# Analyzing the Impact of Variability and Uncertainty on Power System Flexibility

## Abstract

**:**

## Featured Application

## Abstract

## 1. Introduction

## 2. Flexibility Index: Ramping Capability Shortage Probability

#### 2.1. System Ramping Capability (SRC), Ramping Capability Requirement (RCR)

_{t}indicates the extent to which the system can supply the RC during the interval Δt. The larger the SRC secured, the greater the flexibility; however, uncertainty in the system may lower RC availability. In Equation (1), A

_{i,t-Δt}is a random variable denoting whether the unit fails from t−∆t to t.; its value may vary with the failure rate of the generating unit. Further information on the A

_{i,t-Δt}calculation can be found in [30]. Parameters such as O

_{i,t-Δt}and P

_{i,t-Δt}are determined by the generation schedule.

_{t}is the total RC required to respond to unexpected load variations and failures of loaded generating units from t−∆t to t. The forecast errors NLFE

_{t}, LFE

_{t}, and VGFE

_{t}are random variables.

#### 2.2. Ramping Capability Shortage Probability

_{t}is larger than SRC

_{t}, then load shedding is enforced; this situation is denoted as RC shortage. The probability that an RC shortage will happen at time t is defined as the RC shortage probability (RSP

_{t}), is expressed by

_{t}is applied to the worst case, i.e., when calculating the RSP

_{t}of the targeted time, uncertainty cases are considered that occur just before the targeted time and thus cannot be resolved by post-recovery actions. This consideration is helpful in reducing the computational burden in large power systems. The variability relates to FNL

_{t}, O

_{t-Δt}, and P

_{t-Δt}. Any change in FNL

_{t}itself indicates an increase/decrease in variability, and also changes O

_{t-Δt}and P

_{i, t-Δt}. Uncertainty is associated with C

_{t-Δt}, E

_{t}, A

_{t-Δt}, and NLFE

_{t}, and C

_{t-Δt}and E

_{t}represent the set of possible cases for A

_{i,t-Δt}and NLFE

_{t}, respectively.

_{t-Δt}is as follows: two online generating units are considered for 2 h. Each unit has two states, i.e., 0 (failure) or 1 (online). Considering the possible cases of two units, C

_{1}is a set of four elements; i.e., {(0,0), (0,1), (1,0), (1,1)}. The probability of each element can be obtained based on a Markov-chain–based capacity state model; for further information, please refer to [30]. An example for E

_{t}is as follows: the forecast net loads (i.e., FNL

_{t}) are assumed to be 10 MW. If NLFE

_{t}follows a triangular distribution as shown in Figure 1, E

_{t}becomes {(−0.5 MW), (0 MW), (0.5 MW)} by multiplying 10 MW and 5%, and the elements have a probability of 0.25, 0.5, and 0.25, respectively. Therefore, any changes in the parameters relating to the variability and uncertainty can be reflected in the RSP

_{t}calculation.

_{t}, VGFE

_{t}, O

_{i,t-Δt}, and P

_{i,t-Δt}. If the aggregated forecast output of renewable energy resources change with the effects of complementarity, the value of FVG

_{t}varies, and accordingly the value of VGFE

_{t}, leading to changes in O

_{i,t-Δt}, and P

_{i,t-Δt}.

## 3. Scenarios for Variability and Uncertainty

#### 3.1. Variability and Uncertainty and Their Relevance

#### 3.2. Scenario Generation and Sensitivity Analysis

_{t}is used as the selection criterion, i.e., the time interval of the largest RSP

_{t}is chosen.

_{1}and t

_{2}) indicate the forecast net load variation and measured net load variation, respectively. (a) means variability, and (b) and (c) imply uncertainty. Differently with the forecast net load variation, the measured net load variation includes uncertainty.

_{1}and t

_{2}. When adjusting the variability, it is assumed that the net load at time t

_{2}is changed, while the net load at time t

_{1}is fixed. This assumption is useful in clarifying scenario generation, and comparing the result of the uncertainty with that of the variability.

_{2}. A change in the variability is accompanied by a change in the uncertainty, because both parameters are dependent on the net load at time t

_{2}. Therefore, to keep the uncertainty unchanged with the variability, the same value of the uncertainty of the base case is applied to all scenarios for the variability; the “base case” refers to the information of the given system.

