Symbolic computation

4.3 Symbolic computation

As we discussed at the end of Section 4.2.2, to replace constraints ((2.1), (2.7), (4.1), (4.2)) by constraints ((4.3), (4.4), (4.5)), we need to know the function𝑓_𝑗for every node𝑗in a semi-passive network𝐺_𝑠. Consider CNS(𝐺_𝑠) again. For every arc𝑎there is a function𝑔_𝑎with𝑄_𝑎=𝑔_𝑎(𝑄_𝑠), where𝑔𝑎is continuous, decreasing, increasing or constant. The flow direction through arc𝑎, which corresponds to the sign of𝑄_𝑎, is said to be decided with inflow𝑄_𝑠∈ ℐ := [𝑄¹_𝑠, 𝑄²_𝑠]if 𝑄_𝑎 ≤0or𝑄_𝑎 ≥0for all𝑄_𝑠∈ ℐ. Since one of𝑔_𝑎(𝑄¹_𝑠)and𝑔_𝑎(𝑄²_𝑠)is the maximum and the other is the minimum of𝑄𝑎, the flow direction is decided if and only if

𝑔𝑎(𝑄¹_𝑠)·𝑔𝑎(𝑄²_𝑠)≥0.

With this property we make the following definition.

Definition 4.5 (Flow direction decided domain)

A domainℐ := [𝑄¹_𝑠, 𝑄²_𝑠]with two constants𝑄¹_𝑠, 𝑄²_𝑠 ∈Ris said to be aflow direction decided domainif for every arc𝑎∈ 𝒜_𝑠it holds

𝑔_𝑎(𝑄¹_𝑠)·𝑔_𝑎(𝑄²_𝑠)≥0.

Note that such a flow direction decided domain always exists. Because for any𝑄⁰_𝑠 ∈ R a given constant, there exists an𝜀 > 0so that for every arc𝑎, either𝑄_𝑎 = 𝑔_𝑎(𝑄_𝑠) ≤ 0or

as constants or parameter and all𝑄_𝑎,ℎ_𝑗 as variables. Note that𝑄_𝑡can be eliminated since it depends only on value of𝑄𝑠.

Before we solve SPE𝑆(𝐺𝑠), we look at the definitions and theorems concerning varieties and ideals. These are taken from the book of Cox et al. [CLO92].

Definition 4.6 (Ideal)

Let𝑘be a field. A subset𝐼 ⊂𝑘[𝑥1, 𝑥2, . . . , 𝑥𝑛]of the polynomial ring𝑘[𝑥1, 𝑥2, . . . , 𝑥𝑛]is an idealif it satisfies:

• 0∈𝐼;

• if𝑓, 𝑔∈𝐼, then𝑓+𝑔∈𝐼;

• if𝑓 ∈𝐼andℎ∈R[𝑥₁, 𝑥₂, . . . , 𝑥_𝑛], thenℎ𝑓 ∈𝐼. Definition 4.7 (Affine variety and polynomial ideal)

Let𝑘be a field and𝑓₁, 𝑓₂, . . . , 𝑓_𝑠be polynomials in𝑘[𝑥₁, 𝑥₂, . . . , 𝑥_𝑛].Then we set V(𝑓₁, 𝑓₂, . . . , 𝑓_𝑠) ={(𝑎₁, . . . , 𝑎_𝑛)∈𝑘^𝑛:𝑓_𝑖(𝑎₁, . . . , 𝑎_𝑛) = 0for all1≤𝑖≤𝑠}.

We callV(𝑓₁, 𝑓₂, . . . , 𝑓_𝑠)theaffine varietydefined by𝑓₁, 𝑓₂, . . . , 𝑓_𝑠. Further we set

⟨𝑓₁, . . . , 𝑓𝑠⟩= {︃ _𝑠

∑︁

𝑡=1

ℎ𝑡𝑓𝑡:ℎ₁, . . . , ℎ𝑠 ∈𝑘[𝑥₁, 𝑥₂, . . . , 𝑥𝑛] }︃

.

Note that𝐼 :=⟨𝑓₁, . . . , 𝑓_𝑠⟩is an ideal, a proof can be found in the book of Cox et al. [CLO92].

We denote alsoV(𝐼) =V(𝑓₁, 𝑓2, . . . , 𝑓𝑠)as the affine variety defined by𝑓1, 𝑓2, . . . , 𝑓𝑠. From that book we have also the following theorem.

