Voltage-Triggered Flexibility Provision in a Distribution Network with Limited Observability

Abstract

Variable renewable energy sources (VRESs) are essential for decarbonizing the energy sector, but they introduce significant uncertainty into power grids. This uncertainty necessitates increased flexibility to ensure reliable and efficient grid operations, impacting both short-term strategies and long-term planning. Effective management of VRESs is particularly challenging for distribution system operators (DSOs) due to limited grid measurement and other data, complicating state estimation. This paper proposes a tractable framework that operates under low-observability conditions. The framework uses conservative linear approximations (CLAs) to manage grid constraints efficiently, requiring only the ranges of power injections typically available to operators. The objectives are twofold: first, to determine the amount and location of flexibility needed to prevent voltage violations and, second, to establish voltage measurement thresholds to trigger flexibility activation. Key contributions include the formulation of optimization problems to mitigate voltage issues, the introduction of flexibility provision triggered by voltage thresholds, and novel algorithms for determining flexibility and trigger points. The framework’s efficacy is demonstrated on IEEE 33-bus and UK 124-bus test systems, showing it can effectively mitigate grid voltage problems despite limited data.

1. Introduction

Variable renewable energy sources (VRESs), characterized by uncertainty, are expected to play a crucial role in the decarbonization of the energy sector. The integration of these sources into the power grid requires increased flexibility to ensure reliable, efficient, and economical operation. The flexibility of the power system is critical in managing the variability and uncertainty inherent to renewable energy generation, affecting both short-term operational strategies and long-term planning. Several studies highlight the importance of developing flexible systems to accommodate the rapid fluctuations in power supply and demand, mitigate the challenges posed by increased renewable penetration, and maintain system stability and reliability [1,2]. Note here that in this paper, although flexibility can be achieved through various means, including advanced forecasting, improved ramping capabilities, and the strategic deployment of energy storage and responsive load management techniques, we are primarily focused on load and/or generation flexibility facilitated by the technical capability to adjust grid users’ power consumption or generation upon request. While rather abstract, this definition is sufficiently specific, remaining agnostic to technical details of how flexibility is provided or if there are certain temporal or technical constraints regarding the available flexibility. In this paper, we take the perspective of a distribution system operator, while assuming a flexibility provider can always provide an amount of flexibility that has been agreed on previously with the system operator.

The introduction of VRESs into the distribution grid represents a significant operational paradigm shift for the distribution system operators who face significant challenges due to the lack of measurements from the power grid [3]. The lack of measurement data complicates accurate state estimation, making it difficult to verify real-time network constraints and leading to overly cautious operational strategies [4]. With limited network observability, the operators struggle to monitor and control VRESs effectively, which can result in inefficient grid management, increased risk of instability, and sub-optimal decision making. The scarcity of precise, timely data also forces system operators to rely heavily on pseudo-measurements and forecasts, which can introduce significant errors and uncertainties into the state estimation process [5,6].

Using flexibility of load and/or generation to mitigate voltage or current constraint violations has been extensively researched [7]. However, most of these approaches make the assumption of full grid observability, often using exact data for demand and generation in their numerical examples. More recently, the lack of measurements and how this lack can reflect on system operations is being recognized and addressed. In [8], the authors address the challenge of maintaining acceptable voltage levels in distribution grids with increasing solar PV penetration, focusing on optimizing the schedule for transformers and capacitor banks and minimizing the use of responsive inverter-interfaced resources for real-time conditions. It presents a convex optimal power flow (OPF) formulation to minimize the voltage deviations and integrates discrete mechanical devices into the optimization problem. However, the convex inner approximation developed in the paper is unlikely to be sufficiently robust to varying network topology and is not applicable to more realistic grids. In [9], the authors consider the robust AC OPF problem, which minimizes the generation cost while requiring a certain level of system security in the presence of uncertainty. They extend a previously developed convex restriction to a robust convex restriction: a convex inner approximation of the non-convex feasible region of the AC OPF problem that accounts for uncertainty in power injections. However, the proposed method is better suited for transmission networks and, given the nature of convex restriction, could yield overly conservative results in distribution networks. In [10], a two-stage min–max–min robust energy management model considering the correlation constraints and the uncertainty of solar energy is proposed. However, the applicability of the paper to grid management is rather questionable, as certain parameters regarding the treatment of uncertainties are not readily available nor intuitive for system operators (e.g., distribution of uncertainties). Ref. [11] introduces a holistic approach to optimal operational planning and operational management of active distribution systems under uncertainties to avoid line congestion and voltage limit violations using a two-stage stochastic programming model based on weighted scenarios. While comprehensive, the method is computationally demanding.

