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SARO.SIF
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NAME SARO
*
* A discrete-time optimal control problem.
* The model describes the central solar heating plant with seasonal
* water tank store in S"ar"o, south of Gothenburg, Sweden. The plant
* supplies heat energy for 48 apartments in 10 terraced houses.
* About one third of the total annual amount is covered by solar energy.
* The aim of the optimization problem is to find a control strategy
* that minimizes the amount of supplementary heat energy subject to bounds
* on the maximal supplementary heat load. The peak load in the winter can
* be decreased by increasing the storage time at the cost of increased
* storage losses and reduced solar contribution.
* The continuous-time model is described by 9 state variables
* (6 temperatures and 3 heat quantities) and one control variable
* (position of a mixing valve between store and heat load).
* The discrete-time optimization problem is formulated by using
* multistage control parameterization. The time horizon of one year is
* divided into 365 stages. The control trajectory is described in each
* stage with two parameters. The path constraint for the supplementary heat
* load is applied at two points in each stage (every 12 hours).
*
* Reference:
* R. Franke. Object-oriented modeling of solar heating systems.
* Solar Energy, 60(3/4)1997, pages 171-180.
*
* In the formulation given here, Fortran code for the system model has been
* generated with the modeling tool DYMOLA for physical systems.
* ADIFOR has been applied to obtain first order derivatives.
* The structure of the optimization problem is described in SIF.
*
* SIF input: Ruediger Franke, TU Ilmenau, January 1998
*
* classification LOR1-MN-4754-4015
********************************************
* define parameters
* PSMAX is used as upper bound for the supplementary
* heat power [kW].
* For PSMAX greater than 88.3 kW, the according path
* constraint becomes inactive (see data for Psh below).
* For PSMAX lower than 57.6 kW, the optimization problem
* becomes infeasible (see min max formulation SAROMM).
*RE PSMAX 90.0 $ inactive constraint
RE PSMAX 60.0 $ active constraint
*RE PSMAX 55.0 $ infeasible
* define required integer constants
IE KMAX 366 $ number of communication points
IE NX 9 $ number of states
IE NU 2 $ number of controls
IE NG 2 $ constraints
IE 1 1
IE 2 2
IE 3 3
IE 4 4
IE 5 5
IE 6 6
IE 7 7
IE 8 8
IE 9 9
IA KMAX-1 KMAX -1 $ number of time stages
VARIABLES
DO K 1 KMAX-1
* discrete-time states
DO I 1 NX
X X(I,K)
OD I
* control parameters
DO I 1 NU
X U(I,K)
OD I
* slack variables
DO I 1 NG
X S(I,K)
OD I
OD K
* final states
DO I 1 NX
X X(I,KMAX)
OD I
GROUPS
* objective function
* (final value of supplementary heat energy [MWh])
XN OBJ X(7,KMAX) 1.0
* system equations and constraints
DO K 1 KMAX-1
IA K+1 K 1
DO I 1 NX
XE F(I,K) X(I,K+1) -1.0
OD I
DO I 1 NG
XE G(I,K) S(I,K) -1.0
OD I
OD K
BOUNDS
FR BNDS 'DEFAULT'
* initial states
XX BNDS X(1,1) 83.50
XX BNDS X(2,1) 83.42
XX BNDS X(3,1) 82.55
XX BNDS X(4,1) 81.00
XX BNDS X(5,1) 78.00
XX BNDS X(6,1) 68.37
XX BNDS X(7,1) 0.00
XX BNDS X(8,1) 0.00
XX BNDS X(9,1) 0.00
* final states
XX BNDS X(1,KMAX) 83.50
* control bounds
DO K 1 KMAX-1
DO I 1 NU
XL BNDS U(I,K) 0.0
XU BNDS U(I,K) 1.0
ND
* bounds on algebraic variables,
* i.e. supplementary heat power
DO K 1 KMAX-1
DO I 1 NG
XL BNDS S(I,K) 0.0
ZU BNDS S(I,K) PSMAX
ND
START POINT
DO K 1 KMAX
X SPOINT X(1,K) 83.50
