Chapter 5
Application: investment alternatives
In Chapter 3 it was shown that the robustness of a technical system against
infeasibility of the production planning problem can be given in terms of
the commercial scope. Measures of the commercial scope are considered as
key properties of any investment proposal. The application of these measures
to a numerical instance of the matchplus model in Chapter 4 shows that
the described situation provides insucient robustness, and investments are
necessary. In this chapter, within the framework of the matchplus model
some investment alternatives are investigated using the same measures as in
Chapter 4.
Recall that the production planning model matchplus, as its name suggests, is tailored to the problem of matching of the annual ACQ commitments
with daily demand. It is extended with a rude form of the production capacity
restrictions. It does not serve to conclude anything on other possible infeasibilities, like a shortage of transportation capacity. It is therefore obvious that
the analyses of this and the subsequent chapter give only partial information.
In practice the analysis should be complemented by other studies, describing
the production planning problem from other perspectives, before a nal statement about the desirability of investments could be made. The analysis of this
chapter is also partial in the sense that only robustness and risk measures are
considered. Financial measures like investment cost are not taken into consideration. Other criteria that might enter the investment selection process, like
technical, environmental or political arguments, are left out as well.
Like in the previous chapter, special attention is paid to the use of the
induced constraints which are found as a by-product of the calculation of the
robustness characteristics. This proves to be especially useful since it appears
that in this example the commercial scope can very eectively be approached
by only a few of these linear constraints.
145
146
Application: investment alternatives
Table 5.1. Investment alternatives (all capacities in 109 m3 )
Alternative PLUS PLLS
x0
+1
0.9
x1
+1
0.9
x2
+1
0.9
x3
+1
0.9
5.1
NU UGU UGL
0.66
0.66
1.00
1.00
The investment alternatives
0
3.25
0
3.25
0
2:20
0
2:20
The analysis of Section 4.5 does not suce to generate acceptable investment
alternatives. Firstly, the induced constraints that are considered are relevant
under x0 . But, as was already stated in Section 3.3, page 55, it depends on the
investment alternative x which induced constraints are redundant and which
are not. For other investment alternatives, dierent induced constraints might
come in, which severely restrict the eectivity of some investment instrument.
The attractiveness of an investment alternative is not only determined by its
robustness, but also by other criteria like its cost. For a review of usual investment criteria, see Pike (1993). Furthermore, the analysis is only oriented at
feasibility of the matching problem depicted in the matchplus model. Now
matchplus contains many but not all essential elements determining feasibility of all future short term planning problems. An alternative may also inuence
the robustness against infeasibility of other short term planning problems. As
a hypothetic example, consider the inuence of storage outow capacity on the
transportation capacity. Perhaps for the transportation capacity it might be
useful to have a high undergound storage outow capacity UGU , as compared
to its inow capacity UGL, whereas from Figure 4.4 it followed that this is not
cost-ecient for the matchplus problem alone.
For the experiments of this chapter, three investment alternatives are
dened. From the purpose of this thesis it is not interesting to know in all
detail how these alternatives were generated, though some considerations have
passed in review. The alternatives are summarized in Table 5.1. In the analyses of this chapter they are continually set out against the zero investment
alternative x0 which was introduced and investigated in the previous chapter.
Alternative x1 represents the introduction of an underground storage facility,
x2 represents an extension of the nitrogen capacity, and x3 represents the combination of the two. It is assumed that all three investment alternatives can be
realized before the future moment to which the commercial data refer.
5.2 Scenario analysis
147
Table 5.2. Directional scope in coordinate directions under dierent alternatives (in 109 m3 35.17)
Direction d
DH DL ACQ
DH +
1
0
0
DH
1
0
0
DL+
0
1
0
DL
0
1
0
ACQ+
0
0
1
ACQ
0
0
1
5.2
Directional scope under x =
x1 x2 x3
16.80 16.80 16.80 16.80
1.18 3.23 8.99 13.19
x0
1
1
1
1
5.50 15.31 17.40 17.40
1.30 3.58 9.80 14.38
24.89 24.89 24.89 24.89
Scenario analysis
5.2.1 Directional boundary searches in coordinate directions
The central scenario was tested and found to be feasible under all investment
alternatives. The degree of feasibility was analysed by means of the directional
instruments, described in Section 3.4. Where rstly special emphasis was put
on plus and minus the coordinate directions. The results are summarized in
Table 5.2. This table gives a good idea of the robustness of each alternative
and of the capability to take advantage of business opportunities.
