Optimization in Meshed Gas Networks R udiger Schultz (University of - - PowerPoint PPT Presentation

Optimization in Meshed Gas Networks

R¨ udiger Schultz (University of Duisburg-Essen) with Holger Heitsch, Ren´ e Henrion Weierstraß Institute Berlin, and Matthias Claus, Ralf Gollmer, Kai Sp¨ urkel UDE and Klaus Altmann Free University Berlin

Gaswork and Gasworkers

Where is Maths ?

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

◮ Combinatorics underlying the operation of active network

elements,

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

◮ Combinatorics underlying the operation of active network

elements,

◮ Stochastics of uncertain model components (mainly gas

• utput), capacity management

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

◮ Combinatorics underlying the operation of active network

elements,

◮ Stochastics of uncertain model components (mainly gas

• utput), capacity management

◮ Optimization under (probabilistic) Uncertainty,

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

◮ Combinatorics underlying the operation of active network

elements,

◮ Stochastics of uncertain model components (mainly gas

• utput), capacity management

◮ Optimization under (probabilistic) Uncertainty, ◮ Equilibria for trading ,

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

◮ Combinatorics underlying the operation of active network

elements,

◮ Stochastics of uncertain model components (mainly gas

• utput), capacity management

◮ Optimization under (probabilistic) Uncertainty, ◮ Equilibria for trading , ◮ Nonlinear Analysis for Steady-State Networks,

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

◮ Combinatorics underlying the operation of active network

elements,

◮ Stochastics of uncertain model components (mainly gas

• utput), capacity management

◮ Optimization under (probabilistic) Uncertainty, ◮ Equilibria for trading , ◮ Nonlinear Analysis for Steady-State Networks, ◮ Polynomial Algebra for Steady-State Networks,

Where is Maths ?

◮ PDEs of Conservation Laws in Fluid Dynamics (Mass,

Momentum, Energy, Gas Equation),

◮ Network Flows of Grid-bound Commodities,

finite-dimensional models, Kirchhoff Laws

◮ Combinatorics underlying the operation of active network

elements,

◮ Stochastics of uncertain model components (mainly gas

• utput), capacity management

◮ Optimization under (probabilistic) Uncertainty, ◮ Equilibria for trading , ◮ Nonlinear Analysis for Steady-State Networks, ◮ Polynomial Algebra for Steady-State Networks, ◮ Computer Algebra for suitable (simple) Models.

Model Hierarchies from Transient to Steady-State for Power and Gas Conservation Laws in (Gas) Networks Unknowns: gas density ρ, velocity v, pressure p, temperature T

◮ Mass

∂ρ ∂t + ∂ ∂x ρv = 0

◮ Momentum

∂ ∂t (ρv) + ∂ ∂x (p + ρv2) = − λ 2D ρv|v| − gρh′

λ compressibility coefficient, depends on p and T; pipe diameter D; gravitational constant g; height profile of the pipe over ground h

◮ (Energy)

∂ ∂t

• ρ

1 2v2 + e

• + ∂

∂x

• ρv

1 2v2 + e

• + pv
• = −kw

D (T − Tw)

e internal energy, R universal gas constant, heath transfer coefficient kw; pipe wall surface temperatue Tw. ◮ (Constitutive Law)

p = RρTz(p, T) Specification via isothermal to steady-state models.

The challenge to find a good model – Compressor in B¨ unde

Topics

Topics

• 1. Gas Flow

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves);

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes;

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

◮ The probability of those balanced injection and random withdrawals,

for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ??

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

◮ The probability of those balanced injection and random withdrawals,

for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ??

• 3. Explicit Feasibility Representation by Symbolic Computation

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

◮ The probability of those balanced injection and random withdrawals,

for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ??

• 3. Explicit Feasibility Representation by Symbolic Computation

◮ Fundamental cycles of the network imply feasibility system of

“privileged” multivariate polynomials of degree two;

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

◮ The probability of those balanced injection and random withdrawals,

for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ??

• 3. Explicit Feasibility Representation by Symbolic Computation

◮ Fundamental cycles of the network imply feasibility system of

“privileged” multivariate polynomials of degree two; variable elimination;

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

◮ The probability of those balanced injection and random withdrawals,

for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ??

• 3. Explicit Feasibility Representation by Symbolic Computation

◮ Fundamental cycles of the network imply feasibility system of

“privileged” multivariate polynomials of degree two; variable elimination; (reverse) lexicographic order;

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

◮ The probability of those balanced injection and random withdrawals,

for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ??

• 3. Explicit Feasibility Representation by Symbolic Computation

◮ Fundamental cycles of the network imply feasibility system of

“privileged” multivariate polynomials of degree two; variable elimination; (reverse) lexicographic order; parametric system - comprehensive Gr¨

• bner Base;

Topics

• 1. Gas Flow

◮ Stationary gas flow in pipeline systems under additional assumptions:

passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients.

• 2. Probabilistic Nomination Validation

◮ The probability of those balanced injection and random withdrawals,

for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ??

• 3. Explicit Feasibility Representation by Symbolic Computation

◮ Fundamental cycles of the network imply feasibility system of

“privileged” multivariate polynomials of degree two; variable elimination; (reverse) lexicographic order; parametric system - comprehensive Gr¨

• bner Base; Shape Lemma.