Furthermore, the issue of when to use the available flexibility, given the lack of measurements, remains unexplored. In [12], while not considering flexibility, the authors use convex relaxation to certify grid voltage constrain satisfaction. However, their approach results in overly conservative voltage estimation, precisely because of the use of relaxations. In [13], the authors propose a method for constraining the control actions of third-party aggregators to ensure safe operation of the distribution network. While somewhat conceptually similar, our paper differs by both the nature of the problems it addresses and the class of underlying optimization problems. A bi-level optimization-based framework to determine the aggregate power flexibility that can be obtained from an unbalanced distribution grid while ensuring that there is no solution that leads to grid constraint violations is proposed in [14]. While somewhat conceptually similar to our paper, this work is primarily focused on determining the aggregate flexibility of a distribution network, rather than using it to mitigate problems in the distribution grid itself. In a similar vein, authors in [15] formulate a batch reinforcement learning-based demand response approach to prevent distribution network constraint violations in unknown grids. While accounting for uncertainties in the network, the paper’s concern is that of a demand response for frequency regulation.

While developing our optimization framework, we selected a single-objective mixed-integer linear programming (MILP) model due to its computational efficiency, robustness, and practicality. MILP models allow for efficient problem solving using commercial solvers, which is essential given the significant computational time required for large network regression problems. The linear nature of MILP constraints and objective functions simplifies integration with existing grid management systems and ensures scalability. This reliability and ease of implementation are critical for real-world applications, where more complex models may pose significant deployment challenges. Additionally, while learning-based optimization frameworks and multi-objective mixed-integer nonlinear programming (MINLP) frameworks offer potential benefits, they require extensive and high-quality datasets for training and validation—resources that are currently limited in power network flexibility and injection scenarios. A single-objective MILP provides clear, interpretable solutions crucial for decision making in power grid management, facilitating better understanding and trust among operators and stakeholders. Our future research will focus on integrating advanced optimization techniques and data-driven methods as more comprehensive datasets become available and computational capabilities improve. This approach ensures our method remains efficient, robust, and practical for real-world power network optimization.

In order to solve the problem of mitigating grid constraints in an unobservant grid, authors often rely on overly conservative or computationally expensive methods that require certain techniques to circumvent the issue of non-convexity of the AC power flow equations or design methods that accommodate a small subset of distribution grids. This paper tries to address this gap by designing a tractable framework that is robust with regard to demand and generation uncertainties; conservative, as it relies on conservative linear approximations, therefore providing the distribution system operator with sufficiently tight lower and upper bounds on decision variables by construction; data-efficient, in a sense that it only requires knowledge of ranges of power injections, usually readily available to the grid operator; temporally agnostic, in that it is able to encompass an arbitrarily large operational time span, given its nature of dealing with ranges of power injection; and is practical, from an engineering perspective, by effectively avoiding complex mathematical transformations.

The objective of this paper is twofold. Firstly, we want to determine the amount and the location of flexibility that is needed to ensure no voltage constraints are violated in a distribution network with limited observability. Here, we assume that only the range of possible power injections and the range of voltage at the reference node are known. Secondly, we want to determine, given a specific location, a voltage measurement threshold that can serve as a proxy for the voltage condition of the overall network which, if crossed, would mean at least one voltage constraint violation is occurring somewhere in the network. This value then serves as the trigger for activating flexibility.

In summary, the main contributions of this paper are as follows:

1.

This paper leverages recently proposed conservative linear approximations (CLAs) to formulate tractable, optimization-based problems that are able to mitigate voltage problems in a distribution network with low observability. This paper extends the proposed CLAs to include changes in voltage at the reference node.

2.

To the best of the authors’ knowledge, this is the first paper that establishes a concept of flexibility provision triggered by a pre-calculated voltage threshold in a network with limited observability and establishes a tractable framework to derive and validate the solutions within this scheme.

3.

In order to find the amount of flexibility and voltage that triggers it, two novel, optimization-based, iterative algorithms are proposed.

2. Description of Problems

Let us consider a simplified model of a medium-voltage, radial, distribution network from [16]. We assume no measurements are available in the network; however, based on the billing data, typical customer load profiles, and DG historical data, we can assume bounds on node power injections to vary within 50% to 150% of real and reactive power nominal value, as defined by the test case. Voltage at a reference node varies between 0.97 and 1.03 p.u. A reference node is located on the secondary side of a substation that feeds the network. Reference voltage magnitude is independent of power injections and is usually measured and set by the tap changer in a substation.