X SPOINT X(2,K) 83.42
X SPOINT X(3,K) 82.55
X SPOINT X(4,K) 81.00
X SPOINT X(5,K) 78.00
X SPOINT X(6,K) 68.37
ND
DO K 1 KMAX-1
DO I 1 NU
X SPOINT U(I,K) 0.5
ND
ELEMENT TYPE
* system equations and constraints
EP FTYPE RK RIDX
EV FTYPE X1 X2
EV FTYPE X3 X4
EV FTYPE X5 X6
EV FTYPE X7 X8
EV FTYPE X9
EV FTYPE U1 U2
EP GTYPE RK RIDX
EV GTYPE X1 X2
EV GTYPE X3 X4
EV GTYPE X5 X6
EV GTYPE X7 X8
EV GTYPE X9
EV GTYPE U1 U2
ELEMENT USES
* system equations and constraints
DO K 1 KMAX-1
RI RK K
DO I 1 NX
RI RIDX I
XT FE(I,K) FTYPE
ZP FE(I,K) RK RK
ZP FE(I,K) RIDX RIDX
ZV FE(I,K) X1 X(1,K)
ZV FE(I,K) X2 X(2,K)
ZV FE(I,K) X3 X(3,K)
ZV FE(I,K) X4 X(4,K)
ZV FE(I,K) X5 X(5,K)
ZV FE(I,K) X6 X(6,K)
ZV FE(I,K) X7 X(7,K)
ZV FE(I,K) X8 X(8,K)
ZV FE(I,K) X9 X(9,K)
ZV FE(I,K) U1 U(1,K)
ZV FE(I,K) U2 U(2,K)
OD I
DO I 1 NG
RI RIDX I
XT GE(I,K) GTYPE
ZP GE(I,K) RK RK
ZP GE(I,K) RIDX RIDX
ZV GE(I,K) X1 X(1,K)
ZV GE(I,K) X2 X(2,K)
ZV GE(I,K) X3 X(3,K)
ZV GE(I,K) X4 X(4,K)
ZV GE(I,K) X5 X(5,K)
ZV GE(I,K) X6 X(6,K)
ZV GE(I,K) X7 X(7,K)
ZV GE(I,K) X8 X(8,K)
ZV GE(I,K) X9 X(9,K)
ZV GE(I,K) U1 U(1,K)
ZV GE(I,K) U2 U(2,K)
OD I
OD K
GROUP USES
* system equations and constraints
DO K 1 KMAX-1
DO I 1 NX
XE F(I,K) FE(I,K)
OD I
DO I 1 NG
XE G(I,K) GE(I,K)
OD I
OD K
OBJECT BOUND
*LO SOLUTION 232.118 $ (PSMAX = 90 kW)
*LO SOLUTION 236.498 $ (PSMAX = 80 kW)
*LO SOLUTION 240.126 $ (PSMAX = 70 kW)
*LO SOLUTION 252.297 $ (PSMAX = 60 kW)
ENDATA
*
* nonlinear elements
*
ELEMENTS SARO
TEMPORARIES
R SAROFN
F SAROFN
R SAROGN
F SAROGN
R F
R GX1
R GX2
R GX3
R GX4
R GX5
R GX6
R GX7
R GX8
R GX9
R GU1
R GU2
INDIVIDUALS
* system equations
T FTYPE
F SAROFN(RK, RIDX, .TRUE.,
F+ X1, X2, X3, X4, X5, X6, X7, X8, X9,
F+ U1, U2)
A F SAROGN(RK, RIDX, .TRUE.,
A+ X1, X2, X3, X4, X5, X6, X7, X8, X9,
A+ U1, U2,
A+ GX1, GX2, GX3, GX4, GX5, GX6, GX7,
A+ GX8, GX9, GU1, GU2)
G X1 GX1
G X2 GX2
G X3 GX3
G X4 GX4
G X5 GX5
G X6 GX6
G X7 GX7
G X8 GX8
G X9 GX9
G U1 GU1
G U2 GU2
* constraints
T GTYPE
F SAROFN(RK, RIDX, .FALSE.,
F+ X1, X2, X3, X4, X5, X6, X7, X8, X9,
F+ U1, U2)
A F SAROGN(RK, RIDX, .FALSE.,
A+ X1, X2, X3, X4, X5, X6, X7, X8, X9,
A+ U1, U2,
A+ GX1, GX2, GX3, GX4, GX5, GX6, GX7,
A+ GX8, GX9, GU1, GU2)
G X1 GX1
G X2 GX2
G X3 GX3
G X4 GX4
G X5 GX5
G X6 GX6
G X7 GX7
G X8 GX8
G X9 GX9
G U1 GU1
G U2 GU2
ENDATA
C
C FORTRAN ROUTINES
C
C
C CALCULATE FUNCTION VALUE OF DISCRETE TIME SYSTEM EQUATIONS AND CONSTRAINTS.
C THIS FUNCTION PROVIDES AN INTERFACE TO SRFG.
C
DOUBLE PRECISION FUNCTION SAROFN(
1 RK, RIDX, SYSEQ,
2 X1, X2, X3, X4, X5, X6, X7, X8, X9,
3 U1, U2)
C
INTEGER NX, NU, NG
PARAMETER(NX = 9, NU = 2, NG = 2)
DOUBLE PRECISION RK, RIDX
LOGICAL SYSEQ
DOUBLE PRECISION X1, X2, X3, X4, X5, X6, X7, X8, X9
DOUBLE PRECISION U1, U2
INTEGER K, IDX
DOUBLE PRECISION X(NX), U(NU), F(NX), G(NG)
DOUBLE PRECISION RKZBAK
C
SAVE
C
DATA RKZBAK /-1.0D0/
C
IF ((RK .NE. RKZBAK)
* .OR. (X1 .NE. X(1))
* .OR. (X2 .NE. X(2))
* .OR. (X3 .NE. X(3))
* .OR. (X4 .NE. X(4))
* .OR. (X5 .NE. X(5))
* .OR. (X6 .NE. X(6))
* .OR. (X7 .NE. X(7))
* .OR. (X8 .NE. X(8))
* .OR. (X9 .NE. X(9))
* .OR. (U1 .NE. U(1))
* .OR. (U2 .NE. U(2))) THEN
C
X(1) = X1
X(2) = X2
X(3) = X3
X(4) = X4
X(5) = X5
X(6) = X6
X(7) = X7
X(8) = X8
X(9) = X9
C
U(1) = U1
U(2) = U2
C
K = RK
RKZBAK = RK
C
CALL SRFG(K, X, U, F, G)
END IF
C
IDX = RIDX
IF (SYSEQ) THEN
SAROFN = F(IDX)
ELSE
SAROFN = G(IDX)
END IF
C
RETURN
END
C
C
C CALCULATE JACOBIAN OF DISCRETE TIME SYSTEM EQUATIONS AND CONSTRAINTS.
C THIS FUNCTION PROVIDES AN INTERFACE TO GZSRFG THAT HAS BEEN DERIVED
C FROM SRFG USING ADIFOR.