• 4. Spherical-Radial Decomposition of Gaussian Distribution

◮ Splits integration over Gaussian Laws into integration of uniform

distributionn over unit sphere (sampling) and a χ2-distribution in dimension one;

• 4. Spherical-Radial Decomposition of Gaussian Distribution

◮ Splits integration over Gaussian Laws into integration of uniform

distributionn over unit sphere (sampling) and a χ2-distribution in dimension one;

• 5. Performance Gains in Quasi-Monte-Carlo Sampling

◮ Sampling (of Probabilities, Function Values and Gradients) with

implicit representation vs. Sampling with explicit formula vs. Comprehensive Gr¨

• bner bases calculation within Sampling on the

Unit Sphere: Initial experiments with probabilities → variance reduction by two decimals

Graph Theoretic Setting

◮ Gas network model = connected, directed (simple) graph G = (V , E),

Graph Theoretic Setting

◮ Gas network model = connected, directed (simple) graph G = (V , E), ◮ with node-arc incidence matrix A+ and in/out nomination vector b+.

Graph Theoretic Setting

◮ Gas network model = connected, directed (simple) graph G = (V , E), ◮ with node-arc incidence matrix A+ and in/out nomination vector b+. ◮ Kirchhoff 1 (mass preservation at nodes)

A+q = b+ with q ∈ R|E| denoting the gas flow in the pipes.

Graph Theoretic Setting

◮ Gas network model = connected, directed (simple) graph G = (V , E), ◮ with node-arc incidence matrix A+ and in/out nomination vector b+. ◮ Kirchhoff 1 (mass preservation at nodes)

A+q = b+ with q ∈ R|E| denoting the gas flow in the pipes.

◮ after deletion of the first row/component of A+, b+

(slack or reference or root node) Aq = b gives the matrix A with full rank and the vector b. Now “simplex-like” Aq = b iff ABqB + ANqN = b iff qB = (AB)−1 b − (AB)−1 ANqN

Graph Theoretic Setting

◮ Gas network model = connected, directed (simple) graph G = (V , E), ◮ with node-arc incidence matrix A+ and in/out nomination vector b+. ◮ Kirchhoff 1 (mass preservation at nodes)

A+q = b+ with q ∈ R|E| denoting the gas flow in the pipes.

◮ after deletion of the first row/component of A+, b+

(slack or reference or root node) Aq = b gives the matrix A with full rank and the vector b. Now “simplex-like” Aq = b iff ABqB + ANqN = b iff qB = (AB)−1 b − (AB)−1 ANqN The graph behind AB is a spanning tree T = (V , ET) of G, with edge flows qB in ET and edge flows qN in E \ T.

◮ Since T ⊆ G is a spanning tree, for every edge (chord) e ∈ E \ ET

there exists a unique cycle in (V , ET ∪ {e}), called fundamental cycle. → cardinality of |N| = minimum number of edges to be removed from G to obtain a tree.

◮ Since T ⊆ G is a spanning tree, for every edge (chord) e ∈ E \ ET

there exists a unique cycle in (V , ET ∪ {e}), called fundamental cycle. → cardinality of |N| = minimum number of edges to be removed from G to obtain a tree.

◮ Kirchhoff 2: (pressure drops along fundamental cycles sum up to zero)

(A+)⊤(p+)2 = −Φ|q|q with p ∈ R|V |

+

standing for the node pressures; squares as well as moduli understood component-wise.

◮ Since T ⊆ G is a spanning tree, for every edge (chord) e ∈ E \ ET

there exists a unique cycle in (V , ET ∪ {e}), called fundamental cycle. → cardinality of |N| = minimum number of edges to be removed from G to obtain a tree.

◮ Kirchhoff 2: (pressure drops along fundamental cycles sum up to zero)

(A+)⊤(p+)2 = −Φ|q|q with p ∈ R|V |

+

standing for the node pressures; squares as well as moduli understood component-wise.

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values.

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator,

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients

and Jacobian,

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients

and Jacobian, Brouwer’s Fixed Point Theorem.

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients

and Jacobian, Brouwer’s Fixed Point Theorem.

◮ Algebraic View: affine variety,

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients

and Jacobian, Brouwer’s Fixed Point Theorem.

◮ Algebraic View: affine variety, polynomial ideal,

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients

and Jacobian, Brouwer’s Fixed Point Theorem.

◮ Algebraic View: affine variety, polynomial ideal, (comprehensive)

Gr¨

• bner bases,

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients

and Jacobian, Brouwer’s Fixed Point Theorem.

◮ Algebraic View: affine variety, polynomial ideal, (comprehensive)

Gr¨

• bner bases, elimination order,

Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b(ω) (1) (A+)⊤(p+)2 = −Φ|q|q (2) p+ ∈

• p+min, p+max

, (3) Analytical as well as Algebraic Appeal ! Observations:

◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients

and Jacobian, Brouwer’s Fixed Point Theorem.

◮ Algebraic View: affine variety, polynomial ideal, (comprehensive)

Gr¨

• bner bases, elimination order, Shape Lemma.

