# Error in the direct approach for power flow calculation

In the famous paper [teng2003direct] on direct formulation of power flow calculation for power distribution networks, there is a simplification in converting nodal current injections to line current flows. In short, current loss along lines is ignored. I am wondering if the error introduced by this simplification is significant, which is not discussed in detail in the paper.

We can focus on radial distribution feeders only. There is an example in the paper:

The relationship between the bus current injections and branch currents can be expressed as: $$\left[\begin{array}{l} B_{1} \\ B_{2} \\ B_{3} \\ B_{4} \\ B_{5} \end{array}\right]=\left[\begin{array}{lllll} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} I_{2} \\ I_{3} \\ I_{4} \\ I_{5} \\ I_{6} \end{array}\right]$$ where $$\B\$$ represents complex current flows along lines and $$\I\$$ represents complex current injections at buses.

The relationship should be $$\left[\begin{array}{l} B_{1} \\ B_{2} \\ B_{3} \\ B_{4} \\ B_{5} \end{array}\right]=\left[\begin{array}{lllll} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} I_{2} + I_{\text{loss}, 1} \\ I_{3} + I_{\text{loss}, 2} \\ I_{4} + I_{\text{loss}, 3} \\ I_{5} + I_{\text{loss}, 4} \\ I_{6} + I_{\text{loss}, 5} \end{array}\right]$$ where $$\I_{\text{loss}, k}\$$ represents the current loss along line $$\k\$$.

In a related paper [zad2018new], the problem is mentioned when converting nodal power injections to line power flows. $$\left[\begin{array}{l} P_{12} \\ P_{23} \\ P_{34} \\ P_{45} \\ P_{36} \end{array}\right]=\left[\begin{array}{lllll} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} P_{2} + P_{\text{loss}, 1} \\ P_{3} + P_{\text{loss}, 2} \\ P_{4} + P_{\text{loss}, 3} \\ P_{5} + P_{\text{loss}, 4} \\ P_{6} + P_{\text{loss}, 5} \end{array}\right]$$

• [teng2003direct]: Teng, J. H. (2003). A direct approach for distribution system load flow solutions. IEEE Transactions on power delivery, 18(3), 882-887.
• [zad2018new]: Zad, B. B., Lobry, J., & Vallée, F. (2018). A New Voltage Sensitivity Analysis Method for Medium-Voltage Distribution Systems Incorporating Power Losses Impact. Electric Power Components and Systems, 46(14-15), 1540-1553.
• Yes, it's significant. Sep 18, 2020 at 9:57
• OMG. Probably that is the reason why I cannot make progress in a technique based on power flow results from this method. I should have compared with results from OpenDSS. Sep 18, 2020 at 10:05
• Hi, relayman357. I mean there must be some current loss (power loss) along electric lines, which is not considered in [teng2003direct]. Sep 18, 2020 at 10:35