**2.5. Estimation of the confidence interval and risk analysis for wind speed and solar irradiation scenarios**

From the power system planning viewpoint, the important aim is to reduce as far as possi‐ ble the uncertainty and risk associated with generation and power supply. The risk is more crucial when the ex-ante planned generation is less than the ex-post actual generation. The error in the forecast data can be estimated with a level of confidence (LC), in order to deter‐ mine a reliable level of generation to be considered in the planning stage. Here, the confi‐ dence interval method [26], known in risk assessment problems is proposed as a constraint to specify a lower and upper band for the wind speed and solar irradiation scenarios. For example, *LC=90%* means that the probability of forecast error (*Powerrisk*) being less than a definite value obtained from the distribution of forecast error

**Figure 7.** Hourly wind speed data in farm 1

(a)

(b)


**Kendall's tau correlation Freedom degree (υ) Linear correlation (ρ) Hour**

**Table 1.** The parameters of the two-variable t-Student copula distribution for wind speed of two farms at different

As mentioned earlier, besides the spatial correlation among different farms, the wind speed and solar irradiation random variables assigned to the scheduling time steps exhibit tempo‐ ral correlation, i.e., they are dependent on the condition of random variables at previous

**Figure 6.** Correlated samples for solar irradiation of two farms for 7 and 11 AM in fall season

**2.4. Time-series prediction of wind speed and solar irradiation**

hours of the day in fall season

110 New Developments in Renewable Energy

(*Powerrisk,*max) is more than 90%. The value of *Powerrisk,*max is referred to as the "Value at Risk"

Improved Stochastic Modeling: An Essential Tool for Power System Scheduling in the Presence of Uncertain

In order to calculate the *VAR* of forecast based on the defined *LC*, the PDF of forecast error should be analyzed. In fact, the maximum value of *Powerrisk,*max that satisfies equation (11) is chosen as the *VAR*. Given the forecast error PDF to be a normal distribution with mean *μe*

and standard deviation *σe*, the *VAR* is calculated by (Figure 11):

a

[ ] <sup>100</sup> *e e Pe z*a

 ³+ < m s

~ *risk*, *e e Power e z*

a

a = = + m s

is the maximum forecast error at which the condition is satisfied. The error is ex‐

pressed in terms of μe and *σe* where *zα* is the coefficient of σe. Then, this value is subtracted from and added to the forecasted data to give the lower band and upper band of the confi‐ dence interval, respectively. This band limits the scenarios generated for wind speed and so‐ lar irradiation and provides a reliable range for generation of scenarios that will be considered in the stochastic optimization process. Figure 12 and Figure 13 present an exam‐ ple of the distribution of forecast error of wind speed and solar irradiation at 11 AM, respec‐ tively. A representation of the confidence interval method applied to the wind speed data is

The final step of data processing before performing the stochastic optimization procedure is the scenario generation and reduction. This makes the main distinction between the deter‐ ministic programming and the stochastic programming approach. The deterministic pro‐ gramming deals with determined inputs and pre-defined parameters of the system model, whereas the stochastic programming combines the process of assignment of optimum val‐ ues to the control variables with the stochastic models of the existing random variables. The variables with stochastic behavior are indeed represented by a bunch of scenarios reflecting the most probable situations that is likely to occur for them. Here, in the scenario generation step, using the multivariate distribution function obtained, a large set of random vector sam‐ ples will be generated using the Monte Carlo simulation. Then, since all of the generated scenarios may not be useful and some of them may exhibit similarities and correlation with

*risk risk max* , *P Power Power LC* é ù £ > ë û (11)

= - 100 *LC* (12)

(13)

Renewables

113

http://dx.doi.org/10.5772/45849

*max* (14)

(*VAR*).

where *e* ~

depicted in Figure 14.