It appears that the directional scopes of the central scenario s0 in the
directions ACQ+, DH and DL , which were small under x0 , increase under
the alternatives x1 , x2 and x3 . Of course, since S (x0 ) S (xi ) S (x3 ), i = 1; 2,
all new alternatives do better than the zero investment alternative x0 , and
the increase is maximal for x = x3 , the combined alternative. Alternative x2 ,
implying extra nitrogen production capacity, scores better in these directions
than x1 , the underground storage facility.
In the other three coordinate directions the investment alternatives x1 ,
2
x and x3 do not extend the scope. For the direction DL+ this is obvious,
since PLUS = +1 under all alternatives. In the directions DH + and ACQ ,
under alternative x0 the scope was restricted by induced constraint (B ) (see
page 111), which is independent of any investments. Since the scopes under the
other investment alternatives are relaxations of S (x0 ), in the given directions
induced constraint (B ) stays restrictive under all alternatives. Something similar holds for direction DL under alternatives x2 and x3 : beyond a certain
level of nitrogen capacity, investments in underground storage do not make any
dierence on the robustness. This phenomenon will be claried in Section 5.3.
148
Application: investment alternatives
Table 5.3. Boundary points found by going from s0 to si under dierent alternatives
Scenario 0ij found under alternative xj
si
x0
x1
x2
x3
s1
0.06 0.17 0.53 0.74
s2
0.05 0.13 0.38 0.55
s3
0.05 0.15 0.38 0.56
s4
0.04 0.12 0.30 0.44
s5
0.83 0.83 0.83 0.83
s6
1.19 1.19 1.19 1.19
s7
0.83 0.83 0.83 0.83
s8
1.19 1.19 1.19 1.19
Fj
0.04 0.12 0.30 0.44
5.2.2 The extreme scenarios
The feasibility of s0 is further explored, using the extreme scenarios s1 through
s8 . That is, each scope S (xj ) is analysed by following the directions si s0 out of
s0 until the boundary of the scope is reached in the point s0 +0ij (si s0 ). Again,
the technique of Section 3.4, using the optimal solution of (3.6), is applied. The
results are given in Tables 5.3 and 5.6. In Table 5.3, we also gave Swaney and
Grossmann's exibility index1 , dened as Fj = mini 0ij (see Section 3.4).
Like in Section 4.2.3, 0ij is both a measure of the feasibility of the feasible
scenario s0 and of the (in)feasibility of si under alternative xj . As can be
seen in Table 5.3, the scenarios s1 , s2 , s3 and s4 are far from feasible under
alternative x0 . They stay infeasible under the other alternatives, but the degree
of infeasibility decreases. Furthermore, s5 and s7 are infeasible and s6 and s8
are feasible under all alternatives, and their degree of (in)feasibility does not
depend on the alternative.
Under all alternatives, most of the extreme scenarios turn out to be infeasible. Table 5.3 already contains information about their degree of infeasibility,
but more information can be obtained if their infeasibility is measured by their
minimum recourse cost dened in (3.7). That is, in case of an infeasible scenario si , a feasible one si is chosen to be starting point of the production plan.
The damage caused by the infeasibility depends on the deviation z = si si ,
which is determined by cost minimization. Cost coecients were used as given
in Section 4.1. The results are stated in Table 5.4.
1. Swaney and Grossmann (1985). See the remark on the terminology on page 42.
149
5.3 New induced constraints
Table 5.4. Minimum infeasibility costs of infeasible extreme scenarios under
dierent alternatives (in 109 D)
z
DH DL ACQ
Under x0
s1
s2
s3
s4
s5
s7
0
0
0
0
0
0
0
0
:
:
3 56
0
3 56
0
Under x2
s1
s2
s3
s4
s5
s7
0
0
0
0
0
0
0
0
:
:
3 56
0
3 56
0
:
25:68
23:19
30:05
18 75
0
0
:
15:69
15:71
22:58
8 82
0
0
z
cost
:
6:42
5:80
7:51
35:63
35:63
4 69
:
3:92
3:93
5:65
35:63
35:63
2 21
DH DL ACQ
Under x1
s1
s2
s3
s4
s5
s7
0
0
17.41
4.35
0
0
24.41
6.10
0
0
20.41
5.10
0
0
27.34
6.84
3.56
0
0
35.63
3.56
0
0
35.63
Under x3
s1
s2
s3
s4
s5
s7
cost
0
0
5.12
1.28
0
0
12.05
3.01
0
0
13.38
3.35
0
0
19.68
4.92
3.56
0
0
35.63
3.56
0
0
35.63
If the alternatives are compared, the results of the previous section are
conrmed: x3 scores equally or better than x2 , x2 equally or better than x1
and x1 equally or better than x0 . Remark that, where for instance s4 has
a smaller 0ij and (in energy terms) greater distance to the scope (kz k from
Table 5.4) than s5 , on the contrary s5 has larger infeasibility cost. Therefore,
an ordering of extreme scenarios for their `dangerousness' cannot be uniform.
