# Tie-Line Reserve Power Probability Margin for Day-Ahead Dispatching in Power Systems with High Proportion Renewable Power Sources

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## Abstract

**:**

## 1. Introduction

## 2. Grid Area Division Based on Power Source Properties

#### 2.1. Types of Grid Area Division Based on Power Source Properties

#### 2.2. Equivalent Parameters of Power Grid Area

## 3. Prediction Error Probability Optimal Power Flow for Day-Ahead Dispatching

#### 3.1. DPEPOPF Based on Grid Area Division

#### 3.2. Inter-Area DPEPOPF

#### 3.2.1. Prediction Error Probability Optimal Power Flow Mathematical Model

#### 3.2.2. Inter-Area Point Estimation Optimization Algorithm for DPEPOPF

- (1)
- According to Equation (23), the inter-area tie-line power ${P}_{ZL}$ is calculated using the power grid area uncertainty power source predicted value:$${P}_{ZL}=f\left({\theta}_{Z1},{\theta}_{Z2},\cdots ,{\theta}_{Zn}\right)$$
- (2)
- Whether any tie-line exceeds the upper limit power is detected according to Equation (24); if the upper limit is exceeded, the third step will be executed; otherwise, the fourth step will be executed:$${P}_{Zl}\le {P}_{Zl\mathrm{max}},\text{}i=1,2,\cdots ,L$$
- (3)
- According to Equation (12), the mathematical model of the inter-area tie-line power adjustment power flow is used to adjust the tie-line power.
- (4)
- The inter-area tie-line residual power margin is calculated according to Equation (25):$$\Delta {P}_{Zl\mathrm{max}}={P}_{Zl\mathrm{max}}-{P}_{Zl}$$
- (5)
- According to the point estimation optimization algorithm for the DPEPOPF, the variables $\Delta {P}_{Zfi}$ are selected in turn.
- (6)
- The power grid area uncertain power generation probability variable is updated according to Equation (19).
- (7)
- After the power grid area is self-accommodated, the residual power margin $\Delta {P}_{Zfi}{}^{\prime}$ of power grid area is calculated.
- (8)
- According to Equation (11), the inter-area DPEPOPF is calculated.
- (9)
- The mean ${\mu}_{Zl}$ and standard deviation ${\sigma}_{Zl}$ of the inter-area TRPPM are calculated.

#### 3.3. Intra-Area DPEPOPF

#### 3.3.1. Prediction Error Probability Optimal Power Flow Mathematical Model

#### 3.3.2. Intra-Area DPEPOPF

- (1)
- According to Equation (36), the tie-line power ${P}_{L}$ is calculated using the uncertainty power source predicted value:$${P}_{L}=f\left({\theta}_{1},{\theta}_{2},\cdots ,{\theta}_{n}\right)$$
- (2)
- Detecting whether each tie-line power exceeds the upper limit according to Equation (37). If the upper limit is exceeded, the third step will be executed, otherwise the fourth step will be executed.$${P}_{l}\le {P}_{l\mathrm{max}}$$
- (3)
- According to Equation (29), a mathematical model of the intra-area tie-line power adjustment power flow is used to adjust the tie-line power.
- (4)
- Each value of the tie-line residual power margin is calculated according to Equation (38):$$\Delta {P}_{l\mathrm{max}}={P}_{l\mathrm{max}}-{P}_{l}$$
- (5)
- Update the uncertain power generation probability variable according to Equation (35).
- (6)
- According to Equation (28), the intra-area DPEPOPF is calculated.
- (7)
- Calculate the mean ${\mu}_{l}$ and standard deviation ${\sigma}_{l}$ of the intra-area TRPPM.

