# Recent Approaches of Forecasting and Optimal Economic Dispatch to Overcome Intermittency of Wind and Photovoltaic (PV) Systems: A Review

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## Abstract

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## 1. Introduction

## 2. Forecasting of Wind Power Generation

#### 2.1. Wind Power Generation Fundamentals

#### 2.2. Weibull Distribution (WD) for Wind Power Forecasting

#### 2.3. Review of Wind Power Forecasting without NN

#### 2.4. Review of Wind Power Forecasting with the Incorporation of NN

_{t}denotes the forecasted wind power at time t; P

_{t−k}denotes all historical wind power data between t − k and t; W denotes the wind speed data from numerical weather prediction (NWP); h represents all wind speed components, which contains horizontal and vertical information, in general. They concluded that based on the forecasted multidimensional random variables and joint distribution function, the output generation scenario for multiple wind farm power could be achieved.

## 3. Forecasting Solar PV Power Generation

#### 3.1. Solar PV Power Generation Fundamentals

#### 3.2. Weibull Distribution (WD) for Solar (PV) Forecasting

#### 3.3. Review of PV Power Forecasting without NN

#### 3.4. Review of PV Power Forecasting with the Incorporation of NN

^{2}, etc.) for achieving better accuracy of the forecast. The authors concluded that the pure linear transfer function emerged as the worst performer amongst the tested transfer functions, and the proposed model predicted the GHI for the specified period with relatively good accuracy.

## 4. Optimal Economic Dispatching (OED) using PSO

#### 4.1. A Brief Review of PSO Algorithm

_{1}, c

_{2}acceleration coefficient for the cognitive and social components, respectively [95,96,97].

#### 4.2. Review of PSO Applied to OED Incorporating RESs

_{t}, P

_{D}, and P

_{l}are total power, power demand, and power losses respectively, and the second portion of Equation (35) presents the generation limits that must be met to keep the system balanced.

#### 4.3. Constraints Handling by PSO

#### 4.3.1. Compensation for Load and Voltage Variation

#### 4.3.2. Control of Frequency Fluctuations

#### 4.3.3. Regulation of Ramp-Rate Limits

#### 4.3.4. Storage Mechanism as a Solution

- (i)
- Thermal Power Generation is dependable, but it presents major issues such as a rise in carbon emissions, an increase in fuel cost, and special consideration is required in system coordination. We cannot use thermal plants as backup generation only as they take a significant amount of time in their start-up, and fuel cost for spinning reserve contributes to disturbing the economic dispatch that is a major concern in current power system.
- (ii)
- Battery Storage compensation through batteries is a modern age replacement of backup thermal plants. It requires a properly designed storage system to provide adequate power to supply the load when power from RESs is lesser than the demand. The battery storage system also provides an additional benefit of peak shaving as RES starts supplying an excessive supply for storage [117,118]. The major concern is of designing a properly designed storage system to achieve the optimum cost-saving and stability of the power system.

_{phase}of the battery side, V

_{open-circuit}of battery, self-discharge resistance, and over-voltage resistance, respectively. Based on the presented results, the authors concluded that the presented system offered very fast compensation for active power that improved the dynamic stability of the power system. The results also validated the significance of the BESS based PSO mechanism for optimum sizing of the storage system.

^{b}” and state of battery charge “SOC

^{b}”, power requests calculus, and E-broker auction algorithms. The value of P

^{b}was computed using Equation (48).