With these assumptions, we run power flow simulations by varying power injections and reference voltage values within the aforementioned limits. As a result, we obtain the magnitudes of the minimum achievable voltages for each node in the network, as shown in Figure 1. We see that voltages in some nodes in the network can be as low as 0.87 p.u.

Traditionally, the distribution system operator must comply with a 0.90 p.u. at the low-voltage customers’ grid connection points. Since we are dealing with a medium-voltage network, maintaining a 0.92 p.u. at this voltage level allows for an additional voltage drop of 0.02 p.u. at a lower voltage level. In order to maintain the voltage, and in the absence of other voltage control methods (such as on-load tap changing equipment or capacitor banks), the distribution system operator can turn to flexibility providers. These are the customers that have the technical ability to provide ancillary services to the system operator by adjusting their power usage at the point of common coupling. When procuring flexibility, the distribution system operator needs to know in advance where, how much, and when the flexibility should be procured in order not to violate the grid’s voltage limits. To answer the aforementioned question of where and how much, we define the Flexibility Provision Problem as the optimization problem of determining the optimal location and the amount of flexibility that would guarantee no voltage limit violation, irrespective of the power injections in the rest of the distribution network.

Since the operator has to deal with low distribution network observability, it is important to have at least one voltage measurement that can serve as a proxy for overall voltage awareness in the network; this is the key to resolving the issue of ‘when’ to use flexibility. The placement of this voltage measurement, which we refer to as critical voltage, ensures that when the measured value of the critical voltage is below the predefined threshold value, the flexibility, determined previously by solving the Flexibility Provision Problem, will be activated by the distribution system operator. The Critical Voltage Threshold Problem is an optimization problem for determining the threshold value of critical voltage that guarantees no lower (or upper) voltage violation can occur in the entire network if the measured voltage magnitude is above (below) the pre-calculated threshold value. The flexibility provision, which is calculated by solving the Flexibility Provision Problem, should be seen as an upper limit of flexibility that is needed to provide robustness against a realization of the worst-case scenario. Knowing and setting the critical voltage threshold could enable autonomous flexibility control, where critical voltage provides a feedback for power injection control without the need for operator intervention.

3. Conservative Linear Approximations

The solution to the aforementioned optimization problems leverages the recently proposed conservative linear approximations (CLAs) of the power flow equations. The detailed discussion on calculating the CLAs and the associated parameters can be found in Appendix A. Note here that we build upon a previous work and extend the CLAs to account for changes in the reference voltage as these can not be ignored if there is no on-load tap changer at the substation.

4. Flexibility Provision Problem Formulation

In order to find the location and the amount of flexibility provision, we propose a two-stage, optimization-based method that loops between two levels: an upper level, which finds locations and the appropriate amount of flexibility, and a lower level, which finds a combination of power injections that maximizes the number of voltage bound violations. A detailed formulation of each level is described in the following subsections.

4.1. Lower-Level Problem

Given the range of nodal power injections $\left[P_{min}{P}_{max}\right],\left[Q_{min}{Q}_{max}\right]$, and the voltage range at the reference node $\left[V_{max}^{ref}{V}_{min}^{ref}\right]$, this optimization problem maximizes the number of lower or upper voltage limit violations. Depending on which voltage bound we want to violate (upper or lower), appropriate constraints in Equations (3) and (7) are selected. The full optimization problem is structured as follows:

$$\underset{P_{ll},Q_{ll},v,s_1,s_2,V_{ref}}{\max }\sum _{s_1=1}^{N_b}{s}_1$$

(1)

$$s.t.s_2{v}_{min}^2\le v\le v_{min}^2+(1-s_1)\mathbf{or}{s}_1{v}_{max}^2\le v\le v_{max}^2+(1-s_2)$$

