A TRUE UNDERSTANDING OF R-X DIAGRAMS AND IMPEDANCE RELAY CHARACTERISTICS
Manta Test Systems Inc.
(This paper was originally presented in March 2003 at the Hands On Relay School – Spokane WA)
This paper discusses 10 myths or common misunderstandings about R-X diagrams and impedance relay characteristics. Diagrams generated by computer simulations with actual examples are used to dispel each myth and explain the real truth. Understanding why each of these myths are not true, and what the actual relay behavior is will help test technicians to perform tests which correctly simulate real-world conditions, properly interpret test results and troubleshoot impedance relay problems when results are not as expected.
Knowledge of true protective relay characteristics has not been communicated fully & properly from protective relay designers and protection engineers, to field test technicians. As a result, many misunderstandings about R-X diagrams and impedance relay characteristics are widely held. In turn, this leads to either incorrect testing of these devices or incorrect interpretation of the test results and how they relate to the real-world performance.
THE 10 MYTHS
MYTH #1: A mho impedance relay generally has an operating characteristic which is a circle passing through the origin.
Actually, all modern mho impedance relays for phase, ground and 3-phase protection have an expanded characteristic that includes the origin. This, of course, has been presented and discussed many times in different papers. (Ref. 1, 5, 6) The amount of the expansion depends on the element design but is generally proportional to the source impedance of the system (Ref 1, 5).
Only the case in which the mho characteristic does pass through the origin is when the source impedance (Zs) is zero. However, this case would never happen since all real-world sources have some finite non-zero impedance.
So, where is the expanded characteristic hiding? Why don’t we see it when the relay is tested? What most people don’t realize is that even though the source impedance is not set on their test source, there is a source impedance presented to the relay. The simplified explanation is that the relay sees the source impedance from the drop in the voltage from prefault conditions during the fault.
It has been shown (Ref. 2) that traditional fixed voltage or fixed current testing methods actually present a
power system to the relay has the following special characteristics:
Of course, none of these are true in the real world.
When a test of the relay characteristic is done under these conditions, however, we find that a different power system model is presented to the relay for each test point, thereby plotting a point on a different expanded characteristic for each point. (Ref. 2) Because of the nature of the expanding characteristic, the combined test results show a mho characteristic, which passes through the origin
Figure 3. Combined Results
In order to reveal the true expanded characteristic, each test point must be done simulating the same power system model, only changing the fault condition. This requires complex fault calculations and can only be performed practically with computer-aided testing (Ref 4).
So what source should be used to verify the performance of the relay? Source impedance at a particular point will change depending on system conditions. Also, real impedance relay operate times actually vary with source to line impedance ratio (SIR). Generally, the operation times are higher with higher source to line impedance ratio values.
Figure 4. Typical Variation of Operate Time for Different SIR Values
Ideally one should test at maximum and minimum expected source impedance to get a total picture. If we test at the maximum expected source impedance, this will determine the worst case operate times. This will also allow us to confirm that load encroachment is not a problem for phase and 3-phase elements. However, be aware that there may also be situations in communication aided tripping schemes where the minimum expected operate time is of significance because of possible race conditions.
MYTH #2: Points above the directional line represent forward direction faults and points below the directional line represent reverse direction faults.
The confusion here is that there are actually 2 quite different R-X diagrams used:
On both types of graphs, points are plotted by calculating the positive sequence impedance. On a conceptual z-plane graph, these points may be reverse faults or forward faults. Forward faults such as a forward capacitive fault or a forward resistive fault under a high load transfer condition can appear below the “directional line” or the R-axis. Similarly, reverse faults can appear above the “directional line” or the R-axis. Therefore the location of the point does not imply the direction or location of the fault.
A relay operating characteristic graph is shown either for forward faults or for reverse faults. On a relay forward fault characteristic graph, all plotted fault points are forward. Reverse fault points cannot appear at all on this graph.
The so-called “directional” or zero-torque line was used with old impedance relays that had a separate supervising directional element, or with directional overcurrent relays and incorrectly carried over to R-X graphs for modern relays. In fact, the directional line can neither be plotted on the relay forward fault characteristic nor the relay reverse fault characteristic graphs.