^{b}was used to trigger the power requests from the system. The method avoided the scenario faced by researchers where they had to impose a division on the battery peak shaving and energy shifting as it optimized suitable variables and prevented the ESS from capacity wasting. The presented model also prevented power losses and unforeseen peaks occurring due to the supply or absorption of power by optimizing the boundaries of battery behavior. However, the system was not effective for active distribution systems (ADSs).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ANN | Artificial Neural Networks |

ANFIS | Artificial Neuro Fuzzy Inference System |

CNN | Convolutional Neural Networks |

CEED | Combined Emission Economic Dispatch |

DED | Dynamic Economic Dispatch |

DWPSO | Double Weighted Particle Swarm Optimization |

DNI | Direct Normal Solar Irradiation |

ED | Economic Dispatch |

EDP | Economic Dispatch Problem |

ERCOT | Electric Reliability Council of Texas |

ESRMC | Energy and Spinning Reserve Market Cleaning |

EMD | Empirical Mode Decomposition |

GHI | Graphical Horizontal Solar Irradiation |

GP | Genetic Programming |

HANN | Hybrid Artificial Neural Network |

ICA | Independent Component Analysis |

LUBE | Lower-Upper Bound Estimation |

MLFFNN | Multi-layer Feed-forward Neural Networks |

MAPE | Model Predictive Control |

MG | Micro-grid |

MMG | Multi Micro-grid |

MOM | Method of Moments |

NN | Neural Networks |

NWP | Numerical Weather Predictor |

OED | Optimal Economic Dispatch |

PSO | Particle Swarm Optimization |

PV | Photovoltaics |

PDEM | Part Density Energy Method |

PI | Prediction Interval |

PCA | Principal Component Analysis |

RES | Renewable Energy Sources |

RF | Reliability Factor |

RBFNN | Radial Basis Function Neural Networks |

RMSE | Root Mean Squared Error |

SVM | Support Vector Machine |

SD | Standard Deviation |

SVR | Support Vector Regression |

SSER | Small-Scale Energy Resource |

WD | Weibull Distribution |

WT | Wind Turbine |

WTG | Wind Turbine Generator |

WEC | World Energy Council |

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Characteristics | Weibull Distribution (WD) | Rayleigh Distribution (RD) | Gaussian/Normal Distribution (ND) | Ref. No. |
---|---|---|---|---|

Mathematical representation & parameters | $f(v)=\{\begin{array}{l}\frac{\beta}{{v}_{0}}{\left(\frac{v}{{v}_{0}}\right)}^{\beta -1}{e}^{-{\left(\frac{v}{{v}_{0}}\right)}^{\beta}};\hspace{0.17em}v\ge 0\\ 0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}v<0\end{array}$ $\mathrm{where}\beta $$>0\mathrm{is}\mathrm{the}\mathrm{shape}\mathrm{parameter},\mathrm{and}{v}_{0}$> 0 is the scale parameter of the distribution. | $f(v)=\{\begin{array}{l}\frac{v}{{\sigma}^{2}}{e}^{-{v}^{2}/2{\sigma}^{2}},\hspace{0.17em}\hspace{0.17em}v>0\\ 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}v\le 0\end{array}$ $\mathrm{where}\sigma $ is the scale parameter of the distribution. | $f(v)=\frac{1}{\sqrt{2\pi {\sigma}^{2}}}{e}^{-\frac{{\left(v-\mu \right)}^{2}}{2{\sigma}^{2}}}$ $\mathrm{where}\mu $ is the mean, whereas σ is the standard deviation. | [41,42,43,44] |

Flexibility | WD is very flexible as a small sample size; the estimated shape of the distribution may be altered considerably. | Not flexible as a response to the out of range parameters are strict. | Not flexible as the shape doesn’t vary. | [45,46,47,48,49] |

Accuracy | Fatigue test results follow WD, showing it to be more accurate. It is effective for both values above and below the sample size N. | Close to WD. | Effective only for values below the sample size N. | |

Reliability | WD is more reliable even in situations where distribution parameters (shape and scale) tend to vary. | RD loses its effectiveness in situations where variables undergo variation. | Reliability in ND suffers severely at the hands of variation in variables. |

**Table 2.**Wind speed and power prediction with Weibull distribution (WD) and without incorporating neural networks (NN).

