now, there are many indexes to measure the reliability of different types of systems or networks. Table 3.3 gives some examples of reliability indexes, including the component failure probability, the probability of load shaving, and the expected loss of load demand.

Table 3.3 Conventional reliability indexes

The conventional calculation method for the reliability indexes is the Monte-Carlo simulation [44], which generate a set of scenarios and calculate the samples in each scenario. Then the reliability indexes are obtained by the average of samples. There are also plenty of analytical methods based on system approximation [45], which can calculate the reliability indexes more efficiently.

3.6 Lifecycle Cost

After the electrification of maritime grids, fuel cell and energy storage are both important equipment to improve the overall energy efficiency. Many research has investigated their applications and prove their benefits to the maritime grids. Due to the limits of current technologies, the fuel cell and energy storage cannot fully replace current power resources in the maritime grids. Therefore the fuel cell and energy storage need to operate coordinately with the other energy resources to supply the load demand. Additionally, the investment costs of fuel cell and energy storage are still high, and to reduce their overall operating cost, certain operating strategies should be implemented to extend their lifetime, and their lifetime model should be formulated in the first place.

3.6.1 Fuel Cell Lifetime Degradation Model

According to current research, there are many factors to influence the lifetime of fuel cell, including the operating temperature, humidity, and load profiles. Generally, fuel cells are installed in places with an advanced environmental control system, and the temperature and humidity can be sustained within a proper range [46]. Therefore the load profiles are the main factors that influence the lifetime of a fuel cell.

The load profiles which have influences on the fuel cell lifetime include (1) load changing; (2) start-up and shut-down; (3) idling; and (4) high load demand [47]. Then an empirical model for fuel cell lifetime can be shown as:

$$ \Delta De_{FC} = K \cdot p\left( {\left( {k_{1} \cdot t_{1} + k_{2} \cdot n_{1} + k_{3} \cdot t_{2} } \right) + \beta } \right) $$

(3.19)

where $\Delta De_{FC}$ is the degradation percentage of fuel cell; $K$ is the degradation coefficient, which can be obtained by the degradation experiment; $t_{1} , n_{1} ,t_{2}$ are the idle time, start-stop counts, and heavily loading time, respectively; $\beta$ represents the natural decay rate. After defining the $\Delta De_{FC}$, the lifetime of fuel cell can be given by Eq. (3.20), and the operating cost of fuel cell can be given by Eq. (3.21).

$$ L_{FC} = \frac{{1 - EOL_{FC} }}{{\Delta De_{FC} }} $$

(3.20)

$$ Cost_{FC} = \frac{{p_{FC} \cdot E_{FC} }}{{L_{FC} }} $$

(3.21)

where $L_{FC}$ is the lifetime of fuel cell; and $EOL_{FC}$ is the end-of-life rate, generally 10%; $Cost_{FC}$ is the operating cost of fuel cell; $p_{FC}$, $E_{FC}$ are the unit price and energy capacity of fuel cell, respectively.

3.6.2 Energy Storage Lifetime Degradation Model

Among all the energy storage technologies, battery is the most frequently used energy storage technologies for long-term energy management [48]. Furthermore, compared with other energy storage technologies, such as supercapacitors, flywheels, battery is more vulnerable and its lifetime is easier to be influenced by various operating conditions.

Similar to the fuel cell, there are also many factors to influence the lifetime of battery, such as temperature, humidity, and load profiles. Due to the installation of environmental control system, the load profiles are also the main factor on the battery lifetime. Among all the load profiles, the frequent discharging/charging events contribute to significant battery lifetime degradation, which is shown in Fig. 3.21.

Fig. 3.21

Depth of charge and the battery lifetime, reprinted from [9], permission from IEEE

In Fig. 3.21, the Depth of charge (DoD) is defined as $d_{b}$ in a discharging or charging event, which is illustrated in Fig. 3.21a. The discharging or charging event is defined as the process between two concessive state-switching points (charging to discharging or vice versa), i.e., the continuous discharging or charging periods, in which the ESS maintains the single charging or discharging state and lasts for $\Delta T_{SC}$ with the average power $P_{Bat}^{{avg,\Delta T_{sc} }}$. During each charging or discharging event, $d_{b}$ is defined as the difference between the SOCs before and after the event, which can be expressed as Eq. (3.22), where $E^{Bat}$ is the energy capacity of the battery. The relation between the DoD and the battery lifetime, denoted as $L_{b}$, is shown in Fig. 3.21b and the mathematical form in Eq. (3.23), where $a,b,c > 0$ are the fitting coefficients in Fig. 3.21b. At last, the operating cost of battery can be formulated as Eq. (3.24), where $Cost_{Bat}$ is the operating cost of battery, and $p_{ES}$ is the unit investment of battery.

$$ d_{b} \left( {\Delta T_{SC} } \right) = {\raise0.7ex\hbox{${P_{Bat}^{avg} \cdot \Delta T_{SC} }$} \!\mathord{\left/ {\vphantom {{P_{Bat}^{avg} \cdot \Delta T_{SC} } {E^{Bat} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${E^{Bat} }$}} $$

(3.22)

$$ L_{b} \left( {d_{b} } \right) = a \cdot d_{b}^{ - b} \cdot e^{{ - c \cdot d_{b} }} $$

(3.23)

$$ Cost_{Bat} = \frac{{p_{ES} \cdot E^{Bat} }}{{L_{b} }} $$

(3.24)

3.7 Quality of Service

Besides the economic benefits and allocated tasks, the quality of service (QoS) is also a vital management task for the maritime grids. There are many types of QoS, including the on-time rate of ships, the satisfactory level of passengers and ships. The on-time rate can be controlled by the management of navigation task in Sect. 3.2.1 and is not discussed here. The satisfactory levels of passengers and ships are described as below.