Very often we see diagrams such as the following showing multiple relay characteristics on a z-plane graph.
We should keep in mind that these diagrams are conceptual. If we take any point on the graph and state that certain elements will or will not operate, this may be jumping to conclusions that may not be true.
What is misleading is that for the usual assumptions of radial homogeneous systems and simple faults, the theory that points above the directional line are forward faults and points below the directional line are reverse faults does hold up. However real world scenarios do not fall into these simple cases. The actual characteristics must be plotted on separate graphs because the operating quantities used by these elements are different.
Another dangerous graph is one of both offset and non-offset elements plotted together such as the following.
The operating quantities used by the phase comparator in an offset mho element are different from those used in a non-offset mho element (Ref. 9). This type of diagram should be treated as conceptual.
Another illustrative example is the case of faults directly in front of and directly behind the relay. Both fault points plot on top of each other on a z-plane graph. The fault directly in front of the relay can only be plotted on the forward fault characteristic graph. The fault directly behind the relay can only be plotted on the reverse fault characteristic graph. The fault directly in front of the relay cannot be plotted on the reverse fault characteristic graph or vice versa.
MYTH #3: A mho impedance relay bases its operation by determining if the measured impedance is “geometrically” inside its characteristic on the Z-plane
The geometric comparison analogy is used to simplify the explanation of the operating principle. Unfortunately, the geometric analogy is too often taken literally.
False Example 1
In this first example, we have a 2 terminal system with unequal sources. The one of interest is the weak source, that is with high source impedance. A fault in front of the relay may result in an impedance which satisfies the impedance conditions for operation, but the voltage at the relaying point is dragged down and the resulting fault current is below the current supervision level and the element does not operate. This is the common “weak infeed” scenario.
False Example 2
In this second example, we have a series compensated line. There reverse fault directly after the capacitor behind the relay of interest. The calculated impedance at the relaying point is inside the operating characteristic, but the relay does not operate.
The Z-plane graph is plotted using the faulted phase voltage, ignoring the unfaulted phase voltages. However, the relay uses the unfaulted phase voltages for directional comparison, thereby blocking operation for faults in the opposite direction.
False Example 3
In this example, we have a 2 terminal system with some export load flow and an A-B fault behind the relay. The forward-looking C-A phase element uses minus IA in the operating quantity and compares against the phase angle of VB. The result is that the forward-looking C-A phase element can operate for this reverse fault, even though the calculated C-A impedance is well outside the mho characteristic on the R-X diagram.
The truth is that mho impedance relays use a phase comparator between 2 carefully chosen complex quantities to make their operating decisions. Additional supervising elements can block operation, even though the impedance conditions are satisfied.
MYTH #4: Any point on the R-X plane can be simulated simply by applying the formulae:
For ground faults: complex Z= Va / (Ia + k Ir formula) (A-G case)
For phase faults: complex Z = Vab / ( Ia- Ib) (A-B case)
Figure 14. (A-G case)
In actual fact, the fault phasors for a typical system* would look quite different from the normal hand calculated values. Taking the A-G fault case, the fault phasors for a typical (radial) power system would look like the following:
* Although not realistic, a radial system is still used here for simplicity of explanation.
Note the following:
In order to determine the appropriate voltages and currents to simulate a given fault condition, it is necessary to also decide on the power system being simulated. That is:
Fault condition + Power system — Determine -> Voltages & Currents
The power system is defined by:
The circuit configuration (radial, 2-terminal, parallel line etc)
Source + line impedances (positive, negative and zero sequence)
Source voltages (magnitude and angle)
The fault condition is defined by:
Fault section (forward, reverse, adjacent line, beyond remote end)
Position (% of line)
Fault phase(s) and type
Fault impedance (magnitude & angle)
The relay input voltages and currents include
Faulted phase voltage and currents
Unfaulted phase voltages and currents
(Prefault voltages and currents)
Without consciously deciding on power system, some arbitrary assumption must be made about either the voltage or the current or both and we end up with an unknown power system. (Ref Myth #1). Eg:
Fault condition + Power system — Determine -> Voltages & Currents
+ unknown power system
Of course, what an impedance relay does is use some knowledge (read assumption) of the power system which is the line impedances, and the voltages and currents to determine the fault condition. That is:
Fault condition <– Determine — Voltages & Currents
+ power system
Note that the voltages and currents include both the prefault and unfaulted phase voltages and currents.