(2)

$$s_1+s_2=1$$

(3)

$$Q_{min}\le Q_{ll,i}\le Q_{max}\forall i\in N_b\setminus \left\{flex\right\}$$

(4)

$$P_{min}\le P_{ll,i}\le P_{max}\forall i\in N_b\setminus \left\{flex\right\}$$

(5)

$$V_{min}^{ref}\le V_{ll}^{ref}\le V_{max}^{ref}$$

(6)

$$P_{ll,i}=P_{up,i}\forall i\in \left\{flex\right\}$$

(7)

$$Q_{ll,i}=Q_{up,i}\forall i\in \left\{flex\right\}$$

(8)

where

$$v={\underline{\mathbf{a}}}_0+{\underline{\mathbf{a}}}_1^{\top }\left(\begin{array}{c}{V}_{ll}^{ref}\\ P_{ll}\\ Q_{ll}\end{array}\right)\mathbf{or}v={\overline{\mathbf{a}}}_0+{\overline{\mathbf{a}}}_1^{\top }\left(\begin{array}{c}{V}_{ll}^{ref}\\ P_{ll}\\ Q_{ll}\end{array}\right)$$

(9)

is an $N_b$-sized vector of squared voltage variables, while parameters ${\underline{\mathbf{a}}}_0$, ${\underline{\mathbf{a}}}_1^T$, ${\overline{\mathbf{a}}}_0$, and ${\overline{\mathbf{a}}}_1^T$ are CLA linear coefficients, as described in Appendix A. $N_b$ denotes the number of nodes in the network, while $flex$ denotes a set of nodes where flexibility is to be activated. The $up$ index denotes the results from the upper-level problem. $s_1,s_2\in \{0,1\}$ are vectors of size $N_b$, while $v_{min}$ is a vector of nodal lower voltage limits.

If one or several variables in vector $s_1$ are set to one, the corresponding variables in vector $s_2$ are set to zero, causing v to violate the lower $V_{min}$ or upper $V_{max}$ voltage bound. Set of nodes with activated flexibility—$flex$—and the corresponding amount of flexibility—$P_{up},Q_{up}$—are the results of the upper-level problem. The decision variables of the optimization problem that are propagated to the upper-level problem are the nodal power injections and reference voltage, both denoted by the index $ll$ (lower level). The optimization procedure is designed to find the set of nodal power injections that represent the worst-case scenario with regard to the voltage bound violation.

Note here that the optimization problem is structured to find the maximum number of voltage violations. A viable alternative to this would be to minimize (or maximize) each node voltage and then to verify if any voltage violates the limits. In contrast to our method, this would require substantially more computational time on larger networks. Another viable option is to minimize (or maximize) the sum of all nodal voltages. While computationally more efficient, this method is generally less conservative, which makes it less preferred by the distribution system operator.

4.2. Upper-Level Problem

Given the bounds on active and reactive power flexibility $\left[P_{min}^{flex}{P}_{max}^{flex}\right]$ and $\left[Q_{min}^{flex}{Q}_{max}^{flex}\right]$, as well as the power injections determined by solving the lower-level problem, $P_{ll},Q_{ll}$, and $V_{ll}^{ref}$, we formulate the upper-level optimization problem as follows:

$$\underset{P,Q,v}{\min }\sum _{s=1}^{N_b}s$$

(10)

$$s.t.v_{min}^2\le v\le v_{max}^2$$

(11)

$$Q_{ll}+s(Q_{min}^{flex}-Q_{ll})\le Q_{up}\le Q_{ll}+s(Q_{max}^{flex}-Q_{ll})$$

(12)

$$P_{ll}+s(P_{min}^{flex}-P_{ll})\le P_{up}\le P_{ll}+s(P_{max}^{flex}-P_{ll})$$

(13)

where

$$v={\underline{\mathbf{a}}}_0+{\underline{\mathbf{a}}}_1^{\top }\left(\begin{array}{c}{V}_{ll}^{ref}\\ P_{up}\\ Q_{up}\end{array}\right)\mathbf{or}v={\overline{\mathbf{a}}}_0+{\overline{\mathbf{a}}}_1^{\top }\left(\begin{array}{c}{V}_{ll}^{ref}\\ P_{up}\\ Q_{up}\end{array}\right),$$

(14)

$s\in \{0,1\}$, while index $ll$ denotes parameters from the lower-level problem.

The reasoning behind the previous problem is now inverted; instead of trying to violate voltage constraints, we enforce the satisfaction of voltage constraints in Equation (11) by ‘procuring’ the flexibility from the providers. This is accomplished by setting the variables in vector s to one, meaning that the flexibility at a specific node should be used. If a variable in the vector is set to zero, the flexibility is not used, and power injections from lower-level problem are used instead. The decision variables of the optimization problem are nodal power injections for nodes where the flexibility is procured.