However, the ordering of the investment alternatives is uniform, in the sense
that for instance x3 scores equally or better than x2 in all respects.
5.3
New induced constraints
Using Theorems 3.2 and 3.3, it is possible to compute, for each of the table
entries in the previous section, the induced constraint on which the corresponding boundary point is situated. For alternative x1 , having UGL = UGU = 0, in
some cases the dual optimal solution of (3.6) or (3.8) was not unique (namely
when induced constraints (A1) or (A2) were encountered), yet all dual optimum
solutions gave the same induced constraint, as is explained in Appendix 4A. For
the investment alternatives x2 and x3 , in each case the dual optimum solution
appeared to be unique. Therefore all induced constraints found correspond to
a facet of the respective scopes.
150
Application: investment alternatives
In the course of the experiments that led to Tables 5.2, 5.3 and 5.4, only
ve irredundant induced constraints were encountered, among which two new
ones. Three of them, named (A1), (A2) and (B ), already were identied as
determining facets of S (x0 ), and they were extensively discussed in Section 4.3
and Appendix 4A. The two new constraints are denoted by (A0) and (D). It
seems that these ve induced constraints constitute a very complete explicit
description of the scopes S (x) under x = x0 ; x1 ; x2 and x3 . Ad hoc directional
boundary searches conrmed the conjecture that inside the convex hull of the
extreme scenarios convfs1 ; : : : ; s8 g no other induced constraint is active under
the investment alternatives considered. Therefore, for the values of the model
parameters as given in Chapter 2, the available information on the explicit
form of the scope S (x) under x = xi , i = 0; 1; 2; 3 can be characterized by the
following ve constraints:
DH
0:107 DL + 0:900 ACQ 40:331 NU
DH
0:214 DL + 0:909 ACQ
DH
PLLS + 30:248 NU 0:956 UGL
(A1)
0:375 DL + 0:918 ACQ
2 PLLS + 20:166 NU 1:911 UGL
0:297 ACQ 0
0:44 DH
DH
(A0)
DL
+ ACQ 4 PLLS
(A2)
(B )
(D)
In Tables 5.5 and 5.6 it is indicated under which conditions these constraints
were encountered.
In Section 4.3 it appeared to be possible to present the induced constraints
(A1), (A2) and (B ) in their symbolic representation, in which the model parameters are not lled in. This can be done for (A0) and (B ) as well. Actually,
induced constraint (A0) is not new at all. It came up during the sensitivity
analysis in Section 4.6.1, but it played no role in determining S (x0 ) under
the original parameter values. Therefore its symbolic version was given and
interpreted on page 123.
The symbolic formulation of induced constraint (D), which was found to
be binding under x2 and x3 when considering a decrease in DL, turns out to
be rather simple:
5.3 New induced constraints
151
Table 5.5. Induced constraints in coordinate directions under dierent alternatives
Direction
DH +
DH
DL+
DL
ACQ+
ACQ
Constraint encountered under alternative
x0
x1
x2
x3
(B )
(B )
(B )
(B )
(A1)
(A0)
(A2)
(A2)
|
|
|
|
(A1)
(A1)
(D )
(D)
(A1)
(A0)
(A2)
(A2)
(B )
(B )
(B )
(B )
4 PLLS + ACQ DH + DL
(D)
or,
minimum total annual energy production from both Slochteren and non-Slochteren origin may not exceed total annual
energy demand.
Obviously, this is a natural constraint. The symbolic version is derived in
Appendix 5A.
Now that some new induced constraints are found to be irredundant under the
new investment alternatives, it is good to reconsider the eectivity discussion
of Section 4.5.
