**1. Introduction**

The interface between the High-Voltage (HV) transmission grid and the Medium-Voltage (MV) distribution network are HV/MV substations. An example of the structure of such a station is shown in Figure 1. In some HV/MV substations, stations shunt compensators are installed. Most often they are Capacitor Banks (CB), less often reactors.

**Figure 1.** Simple radial Medium-Voltage (MV) grid with High-Voltage (HV)/MV substation.

The variability of active power and reactive power of sources creates the need for voltage regulation. Voltage regulation is done primarily by changing the power transformer voltage ratio in the HV/MV substation and by controlling shunt capacitors (if any). A criterion for the operation of a transformer regulator in a HV/MV substation is to maintain the preset voltage value on the MV side *V*ref. The range of voltage variations is determined by the transformer parameters and the network characteristic. The regulation capacities of a transformer are usually a dozen or so taps that enable voltage change within a range of several dozen percent of the rated voltage. Preset values along with a schedule of changes are defined offline by the operation staff. The voltage regulation is continued either until an extreme maximum or minimum tap position is achieved, or until the voltage value attains one of the limiting values: of over-voltage *V*Tmax or under-voltage *V*Tmin.

An item of the equipment of some HV/MV substations are also capacitor banks. Their regulation involves switching individual sections on or <sup>o</sup>ff. Switching processes are determined by, inter alia, the time schedule and are done by manual switch-on, either remotely or locally, or by an individual controller operating in voltage regulation mode.

To minimize the influence of load variability on the voltage levels in the network, requirements to improve the reactive power managemen<sup>t</sup> are imposed on both the generators and consumers [1]. These are generally limited to the requirement to maintain a constant preset magnitude of reactive power or not to exceed the preset value of the coefficient tg.

Usually, when the volume of distributed generation and daily variability of loads are low, the efficiency of voltage regulation by changing the power transformer ratio is satisfactory. However, in networks with a high saturation of distributed generation, the efficiency of such a method of voltage regulation is increasingly often insufficient. High generation variability (wind, photovoltaic generation) deteriorates the voltage profiles on the grid [2–9]. An equally important problem is periodically too low or too high voltage in the network [10,11]. Too high voltage occurs as a result of a change in the typical direction of power flow (flow—from HV/MV substations to consumers) caused by a large generation volume, e.g., as a result of favorable weather conditions.

The possibilities of improving the quality of regulation, which can be found in the literature, e.g., [2–6,10,12], most often include various methods of coordination of the operation of the transformer regulator with various devices operating in the MV network. This coordination can be decentralized or centralized.

Most of the proposed solutions do not assume a change of the transformer regulator algorithm.

The analysis of the operation algorithms of the transformer regulators supplying the distribution network showed that no solution would take into account the change like the d *Q*/d*V* coe fficient. There is, however, a commercial device called Collapse Prediction Relay (CPR-D) o ffered by A-Eberle [13–15] but its algorithm is very complex. To determine the need to block or properly control the transformer tap changer, the following are used: bifurcation theory [16–18] in combination with neural network elements, determination of Lyapunov exponents, identification of voltage drop and, finally, damping coe fficients.

Bearing the above in mind, an attempt was made to develop an e ffective and simple method of controlling the operation of the transformer supplying the distribution network. The main di fference between the aforementioned CPR-D system and the proposed co-authoring algorithm lies in the specific purpose of operation. In the case of the CPR-D system, the aim is to identify the state of emergency, while the purpose of the proposed adaptive control system for transformers supplying the distribution network is to prevent the emergence of a voltage avalanche hazard.

The paper will present an algorithm for the operation of the regulator of a distribution network-supply transformer, which is protected by the patent [19]. The solution presented in the paper extends the functionality of the typical algorithms of the power transformer regulator. In the described solution, in the normal operating state of the network, the power transformer regulator works according to a typical manufacturer-defined algorithm. The proposed algorithm is activated only when there is a risk of voltage collapse, possibly leading to a blackout.

