Producing Load Shape with Target Peak and Load Factor
Riaz Khan1
Abstract
Modifying a load shape to match specific peak and load factor is a fundamental component for various power system planning and operation studies. This article describes two efficient methods to modify a reference load shape while matching the desired peak and load factor. A validation metrics for assessing the derived load shape in terms of preserving time series properties was also developed.
Electric Load Shape
Load shapes can be defined as the hourly pattern of loads for a given set of variables including equipment, operating characteristics, and other factors such as weather and demographics [1]. Electric load shape is the hourly pattern of electric load. It can be with for a day, week, month or a year with 24, 168, 720 and 8760 hourly load values.
Load Factor
A measure of utilization/efficiency of a given electric load shape is load factor. it is defined as the ratio of the average load of the profile to the maximum load of the system. If \(x_1\), \(x_2\), \(x_3\),…,\(x_n\) are \(n\) hourly values from a system/grid, then load factor, \(f_{load}\) can be derived as:
\[ f_{load} = \frac{\sum^{n}_{i=1}x_i/n}{max(x_1, x_2, x_3,...,x_n)} \] Load factor is a a measure of efficiency and always falls in \([0, 1]\) interval. High load factor indicates high efficiency and vice versa.
Load factor depends on various factors. For example, when considering a regional electric grid, it may vary by month, season, day type (weekday vs weekend), weather, customer class share (residential, commercial, industrial), length of the period and so on. Energy forecast and peak forecasts are used in various studies by the grid planners and preserving the load factor is important as it varies by different factors. Often a base load shape (for month, season, year etc.) is used and forecast energy and peak load (and hence the load factor) is applied to the base load shape. For example, in the grid reliability assessment study [2] conducted by Northeast Power Coordinating Council (NPCC) in 2021, 2002 load shape was as the base load shape and scaled to match the peak demand and energy forecast [3].
Load Shape Scaling
Two methods, namely linear and logistic- have been developed and discussed in this article to scale a reference shape to match the peak and load factor.
Figure 1: Load Shape Scaling Process
Figure 1 shows a general flow diagram of scaling process. The starting point is the per unit load duration curve (LDC). LDC is the load profile ordered in descending order of magnitude, rather than chronologically [4]. For a target load factor smaller than the load factor of the reference profile, a series of multipliers of descending order are calculated, which is multiplied by the reference per unit LDC. The product then becomes the target per unit LDC. This is re-ordered relative to original positional rank to derive the target per unit load shape. The per unit load shape is then scaled to match the target peak. For a target load factor greater than reference, the process is similar except for the multipliers are ordered in ascending order.
The load factor matching process is baked in the stage of deriving the multipliers. The multipliers are derived in such a way so that the product (of multipliers and reference per unit LDC) hits the target peak. Neither re-ordering in later stage, nor multiplying by target peak alters the load factor.
The main difference is two methods (linear and logistic) is the pattern formed by the multipliers. In linear method, the multipliers increased or decrease in linear fashion, whereas the multipliers derived by logistic method form a sigmoid line. The steepness and inflection point of the sigmoid can be controlled using parameters.
Load Shape Scaling - Linear Method
In the linear method, the multipliers form a straight line with positive (case 1) or negative slope (case 2). Suppose \(\textbf{y} = y_1,\ y_2,\ ...,\ y_n\) are the per unit LDC of the reference load shape. A set of multipliers \(\textbf{m}=m_1,\ m_2,\ m_3,\ ...,\ m_n\) are derived so that
\[ m_i = 1-(i-1)\beta \] where \(\beta\) is a constant and can be solved analytically. The \(i_{th}\) element of the series \(\textbf{y} \times \textbf{m}\) is \(y_i m_i=y_i(1-(i-1)\beta)\). So, the load factor formed by \(y_i m_i\) must be equal to the target load factor, \(f_{target}\). So,
\[ \frac{avg(\textbf{y} \times \textbf{m})}{max(\textbf{y} \times \textbf{m})}=f_{target} \] Now, for \(f_{target} < f_{load}\), \(max(\textbf{y} \times \textbf{m})\) is always 1, as both \(\textbf{y}\) and \(\textbf{m}\) are decreasing. For \(f_{target} > f_{load}\), the per unit LDC values are to be multiplied by increasing numbers starting from 1, arising the possibility of resulting a value \(> 1\). However, for small deviance between \(f_{target}\) and \(f_{target}\), the likelihood is low and \(max(\textbf{y} \times \textbf{m})=1\) holds true for most practical scenarios.