C
DOUBLE PRECISION FUNCTION SAROGN(
1 RK, RIDX, SYSEQ,
2 X1, X2, X3, X4, X5, X6, X7, X8, X9,
3 U1, U2,
4 GX1, GX2, GX3, GX4, GX5, GX6, GX7, GX8, GX9,
5 GU1, GU2)
C
INTEGER NX, NU, NG, PMAX
PARAMETER(NX = 9, NU = 2, NG = 2, PMAX = 11)
DOUBLE PRECISION RK, RIDX
LOGICAL SYSEQ
DOUBLE PRECISION X1, X2, X3, X4, X5, X6, X7, X8, X9
DOUBLE PRECISION U1, U2
DOUBLE PRECISION GX1, GX2, GX3, GX4, GX5, GX6, GX7, GX8, GX9
DOUBLE PRECISION GU1, GU2
INTEGER I, J, K, IDX
DOUBLE PRECISION X(NX), U(NU), F(NX), G(NG)
DOUBLE PRECISION GZF(PMAX, NX), GZG(PMAX, NG)
DOUBLE PRECISION GZX(PMAX, NX), GZU(PMAX, NU)
DOUBLE PRECISION RKZBAK
C
SAVE
C
DATA RKZBAK /-1.0D0/
C
IF ((RK .NE. RKZBAK)
* .OR. (X1 .NE. X(1))
* .OR. (X2 .NE. X(2))
* .OR. (X3 .NE. X(3))
* .OR. (X4 .NE. X(4))
* .OR. (X5 .NE. X(5))
* .OR. (X6 .NE. X(6))
* .OR. (X7 .NE. X(7))
* .OR. (X8 .NE. X(8))
* .OR. (X9 .NE. X(9))
* .OR. (U1 .NE. U(1))
* .OR. (U2 .NE. U(2))) THEN
C
X(1) = X1
X(2) = X2
X(3) = X3
X(4) = X4
X(5) = X5
X(6) = X6
X(7) = X7
X(8) = X8
X(9) = X9
C
U(1) = U1
U(2) = U2
C
K = RK
RKZBAK = RK
C
DO 120 I = 1, PMAX
DO 100 J = 1, NX
GZX(I, J) = 0.0D0
100 CONTINUE
DO 110 J = 1, NU
GZU(I, J) = 0.0D0
110 CONTINUE
IF (I .LE. NX) THEN
GZX(I, I) = 1.0D0
ELSE
GZU(I, I - NX) = 1.0D0
END IF
120 CONTINUE
C
CALL GZSRFG(PMAX, K, X, GZX, PMAX, U, GZU, PMAX,
* F, GZF, PMAX, G, GZG, PMAX)
C
END IF
C
IDX = RIDX
C
IF (SYSEQ) THEN
GX1 = GZF(1,IDX)
GX2 = GZF(2,IDX)
GX3 = GZF(3,IDX)
GX4 = GZF(4,IDX)
GX5 = GZF(5,IDX)
GX6 = GZF(6,IDX)
GX7 = GZF(7,IDX)
GX8 = GZF(8,IDX)
GX9 = GZF(9,IDX)
C
GU1 = GZF(10,IDX)
GU2 = GZF(11,IDX)
C
SAROGN = F(IDX)
ELSE
GX1 = GZG(1,IDX)
GX2 = GZG(2,IDX)
GX3 = GZG(3,IDX)
GX4 = GZG(4,IDX)
GX5 = GZG(5,IDX)
GX6 = GZG(6,IDX)
GX7 = GZG(7,IDX)
GX8 = GZG(8,IDX)
GX9 = GZG(9,IDX)
C
GU1 = GZG(10,IDX)
GU2 = GZG(11,IDX)
C
SAROGN = G(IDX)
END IF
C
RETURN
END
C
C
C CALCULATE FUNCTION VALUES OF DISCRETE-TIME SYSTEM EQUATIONS AND CONSTRAINTS.
C THIS SUBROUTINE DIVIDES EACH TIME STEP K INTO TWO SAMPLE PERIODS AND
C INTEGRATES CONTINUOUS-TIME SYSTEM EQUATIONS.
C
SUBROUTINE SRFG(K, X, U, F, G)
C
INTEGER NX, NU, NG, NP, NW
PARAMETER (NX = 9, NU = 2, NG = 2, NP = 1, NW = 1)
INTEGER K
DOUBLE PRECISION X(NX), U(NU), F(NX), G(NG)
INTEGER KK
DOUBLE PRECISION DT, FH(NX), P(NP), W(NW)
C
KK = 2 * (K - 1) + 1
DT = 12.0D0
C
P(1) = U(1)
CALL SRK4(KK, DT, X, P, FH, W)
G(1) = W(1)
C
P(1) = U(2)
CALL SRK4(KK + 1, DT, FH, P, F, W)
G(2) = W(1)
C
RETURN
END
C
C
C RUNGE KUTTA TYPE INTEGRATION PROCEDURE.
C
SUBROUTINE SRK4(KK, DT, X, P, F, W)
C
INTEGER NX, NP, NW
PARAMETER (NX = 9, NP = 1, NW = 1)
INTEGER KK
DOUBLE PRECISION DT, X(NX), P(NP), F(NX), W(NW)
INTEGER I
DOUBLE PRECISION DF(NX), XH(NX), W1(NW)
C
CALL SREQ(KK, X, P, DF, W)
DO 200 I = 1, NX
F(I) = X(I) + DT/6.0D0 * DF(I)
XH(I) = X(I) + DT/2.0D0 * DF(I)
200 CONTINUE
C
CALL SREQ(KK, XH, P, DF, W1)
DO 210 I = 1, NX
F(I) = F(I) + DT/3.0D0 * DF(I)
XH(I) = X(I) + DT/2.0D0 * DF(I)
210 CONTINUE
C
CALL SREQ(KK, XH, P, DF, W1)
DO 220 I = 1, NX
F(I) = F(I) + DT/3.0D0 * DF(I)
XH(I) = X(I) + DT * DF(I)
220 CONTINUE
C
CALL SREQ(KK, XH, P, DF, W1)
DO 230 I = 1, NX
F(I) = F(I) + DT/6.0D0 * DF(I)
230 CONTINUE
C
END
C
C
C CALCULATE RIGHT HAND SIDES OF THE ODES DEFINING THE SYSTEM.
C
SUBROUTINE SREQ(KK, X, P, DF, W)
C
INTEGER KMAX, NX, NP, NW
LOGICAL EXTU
PARAMETER (KMAX = 366, NX = 9, NP = 1, NW = 1, EXTU = .TRUE.)
INTEGER KK
DOUBLE PRECISION X(NX), P(NP), DF(NX), W(NW), V(199)
INTEGER K
DOUBLE PRECISION TAS(KMAX), TRSHS(KMAX), PSHS(KMAX)
DOUBLE PRECISION H0S(KMAX), H1S(KMAX), H2S(KMAX), H3S(KMAX)
DOUBLE PRECISION OP0S(KMAX), OP1S(KMAX), OP2S(KMAX), OP3S(KMAX)
COMMON /SARDAT/
1 TAS, TRSHS, PSHS, H0S, H1S, H2S, H3S, OP0S, OP1S, OP2S, OP3S
C
K = ((KK - 1) / 2) + 1
C
TA = TAS(K)
TRSH = TRSHS(K)
PSH = PSHS(K)
IF (2*(K-1) .EQ. (KK-1)) THEN
H0 = 2.0D0 * H0S(K)
H1 = 2.0D0 * H1S(K)
H2 = 2.0D0 * H2S(K)
H3 = 2.0D0 * H3S(K)
OP0 = 2.0D0 * OP0S(K)
OP1 = 2.0D0 * OP1S(K)
OP2 = 2.0D0 * OP2S(K)
OP3 = 2.0D0 * OP3S(K)
ELSE
H0 = 0.0D0
H1 = 0.0D0
H2 = 0.0D0
H3 = 0.0D0
OP0 = 0.0D0
OP1 = 0.0D0
OP2 = 0.0D0
OP3 = 0.0D0
END IF
C
C SORTED AND SOLVED EQUATIONS, GENERATED BY DYMOLA
C (POSTPROCESSED IN ORDER TO PLACE ALL TEMPORARY VARIABLES IN V)
C M.
