# A Framework for Real-Time Optimal Power Flow under Wind Energy Penetration

^{*}

## Abstract

**:**

## 1. Introduction

- A novel RT-OPF framework is developed to address the conflict between the fast changing wind power and the slow optimization computation and consequently to realize an online optimization of energy systems in a very short sampling time;
- Discrete reference values of the slack bus voltage, wind power curtailment of WSs, and reverse power flow are considered simultaneously, leading to a MINLP problem;
- A scenario generation method is integrated in the RT-OPF framework to represent uncertain wind power for the prediction horizon, which leads to a set of uncoupled MINLP problems solved by parallel computing.

## 2. Problem Formulation

_{s}should be kept as short as possible. However, due to its high complexity, the computation time T

_{OPF}needed to solve the optimization problem can be much higher than the sampling time. To address this conflict, we employ the forecasted data of wind energy which are available in advance of a future time horizon T

_{p}. In this paper, this forecasted time horizon is called a prediction horizon. Since the prediction horizon T

_{p}is usually higher than the sampling time T

_{s}, we can divide the prediction horizon into M sampling times, i.e.,:

_{OPF}is smaller than the prediction horizon T

_{p}. Under this assumption, a prediction-realization approach for RT-OPF will be developed in this paper. In the prediction phase, the optimization problem is solved in advance for a number of probable wind energy scenarios, based on the forecasted data in the prediction horizon and its probability density function (PDF), leading to a lookup table for optional optimal operation strategies. In the realization phase, the actual wind energy data are successively available from one sampling time to the next. In each sampling time, the actual data will be compared with the predefined wind energy scenarios and an optimal operation strategy corresponding to the nearest higher scenario will be selected from the lookup table and realized in the network. In this way, an online update of the operation strategy according to the spontaneously changing wind energy generation is carried out.

## 3. Scenario Generation

## 4. Solution Framework

#### 4.1. Prediction Phase

#### 4.2. Realization Phase

Algorithm 1 Comparing and selection of wind power |

for each WS ${n}_{w}=1,\cdots ,{N}_{w}$ and ${n}_{s}\ge 2$ |

$\begin{array}{l}\mathrm{If}\text{}{P}_{w}({n}_{w},({n}_{s}-1)){P}_{w.A}({n}_{w},m)\le {P}_{w}({n}_{w},{n}_{s})\\ \mathrm{then}\text{}\mathrm{consider}\text{}{P}_{w.A}({n}_{w},m)\text{}\mathrm{as}\text{}{P}_{w}({n}_{w},{n}_{s})\end{array}$ |

end |

Achieve ${P}_{w}({n}_{c})$ |

Based on ${n}_{c}$, set ${\mathsf{\beta}}_{w}(m)={\mathsf{\beta}}_{w}({n}_{c})\text{}\mathrm{and}\text{}{V}_{S}(m)={V}_{S}({n}_{c})$ |

#### 4.3. Implementation of the Real-Time Optimal Power Flow Framework

- (1)
- For the current prediction horizon, provide the forecasted active ${P}_{d}(i)$ and reactive ${Q}_{d}(i)$ demand power and wind power ${P}_{w.M}({n}_{w})$.
- (2)
- Generate ${N}_{c}$ wind power scenario combinations based on the Beta distribution as described in Section 3.
- (3)
- Send the generated scenarios as inputs to formulate ${N}_{c}$ MINLP OPF problems.
- (4)
- Solve the ${N}_{c}$ MINLP OPF problems with parallel computing.
- (5)
- Send the solution results as a lookup table to the selection algorithm.
- (6)
- Provide the actual wind power of WSs, ${P}_{w.A}({n}_{w},m)$, available at the current sampling time $m$ (for $m=1,\cdots ,M$), to the selection algorithm.
- (7)
- Select one of the solutions from the lookup table based on ${P}_{w.A}({n}_{w},m)$ and the selection algorithm (see Section 4.2).
- (8)
- Send the values of the controls ${V}_{S}(m)$ and ${\mathsf{\beta}}_{w}(m)$ to the grid.

## 5. Case Study

#### 5.1. Network and Input Data

#### 5.2. Test Cases

- Forward energy flow: The forward active and reactive energy from the HV network to the MV network is to be minimized based on an energy price model.
- Reverse energy flow: The reverse power flow could have impacts on voltage profiles [53] of the upper-level network and may result in specific operational limits being exceeded at the congested primary substations [54]. However, reverse flows have been considered in many studies [45,55,56,57,58] and in reality, they are likely to happen. Therefore, in this paper we consider the cases with and without reverse power flows.

- Case (1):
- Both reverse active and reactive power to the upstream HV network is not allowed (i.e., ${\mathsf{\gamma}}_{Ps,rev}={\mathsf{\gamma}}_{Qs,rev}=0$), and with a fixed value of the slack bus voltage (${V}_{S}(m)={V}_{S}({n}_{c})=1\text{}\mathrm{pu}$).
- Case (2):
- Both reverse active and reactive power to upstream HV network is not allowed (i.e., ${\mathsf{\gamma}}_{Ps,rev}={\mathsf{\gamma}}_{Qs,rev}=0$), and with the slack bus voltage as a discrete free variable.
- Case (3):
- Both reverse active and reactive power to upstream HV network is allowed (i.e., ${\mathsf{\gamma}}_{Ps,rev}={\mathsf{\gamma}}_{Qs,rev}=1$), and with a fixed value of the slack bus voltage (i.e., ${V}_{S}(m)={V}_{S}({n}_{c})=1\text{}\mathrm{pu}$).
- Case (4):
- Both reverse active and reactive power to upstream HV network is allowed (i.e., ${\mathsf{\gamma}}_{Ps,rev}={\mathsf{\gamma}}_{Qs,rev}=1$), and with the slack bus voltage as a discrete free variable.

