Copyright ® IFAC Control Applications of Optimization, St. Petersburg, Russia, 2000
ALGORITHMS FOR CONTROL OF GAS-LIFT PROCESS OF OIL EXTRACTION Mikhail I. Gusev*, Vasilii A. Leonov**
* Institute
of Mathematics and Mechanics, Ural Branch of RAS, 620219, Ekaterinburg, Russia e-mail: gmi @imm.uran.ru ** NIl GASLIFT, 626440, Nizhnevartovsk, Russia, e-mail:
[email protected]
Abstract: A gas-lift process of oil extraction is based on supply of high pressure gas into oil wells in order to increase well's production. The problems of optimal gas redistribution between gas-lift wells maximizing the wells productivity or reducing the oil cost are studied in this paper. These problems are formulated as nonlinear programming problems. The algorithms for search of global optimum built upon embedding of considered problems into a class of infinite-dimensional variational problems are considered. Copyright @ 2000 IFAC Keywords: Nonlinear programming, Global optimization, Optimal control, Resource allocation.
problem with separable functional
1. INTRODUCTION
n
A gas-lift complex of oil production constitutes a large-scale system, which consists of the oil wells, interacting through the oil gathering, gas transport systems, and pay bed. Operating conditions of gaslift wells, determined by the value of HPG supply, have an impact on wells capacity, the energy consumption, the processes of oil and water transport, the maintenance of reservoir pressure, etc .. Control of the process of oil extraction is realized by means of HPG redistribution between the wells (Leonov, et al., 1986; Zaitsev, et al., 1987; Gusev, 1990; Gusev and Leonov, 1997).
f(u)
=L
J;(Ui) -+ max,
(1)
i=1
under constraints n
L Ui ~ R,
0 ~ Ui ~ bi ,
i
= 1, ... ,n.
(2)
i=1
Here R denotes the summary resource of HPG. The continuous functions J;(Ui), i = 1, ... , n, are assumed to be as follows: J;(u;) = 0, 0 ~ U; < a;, J; (u;) is strictly concave, increasing function on [a;, b;]. Here a; E [0, bi ] are given numbers. These functions, describing the productivity of wells versus HPG supply, are referred to as the characteristic curves and are obtained, practically, by fitting the experimental data with polynomial spline, or as the results of computations with hydrodynamic
The following two problems of optimization of gaslift process are considered in this report. The first one is concerned with the optimal HPG redistribution in order to maximize the overall wells production. It may be written as the resource allocation 131
not peculiar to initial problems. This enables to propose efficient algorithms for solution of initial problems, which in the case of sufficiently large n give the results close to global optimum.
model of gas-lift well. Placing the value of HPG supply for i-th well Ui into the range 0 Ui < ai means switching-off of this well.
:s
Denote I = {i : ai > O} . It is clear that for optimal distribution u* we have either ui = 0 or ui E [ai, bi ], i E I . Thus, the problem (1) is equivalent to the following nonlinear mixed integer programming problem
Note that algorithms for solution of nonlinear mixed integer programming problem based on optimal control approach were considered in (Lee, et al., 1998).
n
F(u, v)
=L
vih(ud -+ max,
(3)
2. MAXIMIZATION OF OVERALL OIL PRODUCTION
i=l under constraints
Consider an infinite-dimensional analog of problem (1), (2)
n
L ViUi :s R, i=l Vi E {O, I} , i E I, Vi
O:S Ui
= 1,
:s bi ,
i ~ I, i
n
= 1, .. . , n .
L
Minimization of a prime cost of oil production by changing the operating conditions of gaslift wells constitutes the second problem, considered in this paper: Fo(u) -+ min, (4)
(6)
h(Ui(t))dt -+ max,
subject to
L Jui(t)dt :s R, n
1
(7)
.=1 0
= {u E Rn : 0 :s Ui :s bi , i =
O:S
1, . .. ,n}.
Ui(t):s b
i,
i = 1, .. . ,n,
in a class of measurable functions.
Here n
Cl
Fo(u)
1
.=1 0
subject to
uEU
J
=
~
i=l
kdi (ud
+ Cz
n
~
i=l ~ !i(Ui) i=l n
Ui
in
and !~ the values of maximum for Denote as problems (I), (2) and (6), (7) respectively. In the case Ui(t) == Ui = const problem (6), (7) transforms into problem (1), (2), and hence, !~. Rewrite (6), (7) in a form of optimal control problem
+ C3 (5)
in :s
ki' i = 1, ... , n, Cj, j = 1,2,3, are given positive numbers. The functions h are the same as in problem (1). The value k i characterizes irrigation of the i-th well, so that kdi(Ui) is the amount of liquid (oil+water) produced by i-th well in a unit of time. The coefficient Cl represents the expense of production of a unit of liquid (oil+water) (it depends on the value of energy consumption, the cost of oil and water transport and oil conditioning). The value of C2 characterizes the cost of production, conditioning and transport of a unit of high pressure gas, C3 is a constant expense.