MTYH #5: An impedance relay will only operate for faults in the direction it is set for.
In actual fact, the traditional forward-looking cross-polarized mho relay has an operating characteristic for reverse faults.
Similarly, the traditional forward-looking quadrilateral element has an operating characteristic for reverse faults.
Only if properly designed and applied will impedance relays not operate in the opposite direction that they are set for. Luckily, under most real-world conditions, the reverse fault characteristic will probably never operate.
MYTH #6: Fault resistance coverage varies greatly depending on load conditions due to mho load compensation.
This diagram is often shown illustrating the shift in the mho characteristic under different load conditions. (Ref 6). People incorrectly assume that the fault resistance coverage varies significantly under these varying conditions.
However, when we plot the fault position versus the actual fault resistance coverage for both the export load case and the import load case, we find that the fault resistance coverage varies very little.
Figure 19. Export Load
Figure 20. Import Load
The R-X graph shows an apparent impedance which has a complex relationship to the actual Rf depending on the power system conditions. The fault position vs. true Rf graph can remove all of those effects and show the true performance no matter what the power system conditions are. It is therefore very useful for comparing performance under different operating conditions. Also, the capacitive faults at 0% are collapsed to the 0% line. Therefore the graph doesn’t mislead us into thinking that the relay will operate for reverse faults (myth #2).
MYTH #7: A quadrilateral element guarantees constant fault resistance coverage.
In actual fact, most distance relay protections are applied to 2 terminal systems. In a 2-terminal system, the remote infeed has the effect of increasing the apparent fault resistance seen by the relay because the remote source contributes to the fault current, raising the voltage at the fault point and thereby decreasing the fault current from the measuring end.
This is easily seen again when we plot the fault position vs. true Rf graphs for a 2 terminal system, and for a radial system for comparison purposes. We can see that in the 2 terminal case the true Rf coverage decreases as the fault location is further. We can also see that the apparent R-X characteristic is the same as the radial case indicating that the relay response to apparent R-X is constant for the 2 cases.
Figure 24. Two Terminal System Case
Note that this trapezoidal characteristic if for a 2 terminal homogeneous system with no prefault load. We will see later what happens when this is not the case.
MYTH #8: A quadrilateral characteristic can be plotted together with a mho characteristic on the same impedance graph for comparison purposes
This diagram is often seen and is very misleading. To see how this is not true, let’s look at a simple example of an A-G fault with 10 ohms of fault resistance as seen by both a quadrilateral and a mho impedance element. Immediately we see that the resistive axis is different, because the apparent resistance seen by the quadrilateral element is 11 ohms, and the apparent resistance seen by the mho element is 7.3 ohms.
A detailed analysis shows that the scales for the 2 types of graphs are actually quite different. Both types of graphs are of apparent impedance because the relays only have access to the voltages and currents at one terminal.
|Graph type||Horizontal Axis||Vertical Axis|
|Positive sequence Z
|Resistive part of the positive
sequence impedance seen by
|Reactive part of the positive sequence
impedance seen by the relay
|The resistive part of the line
section from relay to the
fault + apparent fault
resistance (assuming a radial
system and no load)
|The reactive part of the portion of the
line from relay to the fault + apparent
fault reactance (usually 0)
On the positive sequence impedance graphs used to show mho elements:
(The above 2 statements are true only for a radial system)
On apparent R-X graphs used to show quadrilateral elements:
The y-axis values are equal to each other in the simple case of a homogenous radial system with pure resistive faults. However, this case is probably never encountered in the real world.
In actual fact, when a mho characteristic is plotted on an apparent R-X graph the shape looks like an oval.
Note: The reference quadrilateral element in this figure was chosen to have the same resistive reach at 0% as the mho characteristic for illustrative purposes.
In order to properly compare the performance of a quadrilateral element to a mho element, we can again employ the fault position vs. true Rf graph. A typical result (for the same radial system) appears below:
MYTH #9: A quadrilateral characteristic can be drawn on an R-X diagram from a simple interpretation of the settings.