#### 5.3. Results and Discussions

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

## Sets and Indices

$i,j$ | Indices for buses, i.e., $i,j=1,\cdots ,{N}_{bus}$. |

$m$ | Index for sampling intervals, i.e., $m=1,\cdots ,M$. |

${n}_{c}$ | Index for wind power scenario combinations, i.e., ${n}_{c}=1,\cdots ,{N}_{c}$. |

${n}_{s}$ | Index for wind power scenarios of each individual wind station (WS), i.e., ${n}_{s}=1,\cdots ,{N}_{s}$. |

${n}_{w}$ | Index for WSs, i.e., ${n}_{w}=1,\cdots ,{N}_{w}$. |

$sb$ | Set of buses. |

## Functions

$f$ | Objective function. |

$F$ | Total value of objective function for one day. |

${F}_{1}$ | Total revenue from wind power injection for one day. |

${F}_{2}$ | Total cost of active energy losses in the grid for one day. |

${F}_{3}$ | Total cost of active energy at slack bus for one day. |

${F}_{4}$ | Total cost of reactive energy at slack bus for one day. |

$f({n}_{c})$ | Total value of objective function for scenario combination ${n}_{c}$. |

${f}_{1}({n}_{c})$ | Total revenue from wind power injection for scenario combination ${n}_{c}$. |

${f}_{2}({n}_{c})$ | Total cost of active energy losses in the grid for scenario combination ${n}_{c}$. |

${f}_{3}({n}_{c})$ | Total cost of active energy at slack bus for scenario combination ${n}_{c}$. |

${f}_{4}({n}_{c})$ | Total cost of reactive energy at slack bus for scenario combination ${n}_{c}$. |

${F}_{PDF}$ | Probability distribution function. |

${f}_{P}({n}_{c})$ | Network active power function for scenario combination ${n}_{c}$. |

${f}_{Q}({n}_{c})$ | Network reactive power function for scenario combination ${n}_{c}$. |

$g$ | Equality equations. |

$\rho $ | Density function. |

## Parameters

$L$ | Upper bound on integer variables. |

$M$ | Total number of sampling intervals in each prediction horizon. |

${N}_{bus}$ | Total number of buses. |

${N}_{s}$ | Total number of wind power scenarios for each WS. |

${N}_{pr}$ | Total number of processors. |

${N}_{c}$ | Total number of wind power scenario combinations. |

${N}_{w}$ | Total number of WSs. |

${P}_{d}(i)$ | Active power demand at bus $i$. |

$Pric{e}_{P}$ | Price for active energy. |

$Pric{e}_{O}$ | Price for reactive energy. |

${P}_{w.R}({n}_{w})$ | Rated installed wind power of WS . |

${Q}_{d}(i)$ | Reactive power demand at bus $i$. |

${S}_{l.\mathrm{max}}(i,j)$ | Upper limit of apparent power flow of line between bus $i$ and $j$. |

${S}_{S.\mathrm{max}}$ | Upper limit of apparent power at slack bus. |

${T}_{p}$ | Length of prediction horizon. |

${T}_{OPF}$ | Length of reserved time for computing OPF problems. |

${T}_{s}$ | Length of sampling interval. |

${T}_{r}$ | Length of reserved time for data management. |

${u}_{\mathrm{max}}$ | Upper limits on continuous decision variables. |

${u}_{\mathrm{min}}$ | Lower limits on continuous decision variables. |

${V}_{\mathrm{max}}(i)$ | Upper limit of voltage at bus $i\text{}(i\ne 1)$. |

${V}_{\mathrm{min}}(i)$ | Lower limit of voltage at bus $i\text{}(i\ne 1)$. |

${V}_{S.\mathrm{max}}$ | Upper limit of slack bus voltage. |

${V}_{S.\mathrm{min}}$ | Lower limit of slack bus voltage. |

${x}_{\mathrm{max}}$ | Upper limits on state variables. |

${x}_{\mathrm{min}}$ | Lower limits on state variables. |

${\mathsf{\mu}}_{d}(i)$ | Mean value for demand at bus $i$. |

${\mathsf{\sigma}}_{d}(i)$ | Standard deviation for demand at bus $i$. |

${\mathsf{\sigma}}_{w}({n}_{w})$ | Standard deviation for wind power of WS ${n}_{w}$. |

${\mathsf{\gamma}}_{Ps,rev}$ | Coefficient of reverse boundary on active power at slack bus. |

${\mathsf{\gamma}}_{Qs,rev}$ | Coefficient of reverse boundary on reactive power at slack bus. |

## Random Variables

${P}_{w.A}({n}_{w},m)$ | Actual wind power of WS ${n}_{w}$ in sampling interval $m$. |

${P}_{w}(i,{n}_{c})$ | Wind power of WS located at bus $i$ for scenario combination ${n}_{c}$. |

${P}_{w}({n}_{c})$ | Vector of active power of WSs for scenario combination ${n}_{c}$. |

${P}_{w}({n}_{w},{n}_{s})$ | Wind power of WS ${n}_{w}$ for wind power scenario combination ${n}_{s}$. |

${P}_{w.M}({n}_{w})$ | Mean (forecasted) wind power of WS ${n}_{w}$. |

$\mathsf{\xi}$ | Vector of random variables. |

## Decision Variables

$l$ | Vector of integer decision variables. |

$u$ | Vector of continuous decision variables. |

${V}_{S}(m)$ | Slack bus voltage in sampling interval $m$. |

${V}_{S}({n}_{c})$ | Slack bus voltage for scenario combination ${n}_{c}$. |

${\mathsf{\beta}}_{w}(i,{n}_{c})$ | Curtailment factor of wind power for WS located at bus $i$ for scenario combination ${n}_{c}$. |

${\mathsf{\beta}}_{w}(m)$ | Vector of curtailment factors of wind power for WSs in sampling interval $m$. |

${\mathsf{\beta}}_{w}({n}_{c})$ | Vector of curtailment factors of wind power of WSs for scenario combination ${n}_{c}$. |

$\Delta {V}_{S}({n}_{c})$ | Voltage change at slack bus for scenario combination ${n}_{c}$. |

## State Variables

${P}_{loss}({n}_{c})$ | Active power losses for scenario combination ${n}_{c}$. |

${P}_{S}({n}_{c})$ | Active power injected at slack bus for scenario combination ${n}_{c}$. |

${P}_{S}(m)$ | Active power injected at slack bus in sampling interval $m$. |

${Q}_{S}({n}_{c})$ | Reactive power injected at slack bus for scenario combination ${n}_{c}$. |