V(12) = 0.001D0*PSH
IF (EXTU) THEN
V(13) = P(1)
ELSE
IF (X(1) .GT. 60.0D0) THEN
V(199) = (60.0D0 - TRSH)/(X(1) -
1 TRSH)
ELSE
V(198) = EXP(15.0D0 - 20.0D0
1 *(X(1) - TRSH))
V(199) = 1.0D0/(1.0D0 + V(198))
END IF
V(13) = V(199)
END IF
V(6) = 1000.0D0*V(13)*V(12)/(
1 69666.6666666667D0 - 1161.11111111111D0*TRSH)
V(11) = 1.16111111111111D0*(X(1) - TRSH)*
1 V(6)
V(10) = V(12) - V(11)
DF(7) = 0.001D0*V(10)
V(19) = OP0 + (OP1 + (OP2 + OP3*X(6))*X(6))*
1 X(6)
V(14) = 740.0D0*V(19)
V(16) = 0.000277777777777778D0*V(14)
V(22) = - V(16)
V(26) = - V(16)
V(35) = - V(16)
V(43) = 3600.0D0*V(16)
V(18) = H0 + (H1 + (H2 + H3*X(6))*X(6))*
1 X(6)
V(46) = X(6) + V(18)/(1.16111111111111D-27
1 + 1161.11111111111D0*V(19))
V(8) = 0.000277777777777778D0*V(6)
V(24) = - V(8)
V(28) = - V(8)
V(123) = - V(8)
V(106) = - V(8)
V(89) = - V(8)
V(72) = - V(8)
V(55) = - V(8)
V(37) = - V(8)
V(48) = 3600.0D0*V(8)
V(166) = 47.5428571428572D0*(X(1) -
1 X(2))
V(157) = 10.1587301587302D0*(X(1) - TA)
V(40) = - (V(166) + V(157))
V(138) = 3.76619850066144D0*(X(1) - TA)
V(41) = - V(138)
DF(1) = 9.41985645933014D-06*(
1 1161.11111111111D0*V(43)*(V(46) -
1 X(1)) + 1161.11111111111D0*V(48)*(
1 X(2) - X(1)) + V(40) -
1 V(138))
V(53) = - V(16)
V(61) = 3600.0D0*V(16)
V(65) = 3600.0D0*V(8)
V(167) = - V(166)
V(170) = 39.6190476190476D0*(X(2) -
1 X(3))
V(58) = V(166) - V(170)
V(142) = 4.0D0 +
1 0.5D0*TA
V(141) = 5.64929775099216D0*(X(2) -
1 V(142))
V(59) = - V(141)
DF(2) = 6.27990430622009D-06*(
1 1161.11111111111D0*V(61)*(X(1) -
1 X(2)) + 1161.11111111111D0*V(65)*(
1 X(3) - X(2)) + V(58) -
1 V(141))
V(70) = - V(16)
V(78) = 3600.0D0*V(16)
V(82) = 3600.0D0*V(8)
V(172) = - V(170)
V(175) = 39.6190476190476D0*(X(3) -
1 X(4))
V(75) = V(170) - V(175)
V(145) = 5.64929775099216D0*X(3) -
1 45.1943820079373D0
V(76) = - V(145)
DF(3) = 6.27990430622009D-06*(
1 1161.11111111111D0*V(78)*(X(2) -
1 X(3)) + 1161.11111111111D0*V(82)*(
1 X(4) - X(3)) + V(75) -
1 V(145))
V(87) = - V(16)
V(95) = 3600.0D0*V(16)
V(99) = 3600.0D0*V(8)
V(177) = - V(175)
V(180) = 47.5428571428572D0*(X(4) -
1 X(5))
V(92) = V(175) - V(180)
V(148) = 5.64929775099216D0*X(4) -
1 45.1943820079373D0
V(93) = - V(148)
DF(4) = 6.27990430622009D-06*(
1 1161.11111111111D0*V(95)*(X(3) -
1 X(4)) + 1161.11111111111D0*V(99)*(
1 X(5) - X(4)) + V(92) -
1 V(148))
V(104) = - V(16)
V(112) = 3600.0D0*V(16)
V(116) = 3600.0D0*V(8)
V(182) = - V(180)
V(185) = 79.2380952380953D0*(X(5) -
1 X(6))
V(109) = V(180) - V(185)
V(151) = 3.76619850066144D0*X(5) -
1 30.1295880052915D0
V(110) = - V(151)
DF(5) = 9.41985645933014D-06*(
1 1161.11111111111D0*V(112)*(X(4) -
1 X(5)) + 1161.11111111111D0*V(116)*(
1 X(6) - X(5)) + V(109) -
1 V(151))
V(121) = - V(16)
V(129) = 3600.0D0*V(16)
V(133) = 3600.0D0*V(8)
V(187) = - V(185)
V(161) = 73.1428571428572D0*X(6) -
1 585.142857142857D0
V(126) = V(185) - V(161)
V(154) = 1.88309925033072D0*X(6) -
1 15.0647940026458D0
V(127) = - V(154)
DF(6) = 1.88397129186603D-05*(
1 1161.11111111111D0*V(129)*(X(5) -
1 X(6)) + 1161.11111111111D0*V(133)*(TRSH -
1 X(6)) + V(126) - V(154))
V(196) = 4180.0D0*(X(1) - TRSH)*V(8)
DF(8) = 0.001D0*V(196)
V(197) = 4180.0D0*(V(46) - X(6))*