**2.6. Scenario generation and reduction**

**Figure 8.** Hourly solar irradiation data in farm 1

**Figure 9.** forecast results of wind power along with the actual data for one week

**Figure 10.** forecast results of wind power along with the actual data for one week

(*Powerrisk,*max) is more than 90%. The value of *Powerrisk,*max is referred to as the "Value at Risk" (*VAR*).

$$P\left[Power\_{risk} \le Power\_{risk, max}\right] > LC\tag{11}$$

In order to calculate the *VAR* of forecast based on the defined *LC*, the PDF of forecast error should be analyzed. In fact, the maximum value of *Powerrisk,*max that satisfies equation (11) is chosen as the *VAR*. Given the forecast error PDF to be a normal distribution with mean *μe* and standard deviation *σe*, the *VAR* is calculated by (Figure 11):

$$
\alpha = 100 - L\mathcal{C} \tag{12}
$$

$$P[\varepsilon \ge \mu\_{\varepsilon} + z\_{\alpha}\sigma\_{\varepsilon}] < \frac{a}{100} \tag{13}$$

$$\text{Power}\_{r\&\text{\textquotedblleft},\text{max}} = \stackrel{\text{\textquotedblleft}}{e} = \mu\_e + \text{z}\_a \sigma\_e \tag{14}$$

where *e* ~ is the maximum forecast error at which the condition is satisfied. The error is ex‐ pressed in terms of μe and *σe* where *zα* is the coefficient of σe. Then, this value is subtracted from and added to the forecasted data to give the lower band and upper band of the confi‐ dence interval, respectively. This band limits the scenarios generated for wind speed and so‐ lar irradiation and provides a reliable range for generation of scenarios that will be considered in the stochastic optimization process. Figure 12 and Figure 13 present an exam‐ ple of the distribution of forecast error of wind speed and solar irradiation at 11 AM, respec‐ tively. A representation of the confidence interval method applied to the wind speed data is depicted in Figure 14.

#### **2.6. Scenario generation and reduction**

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>0</sup>

Time (hr)

Actual data Estimated data

Actual data Estimated data

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>0</sup>

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>0</sup>

Hour

Hour

> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

> 0.2 0.4 0.6 0.8 1 1.2 1.4

**Figure 10.** forecast results of wind power along with the actual data for one week

Solar power (p.u.)

**Figure 9.** forecast results of wind power along with the actual data for one week

Wind power (p.u.)

Solar irradiation (W/m

**Figure 8.** Hourly solar irradiation data in farm 1

112 New Developments in Renewable Energy

2

)

The final step of data processing before performing the stochastic optimization procedure is the scenario generation and reduction. This makes the main distinction between the deter‐ ministic programming and the stochastic programming approach. The deterministic pro‐ gramming deals with determined inputs and pre-defined parameters of the system model, whereas the stochastic programming combines the process of assignment of optimum val‐ ues to the control variables with the stochastic models of the existing random variables. The variables with stochastic behavior are indeed represented by a bunch of scenarios reflecting the most probable situations that is likely to occur for them. Here, in the scenario generation step, using the multivariate distribution function obtained, a large set of random vector sam‐ ples will be generated using the Monte Carlo simulation. Then, since all of the generated scenarios may not be useful and some of them may exhibit similarities and correlation with other scenarios, the scenario reduction techniques are applied to eliminate low-probability scenarios and merge the similar ones to extract a limited number of scenarios keeping the whole probable region of the variables covered. Furthermore, the scenario reduction techni‐ que increases the computational efficiency of the optimization process. The most wellknown methods for scenario reduction are fast backward, fast forward/backward and the fast forward method [27-28]. The first step in scenario reduction is clustering. By clustering, the scenarios which are close to each other are put in one cluster. In the following, we present a description on the fast backward method.

The backward method is initialized by selection of scenarios as the candidates to be elimi‐ nated. The selection criterion is based on the minimum value for the product of each sam‐ ple's probability by the probabilistic distance of that sample to others. The probabilistic distance of each sample to others is considered as the minimum distance of that sample to each of the other samples in the same set. In the next step, the same analysis is performed on the remaining scenarios, however, in this step the product of the eliminated scenarios proba‐ bilities in the previous step by the distance of the current sample to other samples are also included. This process continues until the probabilistic distance obtained in each step (itera‐ tion) would be less than a predefined value as the convergence criterion. Mathematically speaking, the algorithm can be summarized as follows:

$$\begin{aligned} &j^{[0]}:=0\\ &j^{[0]}:=0\\ &Z^i = p\_i \min\_{i \neq j} \left\| \boldsymbol{\xi}^i - \boldsymbol{\xi}^j \right\| \; ; \; i = 1, \ldots, N\\ &j^{[1]} = \text{Min } Z^i \quad ; \; i = 1, \ldots, N\\ &\mathbf{STEP} \quad \textbf{[1]}:\\ &j\_i \in \arg\min\_{\boldsymbol{\ell}^{[i-1]} \atop j^{[i-1]} \neq \boldsymbol{1}} \sum\_{j^{[i-1]} \cup \boldsymbol{1}} p\_i \underset{j^{[i-1]} \in \mathbb{J}}{\text{Min}} \left\| \boldsymbol{\xi}^k - \boldsymbol{\xi}^j \right\| \end{aligned} \tag{15}$$
 
