Representative Days for Expansion Decisions in Power Systems

Abstract

Short-term uncertainty needs to be properly modeled when analyzing a planning problem in a power system. Since the use of all available historical data may lead to problems of computational intractability, clustering algorithms may be applied in order to reduce the computational effort without compromising accurate representation of historical data. In this paper, we propose a modified version of the traditional K-means method, seeking to represent the maximum and minimum values of input data, namely, electricity demand and renewable production in several locations of a power system. Extreme values of these parameters must be represented as they are high-impact decisions that are taken with respect to expansion and operation. The method proposed is based on the K-means algorithm, which represents the correlation between demand and wind-power production. The chronology of historical data, which influences the performance of some technologies, is characterized through representative days, each made up of 24 operating conditions. A realistic case study, applying representative days, analyzes the generation and transmission expansion planning of the IEEE 24-bus Reliability Test System. Results show that the proposed method is preferable to the traditional K-means technique.

1. Introduction

Planning problems in power systems require optimal expansion decisions to be made in the power system to be determined, e.g., the building of additional transmission lines or new generating units. These decisions are made with a specific goal, such as minimizing the supply costs or maximizing the profit of a private investor. Therefore, it is crucial to represent the conditions of the power system in the decision-making problem. Power system planning problems are usually handled via historical data, since the more realistic the input data of the problem, the more accurate the solution to the problem. For short-term uncertainty in power systems, the historical data whose variability is most important is that of electrical demand and renewable production. Note that short-term uncertainty associated with renewable generating units increases the complexity of power system planning problems because in a long-term basis weather conditions can only be poorly predicted in advance, unlike the daily evolution of demand, which generally follows a clear pattern. Using storage units in power systems promotes the increased penetration of renewable generating units because energy can be discharged from storage units when needed and stored when there is an excess of energy. In addition, demand and renewable generation are correlated [1,2,3]. Thus, the optimization model used to solve power-system planning problems should properly represent this correlation between demand and renewable production.

Solving a power-system planning problem commonly involves the use of hourly data, especially with technologies that depend on the chronology, such as storage units. Nevertheless, the optimization problem can be intractable because of the large amount of historical data used as input of the model. The amount of historical data used must therefore be reduced, in order to achieve a near-optimal solution in a reasonable time. Several techniques for doing this have been implemented in the technical literature, such as load-duration curves [4] and the K-means method [5,6].

The load-duration curve technique represents the short-term uncertainty of electrical loads through different levels arranged into demand blocks. Then, we build the cumulative distribution function within each block. Subsequently, these functions are divided into a number of segments with an associated probability, and we calculate the average value inside each of them, obtaining different demand levels. This technique can be expanded to jointly consider demand and renewable production [7]; for instance, load- and wind-duration curves in the case of accounting for wind turbines in the power system under study. In this case, the procedure to model the wind-duration curve is identical to the method for load data described above. In addition, both magnitudes share the same blocks, wherein all possible combinations of different levels of demand and renewable production can exist. These combinations, which can be used as input data for expansion planning optimization problems, are known as system operating conditions, and have a probability assigned that is equal to the product of the probability of the demand level and the probability of the wind-production level. The accuracy of the solution obtained using load and wind-duration curves relies on the number of demand blocks and wind-production levels selected in this method, being greater the larger these numbers are. Therefore, a trade-off is required between accuracy and the tractability of the resulting problem. Net load-duration curves have been used in [8,9], while load- and wind-duration curves are considered in [10,11,12]. Note that the ability of duration curves to represent the correlation among historical data is limited.

An alternative to duration curves is the use of clustering algorithms, the K-means technique being one of the most widely used, which applies algorithms arranging data into groups according to similarities, whose centroids are used with the purpose of representing the input data as well as reducing the computational burden. The weight of each centroid is associated with the amount of input data inside its group. This method has the advantage that, in contrast to the load- and wind-duration curve technique, it allows the correlations of demand and renewable production in several locations of the electric energy system under study to be represented. The K-means method is used, for example, in [13,14].

Duration curves and traditional K-means methods are compared in [7]. The main disadvantage of these two methods is that it is not possible to include units with inter-temporal constraints such as storage units in expansion problems. To deal with this issue, the authors in [15] propose using a representative day of each season, while the authors in [16,17] consider a K-means method with some modifications. The main drawback of these methods is that they may not represent extreme values of input data accurately. If demand and renewable production are used as input data, maximum and minimum values may have a great effect on the solution of operation and planning problems for power systems.