Note here that the optimization problem is not structured to find the minimum amount of flexibility; rather, it is trying to limit the number of flexibility providers, a feature which could provide sub-optimal results with regard to the total amount of flexibility needed to mitigate potential voltage issues. While sub-optimal, this design choice is intentional, as the distribution system operators prefer to minimize their interactions to as few entities as possible. Furthermore, once an operator knows when to activate flexibility, they do so in a sequential manner, until the critical voltage is restored to a predefined value. This means that the flexibility calculated here will always be the upper bound to what is actually used by the distribution system operator.

4.3. Flexibility Provision Algorithm

We assume that the CLA parameters have already been determined as outlined in Appendix A. The key steps of the Algorithm 1 that connects the lower and upper optimization problems are as follows:

Algorithm 1 Flexibility Provision Algorithm

1: Input: $N_b$, $flex\in Ø$, ${\overline{a}}_{i,0}$,${\overline{a}}_{i,1}$,${\underline{a}}_{i,0}$,${\underline{a}}_{i,1}$, $P_{min}^{flex},P_{max}^{flex}$, $Q_{min}^{flex},Q_{max}^{flex}$,$P_{min},P_{max},Q_{min},Q_{max}$, $V_{max}^{ref},V_{min}^{ref}$   2: while true do   3:       the lower-level problems (1)–(8)   4:       if $\sum s>=1$ then save the values of $P_{ll},Q_{ll}$ and $V_{ll}^{ref}$.   5:       else   6:             break   7:       end if   8:       Solve the upper-level problems (10)–(13).   9:       Update $\left\{flex\right\}$ with indices of s, where $s\ne 0$. Save $P_{up}$,$Q_{up}$ 10: end while

The algorithm is initialized with an empty set of $flex$, meaning no flexibility is provided. The algorithm then iterates between the lower- and upper-level problems, until the objective of a lower-level problem is zero, meaning that no further voltage bound violations can occur in the network. The algorithm is run once for each (lower and upper) voltage bound violation, with the only difference being the choice of constraints in Equation (2) and CLA parameters used in Equations (9) and (14).

The convergence of the algorithm is not guaranteed. Given the topology, power, and voltage limits, as well as current operating point of the network, it is not unrealistic that there is not enough flexibility in the network that would make the network robust for the entire range of possible power injections and reference voltage variations. However, the algorithm can, even in those cases, provide insight into how this robustness can be achieved. For example, flexibility bounds can be relaxed to ensure convergence.

5. Critical Voltage Threshold Problem Formulation

5.1. Lower-Bound Voltage Threshold Calculation

Here, we formulate an optimization problem that calculates the threshold for the lower bound of critical voltage. The problem is structured so as to ensure that no violation of the lower voltage limit can occur in the network if critical voltage remains above this threshold level.

$$\underset{P,Q,v,s_1,s_2,V_{ref},v_{thr}}{\max }\sum _{s_1=1}^{N_b}{s}_1$$

(15)

$$s.t.s_2{v}_{min}^2\le v\le v_{min}^2+(1-s_1)$$

(16)

$$s_1+s_2=1$$

(17)

$$Q_{min}\le Q\le Q_{max}$$

(18)

$$P_{min}\le P\le P_{max}$$

(19)

$$V_{min}^{ref}\le V^{ref}\le V_{max}^{ref}$$

(20)

$$v_{thr}={(V_{thr}^{init}\ast (1+\eta ))}^2$$

(21)

where

$$v={\underline{\mathbf{a}}}_0+{\underline{\mathbf{a}}}_1^{\top }\left(\begin{array}{c}{V}^{ref}\\ P\\ Q\end{array}\right)$$

(22)

$thr$ is an index of the node for which the threshold is calculated, $V^{init}$ denotes an initial voltage, and $\eta$ is a parameter which increases the initial voltage value with each iteration. Given the arbitrary location of the node ($thr$) and the initial voltage $V_{thr}^{init}$, the program increases the critical voltage threshold value by $\eta$ in each iteration while still trying to violate lower voltage limits ($v_{m}in$).

5.2. Upper-Bound Voltage Threshold Calculation

The threshold for an upper voltage bound of critical voltage is calculated in a similar fashion, as follows:

$$\underset{P,Q,v,s_1,s_2,V_{ref},v_{thr}}{\max }\sum _{s_1=1}^{N_b}{s}_1$$

(23)

$$s.t.s_1{v}_{max}^2\le v\le v_{max}^2+(1-s_2)$$

(24)

$$s_1+s_2=1$$

(25)

$$Q_{min}\le Q\le Q_{max}$$

(26)

$$P_{min}\le P\le P_{max}$$

(27)

$$V_{min}^{ref}\le V^{ref}\le V_{max}^{ref}$$

(28)

$$v_{thr}={(V_{thr}^{init}\ast (1-\eta ))}^2$$

(29)

where

$$v={\overline{\mathbf{a}}}_0+{\overline{\mathbf{a}}}_1^{\top }\left(\begin{array}{c}{V}^{ref}\\ P\\ Q\end{array}\right).$$

(30)

Here, we only change Equations (25), (29) and (30) to account for the upper instead of the lower threshold value.