Since for the investment alternatives x1 and x3 it holds that jUGLj < UGU ,
for these alternatives (A1) and (A2) are irredundant, and the related constraints
(C 1) and (C 2), which are described in Appendix 4A, are redundant. As already
concluded in Section 4.5, induced constraints (A1) and (A2) are shifted by
introducing extra nitrogen capacity and new storage capacity. Induced constraint (B ) is indeed left untouched by investments. However, the eectivity
of the investments considered cannot be read from these three induced constraints alone, since the shift of (A1) and (A2) in some cases the new induced
constraints (A0) and (D) irredundant.
As can be read from both the symbolic version on page 123 and from the
above numerical formulation, induced constraint (A0) is only dependent on
NU . This accounts for the fact that in the direction DL , where under x2 the
directional scope is restricted by (A0), alternative x3 does not improve on x2 .
Constraint (D) is independent of the investment variables involved in the
alternatives. Both (D) and (B ) are not relaxed by the investment alternatives.
There is a natural boundary to the scope, beyond which investments are not
152
Application: investment alternatives
Table 5.6. Induced constraints in direction of extreme scenarios under dierent
alternatives
Scenario
si
s1
s2
s3
s4
s5
s6
s7
s8
Constraint encountered under alternative xj
x0
x1
x2
x3
(A1)
(A1)
(A1)
(A1)
(B )
(B )
(B )
(B )
(A0)
(A0)
(A1)
(A1)
(B )
(B )
(B )
(B )
(A2)
(A2)
(A2)
(A2)
(B )
(B )
(B )
(B )
(A1)
(A1)
(A2)
(A2)
(B )
(B )
(B )
(B )
eective. Constraints (B ) and (D) determine a part of this boundary (at least
if a change in PLLS is left out of question), since they are irredundant under
the given investment alternatives. If only investments in storage capacity are
considered, as under alternative x1 , the boundary is made up by (A0) as well,
since this does not depend on UGU or UGL either. This was not foreseen in
Section 4.5.
Since PLUS = 1, the L-gas system is not involved in the production capacity test constraint (B ) and PLUS does not appear in any of the constraints.
Also as a result of this, the underground storage outow capacity UGU is found
not to be binding in any way, as was predicted in Section 4.5.
5.4
Stochastic results
For all investment alternatives the reliability, dened in (3.9), is estimated using
the four samples, described in Table 4.4. Not only point estimates are given,
but also symmetric 95%-condence intervals are constructed. To compute these
intervals, the binomial distribution of the sample average of f1 (s; x) was approximated by a normal distribution. Table 5.8 is based on the assumption of
stochastic independence between the stochastic variables.
Also the physical and nancial risks, dened in (3.12) and (3.13) respectively, have been estimated using the four samples. Tables 5.9 and 5.10 give the
results. Both types of risk dier in the way the eventual infeasibility is resolved
and in the way eventual infeasibilities are measured. Table 5.9 is based on variation of ACQ only, where only surpluses were counted. Table 5.10 is based on
recourse cost minimization.
5.5 Conclusions on commercial robustness under alternatives
153
Table 5.7. Induced constraints found by minimizing costs in infeasible extreme
scenarios for dierent alternatives
Scenario Constraint encountered under alternative xj
si
x0
x1
x2
x3
s1
(A1)
(A0)
(A2)
(A1)
s2
(A1)
(A0)
(A2)
(A1)
s3
(A2)
(A1)
(A2)
(D )
s4
(A2)
(A1)
(A2)
(D)
s5
(B )
(B )
(B )
(B )
s7
(B )
(B )
(B )
(B )
A comparison of the alternatives learns that the uniform ranking of the
alternatives following from the scenario analyses of Section 5.2 is only conrmed.
On this point it is good to remark that all results of this chapter were
exactly reproduced by optimizing only over the induced constraints, instead of
using the extensive form including y. We already found that for the four investment alternatives considered, the ve induced constraints (A0), (A1), (A2), (B )
and (D) give a perfect description of the commercial scope within the relevant
region, dened by the convex hull of the extreme scenarios convfs1 ; : : : ; s8 g.
Apparently they also give a perhaps not perfect, but anyway very good description of the commercial scope in general under the investment alternatives x0 ,
x1 , x2 and x3 . Unfortunately we are not able to interpolate this conclusion to
all x 2 convfx0 ; x1 ; x2 ; x3 g.