${Q}_{S}(m)$ | Reactive power injected at slack bus in sampling interval $m$. |

$S(i,j,{n}_{c})$ | Apparent power flow from bus $i$ to $j$ for scenario combination ${n}_{c}$. |

$V(i,{n}_{c})$ | Voltage at bus $i\text{}(i\ne 1)$ for scenario combination ${n}_{c}$. |

$x$ | Vector of state variables. |

$\mathsf{\alpha}({n}_{w})$ | First shape parameter of Beta distribution for WS ${n}_{w}$. |

$\mathsf{\beta}({n}_{w})$ | Second shape parameter of Beta distribution for WS ${n}_{w}$. |

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**Figure 1.**Illustration of wind power scenarios (i.e., S

_{1}, …, S

_{7}), for a WS with ${\mathsf{\sigma}}_{w}=0.1$, ${P}_{w.M}$ = 0.15 (

**left**), ${P}_{w.M}$ = 0.5 (

**middle**), and ${P}_{w.M}$ = 0.85 (

**right**).

**Figure 2.**Graphical examples of wind power scenario combinations for (

**a**) ${N}_{w}=2$, ${N}_{s}=7$ and (

**b**) ${N}_{w}=3$, ${N}_{s}=4$. Here, dark to light colors denote high to low wind power scenarios, respectively.

**Figure 3.**Framework of the proposed real-time optimal power flow (RT-OPF) for a prediction horizon. Here, HV, MV and LV denote high-voltage, medium-voltage and low-voltage, respectively.

**Figure 4.**Time allocation for the computational tasks of the 8 steps in Figure 3.

**Figure 7.**Trajectories of one day: (

**a**) total active (blue-solid) and reactive (red-dashed) power demand; (

**b**) forecasted (red-dashed) and actual (blue-solid) wind power of the first WS; and (

**c**) forecasted (red-dashed) and actual (blue-solid) wind power of the second WS.

**Figure 8.**Trajectories of one day for Case 1. (

**a**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for first WS; (

**b**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for second WS; (

**c**) Forecasted (red-dashed) and actual (blue-solid) values of voltage at the slack bus; (

**d**) Forecasted (red-dashed) and actual (blue-solid) slack bus active power; (

**e**) Forecasted (red-dashed) and actual (blue-solid) slack bus reactive power; (

**f**) Forecasted (red-dashed) and actual (blue-solid) total objective function value; (

**g**) Feasibility status of the deterministic (red-dashed) and prediction-realization (blue-solid) approaches when applying actual wind power. Here, 1 denotes feasible and 0 denotes infeasible solution; and (

**h**) computational time of the seven processors.

**Figure 9.**Trajectories of one day for Case 2. (

**a**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for first WS; (

**b**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for second WS; (

**c**) Forecasted (red-dashed) and actual (blue-solid) values of voltage at the slack bus; (

**d**) Forecasted (red-dashed) and actual (blue-solid) slack bus active power; (

**e**) Forecasted (red-dashed) and actual (blue-solid) slack bus reactive power; (

**f**) Forecasted (red-dashed) and actual (blue-solid) total objective function value; (

**g**) Feasibility status of the deterministic (red-dashed) and prediction-realization (blue-solid) approaches when applying actual wind power. Here, 1 denotes feasible and 0 denotes infeasible solution; and (

**h**) computational time of the seven processors.

**Figure 10.**Trajectories of one day for Case 3. (

**a**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for first WS; (

**b**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for second WS; (

**c**) Forecasted (red-dashed) and actual (blue-solid) values of voltage at the slack bus; (

**d**) Forecasted (red-dashed) and actual (blue-solid) slack bus active power; (

**e**) Forecasted (red-dashed) and actual (blue-solid) slack bus reactive power; (

**f**) Forecasted (red-dashed) and actual (blue-solid) total objective function value; (

**g**) Feasibility status of the deterministic (red-dashed) and prediction-realization (blue-solid) approaches when applying actual wind power. Here, 1 denotes feasible and 0 denotes infeasible solution; and (

**h**) computational time of the seven processors.

**Figure 11.**Trajectories of one day for Case 4. (

**a**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for first WS; (

**b**) Forecasted (red-dashed) and actual (blue-solid) curtailment factors for second WS; (

**c**) Forecasted (red-dashed) and actual (blue-solid) values of voltage at the slack bus; (

**d**) Forecasted (red-dashed) and actual (blue-solid) slack bus active power; (

**e**) Forecasted (red-dashed) and actual (blue-solid) slack bus reactive power; (

**f**) Forecasted (red-dashed) and actual (blue-solid) total objective function value; (

**g**) Feasibility status of the deterministic (red-dashed) and prediction-realization (blue-solid) approaches when applying actual wind power. Here, 1 denotes feasible and 0 denotes infeasible solution; and (

**h**) computational time of the seven processors.

n_{c} | Scenario Combination | ||||

P_{w}(n_{w},n_{s}) | P_{w}(n_{w},n_{s}) | … | P_{w}(n_{w},n_{s}) | P_{w}(n_{w},n_{s}) | |

n_{w} = 1 | n_{w} = 2 | n_{w} = N_{w} − 1 | n_{w} = N_{w} | ||

1 | P_{w}(1,N_{s}) | P_{w}(2,N_{s}) | … | P_{w}((N_{w} – 1),N_{s}) | P_{w}(N_{w},N_{s}) |

2 | P_{w}(1,N_{s}) | P_{w}(2,N_{s}) | ... | P_{w}((N_{w} – 1),N_{s}) | P_{w}(N_{w},(N_{s} − 1)) |

. . . | . . . | . . . | . . . | . . . | . . . |

N_{c} − 1 | P_{w}(1,1) | P_{w}(2,1) | … | P_{w}((N_{w} – 1),1) | P_{w}(N_{w},2) |

N_{c} | P_{w}(1,1) | P_{w}(2,1) | … | P_{w}((N_{w} – 1),1) | P_{w}(N_{w},1) |

**Table 2.**Lookup table for the first prediction horizon (with P

_{w.M}(1) = 3.8 MW, P

_{w.M}(2) = 7.05 MW) for Case 1.