1 V(16)
DF(9) = 0.001D0*V(197)
C END OF SORTED AND SOLVED EQUATIONS
C
W(1) = V(10)
C
RETURN
END
C
C
BLOCK DATA SARCOM
C
INTEGER NK1, NK2, NK3, NK4
PARAMETER (NK1 = 108, NK2 = 108, NK3 = 108, NK4 = 42)
DOUBLE PRECISION TA1(NK1), TA2(NK2), TA3(NK3), TA4(NK4)
DOUBLE PRECISION TRSH1(NK1), TRSH2(NK2), TRSH3(NK3), TRSH4(NK4)
DOUBLE PRECISION PSH1(NK1), PSH2(NK2), PSH3(NK3), PSH4(NK4)
DOUBLE PRECISION H01(NK1), H02(NK2), H03(NK3), H04(NK4)
DOUBLE PRECISION H11(NK1), H12(NK2), H13(NK3), H14(NK4)
DOUBLE PRECISION H21(NK1), H22(NK2), H23(NK3), H24(NK4)
DOUBLE PRECISION H31(NK1), H32(NK2), H33(NK3), H34(NK4)
DOUBLE PRECISION OP01(NK1), OP02(NK2), OP03(NK3), OP04(NK4)
DOUBLE PRECISION OP11(NK1), OP12(NK2), OP13(NK3), OP14(NK4)
DOUBLE PRECISION OP21(NK1), OP22(NK2), OP23(NK3), OP24(NK4)
DOUBLE PRECISION OP31(NK1), OP32(NK2), OP33(NK3), OP34(NK4)
C
COMMON /SARDAT/
1 TA1, TA2, TA3, TA4, TRSH1, TRSH2, TRSH3, TRSH4,
2 PSH1, PSH2, PSH3, PSH4,
3 H01, H02, H03, H04, H11, H12, H13, H14,
4 H21, H22, H23, H24, H31, H32, H33, H34,
5 OP01, OP02, OP03, OP04, OP11, OP12, OP13, OP14,
6 OP21, OP22, OP23, OP24, OP31, OP32, OP33, OP34
C
DATA TA1/
1 1.52D+01, 1.60D+01, 1.67D+01, 1.77D+01, 1.54D+01, 1.14D+01,
2 9.10D+00, 1.23D+01, 1.21D+01, 1.46D+01, 1.28D+01, 9.44D+00,
3 1.25D+01, 1.53D+01, 1.51D+01, 1.37D+01, 1.18D+01, 1.26D+01,
4 1.31D+01, 1.36D+01, 1.25D+01, 1.25D+01, 1.14D+01, 1.33D+01,
5 1.35D+01, 1.18D+01, 9.37D+00, 7.50D+00, 8.75D+00, 1.25D+01,
6 1.03D+01, 9.90D+00, 7.45D+00, 1.10D+01, 1.17D+01, 1.14D+01,
7 7.96D+00, 1.34D+01, 1.25D+01, 8.81D+00, 1.14D+01, 1.12D+01,
8 1.04D+01, 1.12D+01, 1.06D+01, 1.07D+01, 9.89D+00, 7.89D+00,
9 5.32D+00, 2.33D-01,-7.92D-02, 2.34D+00, 1.65D+00, 4.45D+00,
1 4.46D+00, 6.70D+00, 6.65D+00, 6.72D+00, 6.22D+00, 5.08D+00,
2 4.08D+00, 4.65D+00, 7.82D+00, 8.24D+00, 6.96D+00, 5.36D+00,
3 4.56D+00, 7.21D+00, 6.74D+00, 4.94D+00, 1.10D+00, 5.45D+00,
4 6.33D+00, 5.92D+00, 5.50D+00, 3.67D+00, 2.60D+00, 1.38D+00,
5 1.59D+00,-2.08D-01,-4.81D+00,-1.77D+00, 6.08D+00, 7.07D+00,
6 6.96D+00, 6.12D+00, 4.92D+00, 5.89D+00, 6.32D+00, 6.60D+00,
7 5.90D+00, 4.59D+00, 4.70D+00, 5.61D+00, 5.00D+00,-5.21D-01,
8 1.70D+00,-1.49D+00,-4.02D+00, 1.21D+00, 1.83D+00, 1.73D+00,
9 1.83D+00, 1.73D+00, 1.83D+00, 1.73D+00, 3.66D+00, 4.05D+00/
DATA TA2/
1 4.98D+00, 4.32D+00, 3.55D+00, 4.17D+00, 4.33D-01, 5.30D+00,
2 -5.08D-01,-2.08D+00, 4.50D+00,-8.17D-01, 3.60D+00, 5.70D+00,
3 3.45D+00,-6.00D-01, 3.83D+00, 3.46D+00, 3.83D+00, 3.46D+00,
4 4.36D+00, 3.72D+00, 3.65D+00, 3.16D+00, 3.56D+00, 4.45D+00,
5 3.87D+00, 1.90D+00,-3.90D+00,-2.36D+00, 4.08D-01, 3.92D-01,
6 -1.97D+00, 3.83D-01, 3.92D-01, 2.60D+00, 2.95D+00,-3.21D-01,
7 3.31D+00, 6.00D-01, 2.40D+00,-1.42D-01,-1.93D+00,-1.48D+00,
8 -2.29D+00,-3.65D+00,-4.23D+00,-5.12D+00,-3.84D+00,-3.91D+00,
9 -5.16D+00,-4.20D+00,-4.86D+00,-4.05D+00,-2.55D+00,-2.36D+00,
1 -2.08D+00,-2.38D+00,-2.43D+00,-5.47D+00,-6.94D+00,-7.25D+00,
2 -4.92D+00,-2.78D+00,-3.52D+00,-4.62D-01, 4.42D-01, 9.37D-01,
3 1.97D+00, 2.65D+00, 3.95D+00, 2.75D+00,-5.33D-01,-3.28D+00,
4 -2.62D+00, 1.12D+00, 1.26D+00, 6.00D-01, 1.45D+00, 8.13D-01,
5 2.48D+00, 2.80D+00, 3.57D+00, 3.33D+00, 2.89D+00, 4.14D+00,
