^{1}

^{1}

With the increased number of PMUs in the power grid, effective high speed, realtime methods to ascertain relevant data for situational awareness are needed. Several techniques have used data from PMUs in conjunction with state estimation to assess system stability and event detection. However, these techniques require system topology and a large computational time. This paper presents a novel approach that uses real-time PMU data streams without the need of system connectivity or additional state estimation. The new development is based on the approximation of the eigenvalues related to the decoupled discreet-time power flow Jacobian matrix using direct openPDC data in real-time. Results are compared with other methods, such as Prony’s method, which can be too slow to handle big data. The newly developed Discreet-Time Jacobian Eigenvalue Approximation (DDJEA) method not only proves its accuracy, but also shows its effectiveness with minimal computational time: an essential element when considering situational awareness.

The traditional power flow Jacobian matrix, utilized in Newton-Raphson method, cannot be calculated given these constraints. Analyzing the load flow Jacobian matrix for singular values has been previously utilized to show system weak areas and assess stability [

For a large scale system with N-number of buses, extensive computational time is needed to perform power flow calculations. Since open PDC data lack system topology, model-free methods are necessary to utilize PMU data. Traditional power flow methods and state estimation require system connectivity and a long computational time to converge, rendering these approaches unsuitable for real-time open PDC data. This paper proposes a novel method, DDJEA that attempts to approximate the eigenvalues of the decoupled Jacobian matrix at each bus by measuring the difference between two neighboring measurements of real and reactive power with bus angle and voltage respectively. DDJEA’s accuracy is analyzed to verify that it performs mathematically the same function as the full power flow Jacobian, while minimizing matrix size to incorporate only the number of PMUs. This reduction and simplification greatly decreases computation time. The DDJEA matrices are analyzed to ascertain parameters for stability and event detection. By looking at the elements of DDJEA as they approach a singularity, accelerating toward zero or infinity, system weak points can be flagged to operators before system instability occurs. On average for real open PDC samples, the speed of the proposed algorithm converges and outputs situational awareness data for 60 PMUs in 27 microseconds on a standard laptop. The DDJEA method is expanded in section 2.3 to ascertain divergent conditions and changes over time to monitor and flag both fast and slow unstable parameters. Prony analysis also is a powerful tool in identifying slow system dynamics that may cause instability if unresolved. Furthermore, dominant mode positive eigenvalues can be used to indicate fast unstable conditions. By comparing the event detection and instability indicators of both methods, strengths of the proposed method standalone are discussed, as well as a combined methodology to enhance situational awareness and stability monitoring.

Over a short time period, assuming that the Jacobian matrix has not changed significantly between two cycles, the decoupled Jacobian matrix can be used to calculate changes in real power with respect to changes in bus angle. When a fault occurs or a significant load change, the error will increase, but this is addressed later for aiding event detection. The below formal definition of the decoupled Jacobian matrix for ΔP_{i} is presented below and derived to show how the DDJEA method simplifies calculations through direct computation. For any calculations, the total number of buses (N) and the total number of PMUs (n) are used. _{ }

P_{i} is the actual real power at a bus at a particular measurement for a specific bus (i). ΔP_{i} expresses a change between the most recent measurement and the prior measurement or iteration. Δδ_{j} represents the difference between the most recent bus angle measurement and prior bus angle measurement where δ_{j} is the bus angle at bus (j).

Y_{ij} denotes the Y_{bus} value for the element relating to the p.u. connectivity between the buses (i) and (j). θ_{ij} is the phase angle relative to Y_{ij} in radians. V_{i} and V_{j} are the relative p.u. voltages of the buses (i) and (j) respectively.

When the time step between samples is decreased, the measured difference due to a bus angle in DDJEA will approach the value of the Jacobian element, as seen in Equation (3) and implemented in Equation (8). As the time step, Δt, is decreased, the effect on the power due to a particular bus angle more closely approximates the power flow Jacobian, similar to how numerical integration becomes more accurate with a smaller step size. For large time steps, this approximation method is highly inaccurate and inapplicable. However, with a 30 Hz rate of new state information from open PDC PMU data, this method becomes very accurate since the Jacobian matrix is unlikely to change almost instantaneously. Only a major change in load or event causes a significant shift in the values between cycles of the Jacobian. Since connectivity is not given in the open PDC format, the above equations cannot be applied per its formal definition. Therefore, the solution is simplified.

It is probable that if 60 PMUs are placed in a 3200 bus system that almost none of those PMUs are directly connected, rendering all non-diagonal terms zero without connectivity. Furthermore, state estimation would need to be applied in order to use the full Jacobian, a very time-consuming, heavily computational process that would waste the speed of the PMU data in large power systems. The proposed method directly calculates the change in power at a bus due to the change in that particular bus’s angle for each diagonal element. Since other buses connected are not known, all other changes are approximated by incorporating all changes into the approximated eigenvalue as shown in Equation (8). By assuming this, significant errors in the DDJEA are expected when there is an event at a nearby connected bus that has no readings on it. However, soon after these errors decrease as the approximation accounts for the new parameters. The approximation of predicted real power for the next iteration is derived in Equation (4) with an expanded definition as presented in Equation (5). In Equation (4), the Jacobian of the previous system state is considered valid for one cycle. If time (