Wind Speed Probability Distribution | Wind Power Distribution | Explanation |
---|---|---|

$f\left(v\right)=\left(\frac{\beta}{{v}_{0}}\right){\left(\frac{v}{{v}_{0}}\right)}^{\left(\beta -1\right)}{e}^{-{\left(\frac{v}{{v}_{0}}\right)}^{\beta}}$ | ${P}_{e}\left(v\right)={P}_{r}\times \{\begin{array}{l}0,v<{v}_{ci}\hspace{0.17em}or\hspace{0.17em}v>{v}_{co}\\ {P}_{cinr}\left(v\right),{v}_{ci}\le v\le {v}_{r}\\ 1,{v}_{r}\le v\le {v}_{co}\end{array}$ | where ${P}_{e}\left(v\right)$ is electric output power of WT; ${v}_{ci}\hspace{0.17em}$, ${v}_{co}$ and ${v}_{r}$ represent cut-in, cut-out, and the rated wind speed, respectively [50,51]. |

$f(v)=\{\begin{array}{l}\frac{v}{{\sigma}^{2}}{e}^{-{v}^{2}/2{\sigma}^{2}},\hspace{0.17em}\hspace{0.17em}v>0\\ 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}v\le 0\end{array}$ | $\begin{array}{l}P(v)=\frac{\beta}{{v}_{0}}{\left(\frac{v}{{v}_{0}}\right)}^{\beta -1}{e}^{-\left(\frac{v}{{v}_{0}}\right)\beta}\\ \mathrm{with}\hspace{0.17em}\beta =1/3;\hspace{0.17em}\hspace{0.17em}{v}_{0}=b;\end{array}$ | The authors suggest that wind velocity follows Rayleigh distribution, whereas the power follows Weibull distribution [52]. |

$f\left(v\right)=\left(\frac{\beta}{{v}_{0}}\right){\left(\frac{v}{{v}_{0}}\right)}^{\left(\beta -1\right)}{e}^{-{\left(\frac{v}{{v}_{0}}\right)}^{\beta}}$ | $P(v)=\{\begin{array}{l}{P}_{r}.\left(\frac{{v}_{co}^{n}-{v}^{n}}{{v}_{ci}^{n}-{v}_{r}^{n}}\right),{v}_{ci}\le v\le {v}_{r}\\ {P}_{r},{v}_{r}\le v\le {v}_{co}\\ 0,\mathrm{otherwise}\end{array}$ | where $P(v)$ is generated power at speed $v$, and ${v}_{ci}$, ${v}_{co}$ and ${v}_{r}$ are wind turbine parameters [53]. |

$f\left(v\right)=\left(\frac{\beta}{{v}_{0}}\right){\left(\frac{v}{{v}_{0}}\right)}^{\left(\beta -1\right)}{e}^{-{\left(\frac{v}{{v}_{0}}\right)}^{\beta}}$ | $P(v)=\{\begin{array}{l}0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}v<{v}_{ci}\hspace{0.17em}or\hspace{0.17em}v>{v}_{co}\\ {P}_{r}\left(\frac{v-{v}_{i}}{{v}_{r}-{v}_{i}}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{v}_{ci}\le v\le {v}_{r}\\ {P}_{r},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{v}_{r}\le v\le {v}_{co}\end{array}$ | Here $\beta $ and ${v}_{0}$ are the shape and scale parameters; $P$ is power output against wind speed [54,56]. |

$f\left(v\right)=\left(\frac{\beta}{{v}_{0}}\right){\left(\frac{v}{{v}_{0}}\right)}^{\left(\beta -1\right)}{\left(e\right)}^{-{\left(\frac{v}{{v}_{0}}\right)}^{\beta}}$ | $P=\frac{\rho A}{2}{v}^{3}{c}_{p}\left(\lambda \right)$ | [55] |

${P}_{m,R}={\displaystyle \sum _{j=1}^{n}\left[\frac{1}{2}\rho {v}_{m,j}^{3}f\left({v}_{j}\right)\right]}$ | [57] | |