$$\begin{aligned} &j \underset{j^{[i-1]} \in \mathbb{J}}{\text{Min}} \sum\_{j^{[i-1]} \in \mathbb{J}} p\_i \underset{j^{[i-1]} \in \mathbb{J}}{\text{Min}} \left\| \boldsymbol{\xi}^k - \boldsymbol{\xi}^j \right\| \leq \boldsymbol{\varepsilon} \\ &j^{[i-1]} = j^{[i-1]} \cup j\_i \\ &\mathbf{E} \boldsymbol{\ell} \boldsymbol{\varepsilon} = \boldsymbol{I}^{[i-1]} \\ &\mathbf{END} \quad \textbf{END} \end{aligned} \tag{16}$$

overall objective function value to be optimized. As a comparison, the mean square error (MSE) of the calculated scenarios with respect to the real data is calculated in two cases. In the first case, the scenarios are obtained from the Copula distribution and within the calcu‐ lated confidence interval, based on the proposed framework. In the second case, the scenar‐ ios are generated from the single-variable distributions without modeling the underlying temporal and spatial correlations. Table 2 exhibits that the error in the first case is less than

Improved Stochastic Modeling: An Essential Tool for Power System Scheduling in the Presence of Uncertain

Renewables

115

http://dx.doi.org/10.5772/45849


Wind power forecast error

that in the second case in three days under study.

**Figure 11.** Value at risk calculation based on the distribution of forecast error

20

**Figure 12.** Distribution of error for wind speed forecast at 11 AM

40

60

Number of samples

80

100

120

where *N* is the total number of scenarios, *n* is the number of remaining scenarios, and |J| = N – n. *ξ <sup>i</sup>* is scenario *i* with probability *pi* . Here, 1000 initial scenarios are generated based on the obtained Copula multi-variable distribution function within the calculated confidence interval. Then, the scenarios are reduced to 20 uncorrelated scenarios using SCENRED func‐ tion in GAMS software. Figure 15 and Figure 16 demonstrate the final set of scenarios for a day of scheduling.

The power generation scheduling problem including units with uncertain and volatile char‐ acteristic is commonly treated as a stochastic optimization problem. In this approach, the ob‐ jective function calculation is repeated per all final scenarios in each iteration of the optimization process, where the summation of these objective values commonly defines the overall objective function value to be optimized. As a comparison, the mean square error (MSE) of the calculated scenarios with respect to the real data is calculated in two cases. In the first case, the scenarios are obtained from the Copula distribution and within the calcu‐ lated confidence interval, based on the proposed framework. In the second case, the scenar‐ ios are generated from the single-variable distributions without modeling the underlying temporal and spatial correlations. Table 2 exhibits that the error in the first case is less than that in the second case in three days under study.

**Figure 11.** Value at risk calculation based on the distribution of forecast error

other scenarios, the scenario reduction techniques are applied to eliminate low-probability scenarios and merge the similar ones to extract a limited number of scenarios keeping the whole probable region of the variables covered. Furthermore, the scenario reduction techni‐ que increases the computational efficiency of the optimization process. The most wellknown methods for scenario reduction are fast backward, fast forward/backward and the fast forward method [27-28]. The first step in scenario reduction is clustering. By clustering, the scenarios which are close to each other are put in one cluster. In the following, we

The backward method is initialized by selection of scenarios as the candidates to be elimi‐ nated. The selection criterion is based on the minimum value for the product of each sam‐ ple's probability by the probabilistic distance of that sample to others. The probabilistic distance of each sample to others is considered as the minimum distance of that sample to each of the other samples in the same set. In the next step, the same analysis is performed on the remaining scenarios, however, in this step the product of the eliminated scenarios proba‐ bilities in the previous step by the distance of the current sample to other samples are also included. This process continues until the probabilistic distance obtained in each step (itera‐ tion) would be less than a predefined value as the convergence criterion. Mathematically

[ 1] [ 1] [ 1]

