1ères JOURNEES NATIONALES SUR LE TRATEMENT, LE STOCKAGE, LE TRANSPORT ET LA DISTRIBUTION DES HYDROCARBURES

OPTIMIZATION OF LNG CHAIN BY STOCHASTIC MODEL Abdelhakim Ainouche Sonatrach-TRC-RTH-Haoud el Hamra-Algerie Abstract Having a park of four liquefaction chains, Algeria is considered as one of the principal natural gas and LNG suppliers of Mediterranean Europe and America. The LNG market, characterized until now by long term contracts of the “take or pay” type, will tend to change. This will result mainly in the quest by the customers of a larger flexibility in the contracts and the multiplication of spot removings. Consequently LNG chains, must anticipate those evolutions and include new organization forms of their exploitation. LNG chain is a complex system, 20 to 30% of the amount that is transited is used for energy consumption .This high value justifies the necessity to elaborate optimization algorithms in order to reduce energy costs. To carry out this objective efficiently, a global optimization of the chain including gas pipelines, liquefaction plants, storage capacity, port and LNG tankers fleet are necessary. For this, a stochastic model has been designed. The input data of the model are given by: • ARMA model for the modeling of the LNG expeditions; • Simulation model of the port functioning and storage state; • Reliability model of the liquefaction plants and gas pipeline compressor stations; • Optimization model of the gas pipeline and the LNG plant using a dynamic programming procedure and MINLP (mixed integer nonlinear programming). An application to Algerian chain Hassi R’mel –Skikda will be given. Key words: LNG Chain, Optimization, Gas pipeline, Storage, Reliability, Stochastic, Dynamic Programming, MINLP, Monte Carlo simulation

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Introduction With the increased demand of natural gas, liquefied natural gas (LNG) is emerging as an important source of natural gas and getting a second look as a fuel option in the world. The Algeria historically played a pioneering role in the development of LNG industry and the global LNG trade. The perspectives of excess liquefaction capacities bound to the coming on line of new trains on a given number of plants through the world will ineluctably influence on the future evolution of the LNG market. The LNG market, characterized until now by long term contracts of the “take or pay” type, will tend to change. This will result mainly in the quest by the customers of a larger flexibility in the contracts and the multiplication of spot removings. This spot, or short-term trade has grown from 1.3% of total world exports in 1992 to nearly 4% in 1999. The emergence of the spot market is due to several factors, including excess capacity in new projects, which is available until purchasers take full volumes; an increasing number of buyers and sellers: and the availability of capacity at terminal. Consequently LNG chains, characterized by a given rigidity because of the strong dependence between the functions production, transport, liquefaction, storage and trading must anticipate those evolutions and include new organization forms of their exploitation. To the market flexibility must respond a larger flexibility of the LNG chain (Law of Requisite Variety of Ashby) 4. The Requisite Variety may be obtained by: • An appropriate re design of interfaces between the different functions; • A larger availability of the chain in its overall. Those operations have a very high cost in term of investment and exploitation. The use of optimization techniques enables to reduce the financial impact of these necessary changes. 1- Model of a port simulation The ports are equipped of many loading platforms which it is question to use at best in function of the demand evolution. The functioning of a port is a complex process. Two removal types will prevail: • The exports by shuttles and rotations to the destination of ports defined in advance. • The spot removals or at trip. While waiting for the confirmation more and more important of the spot deliveries, the simulation remains the only resort for the modeling of the port functioning and of the expeditions. Besides if the spot removals are done with the help of LNG tankers of which the arrivals and destinations are random, shuttles and rotations are programmed with the active competition of man. Which supposes that the use of queuing scheme can not be reasonably recommended. This standpoint finds itself comforted by the following functioning requirements: • A port is composed of loading platforms of different capacities; • An LNG tanker can load only in a platform of higher or equal capacity to its one; • The days of swell, the loading is interrupted momentarily. For LNG tankers organized in shuttles, the arrivals may be supposed regular and spaced by a time equal to the time duration of a rotation. In reality, the variable state of the sea and the random duration of unloading at the expedition port lead to delays which disrupt the arrival programming. The functioning of a port is a dynamic process at the entrance of which a complex harmonic signal acts. Harmonics are constituted by the regular shuttles of different periodicities. Spot arrivals and delays may be assimilated to a noise. -2-