5.5
Conclusions on commercial robustness under
alternatives
The results concerning the analysis of x0 , presented in the previous chapter,
revealed that in this numerical example investments are necessary. Investments
in underground storage (x1 ) shows an improvement of the commercial robustness. The investment alternative concerning extra nitrogen production capacity
(x2 ) does better in guaranteeing feasibility of the (partial) matchplus model.
The best, but also the most expensive option to implement both (x3 ). The
ranking of the alternatives is `uniform' in the sense that if alternative xi is
better than xj , it scores better on all robustness or risk measures. Apparently,
the commercial scope S (xi ) completely includes the commercial scope S (xj )
for i > j .
154
Application: investment alternatives
Table 5.8. Reliability estimates for dierent alternatives (incl. symmetric 95%condence intervals)
Sample
1
2
3
4
Sample
1
2
3
4
Under alternative x0
0:550 (0:519, 0:581)
0:591 (0:560, 0:622)
0:552 (0:521, 0:583)
0:570 (0:539, 0:601)
Under alternative x2
0:859 (0:837, 0:881)
0:875 (0:854, 0:896)
0:862 (0:840, 0:884)
0:889 (0:869, 0:909)
Under alternative x1
0:640 (0:610, 0:670)
0:679 (0:649, 0:709)
0:645 (0:615, 0:675)
0:673 (0:643, 0:703)
Under alternative x3
0:945 (0:931, 0:959)
0:939 (0:924, 0:954)
0:945 (0:931, 0:959)
0:953 (0:940, 0:966)
The four commercial scopes were completely described by only ve induced
constraints. The three new investment alternatives introduced only two new
induced constraints. The other induced constraints had already been encountered in the previous chapter. Clearly, under dierent alternatives dierent
subsets of these ve constraints are irredundant.
An important fact to note is that there is a `natural boundary' on the
scope, and therefore on the commercial robustness, which cannot be inuenced
by the investments in underground storage or nitrogen production capacity.
To compute this superscope, treat the investment decision variable x just like
the production decision variable y and eliminate both. Induced constraints (B )
and (D) determine parts of the boundary of this set. Related to the superscope,
Table 5.9. Estimation of the expected ACQ surplus by variation of ACQ only
under dierent alternatives (in 109 m3 35.17)
Sample Risk under alternative xj
x0 x1 x2
x3
1
3.01 2.15 0.62
0.20
2
2.74 1.94 0.60
0.18
3
3.02 2.15 0.65
0.23
4
2.59 1.77 0.49
0.16
5.6 Sensitivity analysis with respect to the underground storage
155
Table 5.10. Estimation of the nancial risk by recourse cost minimization under
dierent alternatives (in 109 Dutch orins)
Sample Risk under alternative xj
x0 x1 x2
x3
1
0.79 0.57 0.19
0.08
2
0.69 0.49 0.15
0.05
3
0.76 0.54 0.16
0.06
4
0.76 0.55 0.23
0.15
Grossmann and Floudas (1987) dene a structural exibility index by taking
the maximum exibility index over all investment alternatives.
5.6
Sensitivity analysis with respect to the
underground storage
In Chapter 2, some assumptions were made concerning the underground storage. One of these assumptions was that the quality of the gas in the storage
is homogeneous. Now it is reasonable that at least one quality parameter, like
the Wobbe index, is used as a control parameter for the storage inow control.
After all, planners would not like the quality of gas owing out of the storage
to be a random process: : : But if so, other quality parameters like the caloric
value may vary. And what is more: since the storage is not yet in function,
the control value for the control parameter may be dierent from the value
that was assumed in Chapter 2. These are all reasons to perform a sensitivity
analysis with respect to the quality parameters of the gas in the underground
storage.
Furthermore in Chapter 2 it was assumed that the volume capacity of the
underground storage is sucient. At this moment it is interesting to know what
the volume capacity should be, which can be read from production plans.
Finally, note that the new induced constraints that entered the eye do not
involve the underground storage. The scopes under all four alternatives were
well described by only ve induced constraints. The discussion of the eectivity
of the underground storage based on the zero-investment commercial scope
S (x0 ) (see Figure 4.4) was only valid for combinations of (UGL; UGU ) close to
zero. In Section 5.3 however it followed that the induced constraints involving
the undergound storage, if shifted outward by investments, get redundant from
a certain point on, so that investments have no eect any more.