Scenario Combination | Optimal Results | ||||||
---|---|---|---|---|---|---|---|

n_{c} | P_{w}(n_{w},n_{s}) (MW) n_{w} = 1 | P_{w}(n_{w},n_{s}) (MW) n_{w} = 2 | β_{w}(1) - | β_{w}(2) - | Vs (pu) | P_{S} (MW) | Q_{S} (Mvar) |

1 | P_{w}(1,7) = 10 | P_{w}(2,7) = 10 | 0.379 | 0.288 | 1 | 0 | 2.375 |

2 | P_{w}(1,7) = 10 | P_{w}(2,6) = 8.04 | 0.379 | 0.358 | 1 | 0 | 2.375 |

3 | P_{w}(1,7) = 10 | P_{w}(2,5) = 7.55 | 0.379 | 0.382 | 1 | 0 | 2.375 |

4 | P_{w}(1,7) = 10 | P_{w}(2,4) = 7.13 | 0.379 | 0.404 | 1 | 0 | 2.375 |

5 | P_{w}(1,7) = 10 | P_{w}(2,3) = 6.67 | 0.379 | 0.432 | 1 | 0 | 2.375 |

6 | P_{w}(1,7) = 10 | P_{w}(2,2) = 6.07 | 0.379 | 0.475 | 1 | 0 | 2.375 |

7 | P_{w}(1,7) = 10 | P_{w}(2,1) = 0 | 0.669 | 1 | 1 | 0 | 2.406 |

8 | P_{w}(1,6) = 4.79 | P_{w}(2,7) = 10 | 0.792 | 0.288 | 1 | 0 | 2.375 |

9 | P_{w}(1,6) = 4.79 | P_{w}(2,6) = 8.04 | 0.792 | 0.358 | 1 | 0 | 2.375 |

10 | P_{w}(1,6) = 4.79 | P_{w}(2,5) = 7.55 | 0.792 | 0.382 | 1 | 0 | 2.375 |

11 | P_{w}(1,6) = 4.79 | P_{w}(2,4) = 7.13 | 0.792 | 0.404 | 1 | 0 | 2.375 |

12 | P_{w}(1,6) = 4.79 | P_{w}(2,3) = 6.67 | 0.792 | 0.432 | 1 | 0 | 2.375 |

13 | P_{w}(1,6) = 4.79 | P_{w}(2,2) = 6.07 | 0.792 | 0.475 | 1 | 0 | 2.375 |

14 | P_{w}(1,6) = 4.79 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 1.9 | 2.419 |

15 | P_{w}(1,5) = 4.22 | P_{w}(2,7) = 10 | 0.899 | 0.288 | 1 | 0 | 2.375 |

16 | P_{w}(1,5) = 4.22 | P_{w}(2,6) = 8.04 | 0.899 | 0.358 | 1 | 0 | 2.375 |

17 | P_{w}(1,5) = 4.22 | P_{w}(2,5) = 7.55 | 0.899 | 0.382 | 1 | 0 | 2.375 |

18 | P_{w}(1,5) = 4.22 | P_{w}(2,4) = 7.13 | 0.899 | 0.404 | 1 | 0 | 2.375 |

19 | P_{w}(1,5) = 4.22 | P_{w}(2,3) = 6.67 | 0.899 | 0.432 | 1 | 0 | 2.375 |

20 | P_{w}(1,7) = 4.22 | P_{w}(2,2) = 6.07 | 0.899 | 0.475 | 1 | 0 | 2.375 |

21 | P_{w}(1,5) = 4.22 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 2.476 | 2.427 |

22 | P_{w}(1,4) = 3.77 | P_{w}(2,7) = 10 | 1 | 0.29 | 1 | 0 | 2.375 |

23 | P_{w}(1,4) = 3.77 | P_{w}(2,6) = 8.04 | 1 | 0.361 | 1 | 0 | 2.375 |

24 | P_{w}(1,4) = 3.77 | P_{w}(2,5) = 7.55 | 1 | 0.385 | 1 | 0 | 2.375 |

25 | P_{w}(1,4) = 3.77 | P_{w}(2,4) = 7.13 | 1 | 0.407 | 1 | 0 | 2.375 |

26 | P_{w}(1,4) = 3.77 | P_{w}(2,3) = 6.67 | 1 | 0.435 | 1 | 0 | 2.375 |

27 | P_{w}(1,4) = 3.77 | P_{w}(2,2) = 6.07 | 1 | 0.478 | 1 | 0 | 2.375 |

28 | P_{w}(1,4) = 3.77 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 2.927 | 2.435 |

29 | P_{w}(1,3) = 3.33 | P_{w}(2,7) = 10 | 1 | 0.334 | 1 | 0 | 2.376 |

30 | P_{w}(1,3) = 3.33 | P_{w}(2,6) = 8.04 | 1 | 0.416 | 1 | 0 | 2.376 |

31 | P_{w}(1,3) = 3.33 | P_{w}(2,5) = 7.55 | 1 | 0.443 | 1 | 0 | 2.376 |

32 | P_{w}(1,3) = 3.33 | P_{w}(2,4) = 7.13 | 1 | 0.469 | 1 | 0 | 2.376 |

33 | P_{w}(1,3) = 3.33 | P_{w}(2,3) = 6.67 | 1 | 0.501 | 1 | 0 | 2.376 |

34 | P_{w}(1,3) = 3.33 | P_{w}(2,2) = 6.07 | 1 | 0.551 | 1 | 0 | 2.376 |

35 | P_{w}(1,3) = 3.33 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 3.370 | 2.444 |

36 | P_{w}(1,2) = 2.81 | P_{w}(2,7) = 10 | 1 | 0.387 | 1 | 0 | 2.379 |

37 | P_{w}(1,2) = 2.81 | P_{w}(2,6) = 8.04 | 1 | 0.481 | 1 | 0 | 2.379 |

38 | P_{w}(1,2) = 2.81 | P_{w}(2,5) = 7.55 | 1 | 0.512 | 1 | 0 | 2.379 |

39 | P_{w}(1,2) = 2.81 | P_{w}(2,4) = 7.13 | 1 | 0.542 | 1 | 0 | 2.379 |

40 | P_{w}(1,2) = 2.81 | P_{w}(2,3) = 6.67 | 1 | 0.58 | 1 | 0 | 2.379 |

41 | P_{w}(1,2) = 2.81 | P_{w}(2,2) = 6.07 | 1 | 0.637 | 1 | 0 | 2.379 |

42 | P_{w}(1,2) = 2.81 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 3.895 | 2.457 |

