Overall Procedure

P

i edis economic dispatch generation output of thermal unit i at hour t.

E. Duality Gap

The relative duality gap is defi ned as follows

( ) ( ) ( ) ( ) ( ) ( )

where G^(k) is the relative duality gap at iteration k and the values of

( )

are calculated from (3.20).

The relative duality gap is used to measure the solution quality by checking against the stopping criteria. The iteration process stops when either the relative duality gap is less than the specifi ed tolerance or the iteration counter exceeds the maximum allowable number of iterations.

F. Overall Procedure

Step 1: Initialize O^t and P^tas described in Section 3.3.4.B. The con straints (3.2) and (3.3) are satisfi ed by appropriate initial O^t and P^t values.

Step 2: Initialize the ELR iteration counter, k = 1 and J_B = $10⁷.

Step 3: Solve the unit sub-problems considering constraints (3.5), (3.7) and (3.8) as described in Section 3.3.4.A.

Step 4: If the dual solution does not satisfy the constraints (3.3), go to Step 9.

Step 5: Carry out transmission and ramp rate constrained economic dispatch considering constraints (3.2), (3.4), (3.5), (3.7), and (3.8) by linear/

quadratic programming as described in Section 3.3.4.D.

Step 6: Calculate the primal cost J U([ _{i t}^k_,]), the dual cost L P( ^{( )k},U^{( )k},l^{( )k},m^{( )k},g ^{( )k}) and the relative dual gap G^(k) as described in Section 3.3.4.E.

Step 7: If J U([ _{i t}^k_,])< J_B, J_B = J U([ _{i t}^k_,]) and [U_i^B_,t] = [U_i^k_,t]. Step 8: If the relative dual gap G^(k) < H, go to Step 10.

Step 9: If k < K_max, k = k + 1, update Lagrangian multipliers adaptively to satisfy constraints (3.2) and (3.3) as described in Section 3.3.4.C and go to Step 3.

Step 10: Terminate.

where

J_B best total economic dispatch production cost reached ($)

( )

([

_{i t},^k

])

J U

total economic dispatch production cost at iteration k ($) ]

[U_i^B_t, best feasible solution reached ]

[U_i^k_,t feasible solution at iteration k

K_max maximum allowable number of iterations Example 3.5A

This problem is illustrated by the data from system II and solved by the Lagrangian relaxation technique. The spinning reserve is neglected. Sub-problems are solved by DP. Lagrangian multipliers are updated by the adaptive subgradient method.

The initialization starts by sorting the generating units in the ascending order of FLAC_i. At each stage, the units with the least FLAC_i are committed one by one until the power balanced constraint (3.2) is satisfi ed. The corresponding unit schedule and the dispatch power during the initialization process are shown below. The starting Lagrangian multipliers O^t(0), shown below, are set as described in Section 3.3.4.B.

Initialization

Hour Ȝ^t _u^t(0)

1

)

ut₂⁽⁰ u₃^{t(0) (0},1⁾ t

Ped ⁽,⁰2⁾ t

Ped P_ed^t(0_,3⁾ Load

0 - 0 0 1 - - -

-1 11.600 0 0 1 0 0 200 200

2 13.233 0 1 1 0 300 200 500

3 14.100 1 1 1 500 400 200 1100

4 12.500 0 1 1 0 400 200 600

5 13.050 0 1 1 0 125 175 300

Iteration 1

Hour Ȝ^t u₁^t u^t₂ ^u3^t

P1t t

P₂ P₃^{t pdift} Ped^t_,1 P_ed^t_,2 P_ed^t_,3

1 11.600 1 1 1 400 400 200 –800 - -

-2 13.233 1 1 1 700 400 200 –800 - -

-3 14.100 1 1 1 700 400 200 –200 - -

-4 12.500 1 1 1 625 400 200 –625 - -

-5 13.050 1 1 1 700 400 200 –1000 - -

-The corresponding unit schedule is infeasible. -The norm(pdif) in (3.31) for this iteration is 1646.397. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^t(0) 11.600 13.233 14.100 12.500 13.050

pdif^t –800 –800 –200 –624.5 –1000

( ) ( )