156
Application: investment alternatives
5.6.1 Variation of the quality of the gas in storage
We still assume that the quality of the gas owing into the storage is equal to the
quality of the gas coming out of the storage. This assumption gives conservative
results. For if this assumption is loosened, the set of feasible combinations of
gas ows going into the storage is relaxed.
Whatever the assumptions about the quality of the gas in storage are, the
introduction of the underground storage means a relaxation of the commercial
scope compared to S (x0 ). Under x0 the induced constraints in the directions
DL+, DH + and ACQ did not depend on the underground storage. It is
unlikely that they will be dominated by any induced constraint depending on
the underground storage if the quality of gas in storage is varied, as is conrmed
by the following analyses. Of course only the investment alternatives implying
a nonzero underground storage, namely alternatives x1 and x3 , have to be
investigated.
Variation of Wug , with Cug = 35:53
The results of ceteris paribus variations of Wug are given in Table 5.11. The
table contains the directional scopes, but only for those directions that showed
a change. The value of Wug was varied in the L-gas market range [43:8; 44:4].
Table 5.11. Some directional scopes for dierent values of Wug
Wug =
1
Under x :
d = DH
d = DL
d = ACQ+
3
Under x :
d = DH
d = DL
d = ACQ+
43.8
43.95
3 22 ( 1)
3 16 ( 1)
15 58 ( 1)
15 31 ( 1)
15 03 ( 1)
14 76 ( 2)
: A
: D)
14:64 (A2)
: A
: D)
14:51 (A2)
17 40 (
: A
: D)
14:38 (A2)
17 40 (
: A
: D)
14:25 (A2)
17 40 (
13 43 ( 2)
17 40 (
13 31 ( 2)
17 40 (
3 23 ( 0)
13 19 ( 2)
: A
: A
3:55 (A1)
44.4
: A
: A
3:58 (A0)
3 23 ( 0)
: A
: A
3:58 (A0)
44.25
15 86 ( 1)
3 23 ( 0)
: A
: A
3:58 (A0)
44.1
13 08 ( 2)
: A
: A
3:48 (A1)
: A
: D)
14:12 (A2)
12 96 ( 2)
The directional scopes do not vary much, if at all: the given directional
scopes on this interval vary always less than two percent. For large enough
values of Wug , the given directional scopes decrease with Wug . Under x3 the
relationship between the directional scope Sd(x3 ; s0 ; ACQ+) and Wug is linear
on the interval [43:8; 44:4], which is the market range for L-gas. This follows
easily from constraint (A2), page 107, since in the boundary points in these
directions, (A2) holds with equality. The same goes for the relationship between
Sd (x3 ; s0 ; DH ) and Wug .
5.6 Sensitivity analysis with respect to the underground storage
157
Variation of Cug , with Wug = 44:1
The results of ceteris paribus variations of Wug are given in Table 5.11. The
table gives the directional scopes and induced constraints encountered, but only
for those directions that showed a change. The value of Cug is varied in the
range [35:17; 35:89]. The value of Cug will not be outside this interval, if the
Wobbe index is kept within the L-gas market range.
Table 5.12. Some directional scopes for dierent values of Cug
Cug =
1
Under x :
d = DL
3
Under x :
d = DH
d = ACQ+
35.17
35.35
:
A
15 26 ( 1)
13 15 ( 2)
14 34 ( 2)
:
:
A
A
15 21 ( 1)
35.53
:
A
15 31 ( 1)
13 17 ( 2)
14 36 ( 2)
:
:
A
A
35.71
:
A
15 35 ( 1)
13 19 ( 2)
14 38 ( 2)
:
:
A
A
35.89
:
A
15 40 ( 1)
:
A
13 21 ( 2)
:
:
A
A
13 23 ( 2)
14 40 ( 2)
14 42 ( 2)
:
:
A
A
The scope for ACQ+ and DH under x3 increases with the value of Cug ,
as follows from Table 5.12. Under x1 the situation is completely insensitive for
changes in Cug . For x3 the relationships are linear, and again this can be easily
seen from constraint (A2) which holds with equality in the boundary points.
Again, the directional scopes that vary do not vary very much, here even less
than a half percent.