The techniques mentioned above may be used when solving power-system planning problems such as the generation expansion planning problem, the transmission expansion planning problem, or the Generation and Transmission Expansion Planning (G&TEP) problem [18]. In this paper, we solve a G&TEP problem in order to analyze the results of a case study obtained using the traditional and a new modified version of the K-means algorithm.

The G&TEP problem is solved to determine where and when new transmission lines and/or generating units should be built in a power system, in order to meet electrical demand in the coming years. It is motivated by the growth in demand, the increase in the penetration of renewable generating units, and the ageing of transmission facilities.

In most power systems, a central entity is responsible for taking expansion decisions in the transmission network, i.e., determining which transmission lines should be built. The aim of this system operator is to minimize investment and operation costs and preventing load-shedding. However, expansion decisions for generating units are generally taken by private investors, whose purpose is to maximize their own profits. Nevertheless, achieving the optimal solution is not guaranteed by considering the G&TEP problem as two independent problems [19]. This is the reason that we follow the strategy given by [20], which considers a central entity that determines the transmission and generation expansion plan minimizing operation and investment costs. Thus, the central entity is responsible for carrying out the transmission plans and encouraging private investors to build the most appropriate production facilities through incentives [21].

In this context, the contributions of this paper are threefold:

• To propose a modified version of the traditional K-means method to generate system operating conditions that properly represent maximum and minimum values of input data.
• To use this new method to obtain representative days of demand and wind-power production, each composed of 24 operating conditions, in order to characterize the chronology of the historical data allowing the incorporation of technologies that depend on the chronology, such as storage units, in the formulation of expansion planning models.
• To give and analyze the results of a case study to show that the proposed method improves on the outcomes of a traditional K-means method.

The remainder of this paper is organized as follows. Section 2 explains the methodology of the traditional K-means method and the modified version of this technique proposed here. Section 3 gives the results of a case study, where a comparison is made between the outcomes obtained applying the clustering methods mentioned above. Section 4 concludes the paper with some relevant remarks. Finally, Appendix A provides a detailed formulation of the G&TEP problem.

2. Clustering Technique

The K-means method is a clustering algorithm whose aim is to arrange data into groups called clusters according to certain similarities between them. The inputs of this algorithm are historical data of the electricity demand and the wind-power production in several locations of a power system, while the outputs of this method are the cluster centroids, along with the number of observations located at each cluster. Note that cluster centroids, defined by the values of the demand and wind-power production involved, represent the system operating conditions, which can be used as input data for the solution of optimization problems (e.g., a long-term planning problem).

The K-means technique is useful when dealing with a significant amount of data in optimization problems due to the reduction of computational effort. The user defines the K-number of desired operating conditions, i.e., the number of clusters. However, it must be taken into account that a small number of operating conditions may result in inaccuracies in the representation of the input data. In contrast, a high number of clusters may lead to problems of computational intractability.

2.1. Input Data

It is important to normalize the input data before applying the algorithm when working with parameters with different orders of magnitude, to avoid the results being influenced by the weight of each parameter. This is the case of the input data used in this paper, demand and wind-power production data, so they must be normalized beforehand. To do so, we apply the min-max normalization technique to the input data.

In order to characterize the chronology of historical data, representative days comprising 24 operating conditions are considered. Using representative days allows technologies that depend on chronology, such as storage devices, to be modeled in expansion planning problems.

We consider wind-power production data for the sake of simplicity. However, note that the proposed algorithm is general and may be applied to other renewable production data such as solar-power production.

2.2. Traditional K-Means Algorithm

The K-means algorithm traditionally used in technical literature, known from now on as traditional K-means method (TKM), is based on the following steps [5]:

• Step 1: Select the number of required clusters according to the needs of the problem.
• Step 2: Define the initial centroid of each cluster, e.g., by randomly assigning a historical observation to each cluster.
• Step 3: Compute the quadratic distances between each original observation and each cluster centroid.
• Step 4: Allocate each historical observation to the closest cluster according to the distances calculated at Step 3.
• Step 5: Recalculate the cluster centroids using the historical observations allocated to each cluster.

Steps 3–5 are repeated iteratively until there are no changes in the cluster compositions between two consecutive iterations.