Moreover, in this paper, theoretical fundamentals concerning the e ffect of the regulation of the transformer voltage ratio on the risk of an occurrence of a voltage collapse, and the results of the simulation studies and laboratory tests of the proposed algorithm will be described.

#### **2. Voltage Stability Phenomenon**

Voltage stability is the ability of a power system to maintain steady acceptable voltages at all buses in the system under normal operating conditions and after being subjected to a disturbance. The main factor causing instability is the inability of the power system to meet the demand for reactive power [20–25].

A system is voltage unstable if, for at least one bus in the system, the bus voltage magnitude ( *V*) decreases as the reactive power injection ( *Q*) is increased at the same bus. In other words, a system is voltage stable if *V*-*Q* sensitivity is positive for every bus and voltage unstable if *V*-*Q* sensitivity is negative for at least one bus.

Voltage instability is essentially a local phenomenon; however, its consequences may have a widespread impact. Voltage collapse is more complex than simple voltage instability and is usually the result of a sequence of events accompanying voltage instability leading to a low-voltage profile in a significant part of the power system.

The physical cause of a voltage collapse occurring are primarily some phenomena taking place within the complex load. A reduction in load node voltage causes a reduction of the driving torque of asynchronous motors making up the load. The reduction of asynchronous motor torques is accompanied by an increase of slips and thereby a substantial increase in reactive power taken up from the network, and, as a consequence, a further reduction in load node voltage. As the voltage goes down, subsequent motors within the complex load halt. The consequences of this course of events are obvious to the consumers. For the electric power system, the consequences generally do not end up with a voltage collapse in only one load node. At the same time, the voltages of adjacent nodes will go down, and the phenomenon may spread to other loads.

Voltage instability may occur in several di fferent ways. In its simple form, it can be illustrated by considering the two-terminal network of Figure 2 [25]. It consists of a constant voltage source (*E*s) supplying a load ( *P*l + i*Q*l) through the equivalent reactance of the power system ( *X*).

**Figure 2.** Power system as a source of reactive power: (**a**) equivalent circuit; (**b**) a diagram of the nodal currents and voltages of the substitute source. s—generating node; l—load node; *E*s, *X*—fictitious electromotive force and system equivalent reactance.

The properties of the power system as a source supplying a selected load node will be described using the reactive power generation characteristic, hereinafter in abbreviation called the generation characteristic and denoted by *Q*s(*V*). The generation characteristic defines, as a function of voltage, the reactive power given up by the power system to a given load node, with that node being loaded with the preset load active power, *P*l(*V*).

On the right-hand side in Figure 2b, the complex load with a given characteristic is cut o ff, and only the sole equivalent source is separated. For this source, formulae can be written, which define the active power and reactive power supplied to the load node:

$$P\_{\rm s} = V\_{\rm l} \cdot I \cdot \cos \varphi = V\_{\rm l} \cdot I \cdot \frac{E\_{\rm s} \cdot \sin \delta}{I \cdot X} = \frac{E\_{\rm s} \cdot V\_{\rm l} \cdot \sin \delta}{X} \tag{1a}$$

$$Q\_{\rm s} = V\_{\rm l} \cdot I \cdot \sin \phi \,\,=\,\,V\_{\rm l} \cdot I \cdot \frac{E\_{\rm s} \cdot \cos \delta - V\_{\rm l}}{I \cdot X} \,\,=\,\,\frac{E\_{\rm s} \cdot V\_{\rm l}}{X} \cdot \cos \delta - \frac{V\_{\rm l}^2}{X} \,\tag{1b}$$

These formulae can be transformed into the following form:

$$P\_s^2 + \left(Q\_s + \frac{V\_1^2}{X}\right)^2 = \left(\frac{E\_s \cdot V\_1}{X}\right)^2\tag{2}$$

According to the definition of the generation characteristic, the source should be loaded with the active power, that is to substitute *P*s(*V*) = *P*l(*V*). With this assumption, the following will be obtained from Formula (2) after making simple transformations:

$$Q\_s = \sqrt{\left(\frac{E\_s \cdot V\_1}{X}\right)^2 - P\_s^2} - \frac{V\_1^2}{X} \tag{3}$$

Formula (3) defines the generation characteristic, or the relationship of the reactive power *Q*s(*V*) supplied to the load node by the source, with the source being loaded with active power with the preset voltage characteristic *P*l(*V*). The generation characteristic corresponding to Formula (3) is a curve similar to an inverted parabola. The smaller the equivalent reactance of the source *X* and the smaller active power loading the source *P*l, the more the *V*<sup>2</sup>/*X* parabola shifts towards the center of the coordinating system.