L J!i(Ui(t))dt -+ max, n
1
.=1 0
Xi
= Ui,
Xi(O)
= 0, i = 1, ... , n,
n
L xi(l) :s R,
u(t) E U.
i=l The solution of the last problem exists (Lee and Marcus, 1967) and may be found from Pontryagin's maximum principle. Let u*(t) be the optimal control and
Both considered problems are multiextremal, nonsmooth, and have rather high dimension (in practice, n has an order 10 2 -10 3 ). We describe global optimization algorithms for their solutions that are based on infinite-dimensional extensions of considered finite-dimensional problems. We substitute the initial problem (1) by the appropriate optimal control problem, which in the class of constant controls coincides with initial problem. For problem (4) an algorithm of solution is based on the transition from this problem to variational problem in the class of regular Borel measures on U . Both infinite-dimensional optimization problem have convexity properties that are
n
H(x, u, t/J)
=L
!i(Ui)
+ t/J'u
i=l be the Hamiltonian of the problem. There exists the Lagrange multiplier 0* ~ 0 such that
H(x*(t), u*(t), t/J(t)) = = max{H(x*(t) , u,t/J(t)): U E U},
o:s t:S 1,
where t/J(t) is the solution of the adjoint system
-rj; 132
= -oH/ox = 0,
Thus, in the case a· f: ai, i E I, the algorithm gives the global maximum for problem (1), (2) . If a· = ai for some i El, one might expect that = f(u·) will be close to for large n. In particular, if the sequence
satisfying the transversality conditions ""i(l) = -a·,
i = 1, . .. ,n.
in
For a 2: 0 define
in
Vi(a) = argmax{h(ui) - aUi : 0 :; Ui :; bi }. (8) Under ai = 0 (i f/: I) the value vi(a) is uniquely determined due to strict convexity of h(Ui) on
is bounded, and
[ai,bd·
t.pn :=
L
Let i E I. Denote
then relative error n -t 00. In this case Vi (a) is uniquely determined if a f: ai: vi(a) = 0 if a > ai, Vi(a) > 0 if a < ai . Under the condition a = ai the maximum in (8) is attained at two points: Ui = 0 and Ui = vi(ai - 0). From the maximum principle it follows that the optimal control is determined uniquely and does not depend on t provided that a· f: ai, i E I. In this case the values vi(a·), i = 1, . . . , n represent the solution of the initial problem (1), (2). If a = am for some m E I, then u;',,(t) is a bangbang function, taking the values vm(a· - 0) and O. We will further assume that all the values ai, i E I are distinct. This assumption does not imply considerable loss of generality, because it may be achieved by arbitrarily small variations of the functions fi' i E I.
R 100 200 300 400 500 600 700 800 900 1000
L vi(a) .
R 300 600 900 1200 1500 1800 2100 2400 2700 3000
Letting ui = 0, i f/: J, we specify ui, i E J, solving the following convex programming problem
L h (ud -t max, iEJ
subject to Ui :;
bi ,
in 1612 1917 2143 2337 2512 2667 2811 2945 3071 3191
in
in
3390 4193 4790 5292 5720 6115 6453 6758 7035 7282
3390 4194 4804 5307 5735 6115 6453 6758 7035 7283
Relative error (%) 0.00 0.02 0.29 0.28 0.26 0.00 0.00 0.00 0.00 0.01
i E J.
iEJ
(u~,
in 1607 1917 2139 2323 2496 2653 2796 2929 3071 3191
Relative error (%) 0.31 0.00 0.19 0.60 0.64 0.52 0.53 0.54 0.00 0.00
Table 2. Results of maximization, n=292
J = ({I, ... ,n} \1) U {i El : ai > a· }.
ai:;
(in - in)/ in tends to zero as
Table 1. Results of maximization, n=156
Assume that V(O) > R, otherwise problem (1), (2) has an obvious solution Ui = bi , i = 1, . .. ,n. Find the least root of the inequality V (a) :; R, denote it as a·. If a· f: ai, i E I, then the equality V(a·) = R holds and the vector u· = (VI (a·), . . . , Vn (a·)) is a required solution. Otherwise, let a = am for some m E I . Define
Finally, we take u· = of problem (1), (2) .
