Generation Dispatch and Transmission Capacity Assignment

In previous pages, we explained how the grid topology can be generated, bus types assigned, and generation/load capacities set. In this page, we explain how the generation dispatch and transmission line capacity limits are determined for a synthetic grid model, based on [Sadeghian et al., 2018].

As noted in [Sadeghian et al., 2018], a valid synthetic grid model requires at least three components:

the electrical grid topology;
the generation and load settings (correlated placement and sizing);
the transmission constraints (capacity limits of transmission lines and transformers).

This page addresses component (c), together with the generation dispatch needed to compute power flows. The methods are implemented in generation_dispatcher.py, transmission.py, and dcpf.py.

System model

Consider a power grid with $N$ buses and $M$ branches (transmission lines and transformers). The admittance matrix $\mathbf{Y}_{N \times N}$ is

\[\mathbf{Y}_{N \times N} = \mathbf{A}^\top \Lambda^{-1}(z_l) \mathbf{A}\]

where $\mathbf{A}$ is the branch–node incidence matrix, $\Lambda^{-1}(\cdot)$ the diagonal inverse matrix, and $z_l$ the vector of branch impedances (per-unit).

By neglecting power losses (the DC power flow approximation), the power balance and flow distribution follow

\[\begin{split}\mathbf{P}(t) &= \mathbf{B}'(t)\,\boldsymbol{\theta}(t) \\ \mathbf{F}(t) &= \Lambda(y_l)\,\mathbf{A}\,\boldsymbol{\theta}(t)\end{split}\]

where $\boldsymbol{\theta}(t)$ is the vector of bus voltage phase angles, $\mathbf{P}(t)$ the vector of net real power injections, $\mathbf{B}'$ the susceptance matrix, and $\mathbf{F}(t)$ the vector of branch real-power flows.

Grid operations must also satisfy:

\[P_g^{\min} \leq P_g \leq P_g^{\max}, \qquad P_L^{\min} \leq P_L \leq P_L^{\max}, \qquad |F_l| \leq F_l^{\max}\]

Two key normalized parameters are introduced:

Dispatch factor of generator $i$:

\[\alpha_i = \frac{P_{g_i}}{P_{g_i}^{\max}}, \qquad i = 1, \ldots, N_G\]

where $\alpha_i = 0$ means uncommitted and $\alpha_i = 1$ means fully committed.
Transmission gauge ratio of branch $l$:

\[\beta_l = \frac{F_l}{F_l^{\max}}, \qquad l = 1, \ldots, M\]

where $\beta_l \in (0,1]$ under normal operation, and $\beta_l > 1$ indicates overloaded (emergency) operation.

Statistics of generation dispatch

Correlation between capacity and dispatch

Statistical analysis of real-world grids (NYISO-2935, WECC-16994, PEGASE-13659, ERCOT) reveals a non-trivial correlation between generation capacity $P_{g_n}^{\max}$ and actual dispatch $P_{g_n}$, with Pearson coefficient

\[\rho(P_{g_n}^{\max}, P_{g_n}) \in [0.75, 0.95]\]

Three categories of generators

All generators in a grid can be divided into three categories:

(A) Uncommitted units ($\alpha_g = 0$): about 10–20% of total generators. These tend to be small or medium-size units. Reasons include market requirements, annual overhaul, or low load levels.
(B) Partially committed units ($0 < \alpha_g < 1$): about 40–50% of total. Their output power varies between minimum and maximum generation capacity.
(C) Fully committed units ($\alpha_g \approx 1$): the remaining generators operate very close to their maximum capacity.

Normalized capacity and dispatch factor

The normalized generation capacity is defined as

\[\bar{P}_{g_n}^{\max} = \frac{P_{g_n}^{\max}}{\max_i P_{g_i}^{\max}} \in [0, 1]\]

and there exists a significant correlation between $\bar{P}_{g_n}^{\max}$ and $\alpha_n$ with Pearson coefficient

\[\rho(\bar{P}_{g_n}^{\max}, \alpha_n) \in [0.15, 0.55]\]

Small and mid-size units tend to have a wider range of dispatch factor compared with larger units. A small number of units may have negative dispatch factor (e.g., pumped-storage hydro acting as load).

2-D empirical PMF

A joint probability density $f(\bar{P}_{g_n}^{\max}, \alpha_n)$ is estimated from real grid data and discretized into a 2-D probability table Tab_2D_Pg of size $14 \times 10$ (14 capacity bins $\times$ 10 dispatch-factor bins). This table captures the observed correlation structure. Its purpose is to ensure that the synthetic grid reproduces a similar correlation between generation dispatch and capacity as found in real systems.