Although the TKM has advantages over other techniques (e.g., it can represent temporal and spatial correlations between uncertain parameters, while the duration curve technique cannot), it is not without drawbacks. Sometimes, the TKM does not adequately characterize the maximum and minimum values of the parameters analyzed. This may be a problem when we consider electricity demand and wind-power production as input data of the algorithm because their extreme values can have a great impact on the solution of the optimization problem.

In the case of the G&TEP problem, peak values of demand may require the building of new generating units and/or new transmission lines to supply demand in the power system being studied. Moreover, extreme values of wind-power production can also influence investment decisions. Overall, maximum and minimum values of demand and wind-power production can have an impact on the total costs, either through the investment costs associated with the investment decisions made, or by the operation costs related to the power produced by conventional generating units and load-shedding.

2.3. Modified K-Means Algorithm

To overcome these issues, we propose a new clustering method called the modified K-means method (MKM), which tries to characterize the extreme values of the parameters properly. The MKM applies a first clustering to the historical data, and it then applies a second clustering, taking as input data the observations previously obtained for each cluster. Therefore, the second clustering is applied to all the clusters obtained in the first clustering. The steps of the MKM algorithm are set out here:

• Step 1: Arrange the historical data into $K_1$ clusters following the TKM.
• Step 2: Apply the same TKM to the observations allocated to each cluster obtained at Step 1, arranging them into $K_2$ clusters for each of the $K_1$ clusters.

Note that the MKM can only be applied if the number of observations located at each cluster after Step 1 is greater than or equal to the parameter $K_2$. The number of operating conditions that are obtained as the output of this algorithm is equal to the product of $K_1$ and $K_2$. For instance, a first clustering is performed, organizing the input data into five clusters ($K_1=5$). Then, the observations allocated to each cluster are taken as the input data for a second clustering, arranging them into two clusters ($K_2=2$). Thus, the number of operating conditions obtained at the end of the algorithm is 10.

Equation (1) defines the relation that must exist between the parameter K, associated with the TKM, and parameters $K_1$ and $K_2$, linked to the MKM, to make the results of both methods comparable:

$$K=K_1·K_2.$$

(1)

3. Case Study

This section analyzes the results of a case study.

3.1. Data

We apply the G&TEP model described in Appendix A to the modified version of the IEEE 24-bus Reliability Test System (RTS) [22] shown in Figure 1. This power system comprises 11 conventional generating units, 17 loads, 24 buses, two storage units, 38 transmission lines, and two wind-power units. In addition, we consider that seven candidate conventional generating units, five candidate storage units, six candidate transmission lines, and four candidate wind-power units can be built. The data for conventional generating units, storage units, transmission lines, and wind-power units can be seen in [23].

We consider that wind-power production and demand vary according to the area of the power system in which they are located. The historical data of electricity demand and wind-power production have been taken from [24] for the year 2016. The daily evolution of electricity demand for the loads located in the western and eastern zones is shown in Figure 2 and Figure 3, respectively, while the daily evolution of wind-power production in the northern and southern zones of the power system is shown in Figure 4 and Figure 5, respectively. These figures show the minimum and maximum values of electricity demand and wind-power production at different zones. Moreover, note that, while the daily electricity demand follows a clear pattern, the daily wind-power production does not. Note also that each curve of Figure 2, Figure 3, Figure 4 and Figure 5 represents a different day of the year 2016.

We work with hourly data, thus the duration of time steps, $\Delta \tau$, is one hour. The efficiency of the storage units in charging and discharging is considered to be equal to 90%. The energy initially stored in storage units is assumed to be zero for all representative days. Bus 1 is the reference bus of the optimization problem and we consider a base power of 100 MW. We consider a total investment budget of $20,000 million, which is higher than the sum of the investment costs of all candidate units. Therefore, the total investment budget does not limit the investment decisions of the G&TEP problem. Annualized investment costs are 10% of the total costs.

