The mathematical formulae for optimizing the logistics of marine CNG transportation
Natural gas transportation facilities. Design fleets and compatible CNG distribution plans. Optimization of the initial investment project. Mathematical formulas for the selection of optimal structures of ships and power. Cyclic distribution scheme.
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Warsaw University of Technology
The mathematical formulae for optimizing the logistics of marine CNG transportation
Muhammad Akram (Doctoral Candidate)
Prof. Dr. Osiadacz Andrzej
(Professor of Gas Engineering Department)
transportation investment natural gas
Marine compressed natural gas (CNG) has been considered in the past as a means of natural gas transportation but proved to be a non-starter for a number of reasons including long distances or large volumes of gas when compared with liquefied natural gas (LNG). However, marine CNG still figures economically attractive over shorter voyages (up to ~ 4000 km) and medium volumes of gas. Recent advances in containment systems are poised to provide marine CNG with the best opportunity to be resurrected as a major enabler of new and previously stranded hydrocarbons by becoming an important optimization tool to petroleum well performance.
Since the main capital expenditure in a CNG project is on marine transport vessels, careful design of CNG transport fleets and compatible distribution plans is important.
In this work, a structured optimization framework is developed to uncover trends and patterns for optimal selection of the number and capacities of CNG vessels along with corresponding schedules for CNG distribution. An important conclusion from this analysis is that medium-capacity vessels may result in lower capacity for an entire fleet, thus offering significant economic advantages. Two different methods of marine CNG transportation (i.e. hub and spoke pattern and Milk-Run pattern) have been compared in this work as an example.
Keywords: compressed natural gas, stranded gas, hub and spoke pattern, milk-run pattern, optimization
Natural gas transportation over sea by ship currently accounts for about 1/3 of all gas transported. Liquefied Natural gas (LNG) is currently the only technology employed to reduce the volume of gas sufficiently (roughly 600 times) for meaningful quantities to be transportable by LNG tankers. To accomplish this volume reduction, LNG relies on gas liquefaction by refrigeration to about -260? F.  .Even though LNG is a long proven technology, interest has been recently rekindled in compressed natural gas (CNG) as a potentially economical alternative for marine transportation of modest quantities of natural gas over relatively short distances over sea. CNG reduces gas volume about 200 times through compression (and possibly mild chilling but without liquefaction). As a result, LNG and CNG transportation systems have quite different structures. LNG requires a liquefaction terminal at the shipping point and a regasification terminal at the receiving point for storage and supply for consumption. To keep the natural gas liquid during transportation, LNG ships insulate liquefied gas in what are essentially specially designed large thermos containers.
CNG, on the other hand, requires only minimal facilities at the shipping and receiving terminals, while it relies on a fleet of significantly larger volume (capacity). CNG vessels are designed to keep gas pressurized during transportation, and may also serve as temporary storage facilities during the time gas is supplied to consumption. Therefore, the cost structures of LNG and CNG transportation systems are quite different as well: Ship building accounts for 40% only of capital expenditures (liquefaction and regasification facilities accounting for more than half of the fixed cost in LNG systems, as opposed to 80% or more for possible CNG systems.  Operating expenditures are similarly, quite different as well, because of the high-energy cost of liquefaction compared to compression.
1. Marine CNG Transportation
Marine transportation of natural gas as CNG starts with compression of natural gas to about 200 atm or to about 130 atm and mild refrigeration below 30?C at the source. Compressed natural gas is loaded to CNG vessels that can keep the transported gas pressurized during the trip to delivery destinations. At each delivery destination, gas can be offloaded to a local storage facility or a CNG vessel itself can serve as a temporary local storage facility, from which gas is routed to consumption (Wang and Economides, 2009) . After all gas is offloaded from a CNG vessel at delivery destination at its itinerary, the vessel returns to the source, to load and repeat the delivery cycle.
In this paper, a brief overview of marine CNG transportation is provided. Specifically, optimal logistics of CNG transportation fleets and itineraries are determined in terms of required rates of natural gas delivery and travel distances. Two different methods of gas distribution are discussed depending upon the required amount of CNG to be transported.
2. Objective Function
The main goal of this paper is to optimize the initial investment of a CNG project. The capital expenditures consume almost 70-80% budget of the project. To minimize these expenses, mathematical formulas have been developed to select optimal vessel designs and capacities.
Z- is the cost of transportation;
X- vessel's capacities;
Q- is the amount of gas to be transported;
k- is the coefficient.
3. Marine CNG Distribution Schemes
There can be many ways to organize a CNG marine transport. The main distinction regards the kind of path more suitable, which mainly depends on consumption rate at each receiving site and on relative geographical location of them. The preferred schemes are the so-called hub-and-spoke and milk-run. The former is preferred for destinations with sufficient consumption rates and can be better serviced by medium sized ships preferably with storage facility on both hub and receiving points. On the contrary, the milk-and run pattern is compelled when receiving sites have low consumption rates. They will be served by smaller ships in a cyclical path.  In this case, a storage plant at each destination point is mandatory in order to supply gas in adequate quantity for consumption until another CNG ship will visit that receiving site.