## 4. Simulation Results

#### 4.1. Modified IEEE 118 Bus Test System Raw Data

#### 4.2. Simulation Results of Inter-Area TRPPM

#### 4.3. Simulation Results of Intra-Area TRPPM

#### 4.4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

PSHPRPSs | Power systems with a high proportion of renewable power sources |

DPEPOPF | Day-ahead prediction error probability optimal power flow |

TRPPM | Tie-line reserve power probability margin |

DCS | Dispatch control system |

AGC | Automatic generation control |

CDF | Cumulative distribution function |

Probability density function | |

IEEE | Institute of Electrical and Electronics Engineers |

DC | Direct current |

REI | Radial equivalent independent |

AC | Alternating current |

${M}_{l}$ | Equivalent bus |

${P}_{g}$ | Active power of the equivalent bus |

${P}_{g,i}$ | Active power of each power source |

${P}_{l,i}$ | Power of each load |

${P}_{ij}$ | Total power of tie-line |

${P}_{ij,t}$ | Power of tth tie-line |

${X}_{ij,t}$ | Reactance of the tth tie-line |

$\Delta {P}_{ZI}$ | Power variation of equivalent bus |

${P}_{ZS}$ | Active power of the equivalent bus |

$\Delta {\theta}_{ZI}$ | Phase angle variable of the equivalent bus |

${P}_{ZI}^{0}$ | Calculated value of equivalent bus active power |

${y}_{i0}$ | Admittance between bus i and the zero-potential bus |

${G}_{ij}$ | Conductance between bus i and bus j |

${B}_{ij}$ | Represents the reactance between bus i and bus j |

$\Delta {P}_{ZL}$ | Power margin of inter-area tie-line |

$\sum \Delta {P}_{f}$ | Day-ahead power prediction error |

$\sum {P}_{r}$ | Reserve power |

$\Delta {P}_{Zl,i\mathrm{max}}$ | Line residual power margin |

${P}_{ZR,i}$ | Reserve power of the ith power grid area |

$g\left({P}_{ZR,i}\right)$ | Reserve power function |

$\sum {P}_{R,\mathrm{I}}$ | Accommodation capacity of a type I grid |

$\sum {P}_{R,\mathrm{III}}$ | Accommodation capacity of a type III grid |

$\Delta {P}_{Zfi}$ | Uncertain power generation of the ith regional power grid |

${\eta}_{Zi,j}$ | Standard location coefficient |

$M\left(\Delta {P}_{Zfi}\right)$ | Central moments |

$\mu $ | Mean |

${\sigma}_{Zi}$ | Standard deviation |

$f\left(\Delta {P}_{Zfi}\right)$ | PDF of the power grid area uncertainty power generation |

$\Delta {P}_{Zf}$ | Power prediction error of uncertainty power source |

$\Delta {P}_{Zl\mathrm{max}}$ | Inter-area tie-line residual power margin |

$\sum \Delta {P}_{I}$ | Intra-area uncertain generating power |

$\sum {P}_{R,I}$ | Total reserve power |

$error$ | Percentage error of the simulation result |

${P}_{AC}$ | Simulation result of the AC power flow model |

${P}_{L}$ | Simulation result of the DPEPOPF mathematical model |

$\overline{error}$ | Average percentage error |

## Appendix A

**Table A1.**Numerical characteristics of uncertain power source prediction errors for the modified IEEE 118 bus test system.

Bus | Mean | Standard Deviation | Area |
---|---|---|---|

12 | 0.0555 | 0.407 | 1 |

25 | 0.096 | 0.64 | 1 |

26 | 0.0828 | 0.7452 | 1 |

36 | 0.015 | 0.18 | 2 |

42 | 0.03 | 0.205 | 2 |

55 | 0.02 | 0.20 | 2 |

59 | 0.102 | 0.51 | 2 |

66 | 0.0984 | 0.984 | 2 |

74 | 0.02 | 0.20 | 3 |

80 | 0.1154 | 1.154 | 3 |

91 | 0.015 | 0.18 | 3 |

100 | 0.1056 | 0.704 | 3 |

105 | 0.035 | 0.195 | 3 |

110 | 0.04 | 0.17 | 3 |

112 | 0.025 | 0.205 | 3 |

## Appendix B

From-Bus | To-Bus | Mean | Standard Deviation | From-Bus | To-Bus | Mean | Standard Deviation |
---|---|---|---|---|---|---|---|