0-dimensionality of affine variety

0-dimensionality of affine variety Aq = b and F ·

• Φ · |q| · q
• = 0.

F circuit matrix, whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E

0-dimensionality of affine variety Aq = b and F ·

• Φ · |q| · q
• = 0.

F circuit matrix, whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Aq = b, F ·

• Φ · q2

= 0, q ≥ 0

0-dimensionality of affine variety Aq = b and F ·

• Φ · |q| · q
• = 0.

F circuit matrix, whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Aq = b, F ·

• Φ · q2

= 0, q ≥ 0 Denote the set of solutions of the above system by G = Gb,Φ := {q ∈ CE | A q = b, F Φ q2 = 0} and Q = diag(q) Instead of checking the mere dimension of G, we will check when the tangent space at points q ∈ Gb,Φ will be at least one-dimensional.

0-dimensionality of affine variety Aq = b and F ·

• Φ · |q| · q
• = 0.

F circuit matrix, whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Aq = b, F ·

• Φ · q2

= 0, q ≥ 0 Denote the set of solutions of the above system by G = Gb,Φ := {q ∈ CE | A q = b, F Φ q2 = 0} and Q = diag(q) Instead of checking the mere dimension of G, we will check when the tangent space at points q ∈ Gb,Φ will be at least one-dimensional.The tangent space Tq(Gb,Φ) in a point q ∈ Gb,Φ is given by the linear equations A dq = 0 and F Φ Q dq = 0.

0-dimensionality of affine variety Aq = b and F ·

• Φ · |q| · q
• = 0.

F circuit matrix, whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Aq = b, F ·

• Φ · q2

= 0, q ≥ 0 Denote the set of solutions of the above system by G = Gb,Φ := {q ∈ CE | A q = b, F Φ q2 = 0} and Q = diag(q) Instead of checking the mere dimension of G, we will check when the tangent space at points q ∈ Gb,Φ will be at least one-dimensional.The tangent space Tq(Gb,Φ) in a point q ∈ Gb,Φ is given by the linear equations A dq = 0 and F Φ Q dq = 0. Thus, dimCTq(Gb,Φ) ≥ 1 is equivalent to det

• A

FΦQ

• = 0.

0-dimensionality of affine variety Aq = b and F ·

• Φ · |q| · q
• = 0.

F circuit matrix, whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Aq = b, F ·

• Φ · q2

= 0, q ≥ 0 Denote the set of solutions of the above system by G = Gb,Φ := {q ∈ CE | A q = b, F Φ q2 = 0} and Q = diag(q) Instead of checking the mere dimension of G, we will check when the tangent space at points q ∈ Gb,Φ will be at least one-dimensional.The tangent space Tq(Gb,Φ) in a point q ∈ Gb,Φ is given by the linear equations A dq = 0 and F Φ Q dq = 0. Thus, dimCTq(Gb,Φ) ≥ 1 is equivalent to det

• A

FΦQ

• = 0.
• (AB)−1 AN

⊤ ΦB q2

B = ΦNq2 N

and det

• (AB)−1 AN

⊤ ΦBQB

• (AB)−1 AN
• + ΦNQN)
• = 0.

M =

• −1⊤b, b
• : ∃(q, p+) fulfilling (??), (??), (??)
• .

M =

• −1⊤b, b
• : ∃(q, p+) fulfilling (??), (??), (??)
• .

Proposition:

Let A = (AB, AN) be a partition into basis and nonbasis matrices. Let ΦB, ΦN and qB, qN be corresponding partitions of Φ and q. Denote g : R|V | × R|N| → R|V | – a multivariate, piece-wise quadratic mapping – such that g(u, v) :=

• A⊤

B

−1ΦB

• A−1

B (u − ANv)

• A−1

B (u − ANv)

• .

(4)

M =

• −1⊤b, b
• : ∃(q, p+) fulfilling (??), (??), (??)
• .

Proposition:

Let A = (AB, AN) be a partition into basis and nonbasis matrices. Let ΦB, ΦN and qB, qN be corresponding partitions of Φ and q. Denote g : R|V | × R|N| → R|V | – a multivariate, piece-wise quadratic mapping – such that g(u, v) :=

• A⊤

B

−1ΦB

• A−1

B (u − ANv)

• A−1

B (u − ANv)

• .

(4)

Then nomination b+ is valid iff (−1⊤b, b) ∈ M

M =

• −1⊤b, b
• : ∃(q, p+) fulfilling (??), (??), (??)
• .

Proposition:

Let A = (AB, AN) be a partition into basis and nonbasis matrices. Let ΦB, ΦN and qB, qN be corresponding partitions of Φ and q. Denote g : R|V | × R|N| → R|V | – a multivariate, piece-wise quadratic mapping – such that g(u, v) :=

• A⊤

B

−1ΦB

• A−1

B (u − ANv)

• A−1

B (u − ANv)

• .