Setting \(max(\textbf{y} \times \textbf{m})=1\), we get the following:
\[ avg(\textbf{y} \times \textbf{m})=f_{target} \] which is the basis of solving \(\beta\).
\[ avg(\textbf{y} \times \textbf{m})= \frac{\sum_{i=1}^n y_i (1-(i-1)\beta)}{n} = f_{target} \]
\[ \sum_{i=1}^n y_i (1-(i-1)\beta) = n \times f_{target} \] \[ \sum_{i=1}^n y_i - \beta \sum_{i=1}^n (i-1)y_i = n \times f_{target} \]
\[ \sum_{i=1}^n y_i - \beta \sum_{i=1}^n i\ y_i \ + \beta \sum_{i=1}^ny_i = n \times f_{target} \] \[ \beta \left( \sum_{i=1}^n i\ y_i - \sum_{i=1}^ny_i\ \right)= \sum_{i=1}^ny_i - n \times f_{target} \] \[ \beta = \frac{\sum_{i=1}^ny_i - n \times f_{target} }{ \sum_{i=1}^n i\ y_i - \sum_{i=1}^ny_i } \]
Load Shape Scaling - Logistic Method
In the logistic method, the multipliers form a S-shaped (or reversed S-shaped) curve. For case 1, where the target load factor is larger than the reference load factor, a set of increasing multipliers are derived which form a S-shaped curve. For case 2, where the target load factor is smaller than the reference load factor, the derived multipliers form an inverted/reversed S-shaped curve. This is done utilizing logistic function.
A sigmoid/logistic function[5] can be written as,
\[f(x, L, k, x_0)=\frac{L}{1+ exp(-k(x-x_0))}\]
where, \(L\), \(x_0\), \(k\) are three parameters. Parameter \(x_0\) is the inflection point of the sigmoid, parameter \(k\) defines the steepness of the sigmoid and parameter \(L\) is the maximum value of the sigmoid. If two of the parameters are defined, the other parameter can be solved. The method developed takes \(x_0\) and \(k\) as inputs and solves for \(L\) numerically.
Figure 2: Load Shape Scaling Process Using Logistic Method
Figure 2 shows the load shape scaling process of using logistic method. The process, with the input values, calculates a series (of lenght \(n_{iter}\)) of \(L\) values. For each \(L\) values, the process calculates a series of ascending/descending multipliers that follow a general S-shape through the use logistic function. Each multiplier series is then divided by min/max value so that the first multiplier becomes 1. From \(n_{iter}\) set of multipliers, final set is selected which results in load factor closest to the target.
Scoring Load Shape
A diagnostic score was developed to score the derived load shape, in terms of how well time series properties are retained, when a base load shape is projected to a new load shape.
The diagnostic measure is calculated as a weighted mean absolute percent error (MAPE) of auto correlation or partial auto correlation values of the derived series with respect to the original. The values are calculated for given lag. \(Lag = 0\) is omitted from calculation for auto correlation as it would be always 1. If \(o_i\) and \(d_i\) are the correlation values of original and derived load shape at lag \(i\), then weighted MAPE is calculated as:
\[ wmape = \sum _{i=1}^{lag} { w_i * |(o_i - d_i) / o_i|} \] where \(w_i = \frac{|o_i|}{\sum _{i=1}^{lag}|o_i|}\)
Since \(wmape\) is a measure of error, lower value indicates better preservation of time series property.
References
South Dakota State University, mdriazahmed.khan@jacks.sdstate.edu↩︎
Comments on Load Shape Scaling
In the logistic method, parameter \(x_0\) controls position of the inflection point. Lower value of \(x_0\) means smaller portion of the data would be multiplied by closer-to-one multipliers and higher value means higher portion of the data would be multiplied by closer-to-one multipliers.
Parameter \(k\) in the logistic method controls the steepness of the sigmoid curve. As \(k\) decreases, the S-shaped curve gets flatter and results approaches that of linear method.
When \(\beta > 0\), (\(target\ load\ factor< base\ load\ factor\)), the multipliers decrease linearly. If target load factor is too small, it results in negative multipliers (i.e negative load). So, projection of a base load shape to new peak and load factor is limited. It is possible to analytically solve the minimum possible target load factor by setting \(m_n=0\):
\[ m_n=0 \]
\[ 1-(n-1)\beta = 0 \] \[ \beta=\frac{1}{n-1}=\frac{\sum_{i=1}^ny_i - n \times f_{target} }{ \sum_{i=1}^n i\ y_i - \sum_{i=1}^ny_i } \]
\[ \sum_{i=1}^ny_i - n \times f_{target} = \frac{\sum_{i=1}^n i\ y_i - \sum_{i=1}^ny_i}{n-1} \]
\[ n \times f_{target} = \sum_{i=1}^ny_i - \frac{\sum_{i=1}^n i\ y_i - \sum_{i=1}^ny_i}{n-1} \] \[ f_{target} = \frac{\sum_{i=1}^ny_i- \big( \sum_{i=1}^n iy_i\big)/n}{n-1} \]
If the target load factor is much bigger than the base load factor, one/both of the followings can occur:
As a linearly increasing function is multiplied by a decreasing function \((y')\), it is possible that the maximum of the product can exceed the maximum value of the base \((y')\), resulting in a different load factor.
As a linearly increasing function is multiplied by a decreasing function \((y')\), it is possible that the product is not strictly decreasing.
For the latter case, a possible solution is to re-order the product array.