${f}_{R}\left(v\right)=\left(\frac{\pi}{2}\right)\left(\frac{v}{{v}_{m}^{2}}\right)\mathrm{exp}\left[-\left(\frac{\pi}{4}\right){\left(\frac{v}{{v}_{m}^{2}}\right)}^{\beta}\right]$ | ${P}_{R}=\frac{3}{\pi}\rho {v}_{m}^{3}$ |

Resource/Power Forecasting Model | Prediction Error | Description |
---|---|---|

$P=\frac{\rho A}{2}{v}^{3}{c}_{p}$ | $MAPE=\frac{1}{n}{\displaystyle {\sum}_{t=1}^{n}\frac{{y}_{i(ANN)}-{y}_{k(measured)}}{{y}_{k(measured)}}}$ | Here $\overline{v}$, $\sigma $ and $\mathsf{\Gamma}$ are mean wind speed, standard deviation, and gamma function, respectively. Also, $n$, ${y}_{i(ANN)}$ and ${y}_{k(measured)}$ are total input and output pairs, forecasted wind speed, and actual wind speed for one hour, respectively [61]. |

$\stackrel{\_\_}{P}=\hspace{0.17em}\frac{1}{2n}\rho {\displaystyle \sum _{i=1}^{n}\stackrel{\_\_}{{v}^{3}}}$ | $\begin{array}{l}MAPE=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left|\frac{{P}_{i,pred}-{P}_{i,means}}{{P}_{i,means}}\right|}\times 100\\ MABE=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left|{P}_{i,pred}-{P}_{i,means}\right|}\\ {R}^{2}=\frac{{\displaystyle {\sum}_{i=1}^{X}{\left({P}_{i,means}-{P}_{means,avg}\right)}^{2}-{\displaystyle {\sum}_{i=1}^{n}{\left({P}_{i,pred}-{P}_{i,means}\right)}^{2}}}}{{{\displaystyle {\sum}_{i=1}^{X}\left({P}_{i,means}-{P}_{means,avg}\right)}}^{2}}\end{array}$ | Here n denotes the specified time period P_{i,pred} and P_{i,means} are predicted and calculated wind powers [64]. |

$\begin{array}{l}\overline{v}=\frac{1}{n}{\displaystyle \sum _{i=1}^{h}{v}_{i}}\hspace{0.17em}\\ \hspace{0.17em}\sigma =\hspace{0.17em}{\left[\left(\frac{1}{n-1}{\displaystyle \sum _{i=1}^{n}\left({v}_{i}-\overline{v}\right)2}\right)\right]}^{0.5}\end{array}$$P=\frac{\rho A}{2}{v}^{3}{c}_{p}$ | $\begin{array}{l}{R}^{2}=\frac{{\displaystyle {\sum}_{i=1}^{n}{\left({X}_{i,act}-{X}_{act,avg}\right)}^{2}-{\displaystyle {\sum}_{i=1}^{n}{\left({X}_{i,est}-{X}_{i,act}\right)}^{2}}}}{{{\displaystyle {\sum}_{i=1}^{n}\left({X}_{i,act}-{X}_{act,avg}\right)}}^{2}}\frac{{\partial}^{2}\mathsf{\Omega}}{\partial {u}^{2}}\\ MAPE=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left|\frac{{X}_{i,est}-{X}_{i,act}}{{X}_{i,act}}\right|}\times 100\\ MABE=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left|{X}_{i,est}-{X}_{i,act}\right|}\\ RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({X}_{i,est}-{X}_{i,act}\right)}^{2}}}\end{array}$ | [65] |

${P}_{t}=F\left({P}_{t-k},{W}_{t-k}^{h},{W}_{t}^{h},{W}_{t+m}^{h}\right)$ | $\sqrt[\begin{array}{l}\mathrm{min}\\ \theta =weights\end{array}]{{{\displaystyle \sum _{t=1}^{n}\left[{y}_{t}-F\left({X}_{t,i},\theta \right)\right]}}^{2}}$ | Here $n$ denotes the specified time period, and ${P}_{i,pred}$ and ${P}_{i,means}$ are predicted and calculated wind powers, respectively [66]. |