(4)

Then nomination b+ is valid iff (−1⊤b, b) ∈ M

IFF

∃ z such that (b, z) fulfils A⊤

N g(b, z) = ΦN|z|z

(5)

and

min

i=1,...,n

• (pmax

i

)2 + gi(b, z)

• ≥

max

i=1,...,n

• (pmin

i

)2 + gi(b, z)

• (6)

(pmin )2 ≤ min

i=1,...,n

• (pmax

i

)2 + gi(b, z)

• (7)

(pmax )2 ≥ max

i=1,...,n

• (pmin

i

)2 + gi(b, z)

• (8)

After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, |N| equations, and |N| variables.

After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, |N| equations, and |N| variables.

A⊤

N (A⊤ B )−1ΦB

• (AB)−1 b − (AB)−1 ANqN
• (AB)−1 b − (AB)−1 ANqN
• − ΦN|qN|qN = 0

Solve this system in qN, and insert solution(s) below:

After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, |N| equations, and |N| variables.

A⊤

N (A⊤ B )−1ΦB

• (AB)−1 b − (AB)−1 ANqN
• (AB)−1 b − (AB)−1 ANqN
• − ΦN|qN|qN = 0

Solve this system in qN, and insert solution(s) below:

p2 = 1|V |−1p2

• −(A⊤

B )−1ΦB

• (AB)−1 b − (AB)−1 ANqN
• (AB)−1 b − (AB)−1 ANqN

After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, |N| equations, and |N| variables.

A⊤

N (A⊤ B )−1ΦB

• (AB)−1 b − (AB)−1 ANqN
• (AB)−1 b − (AB)−1 ANqN
• − ΦN|qN|qN = 0

Solve this system in qN, and insert solution(s) below:

p2 = 1|V |−1p2

• −(A⊤

B )−1ΦB

• (AB)−1 b − (AB)−1 ANqN
• (AB)−1 b − (AB)−1 ANqN
• Check

p+ ∈

• p+min, p+max

After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, |N| equations, and |N| variables.

A⊤

N (A⊤ B )−1ΦB

• (AB)−1 b − (AB)−1 ANqN
• (AB)−1 b − (AB)−1 ANqN
• − ΦN|qN|qN = 0

Solve this system in qN, and insert solution(s) below:

p2 = 1|V |−1p2

• −(A⊤

B )−1ΦB

• (AB)−1 b − (AB)−1 ANqN
• (AB)−1 b − (AB)−1 ANqN
• Check

p+ ∈

• p+min, p+max

meaning

• max

i=1,...,n

• (pmin

i

)2 + gi(b, z)

• ,

min

i=1,...,n

• (pmax

i

)2 + gi(b, z)

• ∩
• (pmin

)2, (pmax )2 = ∅

M =

• b+ : 1⊤b+ = 0 and ∃(q, p+) : p+ ∈
• p+min, p+max

, Kirchhoff 1 and 2

• .

M =

• b+ : 1⊤b+ = 0 and ∃(q, p+) : p+ ∈
• p+min, p+max

, Kirchhoff 1 and 2

• .

Nomination Validation (passive network) – Decide b+ ∈ M Given a balanced load vector b+. Do there exist arc flows q and node pressures p+ within bounds p+min, p+max fulfilling the Kirchhoff Laws ??

“Regularity of Coefficients no Surprise” – Tailor F(b, qN) = 0 by properly directing G ΦNl · |qNl|∗ =

• j:ej∈E(Cl)

Φj

• i∈Ij

bi −

• h:Ch∈Hj

qNh

• ∗

,

• r

l = 1, . . . , L, ΦNl·|qNl|qNl =

• j:ej∈E(Cl)

Φj

• i∈Ij

bi −

• h:Ch∈Hj

qNh

• i∈Ij

bi −

• h:Ch∈Hj

qNh

• with

◮ variables qN1, . . . , qNL, L = number of fundamental cycles. ◮ Cl denotes the cycle containing ql, ◮ E(Cl) the set of all edges, except for qNl, with both ends in Cl, ◮ Ij = V (T(ehe

j )) the node set of the tree T which is rooted in the head

ehe

j

• f ej.

◮ Hj the set of all cycles Ch, h ∈ {1, . . . , L}, the edge ej belongs to.

ΦN1|qN1|∗ = Φ1|β1 − qN1|∗ + Φ2|β2 − qN1|∗ + Φ3|β3 − qN1 − qN4 − qN5|∗ ΦN2|qN2|∗ = Φ4|β4 − qN2 − qN4 − qN5|∗ + Φ5|β5 − qN2|∗ + Φ6|β6 − qN2|∗ ΦN3|qN3|∗ = Φ7|β7 − qN3 − qN4 − qN5|∗ + Φ8|β8 − qN3|∗ + Φ9|β9 − qN3|∗ ΦN4|qN4|∗ = Φ3|β3 − qN1 − qN4 − qN5|∗ + Φ4|β4 − qN2 − qN4 − qN5|∗ + + Φ7|β7 − qN3 − qN4 − qN5|∗ + Φ10|β10 − qN4|∗ + Φ11|β11 − qN4|∗ ΦN5|qN5|∗ = Φ3|β3 − qN1 − qN4 − qN5|∗ + Φ4|β4 − qN2 − qN4 − qN5|∗ + + Φ7|β7 − qN3 − qN4 − qN5|∗

Two Cycles Sufficient ?? – Gallery of Real Gas Networks (I)