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Under this aspect there is place to note the homomorphism of this representation with speech signals 9. This enables the use of theoretical aspects already developed in this field. The model generally admitted for the speech signal is a structure of AR model (Autoregressive). From this standpoint the extrapolation allows the modeling of expeditions by the following model: qe(t ) =∑ai ⋅qe(t −i ) P

(1)

i =1

where: p : the order of the model; ai : coefficients of prediction The identification of prediction coefficients is done on the basis of simulation results by the least squares method. The model of simulation is constructed on the following hypotheses: • The density of probability of delays in the arrivals of LNG tankers organized in shuttles follows an exponential law of the form: P (θ ) = α ⋅ e −αθ (2) •

For a service duration, which comprises the maneuverings, the de-ballasting, the loading and customs formalities:

G(τ ) =µ⋅e−µ ⋅τ

(3)

Where : α : mean rate of delays; µ : mean rate of service; θ and τ , the time. By analogy with the oil domain 12 the choice of exponential distributions, which are the simplest to manipulate mathematically is the most often confirmed by practice. For the spot arrivals, the adapted model is that of completely random arrivals characterized by a Poisson process: (β ⋅t )m ⋅e − β ⋅t (4) H (t ) = m! Where : β : mean rate of spot arrivals m : average number of LNG tankers t : time The parameters α , β and µ may be estimated by a statistical treatment of data during the normal exploitation of the port. After having admitted an initial state of the system, the use of the Monte Carlo method enables the simulation of a chronology of LNG tankers arrivals and departures as well as the occupation period of each loading platform. According to the real evolution of the spot expeditions the coefficients of prediction will be corrected by the adaptive algorithm use or by Kalman filter. The vectorial form of this algorithm is: a ' (k + 1) = a (k ) + K (k + 1) e (k + 1) (5) a (k ) : vector of prediction coefficients at the modeling step k K (k + 1) : the gain at the step (k + 1) e (k + 1) : the update error at the step (k + 1)

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2- Storage capacities modeling The LNG storage at a terminal is characterized by a continuous supply. If receivings can be assimilated to a determinist “controllable “ process, the expeditions them, because of delays and spot deliveries are most often random. The port can be considered as a complex operator then and the expeditions from the storage park as an exit signal. It becomes possible then to turn into account the theory of the random functions in view to better know the characteristics of the exit signal, its spectral representation and mainly the judicious choice of the planning duration and the receivings volume. Knowing the flow rate of each loading platform, the simulation model of the port may be transformed to furnish a chronic of the exported LNG amounts. Then the result is a chronological series of the amounts sent, that we will assimilate to an ergodic random process. The moments of different orders are defined then by the relationships:

[

][

Ryy(T ) = 1 ∑ y (τ ) −M y ⋅ y (τ +T ) −M y N −T

ρ yy(T ) = M y : mean of the function y (τ )

]

(6)

Ryy(T ) Dy

(7)

Dy : variance

Ryy(T ) : Autocorrelation function The Autocorrelation function is not a random function. Its form enables to judge the nature of the process, which may be harmonic or purely random. In the case of a harmonic process, the planning duration is equivalent to the half period of the correlation function. In the case of a random process, the correlation function weakens when the interval T increases. So one must content himself with defining the interval for which the process may be considered as stationary in the wide sense. The expeditions of LNG being random, a statistical analysis of the evolution of the simulated stocks state enables to define (for a given risk factor) the safety stocks level and the necessary storage capacities, so the possibility of new constructions of tanks. The evolution of the state of stocks may be described by the relationship: (8) ST (τ ) = ST (τ − 1 ) + Q (τ ) − q (τ ) 0

e

Where: Q0(τ ) : sequence of the gas pipelines flow rates chosen so that to minimize the energy costs; qe(τ ) : exit flow rate from the park, random function depending of LNG tankers arrivals and the occupation of the loading platforms. 3- Reliability of LNG chain An LNG chain is a complex system. The modeling of complex systems, from reliability standpoint or other, is always delicate. The systemic approach, or system approach, consists in spliting up the global system in subsystems. Every subsystem being itself susceptible to be decomposed to subsystems of lower level. Of a general way, a subsystem will be defined as a set of elements (or of lower level subsystems) in interaction. The element is considered as the basis of the subsystem in the way that it can not be decomposed to subelements. This results in a hierarchical organization chart having a more or less high number of levels according to the mannered detail degree in -4-

