A Unit Commitment Model with Demand Response for the Integration of Renewable Energies (1112.4909v1)

Abstract: The output of renewable energy fluctuates significantly depending on weather conditions. We develop a unit commitment model to analyze requirements of the forecast output and its error for renewable energies. Our model obtains the time series for the operational state of thermal power plants that would maximize the profits of an electric power utility by taking into account both the forecast of output its error for renewable energies and the demand response of consumers. We consider a power system consisting of thermal power plants, photovoltaic systems (PV), and wind farms and analyze the effect of the forecast error on the operation cost and reserves. We confirm that the operation cost was increases with the forecast error. The effect of a sudden decrease in wind power is also analyzed. More thermal power plants need to be operated to generate power to absorb this sudden decrease in wind power. The increase in the number of operating thermal power plants within a short period does not affect the total operation cost significantly; however the substitution of thermal power plants by wind farms or PV systems is not expected to be very high. Finally, the effects of the demand response in the case of a sudden decrease in wind power are analyzed. We confirm that the number of operating thermal power plants is reduced by the demand response. A power utility has to continue thermal power plants for ensuring supply-demand balance; some of these plants can be decommissioned after installing a large number of wind farms or PV systems, if the demand response is applied using an appropriate price structure.

• The paper introduces a MILP unit commitment framework that integrates thermal, wind, and photovoltaic sources with demand response to maximize profit while maintaining supply-demand balance.
• It demonstrates how forecast errors increase operational costs and reserve requirements, highlighting the impact on spinning reserve and load frequency control margins.
• Simulation results show that demand response can flatten peak loads and reduce the need for additional thermal generation during sudden drops in renewable output.

This paper presents a Unit Commitment (UC) model designed to optimize the operation of thermal power plants in a power system that includes fluctuating renewable energy sources (photovoltaic - PV and wind) and incorporates consumer demand response (1112.4909). The primary goal is to maximize the profit of an electric power utility while ensuring the supply-demand balance, considering the uncertainties associated with renewable energy forecasts.

Model Formulation

The core of the research is a Mixed Integer Linear Programming (MILP) model.

• Objective Function: The model aims to maximize the utility's profit over a time horizon $T$. Profit is defined as total sales revenue minus total operational costs.

  $$F = \sum_{t=1}^T (\text{Revenue}_t) - \sum_{t=1}^T \sum_{i=1}^N (\text{FuelCost}_t^i + \text{StartUpCost}_t^i)$$

  • Revenue depends on the forecasted demand $d_t^{(f)}$, the selected electricity price level $r^l$ (out of $L$ levels), and the price elasticity of demand $\epsilon_d$. Revenue at time $t$ is $$d_t^{(f)} \sum_{l=1}^{L} w_t^l r^l ( \frac{r^l}{\bar{r}} )^{\epsilon_d}$$, where $w_t^l$ is a binary variable indicating if price level $l$ is selected at time $t$, and $\bar{r}$ is the average price.
  • Operational cost includes the fuel cost ($b_i p_t^i$) based on the output $p_t^i$ of thermal plant $i$, and the start-up cost ($S_i z_t^i$) if the plant is started ($z_t^i=1$). $u_t^i$ is a binary variable indicating if plant $i$ is committed (on) at time $t$.
• Key Variables:
  • $p_t^i$: Continuous variable for power output of thermal unit $i$ at time $t$.
  • $u_t^i$: Binary variable for on/off status of thermal unit $i$ at time $t$.
  • $z_t^i$: Binary variable for start-up status of thermal unit $i$ at time $t$.
  • $w_t^l$: Binary variable for selecting price level $l$ at time $t$.
• Constraints:
  • Only one price level can be selected at each time step: $\sum_{l=1}^{L} w_t^l = 1$.
  • The average selected price over the horizon should not exceed the predefined average price: $$\frac{1}{T} \sum_{t=1}^T \sum_{l=1}^{L} w_t^l r^l \le \bar{r}$$.
  • Total demand adjusted by demand response should not decrease overall: $$\sum_{t=1}^T d_t^{(f)} \le \sum_{t=1}^T d_t^{(f)} \sum_{l=1}^{L} w_t^l ( \frac{r^l}{\bar{r}} )^{\epsilon_d}$$.
  • 1. Supply-Demand Balance: This is a crucial constraint ensuring generation meets demand, incorporating uncertainty.