Distribution of committed units

For committed units (category B), the analysis shows:

More than 99% of committed units have capacities following an exponential distribution with parameter $\mu_{\text{committed}}$.
About 1% have extremely large capacities falling outside the normal exponential range.
The dispatch factor distribution depends on the loading level of the system $\alpha^\Sigma = \sum_{i=1}^{N_G} P_{g_i} / \sum_{i=1}^{N_G} P_{g_i}^{\max}$. For high loading levels (e.g., NYISO: $\alpha^\Sigma = 0.74$), it tends toward a generalized extreme value distribution; for low loading levels (e.g., PEGASE: $\alpha^\Sigma = 0.38$), it approaches a uniform distribution.

Algorithm for generation dispatch

The generation dispatch algorithm assigns a dispatch factor $\alpha_i$ to each generator bus, producing the active power output $P_{g_i} = \alpha_i \cdot P_{g_i}^{\max}$. The algorithm proceeds in four stages.

Stage 1: Uncommitted units ($\alpha = 0$)

Select 10–20% of generators as uncommitted. The target capacities are drawn from Uniform$[0, 0.6]$, and units whose normalized capacity is closest to each target are selected. These units receive $\alpha = 0$ (zero dispatch).

Stage 2: Committed units ($0 < \alpha < 1$)

From the remaining units, select 40–50% as partially committed:

Capacity selection: 99% of selected units follow an exponential distribution with parameter $\mu_{\text{committed}}$, and 1% are drawn from the extreme tail (Uniform$[0.5, 1.0]$).
Dispatch factor generation: Generate a random set of $\alpha$ values. When the loading level flag alpha_mod = 0, all values are Uniform$[0, 1]$. Otherwise, 99.5% are Uniform$[0, 1]$ and 0.5% are negative (representing pumped-storage or similar units).
2-D bin-matching assignment: Assign dispatch factors to units using the Tab_2D_Pg table. The table is scaled to integer counts matching the number of committed units. Units are sorted by normalized capacity into 14 bins; alphas are sorted into 10 bins. The assignment loops from high capacity to low, randomly pairing units with alpha values from the corresponding bins. This reproduces the observed 2-D correlation between capacity and dispatch factor.

Stage 3: Fully committed units ($\alpha = 1$)

All remaining generators are assigned dispatch factor $\alpha = 1$.

Stage 4: Balancing

The total generation must match total load (within a tolerance of 1%). An iterative loop adjusts the dispatch:

Excess generation: scale down committed alphas, or turn off fully committed and uncommitted units.
Generation deficit: scale up committed alphas (capped at 1.0), or turn on uncommitted/full-load units.

This guarantees a feasible power balance for the subsequent DC power flow.

DC power flow solver

After generation dispatch, a DC power flow (DCPF) is solved to determine the flow distribution across all branches. The DCPF implementation in dcpf.py proceeds as follows:

Susceptance matrix $\mathbf{B}$: For each branch $(i,j)$ with reactance $x_{ij}$, the susceptance is $b_{ij} = 1/x_{ij}$. Off-diagonals: $B_{ij} = -b_{ij}$; diagonals: $B_{ii} = \sum_j b_{ij}$.
Power injection vector $\mathbf{P}$: Net injection at bus $i$ is $P_i = (P_{g_i} - P_{L_i}) / S_{\text{base}}$, where $S_{\text{base}} = 100$ MVA.
Slack bus selection: The bus with the largest generator is set as the slack (reference angle $\theta_{\text{slack}} = 0$).
Solve: Remove the slack bus row/column from $\mathbf{B}$ and $\mathbf{P}$, then solve the reduced system

\[\mathbf{B}_{\text{red}} \, \boldsymbol{\theta}_{\text{red}} = \mathbf{P}_{\text{red}}\]

using a sparse direct solver.
Branch flows: For each branch $(i,j)$ with reactance $x_{ij}$:

\[F_{ij} = \frac{\theta_i - \theta_j}{x_{ij}} \times S_{\text{base}} \quad \text{(MW)}\]

Branch impedance generation

Before the DCPF can be run, each branch needs an impedance $z_l = r_l + jx_l$. The impedance assignment follows the SynGrid MATLAB toolbox (sg_line.m).