6 3.61D+00, 1.89D+00, 4.30D+00, 4.69D+00, 3.46D+00, 4.04D+00,
7 4.72D+00, 4.74D+00, 4.92D+00, 4.01D+00, 4.04D+00, 3.16D+00,
8 2.78D+00, 3.49D+00, 6.55D+00, 6.41D+00, 2.83D+00, 3.03D+00,
9 4.19D+00, 2.69D+00, 5.51D+00, 6.39D+00, 6.08D+00, 4.89D+00/
DATA TA3/
1 7.48D+00, 7.07D+00, 5.88D+00, 5.92D+00, 4.61D+00, 6.31D+00,
2 9.54D+00, 1.11D+01, 1.11D+01, 9.06D+00, 6.74D+00, 6.02D+00,
3 2.68D+00, 1.36D+00, 1.01D+00, 1.67D+00, 2.49D+00, 3.35D+00,
4 4.59D+00, 4.76D+00, 6.83D+00, 4.51D+00, 6.67D+00, 8.83D+00,
5 7.94D+00, 8.48D+00, 5.20D+00, 7.71D+00, 6.14D+00, 5.75D+00,
6 7.69D+00, 7.78D+00, 7.82D+00, 8.29D+00, 8.85D+00, 8.78D+00,
7 8.93D+00, 1.03D+01, 9.17D+00, 9.85D+00, 8.28D+00, 9.65D+00,
8 8.20D+00, 9.11D+00, 9.03D+00, 1.06D+01, 1.10D+01, 1.09D+01,
9 9.14D+00, 8.81D+00, 9.72D+00, 1.01D+01, 1.17D+01, 1.15D+01,
1 1.18D+01, 1.47D+01, 1.40D+01, 1.11D+01, 1.04D+01, 7.71D+00,
2 7.31D+00, 9.96D+00, 1.05D+01, 1.12D+01, 1.08D+01, 1.16D+01,
3 1.33D+01, 1.22D+01, 1.14D+01, 1.28D+01, 1.13D+01, 1.25D+01,
4 1.16D+01, 1.25D+01, 9.95D+00, 1.19D+01, 1.18D+01, 1.29D+01,
5 1.34D+01, 1.41D+01, 1.39D+01, 1.47D+01, 1.40D+01, 1.43D+01,
6 1.44D+01, 1.23D+01, 1.35D+01, 1.51D+01, 1.73D+01, 1.95D+01,
7 2.03D+01, 2.09D+01, 2.11D+01, 2.11D+01, 2.18D+01, 1.93D+01,
8 1.71D+01, 1.74D+01, 1.80D+01, 1.74D+01, 1.68D+01, 1.66D+01,
9 1.55D+01, 1.63D+01, 1.65D+01, 1.55D+01, 1.67D+01, 1.68D+01/
DATA TA4/
1 1.83D+01, 1.55D+01, 1.66D+01, 1.73D+01, 1.72D+01, 1.69D+01,
2 1.69D+01, 1.72D+01, 2.07D+01, 2.22D+01, 2.24D+01, 2.17D+01,
3 2.00D+01, 1.89D+01, 1.86D+01, 1.75D+01, 1.94D+01, 1.88D+01,
4 1.67D+01, 1.79D+01, 1.68D+01, 1.55D+01, 1.65D+01, 1.66D+01,
5 1.70D+01, 1.66D+01, 1.47D+01, 1.46D+01, 1.59D+01, 1.62D+01,
6 1.65D+01, 1.74D+01, 1.86D+01, 1.58D+01, 1.61D+01, 1.49D+01,
7 1.45D+01, 1.46D+01, 1.45D+01, 1.69D+01, 1.53D+01, 1.52D+01/
C
DATA TRSH1/
1 3.03D+01, 3.02D+01, 3.10D+01, 3.11D+01, 3.11D+01, 3.01D+01,
2 3.19D+01, 3.11D+01, 3.16D+01, 3.08D+01, 3.07D+01, 3.21D+01,
3 2.97D+01, 3.09D+01, 3.01D+01, 3.02D+01, 3.08D+01, 3.07D+01,
4 3.07D+01, 2.93D+01, 3.10D+01, 3.02D+01, 3.04D+01, 3.08D+01,
5 3.06D+01, 3.09D+01, 3.09D+01, 3.28D+01, 3.15D+01, 3.02D+01,
6 3.11D+01, 3.15D+01, 3.30D+01, 3.01D+01, 3.08D+01, 2.98D+01,
7 3.24D+01, 3.15D+01, 3.12D+01, 3.14D+01, 3.09D+01, 3.12D+01,
8 3.04D+01, 3.04D+01, 3.15D+01, 3.08D+01, 3.14D+01, 3.18D+01,
9 3.32D+01, 3.58D+01, 3.64D+01, 3.56D+01, 3.55D+01, 3.43D+01,
1 3.36D+01, 3.27D+01, 3.21D+01, 3.21D+01, 3.32D+01, 3.35D+01,
2 3.37D+01, 3.35D+01, 3.23D+01, 3.10D+01, 3.18D+01, 3.35D+01,
3 3.35D+01, 3.20D+01, 3.24D+01, 3.34D+01, 3.50D+01, 3.25D+01,
4 3.28D+01, 3.27D+01, 3.31D+01, 3.38D+01, 3.45D+01, 3.46D+01,
5 3.48D+01, 3.65D+01, 3.97D+01, 3.83D+01, 3.28D+01, 3.21D+01,
6 3.14D+01, 3.23D+01, 3.34D+01, 3.29D+01, 3.27D+01, 3.25D+01,
7 3.27D+01, 3.30D+01, 3.30D+01, 3.34D+01, 3.37D+01, 3.67D+01,
8 3.53D+01, 3.72D+01, 3.86D+01, 3.49D+01, 3.52D+01, 3.51D+01,
9 3.49D+01, 3.50D+01, 3.50D+01, 3.45D+01, 3.38D+01, 3.42D+01/
DATA TRSH2/
1 3.33D+01, 3.36D+01, 3.39D+01, 3.37D+01, 3.54D+01, 3.28D+01,
2 3.69D+01, 3.80D+01, 3.34D+01, 3.71D+01, 3.47D+01, 3.21D+01,