Theorem 4.8 (Elimination and Extension Theory)

Let𝐼 be a polynomial ideal with𝐼 = ⟨𝑓₁, . . . , 𝑓_𝑠⟩ ⊂ C[𝑥₁, 𝑥₂, . . . , 𝑥_𝑛]where V(𝐼) is not an empty set. Assume there exists𝑝𝑖(𝑥𝑖) ∈C[𝑥𝑖]such that𝑝𝑖(𝑥𝑖) ∈ 𝐼 for one𝑖 ∈ {1, . . . , 𝑛}. It follows then

• if𝑥¯_𝑖∈Cwith𝑝_𝑖(¯𝑥_𝑖) = 0, then there exists(𝑥₁, 𝑥₂, . . . , 𝑥_𝑛)∈V(𝐼)such that𝑥_𝑖 = ¯𝑥_𝑖;

• for any(𝑥1, 𝑥2, . . . , 𝑥𝑛)∈V(𝐼), it holds𝑝𝑖(𝑥𝑖) = 0.

Furthermore,V(𝐼)is a finite set if and only if there exists𝑝𝑖(𝑥𝑖)∈C[𝑥𝑖]with𝑝𝑖(𝑥𝑖)∈𝐼 for any 𝑖∈ {1, . . . , 𝑛}.

Note that Theorem 4.8 may not be true if we replaceCbyR, sinceRis not algebraically closed, e.g., the polynomial𝑥²+ 1∈R[𝑥]has no root inRbut has a root inC. However, the ideal𝐼can be selected so that all𝑓_𝑖 ∈R[𝑥₁, 𝑥₂, . . . , 𝑥_𝑛]⊂C[𝑥₁, 𝑥₂, . . . , 𝑥_𝑛].

Consider the system of polynomial equations SPE𝑆(𝐺𝑠)with a given flow direction decided domainℐ again. Let𝑓˙1,𝑓˙2, . . . ,𝑓˙𝑠be all polynomials in SPE𝑆(𝐺𝑠)which generate an ideal 𝐼 =⟨𝑓˙₁, . . . ,𝑓˙_𝑠⟩. The set𝑉(𝐼)is not an empty set since for parameter𝑄_𝑠∈ ℐthe corresponding CNS(𝐺𝑠)has a solution which is also of solution of SPE𝑆(𝐺𝑠). According to Theorem 4.4, for any fixed given𝑄𝑠the polynomial system SPE𝑆(𝐺𝑠)has a unique solution. We handle𝑄𝑠as symbolic parameter but not a parameter. Then Theorem 4.8 implies that the corresponding set

4.3 Symbolic computation

𝑉(𝐼)is finite while𝐼 is generated by all polynomials in SPE𝑆(𝐺_𝑠)Then for every variableℎ_𝑗 or𝑄𝑎, polynomials of the form𝑝𝑗(ℎ𝑗)or𝑝𝑎(𝑄𝑎)can be found by computing the Gröbner bases with appropriately chosen monomial orderings [Laz83]. There are several software packages, e.g., Maple [Map] and Singular [DGPS12] which can compute Gröbner bases and can check whether the affine variety of a given polynomial system is a finite set. Examples of applications using Maple can be found in [MG94; GDS94; Zac96; RlRmL14; GKZ90].

On the one hand, all of𝑝_𝑗(ℎ_𝑗)and𝑝_𝑎(𝑄_𝑎)can be regarded as univariate functions which contain symbolic parameter𝑄𝑠, every root of𝑝𝑗(ℎ𝑗)and𝑝𝑎(𝑄𝑎)is a function that maps𝑄𝑠

toℎ𝑗 or to𝑄𝑎. On the other hand, for any fixed𝑄𝑠∈ ℐ, the unique solution for CNS(𝐺𝑠)in Theorem 4.4 is a solution for SPE𝑆(𝐺_𝑠)as well. It implies thatℎ_𝑗 =𝑓_𝑗(𝑄_𝑠)is a root for𝑝_𝑗(ℎ_𝑗) for all𝑄_𝑠∈ ℐ and for all nodes𝑗. Similarly,𝑄_𝑎=𝑔_𝑎(𝑄_𝑠)is a root for𝑝_𝑎(𝑄_𝑎)for all𝑄_𝑠∈ ℐ and for all arcs𝑎. According to Abel-Ruffini theorem [Sko11], there is no algebraic solution to the general univariate polynomial equations of degree five or higher with arbitrary coefficients.

It is clear that the roots for a general univariate polynomial can be found if and only if the degree is no more than4. Assume that the degree of𝑝𝑗(ℎ𝑗)is𝑑𝑗 ≤4(in our instance we have 𝑑_𝑗 = 4), then all𝑑_𝑗 roots𝑟₁, 𝑟₂, . . . , 𝑟_𝑑𝑗 can be found by using Maple or Singular. Note that every𝑟_𝑖can be regarded as a function that maps𝑄_𝑠toℎ_𝑗 if all other parameters are given as constants. With any fixed𝑄¯𝑠 ∈ ℐ, the unique solution of CNS(𝐺𝑠)can be found by using any NLP solver, e.g. SCIP.