5.3. Initial Voltage Calculation

The initial voltage ($V^{init}$ ) is calculated by solving one instance of the lower-level problem formulated in Section 4.1, but with the following modifications. The objective function is minimized instead of maximized, and an additional constraint (33) is added to the original problem. The optimization problem now calculates a set of network voltages where only one is below (above) the lower (upper) voltage limit.

$$\underset{P_{ll},Q_{ll},v,s_1,s_2,V_{ref}}{\min }\sum _{s_1=1}^{N_b}{s}_1$$

(31)

$$s.t.s_2{v}_{min}^2\le v\le v_{min}^2+(1-s_1)\mathbf{or}{s}_1{v}_{max}^2\le v\le v_{max}^2+(1-s_2)$$

(32)

$$\sum _{s_1=1}^{N_b}{s}_1\ge 1$$

(33)

$$s_1+s_2=1$$

(34)

$$Q_{min}\le Q\le Q_{max}$$

(35)

$$P_{min}\le P\le P_{max}$$

(36)

$$V_{min}^{ref}\le V^{ref}\le V_{max}^{ref}$$

(37)

5.4. Critical Voltage Threshold Algorithm

We assume that the CLA parameters have already been determined, as outlined in Appendix A. The key steps of the Algorithm 2 are as follows:

Algorithm 2 Critical Voltage Threshold Algorithm

1: Input: $N_b$, $\eta$, $thr\in \left\{N_b\right\}$${\overline{a}}_{i,0}$,${\overline{a}}_{i,1}$,${\underline{a}}_{i,0}$,${\underline{a}}_{i,1}$,$P_{min},P_{max},Q_{min},Q_{max}$, $V_{max}^{ref},V_{min}^{ref}$   2: Initialize: Solve opt. problem from (31)–(37) for $v_i,i\in N_b$   3: Arbitrarily choose $thr$ and set: $V_{thr}^{init}$←$v_{thr}$   4: while true do   5:       Find the lower-bound voltage threshold by solving (15)–(21)   6:       if $\sum s_1>=1$then increase $\eta$   7:       else   8:             break   9:       end if 10: end while 11: Output: $v_{thr}^{low}$←$v_{thr}$ 12: while true do 13:       Find the upper-bound voltage threshold by solving (23)–(29) 14:       if $\sum s_1>=1$then increase $\eta$ 15:       else 16:             break 17:       end if 18: end while 19: Output: $v_{thr}^{upp}$←$v_{thr}$

Note here that the choice of voltage measurement location for which the threshold is being calculated is arbitrary. This allows the operator to calculate the voltage threshold for a location in the network where the measurement can be placed and monitored for a threshold value violation. Although there is no guarantee that the solution to the problem, given the predefined location, can be found, we did not encounter any problems in our test cases. However, further insights into whether this is broadly the case remains an open and interesting direction for further research.

6. Numerical Results

6.1. Test System and Software

The optimization problems and algorithms are coded in MATLAB version 9.5.0 (R2018b) using theYALMIP toolbox [17] version 20230622 and SCIP [18] version 4.0.1, as a (mixed integer) linear programming solver. Test system is a modest PC with 16 GB of RAM. Power flows are calculated using MATPOWER [19] version 7.1.

6.2. IEEE 33 Bus Test Case Analysis

We illustrate the proposed method on a well-known IEEE 33 bus test case from [16]. It is a 12.66 kV medium-voltage-level radial distribution network, which encompasses 33 buses and 32 lines. Node 1 is a reference node with voltage ranging from 0.97 to 1.03 p.u. We assume that every other node in the network (2−33) has independent active/reactive power injections within the range of −100% to 100% of nominal power. The range of possible power injections is shown in Figure 2. The range assumes some form of distributed generation connected in parallel to the load at every node in the network. Lower and upper voltage limits are set to 0.96 and 1.04 p.u., respectively, for all network nodes. Given the aforementioned assumptions, the maximum and minimum achievable node voltages are shown in the first graph in Figure 2, calculated by running ten thousand power flow simulations. We see that, given a certain combination of reference voltage and power injections, both lower and upper voltage limits can be violated.