2 | 1 | 0.60 | 5.28 | 20 | 19 | 1.58 | 17.31 |

3 | 1 | −0.59 | 4.72 | 15 | 19 | −1.40 | 10.62 |

5 | 4 | −2.45 | 5.25 | 21 | 20 | 1.58 | 17.31 |

5 | 3 | −1.35 | 0.64 | 22 | 21 | 1.58 | 17.31 |

5 | 6 | −2.46 | 5.37 | 23 | 22 | 1.58 | 17.31 |

6 | 7 | −2.46 | 5.37 | 25 | 23 | 3.56 | 34.59 |

9 | 8 | −23.43 | 26.57 | 26 | 25 | −3.39 | 4.66 |

8 | 5 | −9.02 | 3.65 | 25 | 27 | 2.65 | 24.75 |

10 | 9 | −23.43 | 26.57 | 27 | 28 | 1.02 | 10.15 |

4 | 11 | −2.45 | 4.75 | 28 | 29 | 1.02 | 10.15 |

5 | 11 | −2.76 | 4.18 | 30 | 17 | −2.74 | 38.83 |

11 | 12 | −4.05 | 15.26 | 30 | 8 | 14.41 | 32.92 |

12 | 2 | 0.60 | 5.28 | 26 | 30 | 11.67 | 79.18 |

12 | 3 | 0.76 | 4.08 | 17 | 31 | −2.42 | 14.71 |

7 | 12 | −2.46 | 5.37 | 31 | 29 | −1.02 | 10.15 |

11 | 13 | −1.16 | 6.34 | 23 | 32 | 1.98 | 17.28 |

12 | 14 | −1.08 | 8.70 | 32 | 31 | 1.39 | 14.57 |

15 | 13 | 1.16 | 6.34 | 27 | 32 | 1.07 | 9.62 |

15 | 14 | 1.08 | 8.70 | 113 | 17 | 2.21 | 7.31 |

16 | 12 | 1.23 | 2.01 | 32 | 113 | 2.21 | 17.31 |

17 | 15 | 0.84 | 40.80 | 32 | 114 | −0.55 | 4.98 |

17 | 16 | 1.23 | 2.01 | 27 | 115 | 0.55 | 4.98 |

17 | 18 | −0.18 | 22.06 | 114 | 115 | −0.55 | 4.98 |

18 | 19 | −0.18 | 22.06 | 12 | 117 | 0.00 | 0.00 |

From-Bus | To-Bus | Mean | Standard Deviation | From-Bus | To-Bus | Mean | Standard Deviation |
---|---|---|---|---|---|---|---|