The term

The term V_{1i} represents the positive sequence voltage at bus (i). Likewise,

Essentially, this method generalizes to approximating the eigenvalues of the actual decoupled Jacobian matrix for each bus that has a PMU. Since it was previously assumed that most of the PMUs are not connected, all off diagonal terms are being calculated as zero. This leaves the DDJEA matrix in Equation (8) in eigenvalue form. Instead of needing to apply Singular Value Decomposition in order to determine singular values as done in [

In

The calculations for reactive power are very similar to the derivation for the decoupled

Calculation Source | Percent Error of Real Power (P_{err}) for Measurements | |
---|---|---|

Mean Percent Error (P_{err}) | Median Percent Error (P_{err}) | |

IEEE 14 Bus Simulation | 0.054% | 9.51 × 10^{−6}% |

Open PDC Measurements | 0.1977% | 0.1003% |

method of estimating real power. In this case, ΔQ_{i} represents the change in reactive power from one measurement or iteration to the next. Q_{i} is the reactive power at bus (i). Otherwise, all other notation in this section is the same as 2.1.

The following derivations present how the PMU data can be used to approximate Equations (10) and (11), similar to the derivations presented for real power.

DDJEA is used to approximate the next reactive power state for PMUs in the same IEEE 14 bus test system mentioned previously. Also the same 147.5 second open PDC measurements are used to ascertain DDJEA’s ΔQ_{i} accuracy in a real-world application.

In

Calculation Source | Percent Error of Predicted and Actual Reactive Power State (Q_{err}) | |
---|---|---|

Mean Percent Error | Median Percent Error | |

IEEE 14 Bus Simulation | 0.1884% | 9.104 × 10^{−7}% |

Open PDC Measurements | 0.8916% | 0.3342% |

simulated and real open PDC PMU measurements are well under 1%. Therefore, the eigenvalue approximation for the change in reactive power due to the change in bus voltage is an effective estimate of the actual Jacobian’s eigenvalue at each bus.

In the traditional power flow Jacobian, Singular Value Decomposition is used in order to identify system weak points as they approach singularity [

In order to determine whether the DDJEA is approaching a singularity, the characteristics of sequential DDJEA outputs are analyzed. A change in system topology or an event causes the DDJEA to change significantly, similar to how the power flow Jacobian changes frequently. If the DDJEA is increasing toward infinity or decreasing toward zero, it can be monitored. A consistent behavior that is sufficiently far from the mean value the either decoupled DDJEA matrix tends to take, accounting for standard deviation, causes system operators to be flagged. However, an additional tool has to be derived to ascertain when the DDJEA method is accelerating toward an instability point. After a significant system event such as a fault or line removal, system parameters that are quickly causing the system to go unstable need to be identified. Therefore, calculations to determine whether a bus’s conditions are increasing/decreasing in Equation (19) and accelerating /decelerating in Equation (20) are developed. For terminology, this is called the acceleration indicator, AI.

Similarly the same equations for reactive power can be derived. With these analytical tools, acceleration toward either an infinite or null singularity can be assessed and patterns can be determined long before a singularity is reached, resulting in voltage or angular instability depending upon which section of the decoupled Jacobian is approaching singularity. For situational awareness, the portion of the DDJEA approximating real power was most advantageous for showing events and fast system changes. The reactive power’s relation to voltage is better for static analysis and assessing voltage stability as the voltage magnitude plays a large role in calculating those elements. Assessing the determinate of the full DDJEA as it approaches 0 is the key to determining a static approach to voltage stability [

ters are immediately sent to system operators since this means unstable conditions are approaching much faster than a slowly divergent system. By looking at the error of DDJEA, as well as how it changes, both event detection and a way to determine if an equilibrium point is being reached can be derived. The reactive portion of the decoupled Jacobian is a good way to check if an increase real power by bus angle is just the system reaching a new equilibrium point, resulting in no flags.

When looking at the DDJEA application, it is important to note that situations which are accelerating outside of the normal system value range are immediately flagged. Looking at both eigenvalues associated with the decoupled matrices yield more information on event identification. However, this paper solely focuses on flagging unstable conditions and the most sensitive regions to changing parameters.

The algorithm in

Code Value | |
---|---|

Interpretation of Value | |

0 | System is Fine/No action Necessary |

5 | Bus Eigenvalue is converging to a new equilibrium point/No action Necessary |

10 | Slightly Divergent Trends Detected in Eigenvalue/No action unless this pattern continues |

15 | Eigenvalue is Converging from Unstable Parameters/No action unless divergence occurs or after Event |

20 | Bus Eigenvalue is Marginally Converging from Unstable Parameters/FLAG |

25 | DDJEA Eigenvalue at Bus is Increasing toward Dynamic Instability/FLAG CRITICAL |

30 | DDJEA Eigenvalue is Accelerating Toward Dynamic Instability/FLAG CRITICAL |

35 | Dynamic and Voltage Instability Parameters Detected/FLAG CRITICAL |

40 | Approaching Singularity/ System will go unstable soon without solution/FLAG CRITICAL |

50 | Major System Event/FLAG HIGHEST PRIORITY |

When the fault is applied at the 1500th measurement, DDJEA detects it as a major event. All other system operations and load changes prior the event are not flagged, being sorted under normal system function. Essentially, any repeated value over 25 will eventually lead to instability if the system cannot converge, so any trend of 25 or over flags operators immediately. Changes in the DDJEA more indicative of dynamic instability and voltage collapse are flagged as a higher priority and the weak point(s) of the system are identified to help aiding situation system operators to avoid any abnormal situation.