Solar Distribution Functions for Prediction | PV Power Production | Reference |
---|---|---|

$\begin{array}{l}f\left(t\right)=\hspace{0.17em}\frac{\beta}{\eta}{\left(\frac{t-\gamma}{\eta}\right)}^{\left(\beta -1\right)}\times {e}^{-{\left(\frac{t-\gamma}{\eta}\right)}^{\beta}}\\ R\left(t\right)={e}^{-{\left(\frac{t-\gamma}{\eta}\right)}^{\beta}}\end{array}$ $f\left(t\right)=\frac{\mathsf{\Gamma}\left(\frac{\nu +1}{2}\right)}{\sqrt{\nu \pi}\mathsf{\Gamma}\left(\frac{\nu}{2}\right)}{\left(1+\frac{{t}^{2}}{\nu}\right)}^{\left(-\frac{\nu +1}{2}\right)}$ | $P=\gamma S\eta \left(1-n\Delta t\right)$ | $\mathrm{Here}R(t)$$,\beta $$,\gamma $$\mathrm{and}\eta $ are the reliability function, slope, location and scale parameters, respectively [78]$;P$$,\gamma $$,\eta $$,S$$,\Delta t$$\mathrm{and}n$ stand for solar active power, amount of solar irradiance, efficiency, the total area of PV modules, PV cell temperature’s forecast error, and co-efficient of the temperature, respectively. |

$f\left(\overline{X};x,k\right)=\{\begin{array}{l}\frac{\beta}{x}{\left(\frac{\overline{X}}{x}\right)}^{\left(k-1\right)}\hspace{0.17em}{e}^{-{\left(\frac{\overline{X}}{x}\right)}^{\beta}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\ge 0\\ 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x<0\end{array}$ | $\begin{array}{l}P=\underset{P\le {P}_{SET}}{\mathrm{max}}\left\{\eta {P}_{PV}\left(V\right)\right\}\\ P=\gamma S\eta \left(1-n\Delta t\right)\end{array}$ | $\mathrm{Here}P$$,{P}_{PV}(V)$ and η are active power, active power-voltage relationship, and converter efficiency, respectively [80]. |

$f(x)=\frac{\beta}{{x}_{0}}{\left(\frac{x}{{x}_{0}}\right)}^{\beta -1}\mathrm{exp}\left[-{\left(\frac{x}{{x}_{0}}\right)}^{\beta}\right]$ $\beta ={\left[\frac{{\displaystyle \sum _{i=1}^{n}{T}_{I}^{k}\mathrm{ln}(x)}}{{\displaystyle \sum _{i=1}^{n}{x}^{k}}}-\frac{{\displaystyle \sum _{i=1}^{n}\mathrm{ln}(x)}}{n}\right]}^{-1}\&\hspace{0.17em}{x}_{0}={\left[\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{x}^{k}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}$ | $P=\gamma S\eta \left(1-n\Delta t\right)$ | [81] |

${f}_{\left(x\right)}=\frac{k\mathsf{\Gamma}\left(1+1/k\right)}{\mu}{\left(\frac{x\mathsf{\Gamma}\left(1+1/k\right)}{\mu}\right)}^{k-1}\times {e}^{-{\left(\frac{k\mathsf{\Gamma}\left(1+1/k\right)}{\mu}\right)}^{k}}$ | $\mathrm{Here}{f}_{(x)}$ is the solar irradiance function [83]. |

Model Used for Power Production or Resource/Power Forecasting | Prediction Error | Reference |
---|---|---|