pdift

k norm pdif a b+ ◊

–0.572 –0.571 –0.164 –0.143 –0.715

Ȝ^t(1) 11.028 12.662 13.957 12.053 12.335

Iteration 2

Hour Ȝ^t t

u₁ u^t₂ u₃^t P₁^t P₂^t P₃^{t pdift Pedt,1 Pedt ,2 Pedt,3}

1 11.028 1 1 1 257.1 338.1 200 –595.2 - -

-2 12.662 1 1 1 665.4 400 200 –765.4 - -

-3 13.957 1 1 1 700 400 200 –200 - -

-4 12.053 1 1 1 513.3 400 200 –513.3 - -

-5 12.335 1 1 1 583.8 400 200 –883.8 - -

-The corresponding unit schedule is infeasible. -The norm(pdif) in (3.31) for this iteration is 1422.951. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^{t(1) 11.028 12.662 13.957 12.053 12.335}

pdif^t –595.2 –765.4 –200 –513.3 –838.8

( ) ( )

pdift

k norm pdif a b+ ◊

–0.380 –0.490 –0.128 –0.328 –0.564

Ȝ^{t(2) 10.648 12.172 13.829 11.725 11.771}

Iteration 3

Hour Ȝ^t u₁^t u^t₂ u₃^t P₁^t P₂^t P₃^{t pdift Pedt,1} Ped^t ,₂ P_ed^t_,3

1 10.648 1 1 1 162 274.7 200 –436.7 - -

-2 12.172 1 1 1 543 400 200 –643 - -

-3 13.829 1 1 1 700 400 200 –200 - -

-4 11.725 0 1 1 0 400 200 0 - -

-5 11.771 0 1 1 0 400 200 –300 - -

-The corresponding unit schedule is infeasible. -The norm(pdif ) in (3.31) for this iteration is 856.962. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^{t(2) 10.648 12.172 13.829 11.725 11.771}

pdif^t –436.7 –643 –200 0 –300

( ) ( )

pdift

k norm pdif a b+ ◊

–0.377 –0.555 –0.173 0 –0.260

Ȝ^{t(3) 10.271 11.617 13.656 11.725 11.511}

Iteration 4

Hour Ȝ^t u₁^t u₂^t u₃^t P₁^{t P}2^t

P₃t ^pdift Ped^t_,1 P_ed^t_,2 P_ed^t_,3

1 10.271 0 1 1 0 211.8 200 –211.8 0 100 100

2 11.617 0 1 1 0 400 200 –100 0 300 200

3 13.656 1 1 1 700 400 200 –200 500 400 200

4 11.725 1 1 1 431.3 400 200 –431.3 140 260 200

5 11.511 0 1 1 0 377.9 200 –300 0 125 175

The total cost calculated from power dispatch is $31,637. The relative duality gap calculated by (3.40) is 0.0323. The norm(pdif ) in (3.31) for this iteration is 609.028. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^{t(3) 10.271 11.617 13.656 11.725 11.511}

pdif^t –211.8 –100 –200 –431.3 –300

( ) ( )

pdift

k norm pdif a b+ ◊

–0.218 –0.103 –0.205 –0.442 –0.308

Ȝ^{t(4) 10.053 11.514 13.451 11.283 11.203}

Iteration 5

Hour Ȝ^t u₁^t u₂^t u₃^{t P1t P2t P}3^t

pdif^{t t}

Ped_,1 P_ed^t _,2 P_ed^t_,3

1 10.053 0 0 1 0 0 200 0 0 0 200

2 11.514 0 1 1 0 400 200 –100 0 300 200

3 13.451 1 1 1 700 400 200 –200 500 400 200

4 11.283 1 1 1 320.7 380.5 200 –301.2 140 260 200

5 11.203 0 1 1 0 367.3 200 –267.3 0 125 175

The total cost calculated from power dispatch is $31,357. The relative duality gap calculated by (3.40) is 0.0185. The norm(pdif ) in (3.31) for this iteration is 460.564. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^{t(4) 10.053 11.514 13.451 11.283 11.203}

pdif^t 0 –100 –200 –301.2 –267.3

( ) ( )

pdift

k norm pdif a b+ ◊

0 –0.117 –0.235 –0.354 –0.313

Ȝ^{t(5) 10.053 11.397 13.216 10.929 10.890}

Example 3.5B

The data is similar to that of Example 3.5A but sub-problems are solved by ODC.

