Refrigeration cycle apparatus and optimization method

A refrigeration cycle apparatus comprising:

• a refrigerant circuit including a compressor, a heat-source-side heat exchanger, a plurality of expansion valves connected in parallel to the heat-source-side heat exchanger, and a plurality of use-side heat exchangers connected in series to the plurality of expansion valves, respectively;
• and
• a controller to control the plurality of expansion valves such that openings of the plurality of expansion valves are set to a plurality of opening command values, respectively, wherein
• the controller has
• a total computing unit to compute a total opening of the plurality of expansion valves,
• a target computing unit to compute a plurality of target openings representing respective target values of openings of the plurality of expansion valves,
• a restriction setting unit to set a plurality of upper limit openings representing respective upper limit values of the openings of the plurality of expansion valves and a plurality of lower limit openings representing respective lower limit values of the openings of the plurality of expansion valves, and
• an optimizer to set the plurality of opening command values by using the total opening, the plurality of target openings, the plurality of upper limit openings, and the plurality of lower limit openings,
• the target computing unit computes the plurality of target openings such that a total of the plurality of target openings is equal to the total opening,
• the optimizer
• in an initial step,
• calculates an adjustment lower limit value vector a, an adjustment upper limit value vector b, and an adjustment target value vector rS by dividing a lower limit value vector having the plurality of lower limit openings as elements, an upper limit value vector having the plurality of upper limit openings as elements, and a target value vector having the plurality of target openings as elements by the total opening, respectively,
• sets as a range vector λrange, a vector obtained by coupling a vector calculated by subtracting the adjustment target value vector rS from the adjustment lower limit value vector a and a vector calculated by subtracting the adjustment target value vector rS from the adjustment upper limit value vector b and sorting the vectors in an ascending order,
• sets as an index vector J, a vector having a first element which is 1 and a second element which is twice as large as the number N of the plurality of expansion valves, and
• defines an adjustment total range vector h in an expression (12) below
• [
• Expression
• ⁢
• 1
• ]
• h
• =
• {
• ∑
• i
• =
• 1
• N
• clip
• (
• (
• r
• S
• [
• i
• ]
• +
• λ
• range
• [
• J
• [
• 1
• ]
• ]
• )
• ,
• a
• [
• i
• ]
• ,
• b
• [
• i
• ]
• )
• ∑
• i
• =
• 1
• N
• clip
• ⁢
• (
• (
• r
• S
• [
• i
• ]
• +
• λ
• range
• [
• J
• [
• 2
• ]
• ]
• )
• ,
• a
• [
• i
• ]
• ,
• b
• [
• i
• ]
• )
• )
• ,
• (
• 12
• )
• in a first step, determines whether J[2]−J[1] is equal to or smaller than 1, J[2]−J[1] being a value calculated by subtracting a value of a first element J[1] from a value of a second element J[2] of the index vector J,
• when J[2]−J[1] is not equal to or smaller than 1, in a second step, sets as a provisional index JP, a value calculated by dropping a fractional portion of an average value of index vectors J and calculates a provisional adjustment total value hP in an expression (14) below
• [
• Expression
• ⁢
• 2
• ]
• h
• P
• =
• ∑
• i
• =
• 1
• N
• clip
• (
• (
• r
• S
• [
• i
• ]
• +
• λ
• range
• [
• J
• P
• ]
• )
• ,
• a
• [
• i
• ]
• ,
• b
• [
• i
• ]
• )
• ,
• (
• 14
• )
• in a third step,
• determines whether the provisional adjustment total value hP is equal to or larger than 1,
• when the provisional adjustment total value hP is equal to or larger than 1, substitutes the provisional index JP into the second element J[2] of the index vector J, substitutes the provisional adjustment total value hP into a second element h[2] of the adjustment total range vector h, and returns to the first step, and
• when the provisional adjustment total value hP is not equal to or larger than 1, substitutes the provisional index JP into the first element J[1] of the index vector J, substitutes the provisional adjustment total value hP into a first element h[1] of the adjustment total range vector h, and returns to the first step,
• repeats processing from the first step to the third step until J[2]−J[1] becomes equal to or smaller than 1, and
• when J[2]−J[1] becomes equal to or smaller than 1, in a fourth step,
• calculates a Lagrange multiplier λ* in an expression (16) below
• [
• Expression
• ⁢
• 3
• ]
• λ
• *
• =
• λ
• range
• [
• J
• [
• 1
• ]
• ]
• +
• (
• 1
• -
• h
• [
• 1
• ]
• )
• ⁢
• λ
• range
• [
• J
• [
• 2
• ]
• ]
• -
• λ
• range
• [
• J
• [
• 1
• ]
• ]
• h
• [
• 2
• ]
• -
• h
• [
• 1
• ]
• ,
• (
• 16
• )
• and
• sets an optimal solution calculated in an expression (17) below as the plurality of opening command values
• [
• Expression
• ⁢
• 4
• ]
• solution
• =
• S
• ×
• clip
• ⁢
• (
• r
• S
• +
• λ
• *
• ,
• a
• ,
• b
• )
• .
• (
• 17
• )

An optimization method causing, when a total value, a restriction vector including an upper limit value and a lower limit value, and a target value vector of which total of elements is equal to the total value are given to a computer or computed by internal computation by the computer, the computer to perform processing comprising:

