This document records the convergence characteristics the PowerApps Optimal Power Flow [OPF]. The load flow [LF] and the OPF are carried out sequentially till there is no further improvement in the performance objective or feasibility of the solution. The cycles of LF and OPF are identified as ADC in the tabular reports.

Two objective Functions are considered for illustration. The first one, only the transmission loss minimization is considered. In the second one, combined objective of economic dispatch and loss minimization are considered.

PowerApps solves the economic dispatch and transmission loss minimization as two phase optimization process as explained in the following.

1. Obtain initial feasible load flow results

2. Step 1 gives total generation requirement with transmission losses. Use this information for economic dispatch solution

3. Use results of step 1 for reactive power optimization with transmission loss minimization [or any other suitable objective] to determine control settings for reactive power controls.

4. Implement the solutions of steps 2 and 3 and go to step 1.

5. Repeat steps 1 to 4 till no further controls can be implemented [optimum point is reached] or till specified number of load flow and optimization cycles covered.

The cost function model used by PowerApps for economic dispatch is as

All generators are assumed to have the same cost function model. All generators are assumed to be running. Consequently only the variable costs associated with MW generation is minimized.d with MW generation is minimized.

2 Notation

The tabular convergence characteristics report for various system has the following headings and corresponding meaning.

Column Heading	Description of the content in tabular column
ADC	Load Flow Cycle Number, 0 is the base case load flow, 1 ,2 etc are after Optimizing the Controllers at the end of previous load flows 0,1 etc
PG Total[MW]	Total MW generation at the end of load flow
PL Total[MW]	Total MW Load
P Loss [MW]	Total MW Loss [ The objective used was minimization of transmission losses, this should be seen as improving]
VMAX [pu]	Maximum bus voltage at a bus in per unit as identified by load flow solution
VMIN [pu]	Minimum bus voltage at a bus in per unit as identified by load flow solution
Sv [pu]	Absolute sum of all over and all under voltage violations in per unit. Reduction in this value implies improved voltage profile.
Nv	Number of bus voltage violations. Reduction in this value implies improved voltage profile.
Qcom[MVAR]	Total MVAR compensation
VariableCost	Variable Cost of Generation [in any specified currency units, depending on cost model]

3 System Data Summary and OPF convergence Characteristics

3.1 6 bus Ward – Hale System

NUMBER OF GENERATOR CONTROLS = 2

NUMBER OF TRANSFORMER CONTROLS = 2

NUMBER OF CAPACITOR CONTROLS = 2

TOTAL NUMBER OF CONTROLS = 6

3.1.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	146.612327	135	11.612327	1.1	0.855219	0.044781	1	0	3693.92537
1	144.307479	135	9.307479	1.1	0.955202	0	0	10.5	3639.027584
2	144.319266	135	9.319266	1.1	0.954126	0	0	10.5	3639.304769

3.1.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total[MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	146.6123	135	11.61233	1.1	0.855219	0.044781	1	0	3693.925
1	143.9538	135	8.953771	1.1	0.955852	0	0	10.5	3599.887
2	143.7244	135	8.724368	1.1	0.955202	0.000013	1	10.5	3571.063

Notes:

1. Variable costs are minimized with economic dispatch

2. With economic dispatch losses also have reduced further.

3.2 24 Bus, Indian Test System

NUMBER OF GENERATOR CONTROLS = 4

NUMBER OF TRANSFORMER CONTROLS = 7

NUMBER OF CAPACITOR CONTROLS = 4

TOTAL NUMBER OF CONTROLS = 15

3.2.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	2680.472	2620	60.47208	1	0.867126	0.656766	15	0	125740.6
1	2671.757	2620	51.75726	1.05	0.948671	0.001329	1	75	124992.5
2	2671.661	2620	51.66116	1.05	0.948813	0.001187	1	75	124984.2

3.2.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	2680.472	2620	60.47208	1	0.867126	0.656766	15	0	125740.6
1	2668.741	2620	48.74097	1.05	0.951578	0	0	75	118690.2
2	2668.759	2620	48.75948	1.05	0.951667	0	0	75	118691

Notes:

1. Variable costs are minimized with economic dispatch

2. With economic dispatch losses also have reduced further.

3.3 89 Bus, Indian Test System

NUMBER OF GENERATOR CONTROLS = 12

NUMBER OF TRANSFORMER CONTROLS = 23

NUMBER OF CAPACITOR CONTROLS = 21

TOTAL NUMBER OF CONTROLS = 56

3.3.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	2203.259	2049.87	153.3887	1.010935	0.749948	5.525923	51	0	54213.98
1	2169.692	2049.87	119.8221	1.05	0.923664	0.065156	6	266.904	53379.55
2	2164.465	2049.87	114.5953	1.05	0.951416	0	0	375.1516	53253.67