## 4. Case Study

#### 4.1. Base Case Information

_{t}was computed through programming in MATLAB (R2012b version) [34]. This simulation was performed on a PC with 16 GB RAM and 3.7 GHz CPU. As shown in Figure 5, VGFE

_{t}and NLFE

_{t}were assumed to follow a normal distribution, with a standard deviation of 5% of the forecast value.

_{t}values are shown in Figure 6. The maximum value of $2.0982\times {10}^{-4}$ occurs at hour 19, because the high net load lowers the available reserve capacity at that time, even though larger values of net load variation occur in other time intervals. The selected time interval was therefore 18–19 h. The values of the net load at 18 and 19 h of the base case were 2507 MW and 2531 MW, respectively. As shown in Table 2, the high (low) net loads were set at 110% and 120% (80% and 90%) of the net loads at hours 18 and 19, respectively. The range of increased variability and uncertainty was varied from 0% to 20%, with a step size of 1%. For reference, 20% of the increased variability is calculated as 506 MW, i.e., multiplying 2531 MW by 30%. As discussed previously, the uncertainty of the base case was set at 5% of the net load; this value indicates the standard deviation of the net load forecast error, and was computed as 127 MW, i.e., multiplying 2531 MW by 5%.

#### 4.2. Results for the Scenarios

_{t}values differ with the level of the net load; the largest RSP

_{t}appears at the highest level of net load, i.e., S1. The largest difference in RSP

_{t}according to the net load is between S5 (i.e., the net load is 100%) and S3 (i.e., the net load is 110%) when the increased variability is 18%. In S7 and S9, the values of RSP

_{t}are almost unchanged with the uncertainty compared to the other scenarios. In the other scenarios, RSP

_{t}increases in a step-wise fashion. This may be related to the failure cases of the generating units.

_{t}is heavily influenced by the net load. The largest value of RSP

_{t}is found when the level of the net load is highest (i.e., S2). S2 (i.e., 120% of the net load) and S4 (i.e., 110% of the net load), when the uncertainty is 4%, make the largest difference in RSP

_{t}. In most scenarios, the value of RSP

_{t}does not increase much further after it increases sharply. This is related to the characteristics of the probability distribution of the uncertainty, which is modeled as a normal random variable. The left and right sides of the normal distribution are symmetrical with respect to its average, and an increase in the left side (right side) decreases (increases) the RSP

_{t}at the same rate. The effects of both sides on the value of RSP

_{t}cancel each other out.

_{t}and the maximum increments (i.e., the change in the RSP

_{t}divided by the change in the uncertainty) in the former scenarios (i.e., S1, S3, and S5) are greater than those in the latter scenarios (i.e., S2, S4, and S6). S7 and S9 are compared to S8 and S10, respectively. The maximum increments in S8 and S10 are greater than those in S7 and S9. The value of RSP

_{t}in S8 (S10) is larger than that in S7 (S9) from 10% (12%) to 21% for each parameter. Regarding flexibility, increased variability is more effective than uncertainty when the level of the net load is equal to or greater than 100% (i.e., high and medium types). For the low type, uncertainty is more influential in terms of the maximum increment; however, the more influential parameter in terms of RSP

_{t}differs by section.

_{t}criterion is set the same as the value of RSP

_{t}of the given system (i.e., $2.0982\times {10}^{-4}$), the allowed ranges of increased variability and uncertainty are then determined as shown in Table 3. S1, S2, S3, S4, and S5 do not have allowed ranges under the given conditions, while S6, S7, S8, S9, and S10 have allowed ranges. For reference, Figure 9 and Figure 10 represent the enlarged graphs of Figure 7 and Figure 8, respectively. On the graphs, each point having the value of RSP

_{t}smaller than the RSP

_{t}criterion is included in the allowed ranges.

## 5. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

A_{i,t} | Random variable representing availability of generator i at time t (1 if available, 0 otherwise) |

c | Element of C_{t−Δt} |

C_{t-Δt} | Set of combinations of A_{i,t−Δt} when O_{i,t−Δt} is nonzero for all i |

e | Element of E_{t} |

E_{t} | Set of NLFE_{t} |

FL_{t} | Forecast load at time t |

FNL_{t} | Forecast net load at time t |

FVG_{t} | Forecast variable generation at time t |

i | Index of generator |

I | Set of generators |

LFE_{t} | Random variable representing load forecast error at time t |

NLFE_{t} | Random variable representing net load forecast error at time t |

O_{i,t} | Value representing whether generator i is online at time t or not |

P_{i,t} | Output of generator i at time t |

P_{max,i} | Maximum output level of generator i |

Prob(·) | Probability in parentheses |

Prob_{c}[·] | Probability of c if condition [∙] is satisfied, 0 otherwise. |

RCR_{t} | Ramping capability requirement at time t |

rr_{i} | Ramp rate of generator i |

RSP_{t} | Ramping capability shortage probability at time t |

SRC_{t} | System ramping capability at time t |

t | Index of time |

Δt | Minimum interval between operating points |

VGFE_{t} | Random variable representing variable generation forecast error at time t |

