Curvature in Electricity Demand Across Temperatures

Why Model Curvature?

Electricity demand may change with temperature in more than one direction. On cold days, demand can rise as heating systems are used; on hot days, it can rise again as air conditioning runs. Between those conditions, demand may be lower. A straight-line model can describe a steady increase or decrease, but cannot describe a curve that changes direction. A quadratic regression gives a simple way to represent that kind of curvature.

A quadratic is not automatically the right model just because a curved pattern seems plausible. Its shape should be compared with the observed data, and residual plots should be checked for structure left behind by the fitted curve. Even a curve that fits the observed pattern well describes an association, not necessarily a causal effect of temperature: other factors, such as day of week or season, may also relate to demand.

The Quadratic Temperature Model

Let \(T_i\) be the temperature in degrees Celsius and \(Y_i\) the electricity demand in megawatts (MW) for observation \(i\). To keep the coefficients interpretable and the arithmetic manageable, choose a reference temperature \(c\) and define centered temperature \(z_i=T_i-c\). The quadratic regression model is

Here, \(\beta_0\), \(\beta_1\), and \(\beta_2\) are model coefficients, and \(\varepsilon_i\) is the error for observation \(i\). The model states that the conditional mean demand at temperature \(T_i\) is \(\beta_0+\beta_1z_i+\beta_2z_i^2\). The fitted regression replaces the unknown coefficients with estimates \(b_0\), \(b_1\), and \(b_2\); its predicted demand is \(\hat{Y}_i=b_0+b_1z_i+b_2z_i^2\).

For regression-based inference, a standard model assumes the errors have mean zero conditional on the temperatures, are independent, and have constant variance. Normal errors are additionally assumed for exact small-sample \(t\)-based inference. These are assumptions about errors around the conditional mean, not claims that temperatures or demands themselves must be normally distributed. In an applied analysis, the adequacy of the mean curve and the error behavior should be assessed using the data and diagnostic plots.

Reading the Curve and Its Turning Point

The fitted mean curve, written as a function of temperature, is

The slope changes with temperature. Differentiating the fitted curve with respect to \(T\) gives

This slope is measured in MW per degree Celsius. A turning point occurs where the slope is zero, if that temperature is relevant to the data. When \(b_2\ne 0\), its temperature is \(T=c-b_1/(2b_2)\). A positive \(b_2\) gives a minimum; a negative \(b_2\) gives a maximum. This point is a feature of the fitted quadratic, not automatically a well-supported feature of demand in the real world. It deserves substantive interpretation only if it falls within, or near, the observed temperature range and the data support the curve there.

Why the Turning-Point Formula Works

The turning point can also be seen by rewriting the fitted curve as a squared term plus a constant. This gives a direct justification for both the turning-point formula and whether the point is a minimum or maximum:

The squared term is smallest when \(z=-b_1/(2b_2)\), which gives \(T=c-b_1/(2b_2)\). If \(b_2>0\), multiplying the squared term by \(b_2\) gives a nonnegative contribution, so the curve reaches a minimum there. If \(b_2<0\), it gives a nonpositive contribution, so the curve reaches a maximum. This algebra establishes the turning point of the fitted function; it does not establish how precisely that point is estimated.

Checking Whether the Curvature Is Adequate

A fitted quadratic can bend, but it may still miss important structure in the data. A residual is the observed demand minus fitted demand: \(e_i=Y_i-\hat{Y}_i\). A residual plot displays these differences against a predictor such as temperature, or against fitted demand. If the quadratic mean curve is reasonably adequate, residuals should generally scatter around zero without a systematic trend or additional curved pattern. The plot does not prove that the model assumptions hold, but it can reveal ways in which the fitted curve fails to describe the observed mean pattern.

An upward or downward trend in residuals across temperature suggests that the fitted curve is systematically underpredicting or overpredicting in parts of the range. A remaining wave or S-shaped pattern suggests that the quadratic has not captured all of the mean curvature. A fan-shaped spread, by contrast, points to changing residual variability rather than necessarily to a wrong mean curve. Isolated large residuals also deserve attention, but they do not by themselves identify why a value differs from its prediction.

Common Pitfalls and Reporting

• Calling \(b_1\) the slope at every temperature. In a quadratic model the slope is \(b_1+2b_2(T-c)\); \(b_1\) is the slope only at \(T=c\).
• Reporting a turning point without checking its location. A calculated minimum far outside the observed temperature range is an extrapolation, not a directly supported feature of the observed data.
• Treating a plausible story as a model check. Heating and cooling may motivate a curved relationship, but residual plots are needed to see whether the fitted quadratic leaves a systematic pattern.
• Interpreting association as causation. An observational temperature–demand pattern can be affected by other variables; the fitted curve alone does not show that temperature caused the predicted differences.

A clear report should state the units and temperature range, the fitted quadratic, the turning point if it is meaningful within that range, and what the residual plots indicate. For instance, the hypothetical fit above has a minimum at 28°C and predicts 480 MW there; that statement should be accompanied by whether the residual pattern gives reasonable support to the quadratic over the observed temperatures.