N N L N N voedingsstation(s) [entry-punten] compressor- en mengstation compressorstation mengstation exportstation installatie ondergrondse opslag installatie voor vloeibaar aardgas stikstofinjectie N L leiding – Groningen-gas leiding – hoogcalorisch gas leiding – laagcalorisch gas leiding – ontzwaveld gas leiding – stikstof

Groningen BBL

Two Cycles Sufficient ?? – Gallery of Real Gas Networks (II)

Two Cycles Sufficient ?? – Gallery of Real Gas Networks (III)

Two Cycles Sufficient ?? – Gallery of Real Gas Networks (III)

Two Cycles Sufficient ?? – Gallery of Real Gas Networks (IV)

Two Cycles Sufficient ?? – Gallery of Real Gas Networks (V)

OGE – High Caloric Grid Northern Germany Essentially 2 Circles !!

OGE - Low Caloric Grid: Always a Matter of Detail

Motivation: Probabilistic Setting Spheric-Radial Decomposition - Procedure for Approximating Gaussian Probabilities

Underlying Probability Distributions and their Decomposition Assume that ξ ∼ N(µ, Σ), i.e., the random vector ξ follows a multivariate Gaussian distribution with mean vector µ and positive definite covariance matrix Σ.

Theorem (spheric-radial decomposition) Let ξ ∼ N(0, R) be some n-dimensional standard Gaussian distribution with zero mean and positive definite correlation matrix R.

Underlying Probability Distributions and their Decomposition Assume that ξ ∼ N(µ, Σ), i.e., the random vector ξ follows a multivariate Gaussian distribution with mean vector µ and positive definite covariance matrix Σ.

Theorem (spheric-radial decomposition) Let ξ ∼ N(0, R) be some n-dimensional standard Gaussian distribution with zero mean and positive definite correlation matrix R. Then, for any Borel measurable subset M ⊆ Rn it holds that P(ξ ∈ M) =

• Sn−1 µχ{r ≥ 0 | rLv ∈ M}dµη(v),

where Sn−1 is the (n − 1)-dimensional sphere in Rn, µη is the uniform distribution on Sn−1, µχ denotes the χ-distribution with n degrees of freedom and L is such that R = LLT (e.g., Cholesky decomposition).

Algorithm:

Algorithm: Let ξ ∼ N(µ, Σ) and L such that LLT = Σ (e.g., Cholesky factorization).

Algorithm: Let ξ ∼ N(µ, Σ) and L such that LLT = Σ (e.g., Cholesky factorization).

• 1. Sample N points {v1, . . . , vN} uniformly distributed on the sphere

Sn−1.

Algorithm: Let ξ ∼ N(µ, Σ) and L such that LLT = Σ (e.g., Cholesky factorization).

• 1. Sample N points {v1, . . . , vN} uniformly distributed on the sphere

Sn−1.

• 2. Compute the one-dimensional sets Mi := {r ≥ 0 | rLvi + µ ∈ M}

for i = 1, . . . , N.

Algorithm: Let ξ ∼ N(µ, Σ) and L such that LLT = Σ (e.g., Cholesky factorization).

• 1. Sample N points {v1, . . . , vN} uniformly distributed on the sphere

Sn−1.

• 2. Compute the one-dimensional sets Mi := {r ≥ 0 | rLvi + µ ∈ M}

for i = 1, . . . , N.

• 3. Set P(ξ ∈ M) ≈ N−1 N
• i=1

µχ(Mi).

Then nomination b+ is valid iff (−1⊤b, b) ∈ M

Algorithm: Let ξ ∼ N(µ, Σ) and L such that LLT = Σ (e.g., Cholesky factorization).

• 1. Sample N points {v1, . . . , vN} uniformly distributed on the sphere

Sn−1.

• 2. Compute the one-dimensional sets Mi := {r ≥ 0 | rLvi + µ ∈ M}

for i = 1, . . . , N.

• 3. Set P(ξ ∈ M) ≈ N−1 N
• i=1

µχ(Mi).

Then nomination b+ is valid iff (−1⊤b, b) ∈ M

IFF

∃ z such that (b, z) fulfils A⊤

N g(b, z) = ΦN|z|z

(9)

and

min

i=1,...,n

• (pmax

i

)2 + gi(b, z)

• ≥

max

i=1,...,n

• (pmin

i

)2 + gi(b, z)

• (10)

(pmin )2 ≤ min

i=1,...,n

• (pmax

i

)2 + gi(b, z)

• (11)

(pmax )2 ≥ max

i=1,...,n

• (pmin

i

)2 + gi(b, z)

• (12)

0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 2000 4000 6000 8000 10000 Spheric-Radial Generic Sampling 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 2000 4000 6000 8000 10000 Spheric-Radial Generic Sampling

Figure : Moving average of computed probability with respect to number of iterations for Mersenne Twister (left) and QMC sampling (right)

Further Considerations:

Further Considerations:

◮ Networks with at most one fundamental cycle can be handled with

the solution formula from highschool maths.

Further Considerations:

◮ Networks with at most one fundamental cycle can be handled with

the solution formula from highschool maths.

◮ For mdels with two or more cycles computational algebra has

something to offer, Gr¨

• bner Bases.