## Appendix A. Failure and Repair Rates in Case Study

Unit # | Failure Rate (occurrences/h) | Repair Rate (occurrences/h) |
---|---|---|

1–5 | 1/2940 | 1/60 |

6–9 | 1/450 | 1/50 |

10 | 1/1960 | 1/40 |

11, 12 | 1/450 | 1/40 |

13 | 1/1960 | 1/40 |

14–16 | 1/1200 | 1/50 |

17–20 | 1/960 | 1/40 |

21–23 | 1/950 | 1/50 |

24 | 1/1150 | 1/100 |

25, 26 | 1/1100 | 1/150 |

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**Figure 5.**NLFE

_{t}and VGFE

_{t}distribution. A normal distribution was described using a seven-step approximation.

**Figure 9.**Results of sensitivity analysis for increased variability: enlarged graph for S7, S9, and RSP criterion.

**Figure 10.**Results of sensitivity analysis for uncertainty: enlarged graph for S6, S8, S10, and RSP criterion.

Scenario # | Net Load | Increased Variability | Uncertainty |
---|---|---|---|

S1 | High (>100%) | Particular range | Fixed |

S2 | Fixed, 0% | Particular range | |

S3 | Medium (100%) | Particular range | Fixed |

S4 | Fixed, 0% | Particular range | |

S5 | Low (<100%) | Particular range | Fixed |

S6 | Fixed, 0% | Particular range |

Scenario, S# | Net Load | Increased Variability of the Net Load at 19 h | Uncertainty of the Net Load at 19 h |
---|---|---|---|

S1 | 120% | 0% to 20% | Fixed, i.e., 5% |

S2 | 120% | Fixed, 0% | 0% to 20% |

S3 | 110% | 0% to 20% | Fixed, i.e., 5% |

S4 | 110% | Fixed, 0% | 0% to 20% |

S5 | 100% | 0% to 20% | Fixed, i.e., 5% |

S6 | 100% | Fixed, 0% | 0% to 20% |

S7 | 90% | 0% to 20% | Fixed, i.e., 5% |

S8 | 90% | Fixed, 0% | 0% to 20% |

S9 | 80% | 0% to 20% | Fixed, i.e., 5% |

S10 | 80% | Fixed, 0% | 0% to 20% |

Scenario, S# | Net Load | Increased Variability of the Net Load at 19 h | Uncertainty of the Net Load at 19 h |
---|---|---|---|

S1 | 120% | N/A | Fixed, i.e., 5% |

S2 | 120% | Fixed, 0% | N/A |

S3 | 110% | N/A | Fixed, i.e., 5% |

S4 | 110% | Fixed, 0% | N/A |

S5 | 100% | N/A | Fixed, i.e., 5% |

S6 | 100% | Fixed, 0% | 0% to 5% |

S7 | 90% | 0% to 11% | Fixed, i.e., 5% |

S8 | 90% | Fixed, 0% | 0% to 10% |

S9 | 80% | 0% to 20% | Fixed, i.e., 5% |

S10 | 80% | Fixed, 0% | 0% to 15% |

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**MDPI and ACS Style**

Min, C.-G.
Analyzing the Impact of Variability and Uncertainty on Power System Flexibility. *Appl. Sci.* **2019**, *9*, 561.
https://doi.org/10.3390/app9030561

**AMA Style**

Min C-G.
Analyzing the Impact of Variability and Uncertainty on Power System Flexibility. *Applied Sciences*. 2019; 9(3):561.
https://doi.org/10.3390/app9030561

**Chicago/Turabian Style**

Min, Chang-Gi.
2019. "Analyzing the Impact of Variability and Uncertainty on Power System Flexibility" *Applied Sciences* 9, no. 3: 561.
https://doi.org/10.3390/app9030561