43 | P_{w}(1,1) = 0 | P_{w}(2,7) = 10 | 1 | 0.67 | 1 | 0 | 2.429 |

44 | P_{w}(1,1) = 0 | P_{w}(2,6) = 8.04 | 1 | 0.833 | 1 | 0 | 2.429 |

45 | P_{w}(1,1) = 0 | P_{w}(2,5) = 7.55 | 1 | 0.887 | 1 | 0 | 2.429 |

46 | P_{w}(1,1) = 0 | P_{w}(2,4) = 7.13 | 1 | 0.939 | 1 | 0 | 2.429 |

47 | P_{w}(1,1) = 0 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | 0.025 | 2.428 |

48 | P_{w}(1,1) = 0 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | 0.619 | 2.414 |

49 | P_{w}(1,1) = 0 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 6.744 | 2.554 |

**Table 3.**Lookup table for the first prediction horizon (with P

_{w.M}(1) = 3.8 MW, P

_{w.M}(2) = 7.05 MW) for Case 2.

Scenario Combination | Optimal Results | ||||||
---|---|---|---|---|---|---|---|

n_{c} | P_{w}(n_{w},n_{s}) (MW) n_{w} = 1 | P_{w}(n_{w},n_{s}) (MW) n_{w} = 2 | β_{w}(1) - | β_{w}(2) - | Vs (pu) | P_{S} (MW) | Q_{S} (Mvar) |

1 | P_{w}(1,7) = 10 | P_{w}(2,7) = 10 | 0.379 | 0.288 | 1.06 | 0 | 2.352 |

2 | P_{w}(1,7) = 10 | P_{w}(2,6) = 8.04 | 0.379 | 0.359 | 1.06 | 0 | 2.352 |

3 | P_{w}(1,7) = 10 | P_{w}(2,5) = 7.55 | 0.379 | 0.382 | 1.06 | 0 | 2.352 |

4 | P_{w}(1,7) = 10 | P_{w}(2,4) = 7.13 | 0.379 | 0.405 | 1.06 | 0 | 2.352 |

5 | P_{w}(1,7) = 10 | P_{w}(2,3) = 6.67 | 0.379 | 0.432 | 1.06 | 0 | 2.352 |

6 | P_{w}(1,7) = 10 | P_{w}(2,2) = 6.07 | 0.379 | 0.475 | 1.06 | 0 | 2.352 |

7 | P_{w}(1,7) = 10 | P_{w}(2,1) = 0 | 0.668 | 1 | 1.06 | 0 | 2.38 |

8 | P_{w}(1,6) = 4.79 | P_{w}(2,7) = 10 | 0.791 | 0.288 | 1.06 | 0 | 2.352 |

9 | P_{w}(1,6) = 4.79 | P_{w}(2,6) = 8.04 | 0.791 | 0.359 | 1.06 | 0 | 2.352 |

10 | P_{w}(1,6) = 4.79 | P_{w}(2,5) = 7.55 | 0.791 | 0.382 | 1.06 | 0 | 2.352 |

11 | P_{w}(1,6) = 4.79 | P_{w}(2,4) = 7.13 | 0.791 | 0.405 | 1.06 | 0 | 2.352 |

12 | P_{w}(1,6) = 4.79 | P_{w}(2,3) = 6.67 | 0.791 | 0.433 | 1.06 | 0 | 2.352 |

13 | P_{w}(1,6) = 4.79 | P_{w}(2,2) = 6.07 | 0.791 | 0.475 | 1.06 | 0 | 2.352 |

14 | P_{w}(1,6) = 4.79 | P_{w}(2,1) = 0 | 1.000 | 1 | 1.06 | 1.896 | 2.391 |

15 | P_{w}(1,5) = 4.22 | P_{w}(2,7) = 10 | 0.898 | 0.288 | 1.06 | 0 | 2.352 |

16 | P_{w}(1,5) = 4.22 | P_{w}(2,6) = 8.04 | 0.898 | 0.359 | 1.06 | 0 | 2.352 |

17 | P_{w}(1,5) = 4.22 | P_{w}(2,5) = 7.55 | 0.898 | 0.382 | 1.06 | 0 | 2.352 |

18 | P_{w}(1,5) = 4.22 | P_{w}(2,4) = 7.13 | 0.898 | 0.405 | 1.06 | 0 | 2.352 |

19 | P_{w}(1,5) = 4.22 | P_{w}(2,3) = 6.67 | 0.898 | 0.433 | 1.06 | 0 | 2.352 |

20 | P_{w}(1,7) = 4.22 | P_{w}(2,2) = 6.07 | 0.898 | 0.475 | 1.06 | 0 | 2.352 |

21 | P_{w}(1,5) = 4.22 | P_{w}(2,1) = 0 | 1 | 1 | 1.06 | 2.471 | 2.398 |

22 | P_{w}(1,4) = 3.77 | P_{w}(2,7) = 10 | 1 | 0.29 | 1.06 | 0 | 2.352 |

23 | P_{w}(1,4) = 3.77 | P_{w}(2,6) = 8.04 | 1 | 0.361 | 1.06 | 0 | 2.352 |

24 | P_{w}(1,4) = 3.77 | P_{w}(2,5) = 7.55 | 1 | 0.384 | 1.06 | 0 | 2.352 |

25 | P_{w}(1,4) = 3.77 | P_{w}(2,4) = 7.13 | 1 | 0.407 | 1.06 | 0 | 2.352 |

26 | P_{w}(1,4) = 3.77 | P_{w}(2,3) = 6.67 | 1 | 0.435 | 1.06 | 0 | 2.352 |

27 | P_{w}(1,4) = 3.77 | P_{w}(2,2) = 6.07 | 1 | 0.478 | 1.06 | 0 | 2.352 |

28 | P_{w}(1,4) = 3.77 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 2.921 | 2.401 |

