Flow Tracing¶

These examples illustrate the flow tracing methodology used to attribute power flows in a simplified power network. The examples demonstrate how generation from different sources is traced through the network to various loads.

They also show how the result of applying the flow tracing algorithm is used in Wattnet to compute the contribution of each generator to the load at each node.

Basic Example¶

Consider a simple network with four nodes (Z₁, Z₂, Z₃, Z₄). Each node has a generator with a specified capacity (power in watts), and there are power flows between the nodes as shown below:

graph LR
    Z1(⚡ Z<sub>1</sub><br>G<sub>1</sub> = 40000 MW)
    Z2(⚡ Z<sub>2</sub><br>G<sub>2</sub> = 50000 MW)
    Z3(⚡ Z<sub>3</sub><br>G<sub>3</sub> = 30000 MW)
    Z4(⚡ Z<sub>4</sub><br>G<sub>4</sub> = 20000 MW)

    Z1 --> |5000 MW| Z2
    Z2 --> |4400 MW| Z3
    Z3 --> |8600 MW| Z4

From this network state diagram we can extract the following information directly:

• Generation Vector: Represents the generation at each node.

  \(\vec{G} = [40000, 50000, 30000, 20000] \, \text{MW}\)

• Import Vector: Represents the total imports at each node.

  \(\vec{I} = [0, 5000, 4400, 8600] \, \text{MW}\)

• Export Vector: Represents the total exports at each node.

  \(\vec{E} = [5000, 4400, 8600, 0] \, \text{MW}\)

Other relevant vectors can also be computed using these basic vectors and their relationships:

• Total Flow Vector: Represents the total power available at each node before accounting for exports (generation + imports).

  Note

  This has to be before exports when using the upstream-looking flow
  tracing algorithm.
  

  \(\vec{P} = \vec{G} + \vec{I} = [40000, 55000, 34000, 28600] \, \text{MW}\)

• Load Vector: Represents the total load at each node after accounting for exports (generation + imports - exports).

  \(\vec{L} = \vec{G} + \vec{I} - \vec{E} = [35000, 50600, 25800, 28600] \, \text{MW}\)

• Contribution Vector: Represents the fraction of power at each node that is used to meet local load.

  \(\vec{C} = \frac{\vec{L}}{\vec{P}} = [0.875, 0.92, 0.75, 1.0]\)

Flow Tracing Upstream Distribution Matrix¶

The flow tracing upstream distribution matrix \(\mathbf{A_u}\) is constructed to represent the contribution of each generator to the power at each node. Each element \(A_{ij}\) in the matrix represents the fraction of power at node \(i\) that originates from generator \(j\). In this case:

• The columns of the matrix represent the nodes of origin (where the power comes from).
• The rows represent the nodes of destination (where the power flows to).

The \((i,j)\) element of the flow tracing matrix is computed as follows:

So for our example, the upstream distribution matrix \(\mathbf{A_u}\) is:

Inverting this matrix gives the downstream contribution of each node's power to every other node it feeds. The inverse of the upstream distribution matrix \(\mathbf{A_u}^{-1}\) is:

Distribution of Power Using the Upstream-Looking Flow Tracing Algorithm¶

Using the upstream-looking flow tracing algorithm, we can compute the contribution of each generator to the load at each node.

This table shows how the load at each node is met by generation from different sources, illustrating the flow tracing methodology in action. The actual numerical values can be computed by performing the matrix multiplication and applying the contribution factors, as follows:

Distribution and Composition Vectors Defined in Wattnet¶

In Wattnet, the distribution and composition vectors are defined as follows:

• Distribution Vector: Represents the distribution (percentage) of generation from each source to each load. This is essentially reading the table above by columns. The percentage is \(\vec{C}_{i} \cdot \mathbf{A_u}^{-1}_{ij}\). In this case:

  • Z₁ distribution vector: \([0.875, 0.115, 0.0075, 0.0025]\)
  • Z₂ distribution vector: \([0.0, 0.92, 0.06, 0.02]\)
  • Z₃ distribution vector: \([0.0, 0.0, 0.75, 0.25]\)
  • Z₄ distribution vector: \([0.0, 0.0, 0.0, 1.0]\)

• Composition Vector: Represents the composition (percentage) of each load that is met by each generation source. This is essentially reading the table above by rows. The percentage is \(\frac{\vec{C}_{i} \cdot \mathbf{A_u}^{-1}_{ij} \cdot G_j}{L_i}\). In this case:

  • Z₁ composition vector: \([1.0, 0.0, 0.0, 0.0]\)
  • Z₂ composition vector: \([0.0909090909090909, 0.9090909090909091, 0.0, 0.0]\)
  • Z₃ composition vector: \([0.0116279069767442, 0.116279069767442, 0.872093023255814, 0.0]\)
  • Z₄ composition vector: \([0.0034965034965035, 0.034965034965035, 0.262237762237762, 0.6993006993006993]\)

These vectors provide insights into how power generation is distributed across the network and how each load is composed of contributions from different generation sources.

Global Footprint Calculation¶

The global footprint for each zone can be calculated by taking a weighted sum of local footprints, weighted by the composition vector for that zone.

Given the local footprints for each zone as follows:

The global footprint for each zone can be calculated as follows:

• Global Footprint for Z₁:

  \[ \text{Global Footprint}_{Z_1} = 1.0 \cdot 100 + 0.0 \cdot 200 + 0.0 \cdot 300 + 0.0 \cdot 400 = 100 \, \text{gCO2/kWh} \]

• Global Footprint for Z₂:

  \[ \text{Global Footprint}_{Z_2} = 0.0909090909090909 \cdot 100 + 0.9090909090909091 \cdot 200 + 0.0 \cdot 300 + 0.0 \cdot 400 = 190.9090909090909 \, \text{gCO2/kWh} \]

• Global Footprint for Z₃:

  \[ \text{Global Footprint}_{Z_3} = 0.0116279069767442 \cdot 100 + 0.116279069767442 \cdot 200 + 0.872093023255814 \cdot 300 + 0.0 \cdot 400 = 272.093023255814 \, \text{gCO2/kWh} \]

• Global Footprint for Z₄:

  \[ \text{Global Footprint}_{Z_4} = 0.0034965034965035 \cdot 100 + 0.034965034965035 \cdot 200 + 0.262237762237762 \cdot 300 + 0.6993006993006993 \cdot 400 = 361.53846153846155 \, \text{gCO2/kWh} \]