With polynomials f1, . . . , fs in C[x1, . . . , xn], the set f1, . . . , fs :=

• s
• i=1

hifi : h1, . . . , hs ∈ C[x1, . . . , xn]

• (of “polynomial linear combinations”) is called the ideal generated by

f1, . . . , fs. The polynomials f1, . . . , fs then are said to form a basis of I.

With polynomials f1, . . . , fs in C[x1, . . . , xn], the set f1, . . . , fs :=

• s
• i=1

hifi : h1, . . . , hs ∈ C[x1, . . . , xn]

• (of “polynomial linear combinations”) is called the ideal generated by

f1, . . . , fs. The polynomials f1, . . . , fs then are said to form a basis of I. With polynomials f1, . . . , fs in C[x1, . . . , xn], the set V (f1, . . . , fs) := {(a1, . . . , an) ∈ Cn : fi(a1, . . . , an) = 0, for all 1 ≤ i ≤ s}

• f common zeros in Cn is called the affine variety defined by f1, . . . , fs.

With polynomials f1, . . . , fs in C[x1, . . . , xn], the set f1, . . . , fs :=

• s
• i=1

hifi : h1, . . . , hs ∈ C[x1, . . . , xn]

• (of “polynomial linear combinations”) is called the ideal generated by

f1, . . . , fs. The polynomials f1, . . . , fs then are said to form a basis of I. With polynomials f1, . . . , fs in C[x1, . . . , xn], the set V (f1, . . . , fs) := {(a1, . . . , an) ∈ Cn : fi(a1, . . . , an) = 0, for all 1 ≤ i ≤ s}

• f common zeros in Cn is called the affine variety defined by f1, . . . , fs.

Fact: If f1, . . . , fs and g1, . . . , gt are bases of the same ideal in C[x1, . . . , xn], i.e., f1, . . . , fs = g1, . . . , gt, then their affine varieties coincide V (f1, . . . , fs) = V (g1, . . . , gt) . That means, the systems f1 = 0, . . . , fs = 0 and g1 = 0, . . . , gt = 0 have the same solution sets (in Cn).

Triangular Form of Polynomial Systems – Variables’ Elimination

Fact: Given I = f1, . . . , fs ⊂ C[x1, . . . , xn], under suitable assumptions, there exists a triangular basis G = {g1, . . . , gn} for I (Reduced Gr¨

• bner Basis) , i.e.,

g1 = g1(x1, . . . , xn), g2 = g2(x2, . . . , xn) . . . gn−1 = gn−1(xn−1, xn), gn = gn(xn) G can be computed by a finite algorithm (Buchberger’s Algorithm).

Shape Lemma

Let I be a zero-dimensional radical ideal in C[x1, . . . , xn] such that all d complex roots of I have distinct xn coordinates. Then the reduced Gr¨

• bner basis G of I in lexicographic monomial order has the shape

x1 − ϕ1(xn) x2 − ϕ2(xn) . . . xn−1 − ϕn−1(xn) ϕn(xn) where ϕn is a univariate polynomial of degree d and the remaining ϕi are polynomials of degree ≤ d − 1.

b0 b2 b3 b1 1 3 2 q02 q23 q01 q13 q12

Now the “red” system (??) of polynomial equations reads φ02|z1|z1 = φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) φ13|z2|z2 = φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) + φ23|b3 − z2|(b3 − z2) 0-dimensionality of variety: We obtain the equations q2

1 + q2 2 = q2 4

and q2

2 + q2 3 = q2 5

(implying (q1 + q3)(q1 − q3) = (q4 + q5)(q4 − q5)) and (q1 + q2 + q4)(q2 + q3 + q5) = q2

2.

Elimination yields a single equation within the variables (b1, b2, b3) which decomposes into a product of three factors: b3 · (b1 + b2 + b3) · (b2

1 + b2 2 + b2 3 − b1b3 + 3b2b3) = 0.

M =                                                                                                                                        b ∈ R3

+

• ∃z :

φ02|z1|z1 = φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) φ13|z2|z2 = φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) + φ23|b3 − z2|(b3 − z2) y1 ≥ y2 + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) y1 ≥ y3 + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) + φ23|b3 − z2|(b3 − z2) y2 ≥ y1 − φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) y2 ≥ y3 + φ23|b3 − z2|(b3 − z2) y3 ≥ y1 − φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) − φ23|b3 − z2|(b3 − z2) y3 ≥ y2 − φ23|b3 − z2|(b3 − z2) y0 ≤ y1 + φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) y0 ≤ y2 + φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) y0 ≤ y3 + φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) + φ23|b3 − z2|(b3 − z2) y0 ≥ y1 + φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) y0 ≥ y2 + φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) y0 ≥ y2 + φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) y0 ≥ y2 + φ01|b1 + b2 + b3 − z1|(b1 + b2 + b3 − z1) + φ12|b2 + b3 − z1 − z2|(b2 + b3 − z1 − z2) + φ23|b3 − z2|(b3 − z2)                                                                                                                                        . (13)

Mi for a Single Sample Point M1 =                                                                                                                      r ≥ 0