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the modeling. For the modeling of the reliability of an LNG chain, of the type of the Hassi Rmel-Skikda chain, we adopted a hierarchical decomposition of five levels. The modeling proceeding is of «bottom-up» type. For systems of production where redundancies of subsystems are the most often numerous, the failure of an element doesn't involve the total stop of the production but a decrease of performance characterized by a matter flux lower than the nominal flow rate. The untimely nature of failures involves by this fact a random fluctuation of the flow rate. These fluctuations can be then expressed as probability values according to a statistical distribution law or assimilated to a random process of discrete phase space. Every phase corresponds to a well definite failure state. In the two cases the reliability coefficient, that characterizes the loss of performance owed to failures, can be quantified as follows: s PQ (9) K R = ∑ k. k k = 0 Q0 Where: • Pk , probability of the failure state k ; • •

Qk , production flow rate associated to the failure state k ; Q0 , design flow rate.

Probabilities of phases Pk are determined by the use: • Of reliability indices of the basis elements, mainly the failure rate and the repair rate; • Of subsystems reliability diagrams; • Of basis models of the systems reliability; • Of the fundamental theorems of the probability theory. The details of this important part of work are given in the second paper presented in this conference3. 4- Optimization of the LNG chain Costs of starts-up/stops of a liquefaction train are very high. The operating at partial load of a liquefaction train is expressed by a decrease of the efficiency and the increase of the fuel consumption. It can be expressed by the relationship: C i (q i )

⎛ qi = a i − bi⎜⎜ ⎝ q i max

n

⎞ ⎟ =α i − β i ⋅q in ⎟ ⎠

(10)

qi max : nominal capacity of the train i qi : the production flow rate of the train i

α i , β i and n : coefficients defined from the real consumption curves The optimization criteria kept consists in minimizing costs of energy and costs of startsup/stops

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Vst+ Vst−

Q0(t )

q1

q2 T1

qe(t)

q3 T2

q4 T3

q5 T4

q6 T5

T6

Qp

T

n

(

)

T

n

Min Z = C ∑ ∑ α i ⋅qijn +1 ⋅Ti + ∑ ∑ ai −1 j − aij ⋅ i =1 j =1

i =1 j = 1

Cd i 2

(11)

C : cost of energy by unit of time in $ T : period of planning n : number of trains aij : binary variable

Cd i : cost of a start-up/stop of the train i

Constraints: 1 - constraints on the production flow rate of the LNG units in link with risks of rupture of stock and of overstock

q (t ) − V

st

(t − 1) − V −st T

e

− (t − 1) (t ) ≤ q e (t ) + V st V st 0i +

≤Q

for all i varying from 1 to T

T

2 - constraints of flows balance n

∑a j =1

ij

⋅qij = Q 0 i (t )

for all i varying from 1 to T

3 - constraints of link q e (t )= ∑ a s ⋅q e (t − s ) P

for all t varying from 1 to T

s =1

4 - constraints on the tolerated operating range of trains q ij 0 , 6 ≤ max ≤1 qj

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Q0(t ) : daily flow rate of produced LNG

Vst+ : maximal storage capacity Vst− : minimal storage capacity

Vst (i ) : state of stocks to the instant i

The model set this way brings is drawn to a nonlinear programming problem with mixed variables. Its solving is obtained by an algorithm of generalized reduced gradient type associated to a procedure of branch and bound type. A failure on a gas chain is a random event, it can be foreseen only from a probabilistic standpoint. In this context, the operator must be able to have at one’s disposal tools enabling him to immediately define the optimal configuration of the optimal parameters of the operation to establish taking into account the failures. In this setting a model of optimization by the criteria of the minimum of energy consumption has been developed. Gas pipeline operates at relatively high pressure. The high operating costs of transmission pipelines (compared to distribution) motivate the study of these ones. For a gas pipeline of an average complexity, the fuel consumption of compressor stations represents approximately 5-10% of the gas entering the pipeline. Because of the compressible nature of the gas, the optimization problem of a gas pipeline operation whatever the criterion chosen comes out to a nonlinear programming problem. The paper by Batey 5 represents one of the early attempts to develop a rational control policy. The application of this simple principle gives very correct results when the flow rate wanted at the end of line is close to the nominal one. However, for intermediate flow rates needing to by pass a given number of compressor stations, the disadvantages of its application become obvious because of the highly combinatorial nature of the problem. Wong and Larson 13 determined the steady-state optimal operating conditions of a gun barrel gas pipeline with compressors in series. Their technical solution was to use dynamic programming to find the optimal suction and discharge pressures of a fixed number of compressor stations but they do not give guidelines on how to accomplish such operation. The objective of the optimization in this case is to minimize the fuel amount which is consumed by the gas turbine drivers 2,10