3 3.40D+01, 3.71D+01, 3.37D+01, 3.42D+01, 3.38D+01, 3.43D+01,
4 3.36D+01, 3.33D+01, 3.33D+01, 3.40D+01, 3.41D+01, 3.34D+01,
5 3.42D+01, 3.53D+01, 3.82D+01, 3.73D+01, 3.55D+01, 3.60D+01,
6 3.72D+01, 3.61D+01, 3.58D+01, 3.38D+01, 3.38D+01, 3.66D+01,
7 3.45D+01, 3.56D+01, 3.49D+01, 3.61D+01, 3.67D+01, 3.64D+01,
8 3.75D+01, 3.86D+01, 3.88D+01, 3.98D+01, 3.85D+01, 3.80D+01,
9 3.94D+01, 3.87D+01, 3.95D+01, 3.90D+01, 3.78D+01, 3.73D+01,
1 3.64D+01, 3.71D+01, 3.73D+01, 3.97D+01, 4.12D+01, 4.14D+01,
2 3.91D+01, 3.71D+01, 3.81D+01, 3.60D+01, 3.55D+01, 3.53D+01,
3 3.47D+01, 3.42D+01, 3.26D+01, 3.39D+01, 3.62D+01, 3.81D+01,
4 3.77D+01, 3.53D+01, 3.49D+01, 3.46D+01, 3.46D+01, 3.50D+01,
5 3.44D+01, 3.42D+01, 3.41D+01, 3.40D+01, 3.33D+01, 3.31D+01,
6 3.38D+01, 3.47D+01, 3.33D+01, 3.35D+01, 3.38D+01, 3.28D+01,
7 3.28D+01, 3.31D+01, 3.35D+01, 3.37D+01, 3.39D+01, 3.41D+01,
8 3.35D+01, 3.40D+01, 3.33D+01, 3.30D+01, 3.45D+01, 3.53D+01,
9 3.37D+01, 3.38D+01, 3.21D+01, 3.24D+01, 3.31D+01, 3.35D+01/
DATA TRSH3/
1 3.26D+01, 3.26D+01, 3.20D+01, 3.20D+01, 3.36D+01, 3.36D+01,
2 3.17D+01, 3.24D+01, 3.17D+01, 3.09D+01, 3.20D+01, 3.26D+01,
3 3.50D+01, 3.55D+01, 3.60D+01, 3.57D+01, 3.43D+01, 3.36D+01,
4 3.38D+01, 3.44D+01, 3.30D+01, 3.40D+01, 3.30D+01, 3.13D+01,
5 3.15D+01, 3.20D+01, 3.34D+01, 3.20D+01, 3.33D+01, 3.27D+01,
6 3.10D+01, 3.16D+01, 3.31D+01, 3.29D+01, 3.26D+01, 3.30D+01,
7 3.14D+01, 3.02D+01, 3.04D+01, 3.15D+01, 3.21D+01, 3.11D+01,
8 3.23D+01, 3.20D+01, 3.14D+01, 3.02D+01, 3.11D+01, 3.14D+01,
9 3.15D+01, 3.23D+01, 3.17D+01, 3.10D+01, 3.01D+01, 3.19D+01,
1 3.14D+01, 3.06D+01, 3.15D+01, 3.08D+01, 3.06D+01, 3.18D+01,
2 3.26D+01, 3.16D+01, 3.13D+01, 3.13D+01, 3.09D+01, 2.99D+01,
3 2.97D+01, 3.11D+01, 3.08D+01, 3.03D+01, 3.08D+01, 3.12D+01,
4 3.03D+01, 2.98D+01, 3.20D+01, 3.06D+01, 3.09D+01, 3.00D+01,
5 3.08D+01, 2.95D+01, 3.04D+01, 3.05D+01, 2.98D+01, 3.16D+01,
6 3.18D+01, 3.05D+01, 2.96D+01, 2.76D+01, 3.07D+01, 3.23D+01,
7 3.05D+01, 3.07D+01, 3.09D+01, 2.98D+01, 2.98D+01, 3.09D+01,
8 3.37D+01, 3.12D+01, 3.22D+01, 3.14D+01, 2.97D+01, 2.86D+01,
9 3.12D+01, 3.11D+01, 3.23D+01, 3.25D+01, 3.16D+01, 2.98D+01/
DATA TRSH4/
1 2.95D+01, 3.09D+01, 3.07D+01, 3.06D+01, 2.93D+01, 3.05D+01,
2 3.00D+01, 2.98D+01, 3.06D+01, 3.11D+01, 3.12D+01, 2.92D+01,
3 3.11D+01, 3.03D+01, 3.03D+01, 3.12D+01, 3.11D+01, 3.12D+01,
4 2.92D+01, 3.12D+01, 3.03D+01, 3.02D+01, 3.11D+01, 3.09D+01,
5 3.11D+01, 2.92D+01, 3.09D+01, 3.01D+01, 3.02D+01, 3.10D+01,
6 3.09D+01, 3.10D+01, 2.92D+01, 3.09D+01, 3.02D+01, 3.04D+01,
7 3.12D+01, 3.10D+01, 3.12D+01, 2.92D+01, 3.09D+01, 3.03D+01/
C
DATA PSH1/
1 2.46D+04, 1.96D+04, 2.12D+04, 1.55D+04, 1.74D+04, 2.89D+04,
2 3.63D+04, 2.99D+04, 2.68D+04, 2.12D+04, 2.41D+04, 3.33D+04,
3 2.50D+04, 1.91D+04, 2.09D+04, 2.11D+04, 2.94D+04, 2.47D+04,
4 2.20D+04, 2.21D+04, 2.61D+04, 2.82D+04, 2.80D+04, 2.48D+04,
5 2.22D+04, 2.62D+04, 3.52D+04, 4.10D+04, 3.92D+04, 2.46D+04,
6 3.39D+04, 3.21D+04, 4.14D+04, 2.98D+04, 2.85D+04, 3.30D+04,
7 3.92D+04, 2.48D+04, 2.44D+04, 3.72D+04, 2.84D+04, 3.11D+04,
8 3.70D+04, 3.22D+04, 3.20D+04, 3.20D+04, 3.29D+04, 3.97D+04,
9 4.92D+04, 6.54D+04, 6.58D+04, 5.78D+04, 5.99D+04, 4.86D+04,