36 | 35 | 0.78 | 1.81 | 49 | 54 | −2.48 | 22.40 |

37 | 35 | −0.78 | 1.81 | 54 | 55 | −0.64 | 6.03 |

37 | 33 | 0.00 | 0.00 | 54 | 56 | −1.41 | 12.95 |

36 | 34 | 0.72 | 16.19 | 56 | 55 | −2.10 | 20.03 |

37 | 34 | −2.90 | 34.62 | 57 | 56 | −1.22 | 11.03 |

38 | 37 | 10.80 | 36.20 | 50 | 57 | −1.22 | 11.03 |

37 | 39 | 7.28 | 1.70 | 58 | 56 | −1.02 | 9.18 |

37 | 40 | 7.21 | 10.01 | 51 | 58 | −1.02 | 9.18 |

39 | 40 | 7.28 | 10.05 | 59 | 54 | 0.89 | 7.55 |

40 | 41 | −6.02 | 10.01 | 59 | 56 | 1.55 | 13.14 |

42 | 40 | 6.04 | 0.81 | 59 | 55 | 0.73 | 6.05 |

42 | 41 | 6.02 | 0.81 | 60 | 59 | −1.31 | 4.54 |

44 | 43 | 2.18 | 0.81 | 61 | 59 | −1.35 | 4.67 |

34 | 43 | −2.18 | 12.73 | 61 | 60 | −0.91 | 3.11 |

45 | 44 | 2.18 | 20.81 | 62 | 60 | −0.40 | 1.43 |

46 | 45 | 0.95 | 16.46 | 61 | 62 | 0.26 | 1.02 |

47 | 46 | 0.53 | 20.81 | 63 | 59 | −4.37 | 15.04 |

48 | 46 | 0.42 | 16.46 | 64 | 63 | −4.37 | 15.04 |

49 | 47 | 0.53 | 20.81 | 64 | 61 | −2.00 | 6.76 |

49 | 42 | 9.05 | 0.43 | 65 | 38 | 10.80 | 36.20 |

49 | 45 | 1.23 | 11.92 | 65 | 64 | −6.37 | 21.80 |

49 | 48 | 0.42 | 16.46 | 66 | 49 | 6.06 | 86.46 |

49 | 50 | −1.22 | 11.03 | 66 | 62 | −0.33 | 1.23 |

49 | 51 | −1.47 | 13.32 | 67 | 62 | −0.33 | 1.23 |

51 | 52 | −0.46 | 4.13 | 65 | 66 | −4.43 | 14.40 |

52 | 53 | −0.46 | 4.13 | 66 | 67 | −0.33 | 1.23 |

54 | 53 | 0.46 | 4.13 |

From-Bus | To-Bus | Mean | Standard Deviation | From-Bus | To-Bus | Mean | Standard Deviation |
---|---|---|---|---|---|---|---|

