Quantity
What residual is being tested?
The criterion may use active- and reactive-power mismatches, current mismatches, voltage corrections, or another residual. Those are not interchangeable.
Direct answer
A tolerance value without a definition is incomplete
A statement such as “the case converged to 0.001” is not fully interpretable until the reader knows what was measured, how the residuals were combined, whether the values are in per unit or physical units, and which equations remained active at the final iteration.
What must be defined
Norm
How are residuals combined?
A maximum absolute mismatch, Euclidean norm, sum, weighted norm, or separate active- and reactive-power tests can produce different stopping behavior.
Units
Per unit or physical quantities?
A numerical threshold in per unit depends on the selected base. A threshold in MW or MVAr carries a different interpretation and may scale differently with system size.
Scaling
Are equations normalized or weighted?
Software may internally scale rows, variables, or residuals to improve numerical conditioning. The reported tolerance may therefore differ from the raw physical mismatch.
Equation set
Which controls are active?
The mismatch vector changes when a PV bus reaches a reactive-power limit and becomes PQ, or when other controls and limits enter or leave the formulation.
Logic
What else stops the solver?
Iteration limits, divergence checks, step damping, voltage-correction limits, control-loop completion, and multiple nested tolerances may all affect the final status.
Why the same number can mean different things
Case A
A solver reports the maximum per-unit active- or reactive-power mismatch over all retained equations. A tolerance of 0.001 means every retained mismatch is below that per-unit threshold.
Case B
Another solver reports an aggregate norm of scaled residuals. The same displayed value of 0.001 does not imply the same largest bus mismatch or the same physical error.
The numbers look identical. The stopping tests are not.
Should the tolerance be made as small as possible?
No automatic benefit
Tighter is not always better
A very tight threshold can add iterations without materially changing the engineering conclusion, especially when model uncertainty is much larger than the remaining numerical mismatch.
Numerical limits
Conditioning matters
Near-singular Jacobians, poor scaling, discontinuous control actions, or finite precision may prevent further meaningful residual reduction.
Study purpose
Required precision depends on use
A screening study, planning case, contingency analysis, and benchmark calculation may justify different numerical requirements and verification steps.
What convergence does establish
Numerical statement
The selected equations were satisfied to the selected stopping rule
That is the defensible claim. It should be accompanied by the residual definition, norm, units or base, active control state, and software status.
What convergence does not establish
The model is correct
A solver can converge on incorrect topology, parameters, load representation, generator data, or control assumptions.
All limits are satisfied
Voltage, thermal, reactive-power, interchange, or other operating limits may still be violated.
The result is usable
The final operating point may be numerically closed but inconsistent with the study objective, scenario definition, or decision standard.
A better reporting checklist
1
State the residual
Identify whether the solver tests power mismatch, current mismatch, state correction, or another quantity.
2
State the norm and units
Report the maximum, aggregate, weighted, or separate criterion and define the base or physical units.
3
State the final controls
Document PV-to-PQ switching, limit enforcement, distributed slack, tap or shunt actions, and other relevant control states.
4
Reconcile the solution
Check generation, load, losses, interchange, and any residual balancing quantity.
5
Check operating limits
Review voltages, branch loading, reactive margins, and study-specific constraints separately from convergence.
6
Compare material outputs
Confirm that a tighter tolerance or independent run does not materially alter the quantities used for the engineering decision.
Decision rule
Do not compare tolerances until you compare definitions
When two software packages use different mismatch equations, norms, scaling, or control logic, their tolerance settings cannot be compared by number alone. Compare the actual residuals, final state, control status, and engineering outputs.
From convergence to acceptance
Newton-Raphson Power Flow Readiness Kit · $9.99
The kit develops mismatch construction, convergence interpretation, debugging, reconciliation, control limits, and the boundary between a solved equation set and a defensible engineering result.