1 5.10D+04, 4.53D+04, 4.71D+04, 4.41D+04, 4.61D+04, 4.81D+04,
2 5.12D+04, 4.99D+04, 4.06D+04, 4.24D+04, 4.58D+04, 4.99D+04,
3 5.12D+04, 4.16D+04, 4.30D+04, 5.25D+04, 6.48D+04, 5.02D+04,
4 4.66D+04, 4.84D+04, 4.77D+04, 5.46D+04, 5.96D+04, 6.40D+04,
5 6.14D+04, 6.67D+04, 8.03D+04, 6.84D+04, 4.54D+04, 4.71D+04,
6 4.76D+04, 4.80D+04, 5.09D+04, 4.80D+04, 4.38D+04, 4.41D+04,
7 5.07D+04, 5.40D+04, 5.23D+04, 4.89D+04, 5.12D+04, 6.48D+04,
8 5.97D+04, 7.27D+04, 7.96D+04, 6.28D+04, 6.04D+04, 6.06D+04,
9 5.98D+04, 6.01D+04, 6.12D+04, 6.28D+04, 5.48D+04, 5.34D+04/
DATA PSH2/
1 5.09D+04, 5.34D+04, 5.46D+04, 5.46D+04, 6.66D+04, 5.02D+04,
2 6.65D+04, 7.16D+04, 5.43D+04, 6.62D+04, 5.51D+04, 5.13D+04,
3 5.83D+04, 6.71D+04, 5.63D+04, 5.50D+04, 5.60D+04, 5.48D+04,
4 5.29D+04, 5.79D+04, 5.73D+04, 5.83D+04, 5.48D+04, 5.42D+04,
5 5.33D+04, 6.11D+04, 8.12D+04, 7.50D+04, 6.66D+04, 6.38D+04,
6 7.31D+04, 6.38D+04, 6.66D+04, 6.14D+04, 6.00D+04, 6.81D+04,
7 5.53D+04, 6.54D+04, 5.78D+04, 6.85D+04, 7.54D+04, 7.28D+04,
8 7.47D+04, 7.60D+04, 8.03D+04, 8.04D+04, 7.96D+04, 8.29D+04,
9 8.39D+04, 8.00D+04, 8.04D+04, 7.93D+04, 7.33D+04, 7.54D+04,
1 7.78D+04, 7.59D+04, 7.52D+04, 8.23D+04, 8.83D+04, 8.78D+04,
2 8.28D+04, 7.96D+04, 7.95D+04, 7.00D+04, 6.58D+04, 6.37D+04,
3 6.15D+04, 6.07D+04, 6.00D+04, 6.21D+04, 7.14D+04, 7.72D+04,
4 7.57D+04, 6.29D+04, 6.53D+04, 7.02D+04, 6.58D+04, 6.72D+04,
5 5.95D+04, 5.83D+04, 5.67D+04, 5.87D+04, 6.28D+04, 5.65D+04,
6 5.82D+04, 6.11D+04, 5.40D+04, 5.21D+04, 5.81D+04, 5.94D+04,
7 5.46D+04, 5.35D+04, 5.12D+04, 5.46D+04, 5.37D+04, 5.89D+04,
8 6.32D+04, 5.80D+04, 4.77D+04, 4.63D+04, 5.94D+04, 5.59D+04,
9 5.55D+04, 6.28D+04, 5.20D+04, 4.83D+04, 4.71D+04, 5.26D+04/
DATA PSH3/
1 4.25D+04, 4.67D+04, 5.17D+04, 5.08D+04, 5.38D+04, 4.57D+04,
2 3.86D+04, 3.20D+04, 3.48D+04, 4.23D+04, 4.80D+04, 5.00D+04,
3 5.71D+04, 6.36D+04, 6.18D+04, 6.15D+04, 6.14D+04, 5.81D+04,
4 5.42D+04, 5.11D+04, 4.70D+04, 5.17D+04, 4.58D+04, 4.24D+04,
5 4.34D+04, 4.21D+04, 5.00D+04, 4.46D+04, 4.62D+04, 4.81D+04,
6 4.60D+04, 4.42D+04, 4.15D+04, 4.09D+04, 4.09D+04, 3.66D+04,
7 4.09D+04, 3.76D+04, 3.97D+04, 3.52D+04, 4.06D+04, 3.90D+04,
8 3.92D+04, 4.04D+04, 4.10D+04, 3.72D+04, 3.19D+04, 3.29D+04,
9 4.04D+04, 3.73D+04, 3.88D+04, 3.79D+04, 3.32D+04, 3.08D+04,
1 3.00D+04, 2.66D+04, 2.34D+04, 3.11D+04, 3.70D+04, 4.28D+04,
2 4.41D+04, 3.57D+04, 3.16D+04, 3.04D+04, 3.60D+04, 3.35D+04,
3 2.75D+04, 2.91D+04, 3.07D+04, 2.53D+04, 3.03D+04, 2.84D+04,
4 3.16D+04, 2.82D+04, 3.38D+04, 2.70D+04, 2.82D+04, 2.38D+04,
5 2.39D+04, 2.53D+04, 2.05D+04, 2.21D+04, 2.15D+04, 2.13D+04,
6 2.30D+04, 2.56D+04, 2.23D+04, 1.88D+04, 1.82D+04, 1.37D+04,
7 1.49D+04, 1.52D+04, 1.53D+04, 1.78D+04, 1.73D+04, 1.57D+04,
8 1.43D+04, 1.70D+04, 1.66D+04, 1.53D+04, 1.83D+04, 1.73D+04,
9 1.65D+04, 1.78D+04, 1.56D+04, 1.93D+04, 1.86D+04, 1.97D+04/
DATA PSH4/
1 1.59D+04, 1.83D+04, 1.88D+04, 1.79D+04, 1.75D+04, 1.89D+04,
2 1.99D+04, 1.85D+04, 1.93D+04, 1.55D+04, 1.44D+04, 1.60D+04,
3 1.68D+04, 1.92D+04, 1.69D+04, 1.77D+04, 1.55D+04, 1.46D+04,
4 1.71D+04, 1.65D+04, 1.90D+04, 1.74D+04, 1.79D+04, 1.65D+04,
5 1.49D+04, 1.61D+04, 1.98D+04, 2.33D+04, 1.76D+04, 1.92D+04,