• ∃z :

|z1|z1 = |r + 3.5 − z1|(r + 3.5 − z1) + |2.5 − z1 − z2|(2.5 − z1 − z2) |z2|z2 = |2.5 − z1 − z2|(2.5 − z1 − z2) + |0.5 − z2|(0.5 − z2) y1 ≥ y2 + |2.5 − z1 − z2|(2.5 − z1 − z2) y1 ≥ y3 + |2.5 − z1 − z2|(2.5 − z1 − z2) + |0.5 − z2|(0.5 − z2) y2 ≥ y1 − |2.5 − z1 − z2|(2.5 − z1 − z2) y2 ≥ y3 + |0.5 − z2|(0.5 − z2) y3 ≥ y1 − |2.5 − z1 − z2|(2.5 − z1 − z2) − |0.5 − z2|(0.5 − z2) y3 ≥ y2 − |0.5 − z2|(0.5 − z2) y0 ≤ y1 + |r + 3.5 − z1|(r + 3.5 − z1) y0 ≤ y2 + |r + 3.5 − z1|(r + 3.5 − z1) + |2.5 − z1 − z2|(2.5 − z1 − z2) y0 ≤ y3 + |r + 3.5 − z1|(r + 3.5 − z1) + |2.5 − z1 − z2|(2.5 − z1 − z2) + |0.5 − z2|(0.5 − z2) y0 ≥ y1 + |r + 3.5 − z1|(r + 3.5 − z1) y0 ≥ y2 + |r + 3.5 − z1|(r + 3.5 − z1) + |2.5 − z1 − z2|(2.5 − z1 − z2) y0 ≥ y2 + |r + 3.5 − z1|(r + 3.5 − z1) + |2.5 − z1 − z2|(2.5 − z1 − z2) + |0.5 − z2|(0.5 − z2)                                                                                                                      . (14)

Parametric Solution of F(b, qN) = 0

ΦNl · |qNl|qNl =

• j:ej ∈E(Cl )

Φj

• i∈Ij

bi −

• h:Ch∈Hj

qNh

• i∈Ij

bi −

• h:Ch∈Hj

qNh

• l = 1, . . . , L,

◮ Case distinction of absolute values identifies validity regions for

quadratic multivariate polynomials.

◮ In every region, reduced Gr¨

• bner basis with lexicographic order

yields triangular representation, at best, Shape Lemma applies.

◮ There is an extension of Buchberger’s Algorithm addressing

parametric polynomial equations and computing a Comprehensive Gr¨

• bner Basis.

◮ The idea is to handle parameters as additional variables and

accompany the Buchberger iterations by a case distinction to separate parameter settings “leading to unwanted eliminations”.

◮ Successful experiments with package SINGULAR for instances

with four pairwise interwoven cycles.

Three Cycles Let us now consider a network with three interconnected cycles given by the following node-arc incidence matrix: A+ =     −1 −1 −1 1 −1 −1 1 −1 1 1 1 1     =   1 −1 − 1 AB AN   b0 b2 b3 b1 1 3 2 q02 q23 q01 q13 q12 q03

0-dimensionality Elimination yields a single equation within the variables (b1, b2, b3) which decomposes into a product of three factors: (b1 + b2) · (b2 + b3) · f (b) = 0. where f (b) is a nasty polynomial of degree 16.

For z2 ≤ 0, 2.5 − z1 − z2 − z3 ≥ 0 and 0.5 − z2 − z3 ≤ 0 equations become = (r + 3.5 − z1 − z3)2 + (2.5 − z1 − z2 − z3)2 − z2

1

= (2.5 − z1 − z2 − z3)2 − (0.5 − z2 − z3)2 + z2

2

= (r + 3.5 − z1 − z3)2 + (2.5 − z1 − z2 − z3)2 − (0.5 − z2 − z3) − z2

3 .

Computing the comprehensive Gr¨

• bner basis of this system yields (note r ≥ 0)

IF r ∈ R \ V

• 16r7 + 176r6 + 616r5 − 236r4 − 8283r3 − 27930r2 − 45995r − 34764
• =

128z7

3 + (−256r − 1088)z6 3 + (192r2 + 1088r + 1856)z5 3 + (416r2 + 3024r + 5168)z4 3

+(−160r4 − 2080r3 − 10272r2 − 23136r − 20120)z3

3 + (96r5 + 1488r4 + 8600r3

+23864r2 + 31888r + 15884)z2

3 + (−16r6 − 320r5 − 2232r4 − 7144r3 − 10660r2

−4844r + 3216)z3 + 8r6 + 72r5 + 76r4 − 1103r3 − 4577r2 − 7224r − 4752 = (64r9 + 1152r8 + 8160r7 + 24752r6 − 10172r5 − 354972r4 − 1363604r3 − 2767556r2 −3181152r − 1668672)z2 + (1280r4 + 11776r3 + 47808r2 + 96512r + 90944)z6

3

+(−2048r5 − 28288r4 − 165312r3 − 520384r2 − 889952r − 690976)z5

3 + (1024r6

+14080r5 + 93120r4 + 368640r3 + 905552r2 + 1275344r + 787680)z4

3 + (512r7

+12544r6 + 128576r5 + 728576r4 + 2514896r3 + 5372256r2 + 6688824r + 3814136)z3

3

+(−1408r8 − 30080r7 − 289680r6 − 1655392r5 − 6169188r4 − 15394072r3 − 25124132r2 −24456980r − 10818988)z2