Prony analysis is another method which does not require system topology to yield viable data. Ultimately the goal is to derive a damping ratio and a dominant frequency from the signal of interest. It has been used before on open PDC measurements to identify poorly damped modes [

For the code, the signal is transformed into a z-domain transfer function. Then that transfer function is converted over to state space. The eigenvalues of the A-matrix determine the frequencies of each mode and associated damping ratios. As mentioned before, one primary novelty of Prony analysis is the ability to determine undamped primary modes leading to such issues as inter-area and tie-line oscillation. On open PDC measurements, the method will determine the dominant mode, frequency aligned with that mode, and the damping ratio by looking at the real and imaginary components of the eigenvalue.

In

cent, the system is not properly damped and some inter-area oscillation may be driving the system to instability.

Although Prony analysis recognizes a change in system topology immediately, it doesn’t fully recognize unstable conditions, while the DDJEA method does. Also, Prony analysis generates a high order pole transfer function from which results are derived. This is computationally intensive, whereas DDJEA can directly calculate values from PMU measurements. Data from 60 PMUs in openPDC for 147.5 seconds took the DDJEA method approximately 0.06 seconds to make all required calculations, easily qualifying it for real-time applications for very large systems. Depending on system dynamics, the computational time comparisons in Matlab placed the DDJEA method at around 6000 - 10,000 times faster than Prony analysis. Although Prony analysis requires a window of 10 seconds of data before it can run, in a real system it is an online application: a 10 second startup is negligible. DDJEA instantaneously reads in and calculates the next system state.

Depending on the data set, Prony analysis can be run to a full order model of every PMU; since DDJEA is so computationally light, running it adjacently would not hinder the process because it converges so quickly to a solution. In large data sets, DDJEA can be run continuously and Prony analysis can be run at a frequency that allows the overall solution to converge in real-time. In this way, damping ratios could be monitored to avoid missing an unstable power flow case which DDJEA can verify by checking if the eigenvalues have been approaching singularity, even if this change is very slow. Despite its computational expense, Prony analysis is a perfect tool for approximating system poles and modes. Due to the variability that a real power flow Jacobian can experience over time, also experienced by DDJEA methods, occasional Prony analysis iterations would be ideal as an adjacent tool to catch a slow poorly damped tie-line/inter-are oscillation. DDJEA is computationally faster algorithm that gives accurate estimates of the Jacobian eigenvalues without any system topology. Due to the speed at which the proposed algorithm can converge, there is plenty of room for more expansion of what the DDJEA algorithm can do and other methods that it could be coupled with.

Topic of Interest | Comparison of Methods | |
---|---|---|

DDJEA | Prony Analysis | |

Real-time Applicability | Yes, the method converges for large systems before the next iteration. | Yes and No The method can be a real-time application if the number of signals inputted is reduced to converge in real-time. It cannot converge for very large systems unless the system order is reduced, adding inaccuracy to the results. |

Immediate Event Detection | Yes | May take a few cycles |

Slow Unstable Power Oscillation Detection | It can detect eigenvalues increasing toward singularity but may fail to flag as immediately as Prony if matrices appear to be approaching an equilibrium point early on. | Yes, this method excels at this type of detection. |

This paper presented an approximate method to be used for the purpose of situational awareness and assessing weak areas of the system without using system topology. The proposed DDJEA method used synchrophasor data to approximate the change in the system by observing the change in the diagonal terms of the DDJEA matrix, the approximation of the eigenvalues for the Jacobian matrix used in the Newton-Raphson power flow method. The accuracy of this novel approach was compared to what the power flow Jacobian matrix would yield, and the percent errors were all under one percent, showing the method as valid. In the case of an event, a comparison between the DDJEA method and the Newton-Raphson power flow calculations reached similar conclusions, giving indication of an unstable case and system weak areas in real time. The developed method was also compared with Prony analysis and results led to similar conclusions. However, the proposed method converged a lot faster and in a very short period after the event, which is a major factor when considering situational awareness. Future research will be extended to use of other techniques to enhance the proposed method by considering Gaussian distributions to give a better estimate of system connectivity and expand the DDJEA method closer to the full power flow Jacobian.

The authors would like to thank the members of Clemson University Electric Power Research Association (CUEPRA) for their financial support and providing PMU data.

Kantra, S.D. and Makram, E.B. (2016) Development of the Decoupled Discreet-Time Jacobian Eigenvalue Approximation for Situational Aware- ness Utilizing Open PDC. Journal of Power and Energy Engineering, 4, 21-35. http://dx.doi.org/10.4236/jpee.2016.49003