Ceteris paribus variation of Cug has only little impact, if at all, on the scope
in the coordinate directions, as was the case with ceteris paribus variation of
Wug .
Simultaneous variation of Cug and Wug
If a gas mixture is composed of the three input gas ows H-gas, L-gas and
nitrogen, there is a high correlation between caloric value and Wobbe index
of the L-gas mixture. Based on the assumptions in Chapter 2, it is possible
to calculate ranges for the L-gas output caloric value, given its Wobbe index
and vice versa. Assume that indeed the L-gas output volume vl is the result of
mixing H-gas, L-gas and nitrogen. For an output Wobbe index Wvl = 43:8 we
have Cvl 2 [35:17; 35:50], for Wvl = 44:1 we have Cvl 2 [35:38; 35:69] and for
Wvl = 44:4 we have Cvl 2 [35:58; 35:88].
Apparently, however interesting the ceteris paribus variations may be, it is
more natural to consider a simultaneous variation of both quality parameters.
Results are given in Table 5.13.
Under x1 , in the directions DH and ACQ+ the directional scope is insensitive to the underground storage parameters, until for high enough parameter
values induced constraint (A1) starts to dominate (A0). The directional scope
158
Application: investment alternatives
Table 5.13. Some directional scopes for simultaneous variation of Wug and Cug
Wug ; Cug ) =
(
1
Under x :
d = DH
d = DL
d = ACQ+
x3 :
d = DH
d = ACQ+
Under
(43.8, 35.17) (43.95, 35.35) (44.1, 35.53) (44.25, 35.71) (44.4, 35.89)
: A
: A
3:58 (A0)
3 23 ( 0)
15 76 ( 1)
:
:
A
A
: A
: A
3:58 (A0)
3 23 ( 0)
15 54 ( 1)
:
:
A
A
: A
: A
3:58 (A0)
3 23 ( 0)
15 31 ( 1)
:
:
A
A
: A
: A
3:58 (A0)
: A
: A
3:50 (A1)
3 23 ( 0)
3 18 ( 1)
15 08 ( 1)
14 85 ( 1)
:
:
A
A
:
:
A
A
13 39 ( 2)
13 29 ( 2)
13 19 ( 2)
13 10 ( 2)
13 00 ( 2)
14 59 ( 2)
14 49 ( 2)
14 38 ( 2)
14 27 ( 2)
14 17 ( 2)
in direction DL varies linearly with the parameters, at least within the interval considered. Again, the induced constraint (A1) is relaxed when the quality
parameters get smaller and vice versa.
Under x3 the directional scope in the direction DL in all cases is determined by induced constraint (D), which is independent from the quality
parameters. The other two directional scopes depend on (A2) and therefore
vary linearly with Wug and Cug .
In the few cases, where the directional scopes do vary, they vary less than
3%.
Final note on the sensitivity to the quality of gas in storage
We may conclude that the scope itself is insensitive to the value of Cug and
Wug . This is a reassuring result.
Investments in storage capacity are most eective with low values of Wug
and Cug . Then the scope is largest under both investment alternatives. (However, under x1 the value of Cug does not matter.) If the lowest values of these
parameters are used as control values, the gas in storage will have a quality
comparable to that from the Slochteren well (Wpl = 43:8, Cpl = 35:17). An
explanation for this phenomenon is not apparent, although it seems to be an
indication towards an optimal control policy for the underground storage.
5.6.2 Volume capacity of the underground storage
To study the necessary volume capacity of the underground storage, an indicative `worst case' analysis is performed. Information about the necessary volume
capacity is derived from the production plans under dierent circumstances. In
this case, it is not enough to use the commercial scope solely, the production
plans y have to be taken into consideration. Given a production plan with the
5.6 Sensitivity analysis with respect to the underground storage
159
quarterly net outows out of storage fug1; ug2 ; ug3; ug4 g, to process this plan
the volume capacity V C should at least be
V C jmax
=1;::4
j
X
j
X
ugi j=1
min
ug
;::4 i=1 i
i=1
That is, for a certain production plan the volume capacity V C should at least be
equal to the maximum amount of gas in storage minus the minimum amount of
gas in storage. The lower bound is optimistic, since the use of the underground
storage is aggregated per quarter. Of course, this is the 'working' volume
P capacity, which should be increased with a certain buer stock (minj=1;::4 ji=1 ugi
is nonpositive), especially to guarantee a minimum outlet pressure. In all cases
studied it turned out, not surprisingly, that there is no ow into the storage in
the cold quarters (1 and 4) whereas there is no ow out of the storage in the
warmer quarters (2 and 3). In that case,
j
X
max
ug = ug1
j =1;::4 i=1 i
and
j
X
min
ug = ug1 + ug2 + ug3
j =1;::4 i=1 i
so that the abovementioned lower bound on V C is given by ug1 + ug4 = ug2
ug3 .