29 | P_{w}(1,3) = 3.33 | P_{w}(2,7) = 10 | 1 | 0.334 | 1.06 | 0 | 2.353 |

30 | P_{w}(1,3) = 3.33 | P_{w}(2,6) = 8.04 | 1 | 0.416 | 1.06 | 0 | 2.353 |

31 | P_{w}(1,3) = 3.33 | P_{w}(2,5) = 7.55 | 1 | 0.443 | 1.06 | 0 | 2.353 |

32 | P_{w}(1,3) = 3.33 | P_{w}(2,4) = 7.13 | 1 | 0.469 | 1.06 | 0 | 2.353 |

33 | P_{w}(1,3) = 3.33 | P_{w}(2,3) = 6.67 | 1 | 0.501 | 1.06 | 0 | 2.353 |

34 | P_{w}(1,3) = 3.33 | P_{w}(2,2) = 6.07 | 1 | 0.551 | 1.06 | 0 | 2.353 |

35 | P_{w}(1,3) = 3.33 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 3.364 | 2.409 |

36 | P_{w}(1,2) = 2.81 | P_{w}(2,7) = 10 | 1 | 0.386 | 1.06 | 0 | 2.355 |

37 | P_{w}(1,2) = 2.81 | P_{w}(2,6) = 8.04 | 1 | 0.480 | 1.06 | 0 | 2.355 |

38 | P_{w}(1,2) = 2.81 | P_{w}(2,5) = 7.55 | 1 | 0.512 | 1.06 | 0 | 2.355 |

39 | P_{w}(1,2) = 2.81 | P_{w}(2,4) = 7.13 | 1 | 0.542 | 1.06 | 0 | 2.355 |

40 | P_{w}(1,2) = 2.81 | P_{w}(2,3) = 6.67 | 1 | 0.579 | 1.06 | 0 | 2.355 |

41 | P_{w}(1,2) = 2.81 | P_{w}(2,2) = 6.07 | 1 | 0.636 | 1.06 | 0 | 2.355 |

42 | P_{w}(1,2) = 2.81 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 3.888 | 2.419 |

43 | P_{w}(1,1) = 0 | P_{w}(2,7) = 10 | 1 | 0.669 | 1.06 | 0 | 2.4 |

44 | P_{w}(1,1) = 0 | P_{w}(2,6) = 8.04 | 1 | 0.832 | 1.06 | 0 | 2.4 |

45 | P_{w}(1,1) = 0 | P_{w}(2,5) = 7.55 | 1 | 0.886 | 1.06 | 0 | 2.4 |

46 | P_{w}(1,1) = 0 | P_{w}(2,4) = 7.13 | 1 | 0.938 | 1.06 | 0 | 2.4 |

47 | P_{w}(1,1) = 0 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.06 | 0.02 | 2.399 |

48 | P_{w}(1,1) = 0 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.06 | 0.615 | 2.387 |

49 | P_{w}(1,1) = 0 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 6.731 | 2.504 |

**Table 4.**Lookup table for the first prediction horizon (with P

_{w.M}(1) = 3.8 MW, P

_{w.M}(2) = 7.05 MW) for Case 3.

Scenario Combination | Optimal Results | ||||||
---|---|---|---|---|---|---|---|

n_{c} | P_{w}(n_{w},n_{s}) (MW) n_{w} = 1 | P_{w}(n_{w},n_{s}) (MW) n_{w} = 2 | β_{w}(1) - | β_{w}(2) - | Vs (pu) | P_{S} (MW) | Q_{S} (Mvar) |

1 | P_{w}(1,7) = 10 | P_{w}(2,7) = 10 | 1 | 1 | 1 | −13.034 | 3.091 |

2 | P_{w}(1,7) = 10 | P_{w}(2,6) = 8.04 | 1 | 1 | 1 | −11.167 | 2.863 |

3 | P_{w}(1,7) = 10 | P_{w}(2,5) = 7.55 | 1 | 1 | 1 | −10.697 | 2.813 |

4 | P_{w}(1,7) = 10 | P_{w}(2,4) = 7.13 | 1 | 1 | 1 | −10.293 | 2.773 |

5 | P_{w}(1,7) = 10 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | −9.85 | 2.731 |

6 | P_{w}(1,7) = 10 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | −9.27 | 2.681 |

7 | P_{w}(1,7) = 10 | P_{w}(2,1) = 0 | 1 | 1 | 1 | −3.301 | 2.439 |

8 | P_{w}(1,6) = 4.79 | P_{w}(2,7) = 10 | 1 | 1 | 1 | −7.96 | 2.755 |

9 | P_{w}(1,6) = 4.79 | P_{w}(2,6) = 8.04 | 1 | 1 | 1 | −6.069 | 2.586 |

10 | P_{w}(1,6) = 4.79 | P_{w}(2,5) = 7.55 | 1 | 1 | 1 | −5.593 | 2.551 |

11 | P_{w}(1,6) = 4.79 | P_{w}(2,4) = 7.13 | 1 | 1 | 1 | −5.184 | 2.524 |

12 | P_{w}(1,6) = 4.79 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | −4.736 | 2.497 |

13 | P_{w}(1,6) = 4.79 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | −4.148 | 2.465 |

14 | P_{w}(1,6) = 4.79 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 1.9 | 2.419 |

15 | P_{w}(1,5) = 4.22 | P_{w}(2,7) = 10 | 1 | 1 | 1 | −7.399 | 2.728 |

16 | P_{w}(1,5) = 4.22 | P_{w}(2,6) = 8.04 | 1 | 1 | 1 | −5.505 | 2.566 |

17 | P_{w}(1,5) = 4.22 | P_{w}(2,5) = 7.55 | 1 | 1 | 1 | −5.029 | 2.533 |

18 | P_{w}(1,5) = 4.22 | P_{w}(2,4) = 7.13 | 1 | 1 | 1 | −4.619 | 2.507 |

19 | P_{w}(1,5) = 4.22 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | −4.17 | 2.481 |

20 | P_{w}(1,7) = 4.22 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | −3.582 | 2.452 |

21 | P_{w}(1,5) = 4.22 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 2.476 | 2.427 |