$\begin{array}{l}{P}_{pv}={P}_{{n}_{pv}}\times \left(\frac{G}{{G}_{STC}}\right)\\ \times \left(1+k\left({T}_{cell}-{T}_{STC}\right)\right)\end{array}$ | $\begin{array}{l}\%V{F}_{k}={\displaystyle \sum _{i=1}^{n-1}\frac{\left|{V}_{k,i+1}-{V}_{k,i}\right|}{(n-1)}}\times 100\\ \%VIF=\sqrt[2]{\frac{1-\sqrt[2]{3-6\beta}}{1+\sqrt[2]{3-6\beta}}}\times 100\end{array}$ | $\mathrm{Here}{P}_{{n}_{pv}}$$,G$$,{G}_{STC}$$\mathrm{and}k$$\mathrm{are}\mathrm{rated}\mathrm{power}\mathrm{of}\mathrm{PV}\mathrm{system},\mathrm{solar}\mathrm{irradiance}\mathrm{on}\mathrm{PV}\mathrm{surface},\mathrm{solar}\mathrm{irradiance}\mathrm{in}\mathrm{standard}\mathrm{test}\mathrm{conditions},\mathrm{and}\mathrm{efficiency}\mathrm{temperature}\mathrm{coefficient},\mathrm{respectively}.\mathrm{Also},VF$$\mathrm{and}VIF$ are voltage fluctuations and voltage imbalance factor, respectively [85]. |

$P=\gamma S\eta \left(1-n\Delta t\right)$ $\begin{array}{l}R={R}_{0}\left(1-0.75{n}^{3.4}\right)\\ {R}_{0}=\hspace{0.17em}990\mathrm{sin}\varphi -30\\ \varphi =\frac{{\varphi}_{tp}+{\varphi}_{p}}{2}\end{array}$ | $\begin{array}{l}MAE=\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}\left|GH{I}_{(measured)}-GH{I}_{(predicted)}\right|}\\ MAPE=\left(\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}\left|GH{I}_{(measured)}-GH{I}_{(predicted)}\right|}\right)\\ RMSE={\left(\frac{1}{n}{{\displaystyle {\sum}_{i=1}^{n}\left|GH{I}_{(measured)}-GH{I}_{(predicted)}\right|}}^{2}\right)}^{\frac{1}{2}}\\ {R}^{2}=\left(1-\left(\frac{{{\displaystyle {\sum}_{i=1}^{n}\left|GH{I}_{(measured)}-GH{I}_{(predicted)}\right|}}^{2}}{GH{I}_{(measured)}}\right)\right)\hspace{0.17em}\end{array}$ | Here ${R}^{2}$$,n$$,{R}_{0}$$,{\varphi}_{tp}$$\mathrm{and}{\varphi}_{p}$$\mathrm{are}\mathrm{solar}\mathrm{radiation},\mathrm{cloud}\mathrm{cover},\mathrm{clear}\mathrm{sky}\mathrm{insolation},\mathrm{solar}\mathrm{elevation}\mathrm{angle}\mathrm{and}\mathrm{for}\mathrm{previous}\mathrm{and}\mathrm{current}\mathrm{hours},\mathrm{respectively}.{R}^{2}$$\mathrm{is}\mathrm{the}\mathrm{coefficient}\mathrm{of}\mathrm{determination}.\mathrm{Also},GH{I}_{(measured)}$$\mathrm{and}GH{I}_{(predicted)}$ are measured and predicted solar irradiations respectively [86]. |