Let𝑟₁, . . . , 𝑟_𝑑𝑗be the𝑑_𝑗roots of polynomial𝑝_𝑗(ℎ_𝑗). Since any𝑟_𝑗can be regarded as univariate function of 𝑄𝑠, Theorem 4.4 and Theorem 4.8 imply together that there exists exactly one

¯𝑗 ∈ {1, . . . , 𝑑_𝑗}such that𝑓_𝑗(𝑄_𝑠) =𝑟¯𝑗(𝑄_𝑠)in domainℐ.

since𝑟¯𝑗 is the unique candidate. We call the solution with𝑄_𝑠= ¯𝑄_𝑠adistinction solution. Simi-larly, all other functions𝑓_𝑗and𝑄_𝑎in domainℐcan be found if the degree of the corresponding polynomial is no more than4and a distinction solution can be found.

Let[𝑄^min_𝑠 , 𝑄^max_𝑠 ]be the domain of𝑄_𝑠. Recall that all constraints ((2.1), (2.7), (4.1), (4.2)) related to 𝐺_𝑠 in the entire MINLP can be replaced by constraints ((4.3), (4.4), (4.5)), where only functions 𝑓𝑗 appear in these new constraints. We design algorithm 4.1 to compute 𝑓𝑗

symbolically for an arbitrarily selected node𝑗. In this algorithm, in the first part, we split the original domain[𝑄^min_𝑠 , 𝑄^max_𝑠 ]to a set of sub-domains such that in every sub-domain is a flow direction decided domain. After that, we compute𝑓𝑗 in every sub-domain separately. The correctness of the theorem is proved.

Theorem 4.9 (Correctness of Algorithm 4.1)

Algorithm 4.1 finds𝑓_𝑗 as defined in Theorem 4.4 if the algorithm terminates with “success”.

Algorithm 4.1:Symbolic computation to get𝑓_𝑗

Input: A semi-passive subnetwork𝐺𝑠, domain[𝑄^min_𝑠 , 𝑄^max_𝑠 ]for𝑄𝑠and node𝑗 Output: Function𝑓_𝑗related to Theorem 4.4 for𝑄_𝑠∈[𝑄^min_𝑠 , 𝑄^max_𝑠 ]

1 Initialize𝒮 :={[𝑄^min_𝑠 , 𝑄^max_𝑠 ]};

2 Get the solution of CNS(𝐺_𝑠)with𝑄_𝑠 =𝑄^min_𝑠 and𝑄_𝑠=𝑄^max_𝑠 using an NLP solver. Let𝑄´_𝑎 and𝑄`_𝑎be the value of𝑄_𝑎for every arc𝑎in the solution with𝑄_𝑠 =𝑄^min_𝑠 and𝑄_𝑠=𝑄^max_𝑠 , respectively.

3 for𝑎∈ 𝒜_𝑠do

4 if𝑄´𝑎·𝑄`𝑎<0then

5 Solve CNS(𝐺_𝑠)with additional constraint𝑄_𝑎= 0, let𝑄¯_𝑠be the value of𝑄_𝑠in the unique solution;

6 Get the intervalℐ_𝑎∈ 𝒮 which is[𝑄₁, 𝑄₂)(or[𝑄₁, 𝑄₂]) and contains𝑄¯_𝑠;

7 if𝑄¯_𝑠∈(𝑄₁, 𝑄₂)then

8 Replaceℐ_𝑎by[𝑄₁,𝑄¯_𝑠)and[ ¯𝑄_𝑠, 𝑄₂)(or[ ¯𝑄_𝑠, 𝑄₂]) in𝒮;

9 end

10 end

11 end

12 foreveryℐ_𝑖 ∈ 𝒮 do

13 Get sign assignment𝑆according to (4.6) for𝑄𝑠∈ ℐ_𝑖, construct the corresponding SPE𝑆(𝐺𝑠);

14 Get𝑝_𝑗(ℎ_𝑗)by computing Gröbner bases ;

15 if𝑝𝑗(ℎ𝑗)doesnotexistthen

16 Report“not a finite set”andstop

17 end

18 Let𝑑𝑗be the degree ;

19 if𝑑𝑗 >4then

20 Report“degree too high”andstop

21 end

22 Compute roots𝑟1, . . . , 𝑟_𝑑𝑗;

23 whiletruedo

24 Choose a𝑄¯_𝑠∈ ℐ_𝑖(randomly) and compute the solution of CNS(𝐺_𝑠)with𝑄_𝑠= ¯𝑄_𝑠;

25 ifthe solution is a distinction solutionthen

26 Get¯𝑗and set𝑓_𝑗(𝑄_𝑠)_{|𝑄𝑠∈ℐ𝑖} :=𝑟¯𝑗(𝑄_𝑠);

27 break

28 end

29 end

30 end

31 Report “success” and return𝑓_𝑗

Im Dokument Optimal Operation of Water Supply Networks by Mixed Integer Nonlinear Programming and Algebraic Methods (Seite 59-63)