3 + (384r9 + 9632r8 + 105520r7 + 680064r6 + 2869488r5

+8242672r4 + 16068894r3 + 20297194r2 + 14763826r + 4474326)z3 + 32r10 + 480r9 +2864r8 + 7232r7 − 646r6 − 22218r5 + 182518r4 + 1322115r3 + 3584458r2 + 4806057r +2727108

= 136799358020z1 + 128752336960z3

2 − 64376168480z2 2 + (806083232r8 + 13618647888r7

+82898479928r6 + 162060396964r5 − 518308769292r4 − 3808651073028r3 −10174970595900r2 − 14424853086608r − 9228142431084)z2 + (16121664640r3 +130502308928r2 + 361391181952r + 504092269248)z6

3 + (−25794663424r4

−327781579328r3 − 1565418697984r2 − 3612517077856r − 3827826973152)z5

3

+(12897331712r5 + 163084706432r4 + 915394807712r3 + 3030354380432r2 +5544162939248r + 4343912761120)z4

3 + (6448665856r6 + 150865511168r5

+1414079940896r4 + 6898087231824r3 + 19087279253392r2 + 29822658225240r +21163974631112)z3

3 + (−17733831104r7 − 359260412704r6 − 3145314400184r5

−15738844137180r4 − 49487035448944r3 − 98520087615620r2 − 114609686785236r −59706410091716)z2

3 + (4836499392r8 + 115970424688r7 + 1171905798608r6

+6716720077000r5 + 24410393301164r4 + 57892047252310r3 + 86645586562762r2 +72847201881234r + 24554853436242)z3 + 403041616r9 + 5600199096r8 + 27469933988r7 +38930154634r6 − 108201562334r5 − 103647156148r4 + 2820005481261r3 +12255711551516r2 + 21397663018155r + 14892245511176

IF r ∈ V

• 16r5 + 64r4 − 24r3 − 836r2 − 2143r − 2897
• =

53248z6

3 + (−2560r4 − 2304r3 + 21632r2 − 18496r − 342592)z5 3 + (3584r4 − 7424r3

+1664r2 + 209600r + 727232)z4

3 + (−2304r4 + 8576r3 + 360256r2 + 1650016r + 2492512)z3 3

+(−34496r4 − 620768r3 − 3500432r2 − 8753272r − 8440632)z2

3 + (174424r4 + 1490844r3

+5204082r2 + 8306255r + 5011335)z3 − 25136r4 − 42424r3 + 488956r2 + 2321922r + 2829042 = 5196940902400z2z3 + (249852928000r4 + 224867635200r3 − 2111257241600r2 −8588694400000r − 10737429580800)z2 + (−12054079616r4 − 45463570560r3 +49483097664r2 + 625675757920r + 1613554380888)z5

3 + (−16007406400r4

+163846520000r3 + 319837949600r2 − 1627154563600r − 3454040788100)z4

3

+(79491469728r4 − 71828135520r3 − 355579762512r2 − 1605442586360r −727366750654)z3

3 + (−426082396912r4 − 1533035118320r3 + 4147395406648r2

+19947135523540r + 24202233764341)z2

3 + (176055463680r4 − 623350876800r3

−1441625814720r2 + 5061688687200r + 25641350852960)z3 + 332666688000r4 +544511212800r3 − 4008412636800r2 − 16625519262400r − 15832695659200

= 10393881804800z2

2 + (−499705856000r4 − 449735270400r3 + 4222514483200r2

+17177388800000r + 16277918259200)z2 + (60169054336r4 + 261984069760r3 −647553185344r2 − 2566496289120r − 6486475458648)z5

3 + (−18172280000r4

−196112760000r3 + 73832501600r2 + 2529651536400r + 2786822266500)z4

3

+(−647902966688r4 − 324894053280r3 + 3970115254352r2 + 16868424099960r +22674594639134)z3

3 + (24464391152r4 + 1298309620720r3 − 2821444436408r2

−5671929608340r + 872131627739)z2

3 + (−499966778880r4 + 700672800000r3

+2965518990720r2 + 4775630676800r + 7891408389440)z3 − 467274777600r4 −839169497600r3 + 5321149267200r2 + 2503535369600r − 1046033360000

= 7876613555200z1 + (−153034918400r4 − 7807904000r3 + 903374492800r2 +2499700465600r + 4838948504000)z2 + (59559801856r4 + 151950492160r3 −593716973824r2 − 1892816667520r − 3986995393408)z5

3 + (−4687524800r4

+80169550400r3 + 185007877600r2 − 763382506800r − 2871241102700)z4

3

+(−312222365248r4 + 38813437120r3 + 1586639032992r2 + 7497521298160r +7818934993964)z3

3 + (−300322513008r4 − 222671941680r3 + 1000859535832r2

+12396509766660r + 6964716740469)z2

3 + (−177176236080r4 + 95974993800r3

+127357268220r2 + 4765514413650r + 22306522877090)z3 − 100899184800r4 −96465152400r3 + 948066550600r2 − 5635797509700r − 8388950934500