3.7.1 Comfort Level of Passengers

A cruise ship should provide heating load and hot water supply to the passengers. Equations (3.25) and (3.26) define the QoS of the above two services in a cruise ship. ${\mathcal{T}}$ is the total time period.

$$ \begin{array}{*{20}c} {\begin{array}{*{20}c} {T_{Air,vio} = T_{Air,vio.1} \cup T_{Air,vio.2} } \\ {T_{Air,vio.1} = \left\{ {T_{{}}^{IN} \ge T_{max}^{IN,RE} } \right\}} \\ \end{array} } \\ {\begin{array}{*{20}c} {T_{Air,vio.2} = \left\{ { T_{{}}^{IN} \le T_{min}^{IN,RE} } \right\}} \\ {T_{Wa,vio} = \left\{ {V_{{}}^{HW} \le \left( {1 + \delta_{HW} } \right)V_{{}}^{SE} } \right\}} \\ \end{array} } \\ \end{array} $$

(3.25)

$$ \begin{array}{*{20}l} {QoS_{Air} = \frac{{\left[ {\begin{array}{*{20}c} {\mathop \smallint \nolimits_{{t \in T_{Air,vio.1} }}^{{}} \left( {\left| {T_{{}}^{IN} - T_{max}^{IN,RE} } \right|} \right)} \\ { + \mathop \smallint \nolimits_{{t \in T_{Air,vio.2} }}^{{}} \left( {\left| {T_{{}}^{IN} - T_{min}^{IN,RE} } \right|} \right)} \\ \end{array} } \right]}}{{\mathop \smallint \nolimits_{{t \in T_{Air,vio} }}^{{}} \left( {T_{max}^{IN,RE} - T_{min}^{IN,RE} } \right)}}} \\ {QoS_{Wa} = \frac{{\mathop \smallint \nolimits_{{t \in T_{wa,vio} }}^{{}} \left| {V_{{}}^{HW} - \left( {1 + \delta_{HW} } \right)V_{{}}^{SE} } \right|}}{{\mathop \smallint \nolimits_{{t \in {\mathcal{T}}}}^{{}} \left| {V_{{}}^{SE} - { \hbox{min} }\left( {V_{{}}^{SE} } \right)} \right|}}} \\ \end{array} $$

(3.26)

where $T_{{}}^{IN}$ and $V_{{}}^{HW}$ represent the indoor temperature and hot water supply; $T_{max}^{IN,RE} ,T_{min}^{IN,RE}$ are the maximal and minimal limits of the indoor temperature; $V_{{}}^{SE}$ is the required hot water demand; $T_{Air,vio}$ and $T_{Wa,vio}$ are defined as the time intervals or sub-intervals which violate the indoor temperature and hot water supply service requirement (tighter than the constraints). Equation (3.26) defines the QoS of indoor temperature and hot water supply, respectively. From the above definitions, the cruise ship with a lower QoS index will better satisfy the thermal load demand of the tourists. When the QoS index equals 0, the thermal load demand is met all the time.

3.7.2 Satisfaction Degree of Berthed-in Ships

For the berthed-in ships, the cold-ironing power and cargo handling are two main services provided by the port. Similar to the QoS of Eq. (3.27), the QoS for berthed-in ships can be defined as follow. ${\mathcal{T}}$ is the total time period.

$$ \begin{array}{*{20}c} {T_{CP,vio} = \left\{ {P^{CP} \le P_{min}^{CP} } \right\}} \\ {QoS_{CP} = \frac{{\mathop \smallint \nolimits_{{t \in T_{CP,vio} }}^{{}} \left( {\left| {P^{CP} - P_{min}^{CP} } \right|} \right)}}{{\mathop \smallint \nolimits_{{t \in T_{CP,vio} }}^{{}} \left( {\left| {P_{min}^{CP} } \right|} \right)}}} \\ {QoS_{CH} = \frac{{T^{CH} - T_{max}^{CH} }}{{T_{max}^{CH} }}} \\ \end{array} $$

(3.27)

where $T_{CP,vio}$ is the time intervals or sub-intervals which violates the cold-ironing power requirement; $P^{CP}$ is the actual cold-ironing power; $P_{min}^{CP}$ is the minimal required cold-ironing power; $QoS_{CP}$ is the QoS of cold-ironing power; $T^{CH}$ is the actual cargo handling time; $T_{max}^{CH}$ is the maximal cargo handling time; $QoS_{CH}$ is the QoS of cargo handling.