Impedance magnitude (LogNormal)

The per-unit impedance magnitude $Z$ is drawn from a LogNormal distribution:

\[Z \sim \text{LogNormal}(\mu_Z, \sigma_Z)\]

with parameters $\mu_Z$ and $\sigma_Z$ extracted from reference system data (e.g., $\mu_Z = -2.38$, $\sigma_Z = 1.99$ for NYISO). Values are clipped to $[0.001, 0.5]$ p.u. for numerical stability.

Line angle via Lévy stable distribution

The impedance angle $\phi$ (determining the X/R ratio) is generated from a Lévy $\alpha$-stable distribution:

\[\phi \sim S(\alpha_s, \beta_s, \gamma_s, \delta_s)\]

with parameters $(\alpha_s, \beta_s, \gamma_s, \delta_s)$ from reference data (e.g., $(1.374, -0.838, 2.965, 85.801)$ for NYISO). The angle is clipped to $[0.01°, 89.99°]$, then the reactance and resistance are computed as:

\[\begin{split}x_l &= Z_l \sin(\phi_l) \\ r_l &= Z_l \cos(\phi_l)\end{split}\]

DCPF-based impedance swapping

To ensure that the impedance assignment is consistent with realistic flow patterns, the following iterative procedure is applied:

Run DCPF to obtain flow magnitudes $|F_l|$.
Sort impedance magnitudes $Z$ in ascending order; sort flow indices in descending order of $|F_l|$.
Assign low-$Z$ to high-flow lines: lines carrying more power should have lower impedance (shorter/thicker conductors). This is the physical principle that lower impedance paths carry more current.
Random perturbation: to avoid an overly deterministic assignment, random swaps are performed among neighboring entries. The swap rate and neighborhood size depend on grid size:
- For $N > 1200$: swap rate $a_s = 0.3$, neighborhood $a_n = 0.8$
- Otherwise: $a_s = 0.2$, $a_n = 0.2$
Reassign $Z_l$, $x_l$, $r_l$ to each branch using the original line angle $\phi_l$.

This process may be iterated (2 times for $N \geq 300$, once otherwise).

Transmission capacity statistics and assignment

Scaling property

The aggregate transmission capacity in a grid follows a scaling law with network size:

\[\log_{10} F_l^{\text{tot}}(N) = 1.03 \log_{10} N + 2.52\]

where $F_l^{\text{tot}} = \sum_{m=1}^{M} F_{l_m}^{\max}$ is the total transmission capacity.

Exponential distribution of gauge ratio

The transmission gauge ratio $\beta_l = F_l / F_l^{\max}$ follows an exponential distribution with parameter $\mu_\beta$ (e.g., $\mu_\beta \approx 0.27$ for NYISO, $0.28$ for WECC):

\[\beta_l \sim \text{Exp}(\mu_\beta)\]

Values are clipped to $(0, 1]$ for normal operation. A small fraction (about 0.08–0.11%) of lines are designated as overloaded with $\beta_l \in (1.0, 1.2]$, representing short-term or emergency ratings.

Correlation between gauge ratio and flow

There exists a considerable correlation between the gauge ratio $\beta_l$ and the normalized flow $\bar{F}_l$, with Pearson coefficient $\rho \in [0.35, 0.65]$. This is captured in a 2-D probability table Tab_2D_FlBeta (size varies by reference system: e.g., $18 \times 16$ for NYISO, $16 \times 14$ for WECC), where rows correspond to flow classes and columns to gauge-ratio classes.

Algorithm for transmission capacity assignment

After DCPF gives the flow distribution $F_l$, the transmission capacity assignment proceeds:

Normalize flows: $bar{F}_l = |F_l| / max_l |F_l| in [0, 1]$.
Generate gauge ratios: Draw $M$ values from $\text{Exp}(\mu_\beta)$, inject overloads, and sort.
2-D bin-matching: Using the Tab_2D_FlBeta table, assign gauge ratios to lines based on their normalized flow magnitude. The procedure is analogous to the dispatch factor assignment: flows are sorted into bins (rows), betas into bins (columns), and pairing proceeds from high to low within each 2-D cell.
Calculate capacity: For each line $l$:

\[F_l^{\max} = \frac{|F_l|}{\beta_l}\]

If $F_l^{\max} \leq 2$ MW (negligible flow), apply a minimum capacity of $5 + 100 \cdot U(0,1)$ MW to avoid degenerate limits.

Note

The paper specifies a scaling check: if $\sum F_m^{\max}$ deviates more than 5% from $F_l^{\text{tot}}(N)$, all capacities should be rescaled proportionally. This is not currently implemented in the code.