6.3. Flexibility Provision

Here, we want to find the amount of flexibility per node that can ensure no voltage violations can occur. Firstly, we calculate CLA parameters as described in Appendix A. We generate one thousand power flow samples ($M=1000$), based on which we derive the parameters. The computational time needed for calculating a thousand power flow samples is approx. 17 s. The time needed for solving the regression problem for all CLA parameter calculations is around two minutes. By far, the most time consuming is the process of creating the optimization model for the regression problem; it takes approximately 10 min, due to YALMIP overhead.

We proceed by calculating nodal flexibility by using Algorithm 1. We assume active/reactive power flexibility availability ($P^{flex},Q^{flex}$) of −50% to 50% of nominal power at all nodes. The algorithm is executed twice: once for lower and once for upper voltage issues mitigation. Total execution time for lower and upper voltage cases is 3 and 6 s, respectively. The results of the calculations can be seen in Figure 3 for the lower voltage case and in Figure 4 for the upper voltage case. From the figures, we can see that in order to elevate the voltages to above a 0.96 p.u. limit, flexibility needs to be used in nodes 7, 8, 10−15, 17–18, 24–25, and 29−32. In order to lower the voltages to below a 1.04 p.u. limit, flexibility needs to be used in nodes 7, 8, 13, 14, 16, 18, 24–25, and 29−33.

6.4. Critical Voltage Threshold

To determine the voltage threshold, we arbitrarily choose node 7 and set $thr=7$. Network parameters and bounds remain the same. We a set step increase of $\eta$ to 0.001. We then proceed by employing the proposed Algorithm 2 to find the upper and lower voltage thresholds for voltage at a specified node.

The algorithm required only one iteration for lower and upper voltage thresholds, each taking less than a second of computational time. The calculated lower and upper voltage thresholds are 0.9816 and 1.018 p.u., respectively.

6.5. Validating Results of IEEE 33 Bus Test Case

We validate the results using MATPOWER simulations again. We create samples by randomly setting reference voltage and power injections within the predefined ranges. Our validation code executes the following logic for every sample: Run power flow and check the voltage magnitude at node 7—if it is lower than the lower threshold of 0.9816 p.u., ‘activate’ pre-calculated flexibility, by setting the value of power injection at node 7. Then, run the power flow again for that sample and check again for the voltage at node 7. If it is still below lower threshold, we ‘activate’ flexibility at node 8 in the same manner as described previously. The process repeats until either the voltage at node 7 is above the threshold value or if there is no more flexibility that can be provided. If at any point the voltage at node 7 is above the threshold, move on to the next power flow sample. The logic is the same for the upper voltage bound. This procedure closely resembles the real-life handling of flexibility, where flexibility needs are determined in advance, reserved (as in our Flexibility Provision Problem), and then activated as needed.

We run five thousand samples in MATPOWER and apply the aforementioned logic to each one. The results of this simulation are shown in Figure 5. As can be seen, there are no instances of voltage magnitude crossing the 0.96. p.u. or 1.04 p.u. voltage limits.

6.6. UK 124 Bus Test Case Analysis

We now proceed by analyzing a more realistic and demanding test case from [20], as shown in Figure 6. It is a medium-voltage 6.6 kV weakly meshed synthetic UK distribution network with 124 buses and 124 lines. Node 7 is a reference node with voltage ranging from 0.98 to 1.01 p.u. We assume active/reactive power injections within the range of 80% to 120% of nominal power. Lower and upper voltage limits are set to 0.93 and 1.04 p.u., respectively, for all network nodes. Given the aforementioned assumptions, maximum and minimum achievable node voltages are shown in the first graph in Figure 7. We see that, given a certain combination of reference voltage and power injections, a lower voltage limit can be violated by a large margin, with some voltage magnitudes decreasing as low as 0.88 p.u.

6.7. Flexibility Provision

Again, we want to find the amount of flexibility per node, which can ensure no lower voltage limit violations can occur. We assume flexibility availability at nodes marked ‘FU’ in Figure 6 (nodes: 14, 20, 79, 83, 86, 99, 100, 106, and 107). Active power flexibility ($P^{flex}$) ranges from 50% to 100% of nominal active power provided in the dataset. Reactive power flexibility ranges from −250 kVAr to 250 kVAr at FU nodes. CLA parameter calculations is carried out in the same manner as for the IEEE 33 bus test case. Given a much larger network, we resort to commercial solver CPLEX [21] version 12.9. to solve the regression problem for CLA parameter calculations. YALMIP takes approximately 120 s to construct the model, while CPLEX takes around 1000 s to solve the regression problem. The flexibility calculation requires only one iteration with an execution time of 4 s. The results of the calculations are presented in Figure 8. In order to increase all node voltage magnitudes to above the 0.93 p.u. limit, flexibility needs to be used in nodes 79, 83, 86, 99, 100, and 105.