3.3.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	2203.259	2049.87	153.3887	1.010935	0.749948	5.525923	51	0	54213.98
1	2165.328	2049.87	115.4578	1.05	0.926642	0.049282	4	266.904	52731.02
2	2158.131	2049.87	108.2615	1.051578	0.954055	0.002966	4	364.3801	52280.68
3	2157.065	2049.87	107.1954	1.054449	0.94652	0.012911	8	364.3801	52210.73
4	2155.865	2049.87	105.9951	1.071999	0.952598	0.03088	8	320.4141	52175.45
5	2154.255	2049.87	104.3852	1.074034	0.946683	0.03885	7	302.552	52124.63
6	2153.81	2049.87	103.9396	1.074893	0.958409	0.048713	12	317.6971	52107.12
7	2153.727	2049.87	103.8568	1.056282	0.958623	0.021778	9	317.6971	52099.53
8	2153.652	2049.87	103.7817	1.057435	0.959246	0.020234	9	317.6971	52097.58
9	2153.562	2049.87	103.6917	1.074952	0.95944	0.045557	10	317.6971	52095.26

Notes:

1. Variable costs are minimized with economic dispatch

2. With economic dispatch losses also have reduced further

3. The algorithm went on for full specified 10 number of load flow and optimization cycles with marginal improvements in the latter cycles..

3.4 125 Bus, Indian Test System [KPTCL]

NUMBER OF GENERATOR CONTROLS = 26

NUMBER OF TRANSFORMER CONTROLS = 25

NUMBER OF CAPACITOR CONTROLS = 18

TOTAL NUMBER OF CONTROLS = 69

3.4.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	4833.519	4643.89	189.6287	1.03	0.740645	3.027959	61	0	139108.1
1	4795.075	4643.89	151.1846	1.05	0.927916	0.127677	10	275	137096.3
2	4789.937	4643.89	146.0468	1.058903	0.94073	0.034106	7	290.5083	136831.9
3	4789.367	4643.89	145.4768	1.058903	0.941461	0.031383	10	295.5083	136802.6
4	4789.521	4643.89	145.6313	1.058903	0.941266	0.032697	9	295.5083	136810.2

3.4.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	4833.519	4643.89	189.6287	1.03	0.740645	3.027959	61	0	139108.1
1	4788.748	4643.89	144.8576	1.05	0.929997	0.102979	9	275	131047.6
2	4785.032	4643.89	141.1423	1.068482	0.942649	0.063537	12	280	130853.6
3	4784.729	4643.89	140.8385	1.068482	0.945397	0.065103	13	280	130857.6
4	4784.716	4643.89	140.8263	1.068482	0.945329	0.063751	13	280	130839.3
5	4784.338	4643.89	140.4477	1.068482	0.945977	0.060669	14	280	130822.6

Notes:

1. Variable costs are minimized with economic dispatch

2. With economic dispatch losses also have reduced further

3.5 319 Bus, Indian System [KPTCL]

NUMBER OF GENERATOR CONTROLS = 22

NUMBER OF TRANSFORMER CONTROLS = 77

NUMBER OF CAPACITOR CONTROLS = 47

TOTAL NUMBER OF CONTROLS = 146

3.5.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	4563.446	4275	288.446	1	0.637032	21.68275	245	0	172591.8
1	4496.877	4275	221.8772	1.05	0.933239	0.039275	5	296.9124	166570.9
2	4492.448	4275	217.4478	1.05	0.950973	0	0	370.7209	166176.5
3	4492.045	4275	217.0445	1.05	0.952069	0	0	394.4941	166140.7

3.5.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	4563.446	4275	288.446	1	0.637032	21.68275	245	0	172591.8
1	4483.542	4275	208.542	1.05	0.93653	0.023548	3	296.9124	151933.7
2	4480.509	4275	205.5087	1.052389	0.951147	0.002389	1	355.0025	151692.3
3	4480.031	4275	205.0313	1.052895	0.952047	0.002895	1	380.7686	151657.1
4	4480.096	4275	205.0962	1.052876	0.949652	0.003224	2	380.7686	151689.3

Notes:

1. Variable costs are minimized with economic dispatch

2. With economic dispatch losses also have reduced further

3.6 1043 Bus, Indian Southern Grid

NUMBER OF GENERATOR CONTROLS = 66

NUMBER OF TRANSFORMER CONTROLS = 138

NUMBER OF CAPACITOR CONTROLS = 590

TOTAL NUMBER OF CONTROLS = 794

3.6.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	13222.2	12545.15	677.0469	1.021436	0.553079	79.19027	829	0	415815.1
1	13222.29	12545.15	677.1386	1.05	0.701053	11.32251	267	5237.901	433129.7
2	13173.37	12545.15	628.2183	1.05435	0.759467	2.142998	85	5706.414	431965.5
3	13168.13	12545.15	622.9804	1.088152	0.766347	1.755495	68	5776.631	431846.7
4	13167.27	12545.15	622.1214	1.086513	0.76956	1.715472	68	5776.631	431827.3
5	13167.53	12545.15	622.3756	1.086294	0.769564	1.695118	64	5776.631	431833.1

Notes:

1. The initial load flow corresponds to infeasible [not converged] load flow and hence its variable cost is not valid. Load flow solution at the point of minimum mismatch was used for optimization solution.