22 | P_{w}(1,4) = 3.77 | P_{w}(2,7) = 10 | 1 | 1 | 1 | −6.959 | 2.708 |

23 | P_{w}(1,4) = 3.77 | P_{w}(2,6) = 8.04 | 1 | 1 | 1 | −5.063 | 2.551 |

24 | P_{w}(1,4) = 3.77 | P_{w}(2,5) = 7.55 | 1 | 1 | 1 | −4.586 | 2.519 |

25 | P_{w}(1,4) = 3.77 | P_{w}(2,4) = 7.13 | 1 | 1 | 1 | −4.176 | 2.495 |

26 | P_{w}(1,4) = 3.77 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | −3.726 | 2.47 |

27 | P_{w}(1,4) = 3.77 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | −3.138 | 2.442 |

28 | P_{w}(1,4) = 3.77 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 2.927 | 2.435 |

29 | P_{w}(1,3) = 3.33 | P_{w}(2,7) = 10 | 1 | 1 | 1 | −6.527 | 2.69 |

30 | P_{w}(1,3) = 3.33 | P_{w}(2,6) = 8.04 | 1 | 1 | 1 | −4.629 | 2.538 |

31 | P_{w}(1,3) = 3.33 | P_{w}(2,5) = 7.55 | 1 | 1 | 1 | −4.151 | 2.508 |

32 | P_{w}(1,3) = 3.33 | P_{w}(2,4) = 7.13 | 1 | 1 | 1 | −3.741 | 2.484 |

33 | P_{w}(1,3) = 3.33 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | −3.29 | 2.461 |

34 | P_{w}(1,3) = 3.33 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | −2.701 | 2.434 |

35 | P_{w}(1,3) = 3.33 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 3.37 | 2.444 |

36 | P_{w}(1,2) = 2.81 | P_{w}(2,7) = 10 | 1 | 1 | 1 | −6.015 | 2.670 |

37 | P_{w}(1,2) = 2.81 | P_{w}(2,6) = 8.04 | 1 | 1 | 1 | −4.115 | 2.524 |

38 | P_{w}(1,2) = 2.81 | P_{w}(2,5) = 7.55 | 1 | 1 | 1 | −3.636 | 2.495 |

39 | P_{w}(1,2) = 2.81 | P_{w}(2,4) = 7.13 | 1 | 1 | 1 | −3.225 | 2.473 |

40 | P_{w}(1,2) = 2.81 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | −2.774 | 2.451 |

41 | P_{w}(1,2) = 2.81 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | −2.184 | 2.427 |

42 | P_{w}(1,2) = 2.81 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 3.895 | 2.457 |

43 | P_{w}(1,1) = 0 | P_{w}(2,7) = 10 | 1 | 1 | 1 | −3.239 | 2.591 |

44 | P_{w}(1,1) = 0 | P_{w}(2,6) = 8.04 | 1 | 1 | 1 | −1.325 | 2.478 |

45 | P_{w}(1,1) = 0 | P_{w}(2,5) = 7.55 | 1 | 1 | 1 | −0.843 | 2.457 |

46 | P_{w}(1,1) = 0 | P_{w}(2,4) = 7.13 | 1 | 1 | 1 | −0.429 | 2.442 |

47 | P_{w}(1,1) = 0 | P_{w}(2,3) = 6.67 | 1 | 1 | 1 | 0.025 | 2.428 |

48 | P_{w}(1,1) = 0 | P_{w}(2,2) = 6.07 | 1 | 1 | 1 | 0.619 | 2.414 |

49 | P_{w}(1,1) = 0 | P_{w}(2,1) = 0 | 1 | 1 | 1 | 6.744 | 2.554 |

**Table 5.**Lookup table for the first prediction horizon (with P

_{w.M}(1) = 3.8 MW, P

_{w.M}(2) = 7.05 MW) for Case 4.

Scenario Combination | Optimal Results | ||||||
---|---|---|---|---|---|---|---|

n_{c} | P_{w}(n_{w},n_{s}) (MW) n_{w} = 1 | P_{w}(n_{w},n_{s}) (MW) n_{w} = 2 | β_{w}(1) - | β_{w}(2)- | Vs (pu) | P_{S} (MW) | Q_{S} (Mvar) |

1 | P_{w}(1,7) = 10 | P_{w}(2,7) = 10 | 1 | 1 | 1.04 | −13.057 | 3.022 |

2 | P_{w}(1,7) = 10 | P_{w}(2,6) = 8.04 | 1 | 1 | 1.05 | −11.187 | 2.798 |

3 | P_{w}(1,7) = 10 | P_{w}(2,5) = 7.55 | 1 | 1 | 1.05 | −10.715 | 2.753 |

4 | P_{w}(1,7) = 10 | P_{w}(2,4) = 7.13 | 1 | 1 | 1.05 | −10.31 | 2.717 |

5 | P_{w}(1,7) = 10 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.05 | −9.866 | 2.679 |

6 | P_{w}(1,7) = 10 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.05 | −9.284 | 2.634 |

7 | P_{w}(1,7) = 10 | P_{w}(2,1) = 0 | 1 | 1 | 1.06 | −3.306 | 2.409 |

8 | P_{w}(1,6) = 4.79 | P_{w}(2,7) = 10 | 1 | 1 | 1.05 | −7.977 | 2.701 |

9 | P_{w}(1,6) = 4.79 | P_{w}(2,6) = 8.04 | 1 | 1 | 1.05 | −6.079 | 2.547 |

10 | P_{w}(1,6) = 4.79 | P_{w}(2,5) = 7.55 | 1 | 1 | 1.05 | −5.602 | 2.516 |

11 | P_{w}(1,6) = 4.79 | P_{w}(2,4) = 7.13 | 1 | 1 | 1.05 | −5.192 | 2.491 |

12 | P_{w}(1,6) = 4.79 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.05 | −4.742 | 2.466 |

13 | P_{w}(1,6) = 4.79 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.06 | −4.155 | 2.432 |