Relay manufacturers have always drawn quadrilateral characteristics on R-X diagrams with some degree of generality and vagueness. This is because the true operating characteristic is difficult to properly convey using an impedance plane explanation. These diagrams are meant to be conceptual but often we try to take them literally.
The actual implementation of a typical quadrilateral element is a composite AND of 4 blinder elements and an overcurrent element. Note that the last 2 elements are really not impedance measuring elements at all.
The true operating characteristic for forward faults on an apparent R-X graph appears like this:
A few points are of note:
In most quadrilateral ground element designs, the resistance blinders operate using phase current, whereas the top reactance blinder operates on residual current. The latter causes a dynamic tilt in the reactance line on R-X or z-plane graphs and compensates for load flow. This also makes the characteristic difficult to properly illustrate the characteristic on a z-plane graph (Ref 7) and again is a good reason to regard these diagrams as conceptual.
In some designs, you will see be a tilt or angle bias setting (Ref 3) for the reactance line. This is to compensate for non-homogeneous systems where the angle of the line and source zero sequence impedances are not equal. Without the angle bias, we see the following, where there is an apparent overreach.
By an appropriate angle bias setting, the reactance line can be tilted back to be more horizontal, resulting in the following:
There are many different variations of quadrilateral elements, and careful study of the manufacturer’s literature is required to properly understand their characteristics. The best tool for determining true performance is computer driven test equipment using a true power system model with the results plotted on a “Fault position versus true Rf” graph.
MYTH #10: Impedance relays will not experience a fault condition with an apparent negative resistance fault in the real world.
False Example 1
One example of how this is not true is for simple phase faults. Take for example an A-B fault. A mho B-G element will see this fault as having negative fault resistance.
False Example 2
Another example is a reverse direction resistive fault. A forward-looking impedance element will see these faults in the negative resistance half of the impedance plane.
False Example 3
A third example is in a 2 terminal system with a tapped load. An in section fault on the main line can also appear as negative resistance.
Therefore there are real-world conditions under which negative resistance can appear and this “half” of impedance relay characteristics are important to proper protection.
IMPORTANT TESTING IMPLICATIONS
Testing using traditional methods only checks the reach setting at MTA, and should not be interpreted by any means as the true performance of the relay.
Two terminal configurations are not tested
Load is not simulated
Source impedance is uncontrolled and unknown
Today’s power system operating environment is much more stringent than before, with narrowing security margins. Protection test procedures should be reviewed in this light and adverse or critical real-world conditions affecting protection should be identified and affected relays tested for their actual performance.
The key lessons learned in this paper are summarized as:
Standard equations for R-X diagrams:
Positive sequence impedance* for mho characteristics
Zag = Va ¸ (Ia + k Ir)
Zbg = Vb ¸ (Ib + k Ir)
Zcg = Vc ¸ (Ic + k Ir)
Where Ir = Ia + Ib + Ic
k = (Zline0 – Zline1)
Zab = (Va – Vb) ¸ (Ia – Ib)
Zbc = (Vb – Vc) ¸ (Ib – Ic)
Zca = (Vc – Va) ¸ (Ic – Ia)
Take the real and imaginary parts of Zxx to get the R and X values respectively
NOTE: All quantities are complex (ie. vectors)
*Sometimes called apparent impedance (for mho elements)
Apparent R-X for quadrilateral characteristics
Rag = Real(Va¸Ia)
Xag = Imag(Va¸Ia) x Imag(Zline1) ¸Imag(c)
Rbg = Real(Vb¸Ib)
Xbg = Imag(Vb¸Ib) x Imag(Zline1) ¸Imag(c)
Rcg = Real(Vc¸Ic)
Xcg = Imag(Vc¸Ic) x Imag(Zline1) ¸Imag(c)
where c= ZLine1(1+k)
Rab = Real ( Va¸Ia + Vb¸Ib)
Xab = Imag ( Va¸Ia + Vb¸Ib) ¸2
Rbc = Real ( Vb¸Ib + Vc¸Ic)
Xbc = Imag ( Vb¸Ib + Vc¸Ic) ¸2
Rca = Real ( Vc¸Ic + Va¸Ia)
Xca = Imag ( Vc¸Ic + Va¸Ia) ¸2