00,
in, (in - in)/ in.
i=1
R,
n -t
in,
n
V(a) =
Ui :;
00,
The following two tables include the results of numerical experiments with real data relevant to gaslift wells for n = 156 and n = 292, respectively. Here the first column contains the resource value R, the second - the value of maximum found by means of described algorithm, the third the "precise" value of maximum found by the dynamic programming method, and the last column - the relative error
Note that function vi(a) is non decreasing as a> O. It is continuous, if i f/: I, or has the unique point of discontinuity a = ai, if i E I. Define a function
L
h(O) -t
i~[
As is seen from the above tables, considered algorithm gives the results close to the value of global optimum.
. . . ,u~) as the solution
133
for i E I, where
3. MINIMIZATION OF PRIME COST OF OIL PRODUCTION BY CHANGING THE OPERATING CONDITIONS OF GASLIFT WELLS
Let us define
Here we describe the algorithm of solution of problem (4). The proposed algorithm is based on transition from this problem to the following variational problem
As in the previous case we can assume without essential loss of generality that ai ¥ a j, i, j E J, i ¥ j. Define
ui(v,a)
where J.L denotes a measure in U,
Fl (J.L)
=
f t.=1
kdi(Ui)
[Cl
+ C2
u
ft
F2(J.L) =
t.=1
Ui] dJ.L
J;(u;)dJ.L,
and M is a set of all a regular, nonnegative measures defined on a Borel sigma-algebra of U such that J.L(U) = 1. Note that M is a convex set in the space C*(U), compact in weak - star topology, where C* (U) is adjoint to a space of continuous functions on U C Rn. The last problem has the convexity properties, which do not inherent in initial problem. For the measure J.Lu concentrated at point U E U, obviously, F(J.Lu) = Fo(u). Thus, problem (9) may be treated as the extension of initial problem.
Let J.La ({U 1 ( a )}) fine
9i(a,{3)
{3)!i(u;(a)),
r (a,{3)
(12) (13)
where
(10)
J.L E M .
r(a,{3):= +C2
L
[{3u~ (a)
+
(1 - {3)u;(a)]
+ C3'
i=l
The above reasoning yields the following equality min {F(J.L) : J.L E P(M)}
(ll)
= (14)
The function G( a, {3) is continuous in a and does not depend on {3, if a ¥ ai, i E I. At a = ai, i El, G (a, {3) has discontinuities in a and is monotone in {3. Find minimum of G(a, {3) in (14). If minimum is attained at a* -:j:. ai, i E I, then u 1 (a*) is a solution of initial problem. If a* = ai for some i E I, then optimal distribution coincides with u 1 (a*), if G(a*, 1) > G(a*, 0), and with u 2 (a*) otherwise.
n
+ C2 L
i=l
Lki9i(a,{3)+
= min{G(a,{3) : 0 ~ a,{3 ~ I}.
a)F2(J.L)}.
n
-a[Cl L kdi(Ui)
Cl
i=l n
a)F2 (J.L*) =
Ui] +
i=l n
+(1- a) L
n
{3. De-
E 9i(a,{3)
For 0 ~ a ~ 1 find a measure J.La, which attains a maximum in (11). It is clear that such measure should be concentrated on the set of points of maximum of the function :=
= {3J;(u~(a)) + (1 -
=1-
n
= IJEM max {-aFl (J.L) + (1 -
p(u)
J.La ({U 2 ( a )})
i=l
It is easily seen that if J.L* is the solution of problem (9), then J.L* E P(M). From the convexity of M and linearity of Fi(J.L) in J.L it follows (see, e.g., Karlin, 1959; Podinovskii and Nogin, 1982) that there exists a, 0 ~ a ~ 1, such that
+ (1 -
= {3,
G(a,{3) =
Let P(M) be the Pareto set in the the following bicriterial optimization problem
-aF1 (J.L*)
(1- a(1 + Ciki))J;(V) - aC2V,
the function Ui(V, a), i = 1, . . . , n, has on [0, bi ] at most two minimum points in v, denoted further as u}{a) and u~(a). Consider that u}{a) ~ u~(a). It is easily to check that u}{a) = uHa), if i ~ I or if i E I and a ¥ ai, i El, that is in this case a maximum point is uniquely determined. The measure J.La is as follows. Under the condition a -:j:. ai, i E I the measure is concentrated in the single point u 1 (a) = (ut (a), .. . , u;'(a)) E U. If a = ai, i El, the measure J.La is concentrated in two points u 1 (a),u 2(a) E U such that u}(a) = u~(a), j -:j:. i.