3.2. Results

As previously explained, the use of representative days reduces the computational effort in power-system planning problems representing historical data without losing accuracy. For that reason, we first normalize the historical data of Figure 2, Figure 3, Figure 4 and Figure 5 and then use the resulting curves as input data for the K-means methods described in Section 2 obtaining the parameters ${\alpha }_{wrh}$ and ${\beta }_{drh}$, which respectively represent the capacity factor of wind-power unit w and the demand factor of load d on representative day r and hour h. Since we normalize the input data, the values of these parameters are between 0 and 1. It is worth mentioning that we obtain different values of parameters ${\alpha }_{wrh}$ for wind-power production in the northern and southern zones as well as different values of parameters ${\beta }_{drh}$ for the loads located in the western and eastern zones of the power system. Furthermore, for the sake of simplicity, we assume that, within each zone, the values of the parameters ${\alpha }_{wrh}$ and ${\beta }_{drh}$ for each representative day and hour are the same for all wind-power units and loads, respectively. In the case of applying the MKM with $K_1=5$ and $K_2=2$ to the historical data in Figure 2, Figure 3, Figure 4 and Figure 5, we obtain the 10 (2 × 5) representative days of electric demand in the western and eastern zones depicted in Figure 6 and Figure 7, respectively, while the 10 representative days of wind-power production in the northern and southern zones are represented in Figure 8 and Figure 9, respectively. Additionally, the values of parameters ${\sigma }_r$, which represent the weight of each representative day, are provided in

Table 1. Case study: weight of representative days obtained using the modified K-means method with $K_1=5$ and $K_2=2$.

Representative Day	${\mathit{\sigma }}_{\mathit{r}}$ [days]
$r_1$	46
$r_2$	80
$r_3$	30
$r_4$	24
$r_5$	24
$r_6$	37
$r_7$	36
$r_8$	22
$r_9$	28
$r_{10}$	39

.

It should be mentioned that the demand values in the western zone are greater than in the eastern zone, as can be observed in Figure 2 and Figure 3. In addition, the wind-power production values are associated with the northern zone, as illustrated in Figure 4 and Figure 5. Both of these results can be observed in Figure 6, Figure 7, Figure 8 and Figure 9, where representative days with the parameters ${\alpha }_{wrh}$ and ${\beta }_{drh}$ are represented, since the normalization process for the input data was carried out considering the historical data from all zones. It is expected that the need to supply the high demands in the western zone will condition the investment decisions of the expansion problem.

3.3. Validation

First of all, we solve the G&TEP problem using all the historical data to find the so-called exact solution in order to compare it with the results obtained using representative days provided by both K-means methods set out in Section 2. For that purpose, we have implemented the approach described in Appendix A with the changes detailed in Appendix B.

The G&TEP problem is solved taking the 366 days of the year 2016 as historical data. The annualized investment cost, ${\widetilde{IC}}^{\mathrm{E}}$, rises to $682 million, while the total annualized cost, ${\widetilde{TC}}^{\mathrm{E}}$, is $3124 million. The results show that 0.14% of the total demand is not supplied. The computational time required to obtain the exact solution is 36 h 56 min.

Next, for assessment purposes, we have implemented the model presented in Appendix A using representative days as follows:

• Step 1: Solve the G&TEP problem using the representative days obtained through clustering methods for different values of the parameter K.
• Step 2: Set the values of the expansion planning decision variables ($m_s^{\mathrm{S}}$, $\forall s\in {\mathsf{\Omega }}^{\mathrm{S}+}$; ${\overline{p}}_g^{\mathrm{G}}$, $\forall g\in {\mathsf{\Omega }}^{\mathrm{G}+}$; ${\overline{p}}_w^{\mathrm{W}}$, $\forall w\in {\mathsf{\Omega }}^{\mathrm{W}+}$; $x_{\ell }^{\mathrm{L}}$, $\forall \ell \in {\mathsf{\Omega }}^{\mathrm{L}+}$) obtained at Step 1 and solve the G&TEP problem using all the historical data.
• Step 3: Calculate the percentage error, ${\epsilon }_{\widetilde{TC}}$, associated with the total annualized cost obtained at Step 2, ${\widetilde{TC}}^{\mathrm{K}}$, with regard to the total annualized cost provided by the exact solution, ${\widetilde{TC}}^{\mathrm{E}}$, using Equation (2):

  $${\epsilon }_{\widetilde{TC}}=\frac{|{\widetilde{TC}}^{\mathrm{K}}-{\widetilde{TC}}^{\mathrm{E}}|}{{\widetilde{TC}}^{\mathrm{E}}}·100.$$

  (2)

These steps are followed in the case study using a set of values of the parameter K which range from 10 to 80, where the maximum value which could be chosen was 366. This means that we work with an equivalent quantity of data ranging from 3 to 22% of all the historical data considered.

Simulations have been implemented on a Gigabyte R280-A3C with 2 Intel Xeon E5-2698 at 2.3 GHz and 256 GB of RAM using CPLEX 12.7.0.0 [25] under GAMS 24.8.3 [26].