The `worst case' lower bound for the working volume follows from the
production plans that most appeal to the storage. Those will be found under
the scenarios that have high H-gas overow values, that is, with low DH or
high ACQ. Now for both x1 and x3 we examined the underground storage
production plans in the boundary scenarios, that are found by going from s0
into the directions DH and ACQ+. Both the boundary scenarios and the
feasible production plan found are the result of optimizing (3.5). To be sure that
there is no other production plan that is feasible under such a scenario, but with
a lower use of the underground storage, one could add a term to the objective
of LP-problem (3.5) penalizing the use of the storage. The resulting boundary
scenario will be the same as before. 2 The production plans were made up in
the same way as were the production plans under x0 in Section 4.2.2, namely
as a feasible (optimum) solution to the LP-problem (3.5). The net outows out
2. Notice that an infeasible scenario will never have a production plan, since its infeasibility
will rst be `absolved' before a production plan is made. In other words, under infeasible
conditions, production plans will still be based on a feasible scenario. The dierence between
the feasible adopted and the infeasible original scenario is interpreted in terms of shortages
and surpluses (see Section 3.5).
160
Application: investment alternatives
of the underground storage under production plans in the boundary scenarios
are given in Table 5.14.
Table 5.14. Storage production plans and minimum working volume for dierent boundary cases
x = Boundary scen.
x1
x3
in direction
ACQ+
DH
ACQ+
DH
ug1
2.17
2.14
3.25
3.25
Production plan
VC ug2
ug3 ug4
0
2:17 0 2.17
0
2:14 0 2.14
2:20 2:20 1.15 4.40
2:20 2:20 1.15 4.40
Notice that for the given directions, under x1 the boundary scenario is on
(A0). Since this induced constraint does not involve UGL or UGU , the storage
in- and outow capacities are not binding in these scenarios. Apparently, the
working volume of the underground storage should at least be 4.40, not to
introduce any new restrictions on the problem.
Appendix 5A The construction of the induced
constraints
Under the same assumptions that were valid in Appendix 4A, it is possible to
derive symbolic representations of the induced constraints incurred in dierent
optimizations, using the zero pattern of the dual multipliers. Constraints (A1),
(A2), (A0) and (B ) were already derived in Appendix 4A. In this chapter, one
new induced constraint came up, namely (D), which is derived in this appendix.
Numerical examples of dual multiplier vectors for the induced constraint
(D), together with its multiplier solution scheme, are given in Tables 5.15
and 5.16. The symbolic version of the multiplier vector is given in Table 5.16.
The symbolic version of (D) follows from applying this solution to the matrices
K , L and M as given in Table 2.7.
5A The construction of the induced constraints
161
Table 5.15. A feasible multiplier vector identifying (D) (dual optimum of the
calculation of Sd (x3 ; s0 ; DL ))
Per period restrictions
restriction reference
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13) upper
(2.13) lower
(2.14)
(2.15) upper
(2.15) lower
(2.16) upper
(2.16) lower
(2.17) upper
(2.17) lower
Aggregating restrictions
restriction reference
(2.18) upper
(2.18) lower
(2.19)
(2.20)
Multiplier in quarter
0
1
2
3
4
0 1.12 1.12 1.12 1.12
0
0
0
0
0
0 0.03 0.03 0.03 0.03
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Multiplier
0
1:00
1:12
1:01
162
Application: investment alternatives
Table 5.16. Symbolic representation of submatrices and dual multipliers for (D)
ph
Matrixcoecients KI
f
pl n ug vl
Matrices KIt, t = 1, 2, 3 and 4
(2.8)
1
1 0 0 0
(2.10)
0 Cph Cpl 0 Cug
Matrix kI
(2.18) lower 0 0 1 0 0
(2.19)
1 0 0 0 0
(2.20)
0 0 0 0 1
Multipliers
I
0
0
Cph
0
0
0
Cpl
Cph
Cug
1