U

Î -

å


*i i*

arg *i i <sup>i</sup>*

*j Min p Min*

*J Min Z i N*

x x

= -= = =

*i k I Ii J i*

*<sup>r</sup> i i <sup>j</sup> <sup>i</sup> i j i*

¹

\

where *N* is the total number of scenarios, *n* is the number of remaining scenarios, and |J| =

the obtained Copula multi-variable distribution function within the calculated confidence interval. Then, the scenarios are reduced to 20 uncorrelated scenarios using SCENRED func‐ tion in GAMS software. Figure 15 and Figure 16 demonstrate the final set of scenarios for a

The power generation scheduling problem including units with uncertain and volatile char‐ acteristic is commonly treated as a stochastic optimization problem. In this approach, the ob‐ jective function calculation is repeated per all final scenarios in each iteration of the optimization process, where the summation of these objective values commonly defines the

min ; 1,..., ; 1,...,

*Z p i N*

\

xx

*<sup>r</sup> k l*

x x


*<sup>r</sup> k l*

(15)

 e

. Here, 1000 initial scenarios are generated based on

[ 1] [ 1] [ 1]

U U



*<sup>k</sup> I Ii J i i i*

*i i <sup>i</sup>*

*if Min p Min*

å

[ 1]

[ ] [ 1]

= =

**END**

N – n. *ξ <sup>i</sup>* is scenario *i* with probability *pi*

day of scheduling.

*JJ j Else J J*

present a description on the fast backward method.

114 New Developments in Renewable Energy

speaking, the algorithm can be summarized as follows:

[0]

**STEP [0] :**

*J*

: 0

=

[1]

**STEP [i] :**

**Figure 12.** Distribution of error for wind speed forecast at 11 AM

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> -2

Improved Stochastic Modeling: An Essential Tool for Power System Scheduling in the Presence of Uncertain

Renewables

117

http://dx.doi.org/10.5772/45849

Time (hr)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>0</sup>

Time (hr)

Natural characteristics of wind and solar energy impose uncertainty in their design and op‐ eration. Hence, considering various possible scenarios in the model of these resources can lead to more realistic decisions. The uncertain parameters are expressed by probability dis‐ tributions, showing the range of values that a random variable could take, and also account‐ ing for the probability of the occurrence of each value in the considered range. Therefore, the way the random processes are modeled in terms of their PDF is a significant problem. The possible spatial correlations have been addressed and shown effective using the Copula

0

Solar irradiation W/m

**3. Conclusion**

2)

**Figure 15.** Final scenarios of wind speed within its confidence interval

**Figure 16.** Final scenarios of solar irradiation within its confidence interval

2

4

Wind speed (m/s)

6

8

10

**Figure 13.** Distribution of error for solar irradiation forecast at 11 AM


**Table 2.** MSE in Case 1 and 2 for the total power of two wind farms

**Figure 14.** Upper and lower band of confidence interval for wind speed data on one day

Improved Stochastic Modeling: An Essential Tool for Power System Scheduling in the Presence of Uncertain Renewables http://dx.doi.org/10.5772/45849 117

**Figure 15.** Final scenarios of wind speed within its confidence interval

**Figure 16.** Final scenarios of solar irradiation within its confidence interval

#### **3. Conclusion**


Solar power forecast error

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> -2

Time (hr)

**Figure 14.** Upper and lower band of confidence interval for wind speed data on one day

1.97e-3 1.4e-3 Day 1 5.99e-3 4.05e-3 Day 2 4.11e-3 3.31e-3 Day 3

**MSE in Case 2 MSE in Case 1**

100

**Figure 13.** Distribution of error for solar irradiation forecast at 11 AM

**Table 2.** MSE in Case 1 and 2 for the total power of two wind farms

Lower band Upper band Measured data Predicted data

0

2

4

Wind speed (m/s)

6

8

10

200

300

400

Number of samples

500

600

700

116 New Developments in Renewable Energy

Natural characteristics of wind and solar energy impose uncertainty in their design and op‐ eration. Hence, considering various possible scenarios in the model of these resources can lead to more realistic decisions. The uncertain parameters are expressed by probability dis‐ tributions, showing the range of values that a random variable could take, and also account‐ ing for the probability of the occurrence of each value in the considered range. Therefore, the way the random processes are modeled in terms of their PDF is a significant problem. The possible spatial correlations have been addressed and shown effective using the Copula method. Similarly, the possible temporal correlations have been taken into account using a time-series analysis. In summary, the overall framework can be listed as follows:

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Improved Stochastic Modeling: An Essential Tool for Power System Scheduling in the Presence of Uncertain

Renewables

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http://dx.doi.org/10.5772/45849

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