    $$\sum_{i=1}^N p_t^i+wd_t^{(f)}+pv_t^{(f)} \ge d_t^{(f)}\sum_{l=1}^{L}w_t^l ( \frac{r^l}{\bar{r}} )^{\epsilon_d} + \text{Reserve}_t$$

    The reserve term accounts for forecast errors ($\sigma_d$, $\sigma_w$, $\sigma_p$). It's formulated probabilistically:

    $$\frac{\text{Available Generation}_t - \text{Adjusted Demand}_t}{\sqrt{\sigma_d^2+\sigma_w^2+\sigma_p^2}} \ge \phi^{-1}(\alpha)$$

    where $\alpha$ is the desired probability of meeting demand (e.g., 95%, 99%), and $\phi^{-1}$ is the inverse cumulative distribution function (CDF) of the forecast error distribution (Normal or Laplace distributions are considered). This ensures sufficient reserve capacity is scheduled based on the combined uncertainty.

  • Thermal Plant Constraints (Standard UC Constraints):

    • Generation Limits: $$\bar{p}_{min}^i u_t^i \le p_t^i \le \bar{p}_{max}^i u_t^i$$.
    • Ramp-Up/Down Limits: Constraints on how quickly a unit can increase or decrease its output ($p_t^i - p_{t-1}^i$).
    • Minimum Up/Down Times: $\tau_+^i$, $\tau_-^i$ constraints ensuring units stay on/off for a minimum duration once started/stopped.
    • Start-up Logic: Relating $z_t^i$ to changes in $u_t^i$.

Implementation and Analysis

The model was applied to a small test system with 12 thermal units, one 30MW PV system, and one 30MW wind farm. A commercial MILP solver was used. Key metrics calculated from the results include:

• Marginal Cost ($C_t^M$): The fuel cost of the most expensive operating unit at time $t$.
• Spinning Reserve ($R_t^S$): The extra generation capacity available from online units to compensate for forecast errors, calculated as the difference in scheduled generation between the uncertainty-aware case (Eq. \ref{eq:DSbalance}) and a deterministic case (Eq. \ref{eq:DSbalanceNoSigma1}).
• Load Frequency Control (LFC) Margin ($R_t^L$): The remaining upward ramping capability of online units (capped at 5% of max capacity per unit).

Simulation Scenarios and Key Findings:

1. Reference Case: Demonstrated the model's ability to schedule units based on merit order, absorbing renewable fluctuations. Higher prices were observed during high-demand daytime hours.
2. Demand Response ($\epsilon_d = -0.30$): Showed that demand response flattens the load profile (reducing peak demand by ~20MW in the example) and smoothens the marginal cost/price profile.
3. Effect of Forecast Error: Increasing the standard deviation of forecast errors ($\sigma_w$, $\sigma_p$) led to:
   • Increased total operation cost.
   • Increased required spinning reserve ($R_t^S$). The level of spinning reserve directly scaled with the magnitude of the forecast error.
   • Decreased LFC margin ($R_t^L$) during daytime, as more capacity is held as spinning reserve.
4. Sudden Decrease in Wind Power: A scenario simulating a rapid drop in wind generation required starting additional, more expensive thermal units during that period (Units 8 and 9 in the example). While the total operation cost didn't change dramatically due to the short duration, it highlights the need to keep thermal capacity available, limiting the direct substitution of thermal plants by renewables.
5. Demand Response during Wind Decrease: Applying demand response during the sudden wind drop reduced the number of thermal units needed online during the event. This suggests that demand response, driven by appropriate price signals reflecting the high marginal cost during such events, can mitigate the impact of renewable intermittency and potentially allow for decommissioning some thermal plants in the long run.

Practical Implications:

• The UC model provides a framework for optimally scheduling thermal generation considering renewable forecasts, their uncertainty, and demand flexibility.
• Accurate forecasting is crucial; higher forecast errors directly translate to higher operational costs and reserve requirements.
• Demand response is a valuable tool for mitigating renewable intermittency, reducing peak loads, and potentially reducing the need for reserve thermal capacity. Effective price signals are key to unlocking this potential.
• Even with high renewable penetration, maintaining sufficient flexible thermal capacity is necessary to handle sudden drops in renewable generation, limiting the extent to which renewables can fully substitute thermal plants without other flexibility measures (like storage or enhanced demand response).