$P=\gamma S\eta \left(1-n\Delta t\right)$ $\begin{array}{l}{k}_{t}=\frac{H}{{H}_{0}}\\ {H}_{0}={I}_{SC}{E}_{0}\\ \times \left(\mathrm{sin}\delta \mathrm{sin}\phi +\mathrm{cos}\delta \mathrm{cos}\phi \mathrm{cos}\omega \right)\end{array}$ | $\begin{array}{l}RMSE=\sqrt{{\displaystyle {\sum}_{i=1}^{N}\frac{{\left({y}_{i}-{x}_{i}\right)}^{{}_{2}}}{N}}}\\ MBE={\displaystyle {\sum}_{i=1}^{N}\frac{\left({y}_{i}-{x}_{i}\right)}{N}}\\ MPE={\displaystyle {\sum}_{i=1}^{N}\left(\frac{\left({y}_{i}-{x}_{i}\right)}{N{x}_{i}}\right)}\times 100\\ {R}^{2}=\frac{{\displaystyle {\sum}_{i=1}^{N}{\left({y}_{i}-{x}_{i}\right)}^{{}_{2}}}}{{\displaystyle {\sum}_{i=1}^{N}{\left({y}_{i}-{\overline{y}}_{i}\right)}^{{}_{2}}}}\hspace{0.17em}\end{array}$ | Here ${k}_{t}$$,H$$\mathrm{and}{H}_{0}$$\mathrm{are}\mathrm{clearance}\mathrm{index},\mathrm{global}\mathrm{ground}\mathrm{radiation},\mathrm{and}\mathrm{extraterrestrial}\mathrm{global}\mathrm{radiation},\mathrm{respectively}.\mathrm{Also},{y}_{i}$$\mathrm{and}{x}_{i}$$,\overline{y}$$\mathrm{and}\overline{x}$ are estimated, measured, and average estimated and measured values, respectively [87]. |

$P=\gamma S\eta \left(1-n\Delta t\right)$ | $RMSE=\sqrt{\frac{1}{n}\left({\displaystyle \sum _{i=1}^{n}{\left({X}_{hist,i}-{X}_{pred,i}\right)}^{2}}\right)}$ $\begin{array}{l}MAPE=\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left|\frac{{X}_{hist,i}-{X}_{pred,i}}{{X}_{hist,i}}\right|}\right)\times 100\%\\ R=\frac{{\displaystyle \sum _{i=1}^{n}\left({X}_{hist,i}-{\overline{X}}_{hist}\right)\left({X}_{pred,i}-{\overline{X}}_{pred,i}\right)}}{\sqrt{{\displaystyle \sum _{i=1}^{n}{\left({X}_{hist,i}-{\overline{X}}_{hist}\right)}^{2}{\left({X}_{hist,i}-{\overline{X}}_{hist,i}\right)}^{2}}}}\end{array}$ | hist and pred historical and predicted results [88]$.P$$,\gamma $$,\eta $$,S$$,\Delta t$$\mathrm{and}n$ stand for solar active power, amount of solar irradiance, efficiency, total area of PV modules, PV cell temperature’s forecast error, and co-efficient of the temperature, respectively. |

**Table 6.**Constraints and their solutions for optimal economic (OED) of Renewable Energy Sources (RESs).

Constraints | Presented Model | Objective Function | Reference |
---|---|---|---|

Load and Voltage Variations | $\begin{array}{l}{P}_{D}^{a}+{P}_{L}-{\displaystyle \sum _{i=1}^{{N}_{G}}{P}_{Gi}}=0\\ {P}_{D}^{a}={P}_{D}^{t}-\left({P}_{s}+{P}_{W}\right)\end{array}$ | $\begin{array}{l}\mathrm{min}{F}_{f}\left({P}_{Gi}\right)\\ ={\displaystyle \sum _{i=1}^{{N}_{G}}\left({a}_{i}+{b}_{i}{P}_{Gi}+{c}_{i}{P}_{Gi}^{2}\right)}\end{array}$ | $\mathrm{Here}{P}_{D}^{a}$$,{P}_{L}$$,{P}_{s}$$\mathrm{and}{P}_{W}$ are the load demand, transmission losses, solar and wind powers, respectively [103,104]. |