• in an initial step, setting a provisional value vector as the target value vector;
• in a first step, determining whether an inequality constraint is satisfied, the inequality constraint being that the provisional value vector is within a restricted range defined by the restriction vector;
• when the inequality constraint is not satisfied, in a second step, computing an upper limit excess indicator and a lower limit excess indicator and comparing the upper limit excess indicator and the lower limit excess indicator with each other, the upper limit excess indicator indicating a degree of excess over an upper limit value of an element of the provisional value vector, the lower limit excess indicator indicating a degree of excess over a lower limit value of the element of the provisional value vector, and fixing an element which deviates most from a limit value larger in indicator to the upper limit value or the lower limit value of the element;
• in a third step, setting again as the target value vector, a point obtained by orthogonally projecting a point representing the provisional value vector on an affine hyperplane in a space reduced in order by exclusion of an already fixed element from the provisional value vector, the affine hyperplane having as an intercept, a subtraction total value calculated by subtracting the already fixed element from the total value, and returning to the first step;
• repeating processing from the first step to the third step until the inequality constraint is satisfied;
• and
• when the inequality constraint is satisfied, in a fourth step, outputting as a solution vector, the provisional value vector at time when the inequality constraint is satisfied, and quitting.

An optimization method causing, when a total value, an upper limit value vector, a lower limit value vector, and a target value vector of which total of elements is equal to the total value are given to a computer or computed by internal computation by the computer, the computer to perform processing comprising:

• in an initial step,
• calculating an adjustment lower limit value vector a, an adjustment upper limit value vector b, and an adjustment target value vector rS by dividing the lower limit value vector, the upper limit value vector, and the target value vector by the total value, respectively,
• setting as a range vector λrange, a vector obtained by coupling a vector calculated by subtracting the adjustment target value vector rS from the adjustment lower limit value vector a and a vector calculated by subtracting the adjustment target value vector rS from the adjustment upper limit value vector b and sorting the vectors in an ascending order,
• setting as an index vector J, a vector having a first element which is 1 and a second element which is twice as large as the number N of elements of the target value vector, and
• defining an adjustment total range vector h in an expression (12) below
• [
• Expression
• ⁢
• 5
• ]
• h
• =
• {
• ∑
• i
• =
• 1
• N
• clip
• (
• (
• r
• S
• [
• i
• ]
• +
• λ
• range
• [
• J
• [
• 1
• ]
• ]
• )
• ,
• a
• [
• i
• ]
• ,
• b
• [
• i
• ]
• )
• ∑
• i
• =
• 1
• N
• clip
• ⁢
• (
• (
• r
• S
• [
• i
• ]
• +
• λ
• range
• [
• J
• [
• 2
• ]
• ]
• )
• ,
• a
• [
• i
• ]
• ,
• b
• [
• i
• ]
• )
• )
• ;
• (
• 12
• )
• in a first step, determining whether J[2]−J[1] is equal to or smaller than 1, J[2]−J[1] being a value calculated by subtracting a value of a first element J[1] from a value of a second element J[2] of the index vector J;
• when J[2]−J[1] is not equal to or smaller than 1, in a second step, setting as a provisional index JP, a value calculated by dropping a fractional portion of an average value of index vectors J and calculating a provisional adjustment total value hP in an expression (14) below
• [
• Expression
• ⁢
• 6
• ]
• h
• P
• =
• ∑
• i
• =
• 1
• N
• clip
• ⁢
• (
• (
• r
• S
• [
• i
• ]
• +
• λ
• range
• [
• J
• P
• ]
• )
• ,
• a
• [
• i
• ]
• ,
• b
• [
• i
• ]
• )
• ,
• (
• 14
• )
• in a third step,
• determining whether the provisional adjustment total value hP is equal to or larger than 1,
• when the provisional adjustment total value hP is equal to or larger than 1, substituting the provisional index JP into the second element J[2] of the index vector J, substituting the provisional adjustment total value hP into a second element h[2] of the adjustment total range vector h, and returning to the first step, and
• when the provisional adjustment total value hP is not equal to or larger than 1, substituting the provisional index JP into the first element j[1] of the index vector J, substituting the provisional adjustment total value hP into a first element h[1] of the adjustment total range vector h, and returning to the first step;
• repeating processing from the first step to the third step until J[2]−J[1] becomes equal to or smaller than 1;
• and
• when J[2]−J[1] becomes equal to or smaller than 1, in a fourth step,
• calculating a Lagrange multiplier λ* in an expression (16) below
• [
• Expression
• ⁢
• 7
• ]
• λ
• *
• =
• λ
• range
• [
• J
• [
• 1
• ]
• ]
• +
• (
• 1
• -
• h
• [
• 1
• ]
• )
• ⁢
• λ
• range
• [
• J
• [
• 2
• ]
• ]
• -
• λ
• range
• [
• J
• [
• 1
• ]
• ]
• h
• [
• 2
• ]
• -
• h
• [
• 1
• ]
• ,
• (
• 16
• )
• and
• outputting a vector calculated in an expression (17) below as a solution vector and quitting
• [
• Expression
• ⁢
• 8
• ]
• solution
• =
• S
• ×
• clip
• ⁢
• (
• r
• S
• +
• λ
• *
• ,
• a
• ,
• b
• )
• .
• (
• 17
• )