68 | 69 | 12.60 | 54.03 | 93 | 94 | 0.53 | 17.92 |

69 | 70 | 16.15 | 41.86 | 94 | 95 | 4.37 | 11.13 |

70 | 24 | 17.80 | 21.16 | 96 | 80 | 2.62 | 5.11 |

70 | 71 | 17.80 | 63.24 | 96 | 82 | 4.06 | 33.91 |

72 | 24 | 17.80 | 6.75 | 94 | 96 | 4.93 | 12.56 |

71 | 72 | 17.80 | 43.25 | 97 | 80 | 2.62 | 5.11 |

71 | 73 | 0.00 | 20.00 | 98 | 80 | 4.82 | 12.60 |

70 | 74 | −8.99 | 30.68 | 99 | 80 | 4.82 | 12.58 |

75 | 70 | 10.45 | 31.86 | 92 | 100 | −1.34 | 21.31 |

69 | 75 | 4.73 | 6.75 | 94 | 100 | −8.24 | 59.54 |

75 | 74 | 6.99 | 10.68 | 95 | 96 | 4.37 | 11.13 |

77 | 76 | 5.64 | 15.89 | 96 | 97 | 2.62 | 5.11 |

77 | 69 | 8.28 | 31.25 | 100 | 98 | 4.82 | 12.60 |

77 | 75 | 7.07 | 19.91 | 100 | 99 | 4.82 | 12.58 |

77 | 78 | −3.27 | 18.04 | 101 | 100 | −1.34 | 21.37 |

79 | 78 | 3.27 | 18.04 | 92 | 102 | −1.34 | 21.37 |

80 | 77 | 10.56 | 58.29 | 102 | 101 | −1.34 | 21.37 |

80 | 79 | 3.27 | 18.04 | 100 | 103 | −5.95 | 33.98 |

81 | 68 | 12.60 | 54.03 | 100 | 104 | −2.11 | 11.98 |

80 | 81 | 12.60 | 54.03 | 103 | 104 | −0.74 | 4.17 |

82 | 77 | 7.18 | 9.29 | 103 | 105 | −1.38 | 7.82 |

83 | 82 | 3.11 | 43.20 | 100 | 106 | −1.94 | 11.04 |

84 | 83 | 1.34 | 18.58 | 104 | 105 | −2.85 | 16.15 |

85 | 83 | 1.78 | 24.62 | 105 | 106 | 1.69 | 9.61 |

85 | 84 | 1.34 | 18.58 | 105 | 107 | 0.25 | 1.44 |

85 | 86 | 0.00 | 50.00 | 105 | 108 | −2.67 | 15.50 |

87 | 86 | 0.00 | 50.00 | 106 | 107 | −0.25 | 1.44 |

88 | 85 | 1.56 | 13.39 | 108 | 109 | −2.67 | 15.50 |

89 | 85 | 1.56 | 13.41 | 103 | 110 | −3.83 | 22.00 |

89 | 88 | 1.56 | 13.39 | 109 | 110 | −2.67 | 15.50 |

89 | 90 | −0.99 | 19.04 | 111 | 110 | 0.00 | 0.00 |

90 | 91 | −0.99 | 19.04 | 110 | 112 | −2.50 | 20.50 |

89 | 92 | −2.13 | 77.48 | 68 | 116 | 0.00 | 0.00 |

91 | 92 | 0.51 | 1.04 | 75 | 118 | −5.64 | 15.89 |

92 | 93 | 0.53 | 17.92 | 76 | 118 | 5.64 | 15.89 |

92 | 94 | 0.53 | 17.92 |

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**Figure 1.**Schematic diagram of the power grid area division process. (

**a**) Power grid area schematic diagram; (

**b**) equivalent grid area schematic diagram.

**Figure 4.**Flow chart of day-ahead prediction error probability optimal power flow (DPEPOPF) calculation.

**Figure 7.**Modified Institute of Electrical and Electronics Engineers (IEEE) 118 bus test system diagram. The orange power area grid is area 1; the blue power area grid is area 2; and the purple power area grid is area 3. The thick black line is the inter-area tie-line.

**Figure 8.**Probability density function (PDF) and cumulative distribution function (CDF) of the TRPPM between bus 49 and bus 69. (

**a**) PDF; (

**b**) CDF.

**Figure 10.**PDF and CDF of the standard deviation minimum corresponding to the TRPPM. (

**a**) PDF; (

**b**) CDF.

From-Bus | To-Bus | Mean | Standard Deviation |
---|---|---|---|

15 | 33 | −1.36 | 21.17 |

19 | 34 | −0.68 | 10.66 |

30 | 38 | −3.12 | 48.77 |

23 | 24 | −2.65 | 41.38 |

47 | 69 | −0.14 | 2.15 |

65 | 68 | −2.39 | 37.38 |

49 | 69 | −0.12 | 1.85 |

**Table 2.**Numerical characteristics of the mean and standard deviation corresponding to the minimum TRPPM.

Condition | From-Bus | To-Bus | Mean | Standard Deviation |
---|---|---|---|---|

Minimum mean | 18 | 19 | −0.1840 | 22.0614 |

Minimum standard deviation | 68 | 116 | 0 | 0.0013 |

Power Flow Model | DPEPOPF Model | DC Power Flow Model |
---|---|---|

$\overline{error}$ | 0.396% | 11.05% |

Model | DPEPOPF Model | DC Power Flow Model | AC Power Flow Model |
---|---|---|---|

Operation time (s) | 0.076 | 0.023 | 0.270 |

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**MDPI and ACS Style**

Chen, Y.; Guo, Z.; Tadie, A.T.; Li, H.; Wang, G.; Hou, Y.
Tie-Line Reserve Power Probability Margin for Day-Ahead Dispatching in Power Systems with High Proportion Renewable Power Sources. *Energies* **2019**, *12*, 4742.
https://doi.org/10.3390/en12244742

**AMA Style**

Chen Y, Guo Z, Tadie AT, Li H, Wang G, Hou Y.
Tie-Line Reserve Power Probability Margin for Day-Ahead Dispatching in Power Systems with High Proportion Renewable Power Sources. *Energies*. 2019; 12(24):4742.
https://doi.org/10.3390/en12244742

**Chicago/Turabian Style**

Chen, Yue, Zhizhong Guo, Abebe Tilahun Tadie, Hongbo Li, Guizhong Wang, and Yingwei Hou.
2019. "Tie-Line Reserve Power Probability Margin for Day-Ahead Dispatching in Power Systems with High Proportion Renewable Power Sources" *Energies* 12, no. 24: 4742.
https://doi.org/10.3390/en12244742