The source and the load may only operate with each other at such a voltage, where the power supplied by the source is equal to the power taken up by the load *Q*s(*V*) = *Q*l(*V*). In Figure 3, this corresponds to the points of intersection of both characteristics. Obviously, the point of the stable

operation of the system may only be a locally stable point of equilibrium. The condition of equilibrium is as follows:

$$
\frac{\mathrm{d}Q\_{\mathrm{S}}}{\mathrm{d}V} < \frac{\mathrm{d}Q\_{\mathrm{l}}}{\mathrm{d}V} \qquad\qquad\text{or}\qquad\frac{\mathrm{d}\Lambda\underline{Q}}{\mathrm{d}V} < 0\tag{4}
$$

whereas Δ*Q* = *Q*s − *Q*l is the excess of the source's reactive power over the load demand.

**Figure 3.** Illustration of the mutual location of the production and reception characteristics: (**a**) two points of equilibrium; (**b**) one point of equilibrium.

In the case shown in Figure 3a, the stability condition (4) is met at point "b" only. At that point, the system can operate in a stable manner. Electrically, the behavior of the system is as follows. A momentary increase in voltage by a small magnitude of Δ*V* is accompanied by an excess of received power over the generated power, which entails a drop of voltage and the return to the point of equilibrium. Similarly, a momentary decrease in voltage is accompanied by an excess of generated power over the received power, which results in an increase in voltage and the return to the point of equilibrium. A special case of equilibrium point (the point "a") is the situation illustrated in Figure 3b. At point "a", the stability condition (3) is not satisfied. At this point, the system could not operate in a stable manner. Any change in voltage causes an excess of receiver power over the generated power, which results in a further decrease in voltage. The system departs from the point of equilibrium with continuously decreasing voltage. The loss of the system's stability, manifesting itself in a voltage breakdown due to a small disturbance, is called the voltage collapse.

#### **3. The Voltage Dependency of Loads**

The effect of operation and some regulation properties of any electric device taking up active and/or reactive power can be described using, e.g., static characteristics. Typical characteristics describe the variability of active *P*L and reactive *Q*L power take-up as a function of variations in voltage and frequency. Simulation studies concerning the analyses of the operation of electric power systems rarely consider single loads. More often, aggregated load models are used, in which the load power is the sum of the powers of electric loads of different characteristics [26–34]. This sum includes the proportionality factors *k*P and *k*Q, defined as the percentage fraction of individual groups of loads of the total load power, that are determined at the point of common connection (PCC) (e.g., a specific outlet field in a HV/MV substation), which, assuming a constant frequency level, can be described in a general form using the following relationships:

$$P\_{\rm L.} = P\_{\rm L.n} \cdot \sum\_{i} k\_{\rm P,i} \cdot \left(\frac{V\_{\rm i}}{V\_{\rm n}}\right)^{\alpha\_{\rm P,i}} \tag{5a}$$

$$Q\_{\rm L} = Q\_{\rm Ln} \cdot \sum\_{j} k\_{\rm Q,j} \cdot \left(\frac{V\_j}{V\_{\rm n}}\right)^{\alpha\_{\rm Q,j}} \tag{5b}$$

where:


From the point of view of the regulation of voltage in HV/MV substation, power variations are the most important, as they determine the greatest voltage variability. With small voltage deviations from the rated voltage of ±10%, it can be assumed that, in the vast majority of instances, the coefficient of voltage sensitivity of the reactive power of loads, defined as d*Q*/d*V*, has a positive value. This is an advantageous situation since at that case one has the so-called self-regulation of the power system—where a decrease in voltage results in a reduction in reactive power take-up by the loads. It should be noted, however, that the value of the d*Q*/d*V* coefficient is variable in time. This is caused by the variation of network voltages, the variation of the fractions of individual groups of loads on the total node power (the variation of the coefficients *k*P and *k*Q), etc. The variation of the d*Q*/d*V* coefficient applies not only to the absolute value but also to the sign. In a situation where the d*Q*/d*V* coefficient is less than 0, lowering the voltage at the load's terminals will result in an increase in reactive power uptake. Such behavior of loads is undesirable in a situation where a voltage drop is due to a deficit of reactive power in the system, as it will contribute to a deepening of that deficit. A negative value of the d*Q*/d*V* coefficient may occur, e.g., in networks with large participation on asynchronous motors [26]. Examples of static characteristics *V-Q* for selected cases are shown in Figures 4 and 5.

**Figure 4.** Characteristics of a drive-controlled (elaborated base on [26,27]): (**a**) closed-loop operated three-phase motor for nominal speed; (**b**) closed-loop operated single-phase motor.

It should be noted that the value of the d*Q*/d*V* coefficient depends not only on the voltage level but also on the machine loading. The examples shown (Figures 4 and 5) include the cases of loading the machine with a constant moment, *T* = const.; a constant power, *P* = const.; and a moment dependent on the square of velocity, *T* = f(ω2).

**Figure 5.** Reactive power characteristics of a residential air conditioner (RAC) during normal conditions (elaborated base on [29]): (**a**) research by the Electric Power Research Institute (EPRI); (**b**) research by Southern California Edison.

Even though the results of the investigation [35] carried out by the author in the years 2007–2008 show that instances where the d*Q*/d*V* is negative or close to 0 are not frequent, this problem should be expected to exacerbate in the nearest future. One of the main causes may be the increased number of refrigerating equipment units—air conditioners. The International Energy Agency (IEA) has published a report entitled The Future of Cooling [36] which forecasts that the growing use of air conditioners in households and offices worldwide will become, over the nearest three decades, one of the main factors of the global demand for electric energy. This problem is addressed also in other publications [29,37]. As has been shown by the investigation results reported in [29,33,38–40], the coefficient of reactive power voltage susceptibility, d*Q*/d*V*, of the above-mentioned equipment is negative at lower voltages, as shown in Figure 5. It is also worth noting that the demand for reactive power increases with increasing ambient temperature.

The characteristic of voltage sensitivity d*Q*/d*V*, as seen from the terminals of the distribution network supply transformer, is a resulting characteristic. In a situation, where a large number of loads with an "unfavourable" characteristic of voltage susceptibility are present in the network under analysis (Figures 4 and 5), this may pose a threat to the voltage stability in a specific area. The problems may deepen in a situation, where capacitor banks are installed in a large number of network nodes, or the network has a large number of cable lines.

#### **4. The Influence of Transformer Tap Position Changes on the** *V***-***Q* **Characteristics**

The characteristics shown in Figures 4 and 5 apply to the situation where the voltage ratio of a transformer is constant. At a constant voltage ratio, the reduction of voltage on the primary side *V*1 results in a corresponding reduction of voltage on the secondary side *V*2. In the situation where the transformer voltage controller would maintain a constant voltage value on the secondary side, the reactive power *Q*2 taken up by the loads would not change (Figure 6).

The discrete nature of voltage regulation by changing the voltage ratio, associated with the stepwise change of the number of coils and the dead zone, also causes a discrete change in the reactive power of the loads.

The process of voltage regulation by changing the transformer voltage ratio is slow. In real power systems, the aim is to reduce the number of tap switchovers during 24 h. It is achieved, e.g., by increasing the settings of switchover delay times or by increasing the dead zone. This is an intentional action, dictated, e.g., by economical and operational reasons. The characteristic presented

in Figure 6 proves itself excellently in the analysis of steady states or in the case of slow voltage changes—slower than the action of transformer control systems.

**Figure 6.** Influence of transformer tap position changes on the *V*-*Q* characteristics: (**a**) the situation corresponds to work in point "b" shown in Figure 3a; (**b**) the situation corresponds to work in point "a" shown in Figure 3a.