Figure 7. Initial state of the UK 124 bus test network; (a) minimum (orange) and maximum (blue) achievable nodal voltage magnitudes. (b) Nodal minimum (grey) and maximum (orange) active power injection ranges. (c) Nodal minimum (grey) and maximum (orange) reactive power injection ranges.

Figure 7. Initial state of the UK 124 bus test network; (a) minimum (orange) and maximum (blue) achievable nodal voltage magnitudes. (b) Nodal minimum (grey) and maximum (orange) active power injection ranges. (c) Nodal minimum (grey) and maximum (orange) reactive power injection ranges.

6.8. Critical Voltage Threshold

We arbitrarily choose node 28, and set $thr=28$ and $\eta$ to 0.001. Network parameters and bounds remain the same. Algorithm 2 finds that the lower voltage threshold for the voltage at node 28 is 0.9799 p.u. The algorithm required only one iteration, which took less than a second of computational time.

6.9. Validating Results of UK 124 Bus Test Case

The methodology used for validating results remains the same as for the IEEE 33 bus test case. We create random power flow samples with reference voltage and power injections within predefined ranges of 0.98 to 1.01 p.u. for the reference voltage and 80% to 120% of nominal active and reactive power injections, as defined in the dataset. We run power flow simulations in MATPOWER where, for each sample, we check if the calculated voltage magnitude at node 28 is below threshold of 0.9799 p.u. If so, we activate the flexibility sequentially, until the voltage is restored to above the threshold value. The results of this simulation are shown in Figure 9. As can be seen, there are no instances where the voltage magnitude is below the 0.93 p.u. limit.

7. Discussion

If we disregard YALMIP overhead, the computational time needed for solving the regression problem of (A3)–(A5) for larger networks can be substantial; hence, using a commercial MILP solver is recommended in this case. This calculation can be completed offline, so we do not consider it a major drawback. However, further research is needed to reduce the number of samples used in the regression problem (M), while maintaining the accuracy of the CLAs.

Realistic data regarding flexibility availability and ranges of power injections are needed to conduct further research into the potential downsides of the proposed method.

In order to avoid overly conservative results, a more complex construction of an uncertainty set, which is specially tailored to operators intuition and grid data availability, such as in [22,23], is needed and is an interesting avenue for future research.

We emphasize the need for further research into selecting the appropriate location for voltage measurement placements that can serve as a proxy for voltage conditions of the entire network. The problem with arbitrarily choosing a location is that a voltage measurement is not guaranteed to be sensitive to changes in power injections stemming from the use of flexibility. This could lead to a situation where the activation of flexibility mitigates voltage problems in a part of the network, but this is not reflected in voltage measurements that are monitored, leading the operator to unnecessarily activate additional flexibility.

8. Conclusions

The provision of flexibility will likely play a key role in future power grids. From a technical perspective, operators are currently unable to exploit the flexibility potential of the consumer, mostly because of the lack of tools that incorporate system uncertainties, some of which stem from the lack of medium- and low-voltage measurements. To address the issue, this paper proposes a method to determine the location and the amount of flexibility needed to mitigate potential voltage problems, given limited network observability. The method leverages recently proposed conservative linear approximation to establish tractable, mixed-integer linear optimization problems that are used to determine the required flexibility. Furthermore, this paper proposes a framework that enables the operator to know when to use the required flexibility by determining the critical voltage threshold value. As shown on 33 and 124 bus distribution system test cases, the iterative algorithms proposed are able to determine the amount of flexibility and the voltage threshold at which this flexibility is to be activated efficiently and accurately, requiring only the knowledge of the ranges of nodal power injections.

To enhance the robustness of the proposed method, future research should focus on optimizing the sample size used in the regression problem while ensuring the accuracy of the CLAs, as well as on developing advanced techniques for constructing uncertainty sets tailored to grid operators’ intuition and data availability. Additionally, further research is required to identify optimal voltage measurement locations that accurately reflect network-wide voltage conditions, thereby preventing the unnecessary activation of flexibility and ensuring efficient grid management.