Iteration 1

Hour Ȝ^t u₁^t u₂^t u^t₃ P1^t

P2t

P3t

pdif^t P_ed^t_,1 P_ed^t_,2 P_ed^t_,3

1 11.600 0 1 1 0 400 200 –400 0 100 100

2 13.233 1 1 1 700 400 200 –800 100 200 200

3 14.100 1 1 1 700 400 200 –200 500 400 200

4 12.500 1 1 1 625 400 200 –625 140 260 200

5 13.050 1 1 1 700 400 200 –1000 100 100 100

The total cost calculated from power dispatch is $32,682. The relative duality gap calculated by (3.40) is 0.1815. The norm(pdif ) in (3.31) for this iteration is 1493.52. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^{t(0) 11.600 13.233 14.100 12.500 13.050}

pdif^t –400 –800 –200 –625 –1000

( ) ( )

pdift

k norm pdif a b+ ◊

–0.315 –0.627 –0.16 –0.492 –0.788

Ȝ^{t(1) 11.285 12.603 13.94 12.008 12.262}

Iteration 2

Hour Ȝ^t _u^t

1

u₂t u₃^t P₁^{t P}2^t

P₃t pdif^{t t}

Ped_,1 P_ed^t_,2 P_ed^t_,3

1 11.285 0 1 1 0 381 200 –381 0 100 100

2 12.603 1 1 1 650.8 400 200 –750.8 100 200 200

3 13.942 1 1 1 700 400 200 –200 500 400 200

4 12.007 1 1 1 0 400 200 0 140 100 200

5 12.262 1 1 1 0 400 200 –300 100 100 100

The total cost calculated from power dispatch is $32,682. The relative duality gap calculated by (3.40) is 0.113. The norm(pdif ) in (3.31) for this iteration is 1322.818. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^{t(0) 11.285 12.603 13.94 12.008 12.262}

pdif^t –381 –750.8 –200 0 –300

( ) ( )

pdift

k norm pdif a b+ ◊

–0.378 –0.745 –0.198 0 –0.298

Ȝ^{t(1) 10.907 11.858 13.744 12.007 11.964}

Iteration 3

Hour Ȝ^t u₁^t u₂^t u₃^t P₁^t P₂^t P₃^t pdif^t P_ed^t_,1 P_ed^t_,2 P_ed^t_,3

1 10.907 0 0 1 0 0 200 0 0 0 200

2 11.858 0 1 1 0 400 200 –100 0 300 200

3 13.744 1 1 1 700 400 200 –200 500 400 200

4 12.007 1 1 1 415.7 400 200 –415.7 140 2600 200

5 11.964 0 1 1 0 400 200 –300 0 125 175

The total cost calculated from power dispatch is $31,357. The relative duality gap calculated by (3.40) is 0.032. The norm(pdif ) in (3.31) for this iteration is 559.28. The Lambda Lagrangian multipliers are updated each hour by (3.29) as shown below.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^{t(0) 11.023 12.017 13.805 11.663 11.667}

pdif^t 0 –100 –200 –502 –300

( ) ( )

pdift

k norm pdif a b+ ◊

0 –0.119 –0.237 –0.593 –0.355

Ȝ^{t(1) 11.023 11.955 13.540 11.112 11.270}

Example 3.6

The data is similar to that of Example 3.5A and 3.5B but the spinning reserve requirement is set at 10%. The results are shown as follows.

Hour 1 Hour 2 Hour 3 Hour 4 Hour 5

Ȝ^t(0) 10.00 10.80 12.00 13.40 9.75

ȝ^t(0) 1.075 0.575 0.214 0.704 1.133

No. of iterations

CPU time (s) k^th iteration solution found.

Total cost of best solution Unit sub-problems

solved by DP

100 1.625 33 31,637

Unit sub-problems solved by ODC

100 0.547 32 32,212

3.3.5 Enhanced Augmented Lagrange Augmented Hopfi eld Method

The UC problem to be solved by the enhanced augmented Lagrange augmented Hopfi eld network (ALAHN) method includes objective functions (3.1) subject to the constraints of power balance (3.2), spinning reserve (3.3), generation limit (3.4), and minimum up/down times (3.5). The enhanced ALAHN method for solving UC problems explained in this section consists of three stages. In the fi rst stage, ALAHN is used to commit units to satisfy load demand and spinning reserve neglecting minimum up and down time constraints. In this step, the spinning reserve is handled by constraint neurons. Mathematically, the minimum up/down times of all units are set to one hour. In the second stage, a heuristic search based algorithm is applied to repair the minimum up and down time constraint violations as well as de-commit excessive units based on the unit schedule of the fi rst stage. In the last stage, ALHN is used to solve ED and obtain the fi nal UC schedule. A model of ALAHN is given in Appendix A.

In document artificial-intelligence-in-power-system-optimization.pdf (Page 164-170)