The overall capacity of a CNG fleet, Gf , substantially has to satisfy shipping of the natural gas available at the origin. At destination site the gas can be delivered by a different number of sister ships sailing at optimal speed with adequate deadweight, e.g.
Gf = m. n. Gn (3) - 
Gf = capacity of CNG Fleet
Gn = capacity of a ship (mmscf);
n = number of ships in each fleet;
m = number of terminals at delivery site, necessitated when the utilization rate per day, Qu, at the receiving point is higher than the maximum daily offloading rate that is technically feasible; this number (trains of ships) has to satisfy the inequality m ? [Qu?Qoff] , where m is the smallest integer larger than Qu?Qoff.
The primary criterion to apply in selecting the best fleet composition shall be minimization of the tariff to transport unit of gas volume (USD/Mmscf] or unit of gas energy [USD/MMBtu] per nautical mile.
Below is the explanation of both of these methods:
4. The Hub and Spoke CNG Distribution pattern
In the Hub-and-spoke pattern, a natural gas source (Hub) feeds multiple destinations (the spokes), each destination served by a single or multiple cycles (chains) of vessels. Each transportation vessel from which gas is offloaded at a delivery site also serves as a temporary floating storage facility for that site during the offloading period. No additional storage is used at the delivery sites.
Graphical illustration of Hub-and-Spoke CNG pattern
Figure 1. Hub-and-Spoke Pattern for CNG distribution to N receiving sites (terminals T1,…..Tn)
CNG vessel capacity in a hub-and-spoke configuration
The time taken by a transportation vessel to complete the cycle travel-load is:
(4) - 
Where L/V is travel time from source to delivery and Gn/q load is the time needed for loading. During T round trip, n-1 vessels must successively offload, taking a total time:
(5) - 
So, we get,
(6) - 
Minimum fleet capacity in a hub-and-spoke configuration
Using Gn, minimum from above equation yields,
(7) - 
Upper bound on minimum fleet capacity in a hub-and-spoke configuration
(8) - 
which provides the upper bound in equation.
(9) - 
The graphical summary of Hub and Spoke Pattern is shown below:
offloading at delivery site
Site > Source
Loading at source
Site > Source
offloading at delivery site
offloading at delivery site
Site > Source
offloading at delivery site
5. Cyclical Milk-Run CNG distribution pattern
For very small consumption rates, it implies that n cycles, min = 1, which suggests that minimum vessel capacity will be approximately equal to T qc/(n-1). Such capacity may be below the practical lower bound for CNG vessels. In that case, the use of a vessel of capacity larger than T qc/(n-1) in a hub-and-spoke pattern would be logistically feasible, but economically wasteful, because of the unused excess capacity. To avoid leaving vessel capacity unused, a vessel with excess capacity with respect to a single site can be used to deliver gas to additional sites successively in a single trip, assuming that these additional sites also have fairly low consumption needs. Each site will have to have some local storage capacity, to feed stored gas for consumption during the time the CNG vessel completes its distribution trip. We will use the term Milk-Run to denote the resulting plan of a vessel following a cyclical path to deliver gas to multiple destinations successively. A Milk-Run distribution scheme is preferable when multiple destination sites, each of fairly low consumption rates must be accommodated with natural gas from a certain source. The graphical illustration is shown below:
Figure 2. Cyclical milk-run path for CNG distribution to N receiving sites (terminals T1,…Tn)
In the above mentioned figure, N sites (terminal T1, …Tn) are receiving Natural gas, each consuming gas at a rate qc, k, k=1,….N. Gas is to be provided to each of these sites successively by n CNG vessels of capacity G n each. Each vessel will deliver a gas load G load, k to each receiving site k=1,…N per visit. Each receiving site has local gas storage capacity G storage, k, k=1,…N. All vessels can load or offload gas at a rate q load = q offload, max qc, k and travel at speed v.
A gas delivery schedule for each vessel involves gas loading at the source, travel and offloading to each destination Tk, k=1,…N, successively, and return to the source, to repeat the cycle as shown in figure above. The cyclical route shown in figure above follows the minimum closed path from the source through the delivery sites and back. While, finding this minimum path through numerical optimization is a classic challenging problem for larger values of N. Probabilistic methods, such as simulated annealing or genetic algorithms can be used.