2. Losses and Variable Cost however, steadily improve from converged load flow solution of ADC 1 and above.

3.6.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	13222.2	12545.15	677.0469	1.021436	0.553079	79.19027	829	0	415815.1
1	13234.75	12545.15	689.5994	1.050258	0.702001	11.97724	276	5237.901	416831.6
2	13186.04	12545.15	640.8946	1.050674	0.758889	2.236873	86	5671.229	416152.7
3	13179.77	12545.15	634.6243	1.051316	0.7699	1.72662	63	5642.4	414490.6
4	13178.79	12545.15	633.6431	1.053028	0.772968	1.707763	62	5642.4	414252.3
5	13178.72	12545.15	633.5735	1.052937	0.772805	1.709201	62	5642.4	414220.1
6	13178.68	12545.15	633.5295	1.052962	0.77285	1.707772	63	5643.69	414219.2
7	13178.73	12545.15	633.584	1.052942	0.772815	1.707775	61	5642.09	414220.5
8	13178.7	12545.15	633.5527	1.052969	0.772863	1.705596	63	5643.803	414219.7

Notes:

2. Losses and Variable Cost however, steadily improve from converged load flow solution of ADC 1 and above.

3. Variable Costs Improve better with this objective compared to plain loss minimization objective.

4. Over voltages are controlled better with this combined objective compared to plain loss minimization objective.

3.7 Gujarat Electricity Board Data [1991-94]

NUMBER OF GENERATOR CONTROLS = 20

NUMBER OF TRANSFORMER CONTROLS = 23

NUMBER OF CAPACITOR CONTROLS = 11

TOTAL NUMBER OF CONTROLS = 54

3.7.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	4047.642	3933	114.6419	1.05	0.713986	3.145116	54	0	107107
1	4023.298	3933	90.29814	1.05	0.944865	0.00914	2	260	106210.6
2	4023.792	3933	90.7917	1.05	0.971693	0	0	280	106228.5
3	4023.614	3933	90.61445	1.05	0.969762	0	0	280	106222.1
4	4023.245	3933	90.24471	1.050238	0.973029	0.000238	1	280	106208.5

3.7.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	4047.642	3933	114.6419	1.05	0.713986	3.145116	54	0	107107
1	4018.345	3933	85.3453	1.05	0.947386	0.004102	2	260	104144.6
2	4019.783	3933	86.78281	1.05	0.973307	0	0	280	104185.7
3	4019.389	3933	86.38881	1.05	0.972943	0	0	280	104165.9
4	4019.641	3933	86.64112	1.05	0.970029	0	0	280	104180.1
5	4019.269	3933	86.26866	1.05	0.97416	0	0	280	104163.7

Notes:

1. Variable costs are minimized with economic dispatch

2. With economic dispatch losses also have reduced further

3.8 Orissa State Electricity Board System [Around 1995]

NUMBER OF GENERATOR CONTROLS = 14

NUMBER OF TRANSFORMER CONTROLS = 19

NUMBER OF CAPACITOR CONTROLS = 13

TOTAL NUMBER OF CONTROLS = 46

3.8.1 Loss Minimization Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	3955.887	3770	185.8874	1.04	0.654134	8.09204	82	0	119355.2
1	3922.079	3770	152.0789	1.05	0.908798	0.298497	25	185.1513	117368.7
2	3916.141	3770	146.141	1.05	0.925086	0.061712	8	185.1513	117024.5
3	3915.149	3770	145.1491	1.05	0.928355	0.038809	7	193	116967.1

3.8.2 Loss Minimization and Economic Dispatch Objective

ADC	PG Total [MW]	PL Total [MW]	P Loss [MW]	VMAX [pu]	VMIN [pu]	Sv [pu]	Nv	Qcom [MVAR]	Variable Cost
0	3955.887	3770	185.8874	1.04	0.654134	8.09204	82	0	119355.2
1	3933.614	3770	163.6135	1.05	0.903117	0.415851	31	185.1513	112740.3
2	3928.594	3770	158.5938	1.05	0.921461	0.134894	14	193	112511
3	3925.758	3770	155.758	1.05	0.927761	0.026561	4	211.2418	112379.5

Notes:

1. Variable costs are minimized with economic dispatch

2. With economic dispatch losses also have NOT reduced further

4 Conclusion

The results for 8 systems starting from 6 bus test system to 1043 bus practical Indian Grid system, indicates the robustness and dependability of the PowerApps OPF algorithm

In most cases as observed by this author , the combined objective of economic dispatch and loss minimization was more effective, as compared to only transmission loss minimization objective.

Optimal Power Flow Algorithm

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Mail your questions, enquiries to raghunatha@powerapps.org