14 | P_{w}(1,6) = 4.79 | P_{w}(2,1) = 0 | 1 | 1 | 1.06 | 1.896 | 2.391 |

15 | P_{w}(1,5) = 4.22 | P_{w}(2,7) = 10 | 1 | 1 | 1.05 | −7.414 | 2.676 |

16 | P_{w}(1,5) = 4.22 | P_{w}(2,6) = 8.04 | 1 | 1 | 1.05 | −5.515 | 2.529 |

17 | P_{w}(1,5) = 4.22 | P_{w}(2,5) = 7.55 | 1 | 1 | 1.05 | −5.037 | 2.499 |

18 | P_{w}(1,5) = 4.22 | P_{w}(2,4) = 7.13 | 1 | 1 | 1.05 | −4.626 | 2.475 |

19 | P_{w}(1,5) = 4.22 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.05 | −4.176 | 2.452 |

20 | P_{w}(1,7) = 4.22 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.06 | −3.588 | 2.42 |

21 | P_{w}(1,5) = 4.22 | P_{w}(2,1) = 0 | 1 | 1 | 1.06 | 2.471 | 2.398 |

22 | P_{w}(1,4) = 3.77 | P_{w}(2,7) = 10 | 1 | 1 | 1.05 | −6.974 | 2.658 |

23 | P_{w}(1,4) = 3.77 | P_{w}(2,6) = 8.04 | 1 | 1 | 1.05 | −5.072 | 2.515 |

24 | P_{w}(1,4) = 3.77 | P_{w}(2,5) = 7.55 | 1 | 1 | 1.05 | −4.594 | 2.487 |

25 | P_{w}(1,4) = 3.77 | P_{w}(2,4) = 7.13 | 1 | 1 | 1.05 | −4.183 | 2.464 |

26 | P_{w}(1,4) = 3.77 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.06 | −3.733 | 2.437 |

27 | P_{w}(1,4) = 3.77 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.06 | −3.143 | 2.412 |

28 | P_{w}(1,4) = 3.77 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 2.921 | 2.401 |

29 | P_{w}(1,3) = 3.33 | P_{w}(2,7) = 10 | 1 | 1 | 1.05 | −6.541 | 2.642 |

30 | P_{w}(1,3) = 3.33 | P_{w}(2,6) = 8.04 | 1 | 1 | 1.05 | −4.637 | 2.504 |

31 | P_{w}(1,3) = 3.33 | P_{w}(2,5) = 7.55 | 1 | 1 | 1.05 | −4.158 | 2.476 |

32 | P_{w}(1,3) = 3.33 | P_{w}(2,4) = 7.13 | 1 | 1 | 1.05 | −3.747 | 2.455 |

33 | P_{w}(1,3) = 3.33 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.06 | −3.297 | 2.428 |

34 | P_{w}(1,3) = 3.33 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.06 | −2.706 | 2.405 |

35 | P_{w}(1,3) = 3.33 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 3.364 | 2.409 |

36 | P_{w}(1,2) = 2.81 | P_{w}(2,7) = 10 | 1 | 1 | 1.05 | −6.028 | 2.624 |

37 | P_{w}(1,2) = 2.81 | P_{w}(2,6) = 8.04 | 1 | 1 | 1.05 | −4.122 | 2.491 |

38 | P_{w}(1,2) = 2.81 | P_{w}(2,5) = 7.55 | 1 | 1 | 1.05 | −3.643 | 2.465 |

39 | P_{w}(1,2) = 2.81 | P_{w}(2,4) = 7.13 | 1 | 1 | 1.06 | −3.232 | 2.439 |

40 | P_{w}(1,2) = 2.81 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.06 | −2.78 | 2.42 |

41 | P_{w}(1,2) = 2.81 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.06 | −2.189 | 2.398 |

42 | P_{w}(1,2) = 2.81 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 3.888 | 2.419 |

43 | P_{w}(1,1) = 0 | P_{w}(2,7) = 10 | 1 | 1 | 1.05 | −3.249 | 2.551 |

44 | P_{w}(1,1) = 0 | P_{w}(2,6) = 8.04 | 1 | 1 | 1.06 | −1.332 | 2.443 |

45 | P_{w}(1,1) = 0 | P_{w}(2,5) = 7.55 | 1 | 1 | 1.06 | −0.849 | 2.425 |

46 | P_{w}(1,1) = 0 | P_{w}(2,4) = 7.13 | 1 | 1 | 1.06 | −0.435 | 2.411 |

47 | P_{w}(1,1) = 0 | P_{w}(2,3) = 6.67 | 1 | 1 | 1.06 | 0.02 | 2.399 |

48 | P_{w}(1,1) = 0 | P_{w}(2,2) = 6.07 | 1 | 1 | 1.06 | 0.615 | 2.387 |

49 | P_{w}(1,1) = 0 | P_{w}(2,1) = 0 | 1 | 1 | 1.07 | 6.731 | 2.504 |

Case | P_{loss} Average (kW) | F_{1} ($/day) | F_{2} ($/day) | F_{3} ($/day) | F_{4} ($/day) | F ($/day) |
---|---|---|---|---|---|---|

1 | 29.61 | 7654.76 | 35.54 | 1540.61 | 773.73 | 5304.88 |

2 | 26 | 7651.96 | 31.2 | 1539.07 | 766.30 | 5315.39 |

3 | 97.94 | 14,007.64 | 117.52 | −4730.28 | 821.61 | 17,798.78 |

4 | 88.95 | 14,007.63 | 106.74 | −4741.06 | 811.02 | 17,830.94 |

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## Share and Cite

**MDPI and ACS Style**

Mohagheghi, E.; Gabash, A.; Li, P. A Framework for Real-Time Optimal Power Flow under Wind Energy Penetration. *Energies* **2017**, *10*, 535.
https://doi.org/10.3390/en10040535

**AMA Style**

Mohagheghi E, Gabash A, Li P. A Framework for Real-Time Optimal Power Flow under Wind Energy Penetration. *Energies*. 2017; 10(4):535.
https://doi.org/10.3390/en10040535

**Chicago/Turabian Style**

Mohagheghi, Erfan, Aouss Gabash, and Pu Li. 2017. "A Framework for Real-Time Optimal Power Flow under Wind Energy Penetration" *Energies* 10, no. 4: 535.
https://doi.org/10.3390/en10040535