The minimum vessel capacity in M-R pattern
It follows that the minimum capacity of each vessel, G, n, min, in a cycle of n similar vessels satisfies:
(10) - 
The minimum total capacity of fleet
The minimum total capacity of the fleet satisfies,
(11) - 
Minimum number of cycles of a vessel
The corresponding cycle time for a vessel satisfies,
(12) - 
The amount of gas to be delivered by a vessel
The amount of gas to be delivered by a vessel to each receiving site per visit satisfies,
(13) - 
The minimum gas storage capacity
The minimum gas storage capacity at each receiving site, G storage, min, k = G load, k - qc, k (G load, k/q offload, max) satisfies,
(14) - 
If all receiving sites have the same consumption rates, then xk=x/N, which immediately yields,
(15) - 
The graphical illustration of Milk-Run pattern is shown below:
6. Comparison between hub-and-spoke and milk-run CNG distribution patterns
In this section, the comparison of the fleet capacities required for hub-and-spoke or milk-run distribution patterns capable of serving a number of delivery sites. Of course, the milk-run pattern is required for small consumption rates, for which the hub-and-spoke pattern would either require vessel capacities below practical bounds or result in unused capacity in vessels of reasonable capacity. Nevertheless, a comparison between the two distribution patterns for a broader range of consumption rates is useful for establishing their relative advantages and disadvantages. To do such a comparison, let's assume N similar delivery sites, each with the same consumption rate qc, k, k=1,…N.
For hub-and-spoke pattern, the minimum capacity for a fleet serving each of these cites is given below. Therefore, the total fleet capacity for all N sites at the end of N spokes emanating from the same hub is given by:
(16) - 
And the entire fleet contains,
(17) - 
Vessels in total, corresponding to [xk] cycles per site and nHAS vessels per cycle per site.
For a milk-run pattern, there is a need for a number of cycles through all N sites, namely ncycles=[xk], and the consumption that each cycle must accommodate at each site is
(18) - 
Furthermore, for a meaningful comparison to the hub-and-spoke pattern, it is assumed that n fleet, HAS = n fleet, MR, to ensure that both hub-and-spoke and milk-run patterns have the same number of vessels; hence comparable operating costs (provided comparable travel distances).
To compute the total capacity of a fleet serving all N sites in a milk-Run pattern with the same number of vessels, corresponding to n MR=nHAS N vessels per milk-run cycle, yields that,
(19) - 
(20) - 
If T,N = T, the capacity of a fleet in a milk-run distribution pattern can be as little as half the capacity of a fleet in a hub-and-spoke pattern with as many vessels. The advantage decreases as the number of vessels in the fleet increases.
The smaller capacity of the milk-run fleet is due to the availability of stationary local storage capacity, given by:
(21) - 
This allows vessels to travel while gas from storage is fed to consumption. By contrast, hub-and-spoke vessels must provide both transportation and storage, hence the need for additional fleet capacity. This result has the following important implications:
Similarly, to fleet capacity, storage capacity also decreases as the number of vessels in a fleet increases. For the number of vessels going to infinity, the require storage tends to zero, as the two-way pipeline continuum is approached.
For a single destination with capability to provide local storage, the milk-run pattern degenerates to a single spoke of a hub-and-spoke pattern, but with the localized storage capacity. Because T,N=T for a single destination, the milk-run pattern offers clear advantages in terms of required fleet capacity. However, building and maintaining local storage capacity requires additional cost. Therefore, an economic optimization should decide the final design.
As T,N increases, the hub-and-spoke pattern requires lower fleet capacity than the milk-run pattern.
7. Summary of CNG-Distribution system design
The preceding results can be implemented towards the conceptual design of a CNG distribution system using the following algorithm.
Enter data related to the natural gas source and delivery destination, namely values for the following parameters:
T connect, [Lij], v, qc, q load, q offload, Max, N, Gmin, Gmax
For each of the N, destinations:
Compute n cycles and Gn, min with n=2 for the hub-and-spoke distribution pattern.
If G2,min ? G min, assign that destination to the milk-run distribution list. Otherwise, assign the destination to the hub-and-spoke list of destinations.
For each destination in the hub-and-spoke list, to calculate Gn, min for n ? 2. Retain as feasible values that satisfy Gmin ?Gn,min ?G,max.
Calculate the fleet capacity corresponding to the hub-and-spoke list.
Use a minimum-path algorithm to calculate the minimum closed path starting from the natural gas source and visiting successively each destination in the milk-run list once.
To calculate Gn,min,n?1, for a fleet that serves all destinations in the milk-run list. Retain as feasible values that satisfy Gmin ?Gn,min ?G,max., and calculate the corresponding fleet capacity.
Calculate the storage capacity for each destination in the milk-run list.
The above algorithm determines optimal vessel capacities and numbers. These results may be used further in economic optimization that balances capital and operating expenditures, to yield the overall economically optimal design.
The preceding analysis indicates that either hub-and-spoke or milk-run patterns may be preferred, depends on the market sizes and distances from the source. Unquestionably, more elaborate schemes can be conceived, and detailed numerical optimization can be used to fine-tune the final design. However, the calculations presented here provide both a quantitatively useful design, and a framework for intuitive thinking about more elaborate solutions.
In addition, to being useful for quick calculations relevant to economic optimization, the formulas developed here provide insight into the trend of the capacity (hence cost) of a CNG fleet as a function of various external variables and parameters. As perhaps, the most important conclusion is that, “Large CNG ships are logistically less efficient than ships of medium size, which may be preferable, namely result in fleets of lower capacity, hence capital cost”. Such insight may prove useful in shaping the fundamental thinking behind future natural gas transportation technologies and ventures.
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