Frequency Fluctuations | $\begin{array}{l}{P}_{force,t,h}={\displaystyle \sum _{k=1}^{{K}_{t,h}}{P}_{rate,k}^{{U}_{n}}+{\displaystyle \sum _{j=1}^{{M}_{t,h}}{P}_{rate,j}^{M}}}\\ {P}_{c,\mathrm{max},t,h}={\displaystyle \sum _{i=1}^{{N}_{t,h}}{P}_{rate,i}}\end{array}$ | $\mathrm{min}{\displaystyle \sum _{t=1}^{T}\left\{{\displaystyle \sum _{g=1}^{{N}_{g}}\left(\begin{array}{l}{a}_{g}\cdot {\left({P}_{g,t}^{G,ref}\right)}^{2}+\\ {b}_{g}\cdot {P}_{g,t}^{G,ref}+{c}_{g}\end{array}\right)}\right\}}$ | [105,106,107,108,109] |

Ramp-Rate Limits | $\begin{array}{l}{F}_{i}\left({P}_{i}\right)={\alpha}_{i}{P}_{i}^{2}+{\beta}_{i}{P}_{i}+{\gamma}_{i}+\\ {\epsilon}_{i}\mathrm{exp}\left({\delta}_{i}\times {P}_{i}\right)\end{array}$ | $\begin{array}{l}\underset{{P}_{i},U{s}_{j},P{s}_{k}}{\mathrm{min}}{\displaystyle \sum _{i=1}^{n}\left({F}_{i}\left({P}_{i}\right)+{E}_{i}\left({P}_{i}\right)\right)}+\\ {\displaystyle \sum _{j=1}^{m}{G}_{j}-U{s}_{j}}-{\displaystyle \sum _{k}^{NB}P{s}_{k}}-{\displaystyle \sum _{j=1}^{m}U{s}_{j}}\end{array}$ | [110,111,112,113] |

Storage Mechanism | $\begin{array}{l}{P}_{i}-{V}_{i}{\displaystyle \sum _{j=1}^{N}{V}_{j}\left({G}_{ij}\mathrm{cos}{\delta}_{ij}+{B}_{ij}\mathrm{sin}{\delta}_{ij}\right)}=0\\ {Q}_{i}-{V}_{i}{\displaystyle \sum _{j=1}^{N}{V}_{j}\left({G}_{ij}\mathrm{sin}{\delta}_{ij}-{B}_{ij}\mathrm{cos}{\delta}_{ij}\right)}=0\end{array}\}$ | $\begin{array}{l}\mathrm{min}{f}_{1}={\displaystyle \sum _{i=1}^{5}\mathrm{Pr}o{b}_{i}\cdot Cos{t}_{i}}\\ \mathrm{min}{f}_{2}={\displaystyle \sum _{k=1}^{n}{\left(\frac{{V}_{k}-{V}_{k}^{spec}}{\Delta {V}_{k}^{\mathrm{max}}}\right)}^{2}}\end{array}\}$ | [114,115,116,117,118,119,120,121,122,123,124,125] |

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**MDPI and ACS Style**

Ellahi, M.; Abbas, G.; Khan, I.; Koola, P.M.; Nasir, M.; Raza, A.; Farooq, U.
Recent Approaches of Forecasting and Optimal Economic Dispatch to Overcome Intermittency of Wind and Photovoltaic (PV) Systems: A Review. *Energies* **2019**, *12*, 4392.
https://doi.org/10.3390/en12224392

**AMA Style**

Ellahi M, Abbas G, Khan I, Koola PM, Nasir M, Raza A, Farooq U.
Recent Approaches of Forecasting and Optimal Economic Dispatch to Overcome Intermittency of Wind and Photovoltaic (PV) Systems: A Review. *Energies*. 2019; 12(22):4392.
https://doi.org/10.3390/en12224392

**Chicago/Turabian Style**

Ellahi, Manzoor, Ghulam Abbas, Irfan Khan, Paul Mario Koola, Mashood Nasir, Ali Raza, and Umar Farooq.
2019. "Recent Approaches of Forecasting and Optimal Economic Dispatch to Overcome Intermittency of Wind and Photovoltaic (PV) Systems: A Review" *Energies* 12, no. 22: 4392.
https://doi.org/10.3390/en12224392