The results are summarized in

Table 2. Case study: analysis of the results obtained from the clustering methods.

	${\widetilde{\mathit{IC}}}^{\mathbf{K}}\left({10}^6\mathbf{\$}\right)$		${\widetilde{\mathit{TC}}}^{\mathbf{K}}\left({10}^9\mathbf{\$}\right)$		${\mathit{\epsilon }}_{\widetilde{\mathit{TC}}}$ (%)		Computational Time (min)
$\mathit{K}$	TKM	MKM	TKM	MKM	TKM	MKM	TKM	MKM
10	279.63	298.81	4.69	4.43	48.44	41.97	1	1
20	266.84	330.76	4.89	4.07	56.51	30.29	4	3
30	331.90	539.58	3.94	3.30	26.18	5.80	10	8
40	594.77	654.69	3.30	3.19	5.55	2.04	12	16
50	544.38	611.58	3.34	3.27	7.08	4.78	20	21
60	579.12	665.89	3.21	3.14	2.70	0.66	22	25
70	564.56	682.69	3.24	3.12	3.58	0.01	43	47
80	633.89	667.48	3.17	3.14	1.40	0.63	55	68

, where ${\widetilde{IC}}^{\mathrm{K}}$ and ${\widetilde{TC}}^{\mathrm{K}}$ denote the annualized investment cost obtained at Step 1 and the total annualized cost obtained at Step 2, respectively, ${\epsilon }_{\widetilde{TC}}$ represents the error associated with the total annualized cost defined in (2), and the last two columns show the computational time required for solving the problem set out at Step 1. It should be noted that the investment costs shown in Table 2 are different depending on the value of K and the clustering method selected, which means that the investment decisions obtained at Step 1 are different for each of the cases analyzed.

Figure 10 depicts the total annualized cost obtained using several values of K and for each of the two clustering methods. It is worth mentioning that we define the value of K when we use the TKM while we fix the values of $K_1$ and $K_2$ when we apply the MKM. In order to compare the results obtained using the two clustering methods, we have chosen the values of $K_1$ and $K_2$, which can be seen in [23], taking into account Equation (1). As can be observed, the total annualized costs for the MKM are closer to the exact solution than those obtained using the TKM for all the cases evaluated. Note that the differences between the results obtained using the clustering methods and the exact solution generally decrease at the same time that the value of K increases.

As observed in Table 2, the errors associated with the MKM are always lower than those obtained using the TKM for all the cases analyzed, especially in those where the parameter K presents a small value. In particular, the case analyzed using $K=70$ with the MKM returns an error of 0.01%. This is because the expansion plan of the solution obtained in this case is to build all candidate conventional generating units and wind-power units at maximum capacity, all candidate storage units, and all transmission lines except for one of them, while the expansion plan of the exact solution is the same, with the difference that two candidate transmission lines are not built.

Regarding the resolution of the G&TEP problem using representative days (Step 1), the TKM generally provides shorter computational times, especially in those cases where parameter K presents a high value, although the differences are in no case significant.

Taking into account the results shown in Table 2, the MKM leads to lower values of total annualized cost and ${\epsilon }_{\widetilde{TC}}$ than the TKM. Although the computational times obtained using the MKM are generally greater than those acquired using the TKM, the difference between the two methods is not very large, and in fact they are reasonable times for a planning problem. Thus, the expansion plan using MKM outperforms the investment scheme considering TKM.

4. Conclusions

This paper proposes a new clustering method to generate operating conditions for making expansion planning decisions in power systems. As a distinctive feature, the proposed method allows an adequate characterization of the maximum and minimum values of the input data. In addition, we arrange the operating conditions obtained using the K-means method into representative days in order to represent the chronology of the historical data. This allows us to include storage units in expansion planning problems.

From numerical results of using the MKM to solve the G&TEP problem, we conclude that for a given number of representative days the MKM provides a solution closer to the exact solution than that provided by the TKM. Note that the origin of these differences is in the fact that investment decisions change depending on the clustering method chosen to obtain the representative days. Furthermore, for a given clustering technique, either TKM or MKM, investment decisions change if we modify the value of parameter K or parameters $K_1$ and $K_2$, respectively. This is expected since the use of a large number of representative days usually leads to a more accurate solution. Additionally, the MKM also outperforms the TKM in terms of computational times needed to achieve the solution of the G&TEP problem for a given error with regard to the total annualized cost provided by the exact solution.