⎡ N C Min Z = ⎢ ∑ 1 ⋅ 0 i ⎢⎣ i =1 η ST i LHV

⎤ ⎛ ⎛ Pd ⎞ γ ⎞ i ⎜⎜ ⎟ −1 ⎟⋅Q p ⎥ ⎜ ⎝ Ps i ⎠ ⎟ ⎥⎦ ⎝ ⎠

(12)

Pdi : discharge pressure Psi : suction pressure LHV

: low heating value

ηST : global efficiency of station i i

Q p : gas pipeline flow rate C0i : coefficient

N : number of compressor stations in the system γ : ratio of specific heat

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Operating constraints include constraints on stations and on individual turbo compressor units. At station level constraints are imposed on suction and discharge pressures, maximum discharge temperature and horsepower of coolers. Constraints imposed on individual turbo compressor are sufficient to ensure a feasible operating profile at all times during the solution. Checks are made of surge, stonewall lines and maximum horsepower for each turbo compressor. The problem set this way is solved by a dynamic programming procedure. The dynamic programming is dominated by the principle of optimality of Bellman according to which: « A policy is optimal if, at a stated stage, whatever the preceding decisions may have been, the decisions still to be taken constitute an optimal policy when the result of the previous decisions is included ». For the case of GK1 gas pipeline Hassi R’mel-Skikda feeding the liquefaction plant GL1K, enables to reduce the fuel consumption of 11%.

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Conclusion The recent development of the spot transactions is the proof of the gas markets mutation. These circumstantial sales, represented 5% of gas transactions in 2000. This number will have tendency to increase according to a given number of specialists. This activity will represent a complementary element of the Algerian gas strategy based on the long term. Sonatrach, one of the most active gas operators, has already started to adapt its production tools to those new perspectives. Consequently the LNG chains, characterized by a given rigidity must anticipate these evolutions and include new forms of organization of their exploitation. It can be obtained by the maximization of the availability and the minimization of production costs on the whole of the LNG chain. We presented in this article the necessary modeling tools enabling: • the evaluation of the quality of response of an LNG chain facing those mutations; • the quantitative analysis and the improvement possibilities of an LNG chain availability by the implementation of a systemic model of reliability. • the reduction of production costs by the optimization of the whole chain taking into account the stochastic nature of expeditions that will result from the more and more important part of the short term market. For this last point an approach associating dynamic programming and MINLP was implemented.

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References 1. Ainouche, A., Smati, A., Global optimization of the Algerian gas pipeline network, 16th World Petroleum Congress, Calgary, Canada, June 2000. 2. Ainouche, A., Smati, A., and Younsi, K., Optimisation des régimes d’exploitation d’un gazoduc par un modèle de programmation non linéaire, Conférence internationale sur la productique juin 2001 Sheraton Algérie. 3. Ainouche, A., Reliability of LNG and natural gas transmission chain, 17th World Petroleum Congress, Rio de Janeiro, September 2002 4. Ashby,W.R., Introduction to cybernetics, Wiley, New York 1963. 5. Batey, E., Courts H.R. and Hannah K.W., Dynamic approach to gas-pipeline analysis, Oil & Gas J., N°59,1961. 6. Bellman, R., Dynamic programming, Princeton University Press, 1957. 7. Duran, M.a., Grossman I., A mixed integer nonlinear programming algorithm for process systems synthesis, AICHE j.,32(4), 1986 8. Edgar, T.F, Himmelblau, D.M, Optimization of chemical processes, McGRAW-HILL 1988. 9. Markel,J.D, Gray, Jr.A.H., Linear prediction of speech. Springer Verlag, New York 1976 10. Smati A., Ainouche A., Algorithmes adaptatifs d’optimisation d’un gazoduc, Proceeding of the MCEA 95, Grenoble (France). 11. Smati A., Zeraibi N., and Touabti M C., Optimisation du réseau Algerien de transport de brut et de condensat, Oil and Gas science and technology, Rev IFP Vol 55 (2000) 12. Smati, A., A systemic approach in optimization of the Algerian crude oil network. AMSE, Lyon, France 13. Wong P. J ., Larson R. E., Optimization of natural gas pipeline systems via dynamic programming, IEEE trans. automatic control, vol . AC-13, N° 5, pp. 475-481, 1962.

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