In the case of large overloads, the rate of voltage changes might be greater than the transformer voltage ratio change rate. In such instances, load characteristics should be taken for consideration, which do not allow for the transformer regulation action.

Depending on the phenomena under consideration, the voltage ratio value should remain either unchanged or equal to the extreme voltage ratio values corresponding to the outermost position of the tap changer.

#### **5. The Effect of Transformer Voltage Ratio Regulation on the Risk of a Voltage Collapse Occurring**

The mechanism of a voltage collapse occurrence is very complex and comprises many components of the electric power system. Regardless of the causes of its occurrence, the loss of voltage stability is closely related to a disturbance of the reactive power balance in the system. It can either be a local phenomenon or cover a larger part of the electric power system. The causes of the overloading of an electric power system with reactive power and the methods of analysis of the voltage stability of electric power subsystems were addressed in numerous publications, including [20,21,41–45]. This section discusses phenomena that pose a threat to voltage stability and result from transformer voltage ratio regulation. The following two extreme situations are taken for analysis: the first one, in Figure 7a, in which the coefficient of voltage susceptibility of the power of loads is positive (d*Q*/d*V* > 0); and the second one, in Figure 7b, where the coefficient is negative (d*Q*/d*V* < 0). It has also been assumed that the voltage change rate is greater than the speed of action of transformer control systems.

The effect of voltage ratio changes with a reactive power deficit in a situation, where (d*Q*/d*V* > 0), is illustrated in Figure 7a. As a result of a lowering of voltage in the MV network, the transformer controller changes the voltage ratio magnitude to maintain the preset voltage value on the transformer secondary side. The change of the voltage ratio entails a change in the reactive power uptake characteristic. The fixed working point OP1 moves toward the increased reactive power uptake up to the point OP2, in which, in an extremely unfavorable case, limiters in the generator controller might be activated. The activation of any current limiter causes a reduction of the magnitude of excitation voltage and, as a consequence, of the generated reactive power, contributing thereby to an aggravation of the reactive power deficit in the electric power system. In the case under discussion, from the point of view of voltage stability, it would be the most favorable to halt the voltage regulation (a constant transformer voltage ratio).

**Figure 7.** Impact of the transformer tap-ratio control on the risk of voltage collapse: (**a**) positive effect of blocking transformer ratio changes; (**b**) unfavorable effect of blocking transformer ratio changes. Explanation of abbreviations: *<sup>V</sup>*g—generator voltage (Vg0—reference Generator Voltage), —generator current, AVR—Automatic Voltage Regulator, OEL—Over Excitation Limiter, OCL—Over Current Limiter.

The constant transformer voltage ratio will, however, not always bring about desirable effects. As has been mentioned in the introduction, in distribution networks with a large number of asynchronous drives, including air conditioning devices, and a considerable power of capacitors, the d*Q*/d*V* coefficient may take on negative values. This means that—with decreasing voltage—the input power increases. In that case, the natural characteristic of loads is distinctly less advantageous than the characteristic determined by the action of the transformer voltage controller that maintains, in a certain range, a constant voltage and the resulting constant reactive power uptake Figure 7b.

When the coefficient d*Q*/d*V* < 0 of the characteristics of reactive power, operating transformer voltage controller leads to achieving a stable, new working point OP2. Blocking the controller operation could, in this case, result in a permanent deficit in reactive power (the working point OP3), causing aperiodic instability—a voltage collapse.

To sum up, if at a given voltage, the voltage susceptibility coefficient d*Q*/d*V* determined in the HV/MV substation is negative, while the voltage is lowering, stopping the regulation by changing the power transformer voltage ratio is unfavourable. A stop in transformer regulation may occur, e.g., in a situation when the transformer controller's under-voltage interlock is activated. A similar effect will be observed when the voltage change rate is greater than the rate of voltage correction as a result of transformer voltage ratio regulation (e.g., a large time delay value).

#### **6. New Algorithm of the Transformer Controller Reducing the Risk of Voltage Collapse**