+ C3,
U .=1
Fl (J.L) --+ min, F2(J.L) --+ max,
:=
fi(ui).
Consider problem (4) with additional constraint E7=1 Ui ~ R . Corresponding infinite-dimensional extension takes the following form
i=l
Denote
(15) 134
h( 0:, A, (3)
where
~
R.
Let 0:*, A*, f3* be such that C (o:*, A*, f3*) "f(o:*,A*) f; "fi, i E I, then Let J-L* be the solution of problem (15). Passing to equivalent bicriterial convex programming problem and writing for the last one the optimality conditions we get that there exist 0 ~ "f ~ 1 and A ~ 0 such that J-L* maximizes the functional -o:Fl(J-L) + (1- 0:)F2 (J-L) + AF3 (J-L) on the set M . For i
= 1, .. . , n
where u!b)
gives the solution of considered problem. Unlike the optimization problem (9) the optimal measure may be concentrated here in two distinct points, if "f(o:*,A*) = "fi for some i E I . In the last case u* = (WHo:*,A*), ... ,W~(o:*,A*)) gives a result close to optimal for sufficiently large n (assuming all "fi, i E I are distinct).
= u;b) = 0 as "f < 0,
4. CONCLUSION
{u} ("f), u; ("f)} =
Considered algorithms were implemented as a part of decision support system used for control of gaslift process of oil extraction in Samotlor oil field of Western Siberia. They are realized for rather general situation that considered here, provided that restriction on Ui are given as a sum of few disjoint intervals. This permits to exclude from consideration the "unstability zones" of gas-lift wells. The practical implementations show that algorithms ensure finding of global optimum with acceptable accuracy in a small time.
"fUi : 0 ~ Ui ~ bi }.
Note, that for "f > 0 u!b)
= u;("(),
if i ~ I or if
i E I and "f f; "fi, where
"fi
= max {j;(Ud/Ui : 0 < Ui ~ bi } .
= max{"ti : i
Let t
E I}. Define
A + O:C2 "f(0:, A) := 1 _ 0:(1 + clk i )' . .
Wf{o:,A):= uib(o:,A)) if 0: f; 1
1
+ Cl
k '
REFERENCES
i
w1 (0:, A)
Gusev, M.1. (1990). Recourse allocation problem under weak subsystem's interconnection. Engineering cybernetics, 6, 58-63. (in Russian) Gusev, M.1. and V. A. Leonov (1997). Method of optimization of the system of gas-lift wells. Patent, no. 2081301, pp. 1-12, Rospatent, Moscow. (in Russian) Karlin, S. (1959). Mathematical Methods in Games Theory, Programming and Economics. Reading Mass, Addison-Wesley. Lee, E. B. and L. Markus (1967). Foundation of Optimal Control Theory. Wiley, New York. Lee, H. W., K. L. Teo and X. Q. Cai (1998). An Optimal Control Approach to Nonlinear Mixed Integer Programming Problems. Computer Math. Applic., 36, 87-105. Leonov, V. A., G. M . Dolgih and R. R. Shigapov (1986). Optimization of the basic objects of gas-lift oil production. VNIIOAN, Moscow. (in Russian) Podinovskii, V.V. and V.D. Nogin (1982). ParetoOptimal Solutions of Multicriteria Problems. Nauka, Moskow. (in Russian) Zaitsev, Yu. V., et. al. (1987) . Theory and Practice of Gas-lift. Nedra, Moscow. (in Russian)
= 0 otherwise (i = 1, ... , n, j = 1,2). Consider the following functions, analogous to (12), (13) : gi(O:,
A,(3)
= f3j;(wt{o:, A)) + (1 C(o:, A, (3) =
(3)fi(w;(o:, A)),
nr(o:, A, (3)
2: gi (0:, A, (3) i=l n
r(o:,A,f3):=
Cl
Lki9i(0:,A,f3)+ i=l
n
+C2 L [f3w; (0:, A) + (1 - (3)w~(o:, A)]
+ C3,
i=l
and n
h(o:,A,f3)
=L
[f3w; (0:, A)
+ (1- (3)W~(o:,A)].
i=l
The following equality holds min {F(J-L) : J-L E P(M), F3 (J-L) ~ R} where C*
= C*,
= infC(o:,A,f3),
subject to
o ~ 0: ~ 1,
If
u* = (Wi(o:*,A*), . .. ,W~(o:*,A*))
and "f E R define
= arg max {j; (Ui) -
= C*.
0 ~ A ~ t, 0 ~ f3 ~ 1, 135