# Flow decomposition

## Introduction

Flow decomposition methodology has been formerly described in a decision published by ACER (European Union Agency for the Cooperation of Energy Regulators).

European power system is based on zonal management. Flow decomposition is a tool designed to give insights on impacts of internal and cross zonal exchange of power on the flows (and associated constraints) on the network devices. It is an important part of the cost sharing methodolgy for remedial actions costs sharing between TSOs.

The aim of flow decomposition algorithm is to provide for each network element a decomposition of the active flow into different parts:

• Allocated flow: flow due to electricity market exchanges. This includes import/export flows and transit flows.
• Internal flow: flow due to electricity exchange inside the network element’s zone.
• Loop flow: flow due to electricity exchange inside another zone.
• PST flow: flow due to a shift commanded by the action of an active phase shifting transformer on the network.
• HVDC flow: flow due to a shift commanded by the action of an active HVDC line on the network.

Current algorithm does not model HVDC flow yet.

This decomposition does not reflect the exact reality of how electric flows act on a real network, but it is a useful approximation needed for some cross zonal coordination processes.

## Algorithm description

The flow decomposition algorithm is based on the DC approximation, in which the losses in the network branches are neglected, and that allows to rely on the superposition principle to assess which is the impact of any injection on any branch flow by simple sensitivity analysis.

Below is the concrete description of the algorithm implemented in PowSyBl.

### Net positions computation

Countries’ net position computation is done using AC loadflow in the initial network, before any other alteration of the input.

The net position of a country is calculated as the sum of :

• The injection of all dangling lines connected in the country
• The mean leaving flow of all AC and HVDC line interconnections (losses are shared equally between both countries)

### Losses compensation

In order to mitigate the impact of DC approximation in the flow decomposition process, a dedicated step of losses compensation is implemented.

Instead of using standard power flow compensation methodology, a standard full AC power flow is run on the input network that allows to calculate the losses on each network element.

These losses are then compensated on the sending side of each network element.

A special treatment is done on tie lines, where instead of compensating the losses on the sending terminal, losses are compensated at both sides proportionally to the resistance of each half line.

### Nodal Injections partitioning

In order to distinguish internal/loop flows and allocated flows, the nodal injections in each zone must de decomposed in two parts:

• Nodal injections for allocated flows
• Nodal injections for loop flows and internal flows

This decomposition is based on GLSK (Generation and Load Shift Keys). It is an input of the process that provides, for each zone of the study a list of injections and associated factor to be used to scale the zone to a given net position.

By default, the algorithm uses so-called “Country GSK”, which is an automatic GLSK that scales on all generators proportionally to their target power setpoint.

Nodal injection decomposition is done as follows:

$\begin{array}{l} \mathrm{NI}_\mathrm{AF} = \mathrm{GLSK} \cdot \mathrm{NP} \\ \mathrm{NI}_\mathrm{LIF} = \mathrm{NI} - \mathrm{NI}_\mathrm{AF} \end{array}$

where:

• $$\mathrm{NI}$$ is the vector of the network injections,
• $$\mathrm{NI}_\mathrm{AF}$$ is the vector of allocated flow part of the network injections,
• $$\mathrm{NI}_\mathrm{LIF}$$ is the vector of loop flow and internal flow part of the network injections,
• $$\mathrm{NP}$$ is the vector of the zones’ net position,
• $$\mathrm{GLSK}$$ is the matrix of the GLSK factors for each injection in each zone,

### Sensitivity analysis

In order to assess the linear impact (implied by the DC approximation) of each nodal injection and phase shift transformer on the network elements’ flow, a sensitivity analysis is run.

The following matrices are calculated using sensitivity analysis API:

• $$\mathrm{PTDF}$$ is the matrix of the sensitivity of the network element flow to each network injection shift,
• $$\mathrm{PSDF}$$ is the matrix of the sensitivity of the network element flow to each phase shift transformer tap angle change,

### Flow partitioning

Based on previously calculated elements, flow partitioning can now be calculated as follows:

$\begin{array}{l} \mathrm{F}_\mathrm{AF} = \mathrm{PTDF} \cdot \mathrm{NI}_\mathrm{AF} \\ \mathrm{F}_\mathrm{LIF} = \mathrm{PTDF} \cdot \mathrm{diag}(\mathrm{NI}_\mathrm{LIF}) \cdot \mathrm{AM} \\ \mathrm{F}_\mathrm{PST} = \mathrm{PSDF} \cdot \mathrm{\Delta}_\mathrm{PST} \end{array}$

where:

• $$\mathrm{F}_\mathrm{AF}$$ is the vector of the network element allocated flow,
• $$\mathrm{F}_\mathrm{LIF}$$ is the matrix of the network element loop flow or internal flow for each zone,
• $$\mathrm{F}_\mathrm{PST}$$ is the vector of the network element PST (phase shift transformer) flow,
• $$\mathrm{AM}$$ is the allocation matrix, which associates each injection to its zone. $$\mathrm{AM}_{ij}$$ = 1 if node i is in zone j, 0 otherwise,
• $$\mathrm{\Delta}_\mathrm{PST}$$ is the phase shift transformers angle vector,

### Flow parts rescaling

Due to superposition principle, the sum of all the flow parts calculated previously is equal to the flow that was calculated by the DC power flow.

However, the flow reference is the one calculated using AC power flow which is different. The final step of the algorithm is though to rescale the different flow parts in order to ensure that the sum of the parts is equal to the initially calculated AC flow.

The difference between reference AC flow and the sum of the parts of the decomposition is redispatched on the different parts proportionally to their rectified linear unit ($$\mathrm{ReLU}(x) = \mathrm{max}(x, 0)$$).

## Flow decomposition inputs

### Network

The first input of the flow decomposition algorithm is a network. As this simulation uses power flow simulations for losses compensation, this network should converge.

### GLSK

The second input of the flow decomposition algorithm are the GLSK to be used for each zone.

Current implementation of the algorithm only allows automatic generation of Country GSK for each country of the network.

### Network elements

The third input of the flow decomposition algorithm are the network elements of interest which flow is to be decomposed into the parts listed in introduction - called XNEC (cross-border relevant network element with contingency) in the flow decomposition methodology.

Current implementation of the algorithm is based on a XnecProvider interface. This interface should provide XNECs in the base case. Basic implementations of this interface are available:

• Set of all branches.
• Set of branches selected by IDs.
• Set of all interconnections on the network (i.e. branches which have different country attribute in their source and destination substation).
• Set of all interconnections on the network with the addition of all branches that have a maximum zonal PTDF greater than 5%.

## Flow decomposition outputs

### Network element parts

For each network element of interest, contains the following elements:

• Reference flow : active power flow that is considered as the reference for the decomposition. It is actually equal to the sum of all the flow parts calculated by the algorithm.
• Allocated flow : allocated flow part of the network element’s flow.
• Internal flow : internal flow part of the network element’s flow. It is calculated as the loop flow from the country which network element is part of (interconnections are considered as part of no specific country, so will always have an internal flow to 0).
• Loop flows : map of the loop flow part of the network element’s flow for each zone.
• PST flow : PST flow part of the network element’s flow.

## Configuration

### Dedicated parameters

Name Type Default value Description
enable-losses-compensation boolean false When set to true, adds losses compensation step of the algorithm. Otherwise, all losses will be compensated using chosen power flow compensation strategy.
losses-compensation-epsilon double 1e-5 Threshold used in losses compensation step of the algorihm. If actual losses are below the given threshold on a branch, no injection is created in the network to compensate these losses. Used to avoid creating too many injections in the network. May have an impact in overall algorithm performance and memory usage.
sensitivity-epsilon double 1e-5 Threshold used when filling PTDF and PSDF matrices. If a sensitivity is below the given threshold, it is set to zero. Used to keep sparse matrices in the algorithm. May have an impact in overall algorithm performance and memory usage.
rescale-enabled boolean false When set to true, rescaling step is done to ensure that the sum of all flow parts is equal to the AC reference flow.
dc-fallback-enabled-after-ac-divergence boolean true Defines the fallback behavior after an AC divergence Use True to run DC loadflow if an AC loadflow diverges (default). Use False to throw an exception if an AC loadflow diverges.
sensitivity-variable-batch-size int 15000 When set to a lower value, this parameter will reduce memory usage, but it might increase computation time

### Impact of existing parameters

Flow decomposition algorithm relies on load flow parameters and sensitivity analysis parameters.